In this paper we prove that extremal black holes arise on the threshold of gravitational collapse. More precisely we construct smooth oneparameter families of smooth spherically symmetric solutions to the EinsteinMaxwellVlasov system which interpolate between dispersion and collapse and for which the critical solution is an extremal black hole. Physically these solutions can be understood as beams of gravitationally selfinteracting collisionless charged particles fired into Minkowski space from past infinity. Depending on the precise value of the parameter we show that the Vlasov matter either disperses due to the combined effects of angular momentum and electromagnetic repulsion or undergoes gravitational collapse. At the critical value of the parameter an extremal ReissnerNordstrm black hole is formed. No naked singularities occur as the extremal threshold is crossed. We call this critical phenomenon extremal critical collapse and the present work constitutes the first rigorous result on the black hole formation threshold in general relativity.
['General Relativity and Quantum Cosmology', 'Mathematical Physics', 'Mathematics - Analysis of PDEs', 'Mathematics - Differential Geometry']
Extremal black hole formation as a critical phenomenon
{'Extremal black hole formation as a critical phenomenon': 'Christoph Kehle ∗1 and Ryan Unger †2  \n1 Institute for Theoretical Studies & Department of Mathematics, ETH Zürich, Clausiusstrasse 47, 8092 Zürich, Switzerland  \n2 Department of Mathematics, Princeton University, Washington Road, Princeton NJ 08544, United States of America  \nFebruary 15, 2024', 'Abstract': 'In this paper, we prove that extremal black holes arise on the threshold of gravitational collapse. More precisely, we construct smooth one-parameter families of smooth, spherically symmetric solutions to the Einstein-Maxwell-Vlasov system which interpolate between dispersion and collapse and for which the critical solution is an extremal black hole. Physically, these solutions can be understood as beams of gravitationally self-interacting collisionless charged particles fired into Minkowski space from past infinity. Depending on the precise value of the parameter, we show that the Vlasov matter either disperses due to the combined effects of angular momentum and electromagnetic repulsion, or undergoes gravitational collapse. At the critical value of the parameter, an extremal Reissner-Nordström black hole is formed. No naked singularities occur as the extremal threshold is crossed. We call this critical phenomenon extremal critical collapse and the present work constitutes the first rigorous result on the black hole formation threshold in general relativity.', 'Contents': 'References  \n91', '1 Introduction': "One of the most spectacular predictions of general relativity is the existence and formation of black holes. Solutions of the Einstein field equations,  \nRic( g ) -1 2 R ( g ) g = 2 T , (1.1)  \ncan undergo gravitational collapse to form a black hole dynamically, starting from regular, one-ended Cauchy data. Building on earlier work of Lemaître [Lem33], Oppenheimer and Snyder [OS39] produced the first mathematical example of gravitational collapse with the collapse of a homogeneous dust cloud. In contrast, for reasonable matter models, solutions with small initial data disperse without a black hole forming. It is a fundamental problem in classical general relativity to understand how these different classes of spacetimescollapsing and dispersing-fit together in the moduli space of solutions. The interface between collapse and dispersion is known as the black hole formation threshold and families of solutions crossing this threshold are said to exhibit critical collapse . Spacetimes lying on the threshold are called critical solutions .  \nCritical collapse has been extensively studied numerically, starting with the influential work of Choptuik [Cho93] on the spherically symmetric Einstein-scalar field model, in a regime where the critical solutions are believed to be naked singularities. The Einstein-Vlasov system is believed to have static 'star-like' critical solutions [RRS98; OC02], but critical collapse involving naked singularities has so far not been observed. These numerical studies on critical collapse (see also the survey [GM07]) have yet to be made rigorous.  \nAt first glance, the Reissner-Nordström family of metrics (indexed by the mass M > 0 and charge e ) appears to exhibit a type of critical behavior: the solution contains a black hole when | e | < M ( subextremal ) or | e | = M ( extremal ) and does not contain a black hole when | e | > M ( superextremal ). However, the ReissnerNordström black holes are eternal and arise from two-ended Cauchy data, while the superextremal variants contain an eternal 'naked singularity' that has historically caused much confusion. Moreover, it was long thought that extremal black holes could not form dynamically (a consideration closely related to the third law of black hole thermodynamics [BCH73; Isr86]). Were this true, it would seem to rule out extremal ReissnerNordström as the late-time behavior of any critical solution. In [KU22], the present authors disproved the third law in the Einstein-Maxwell-charged scalar field model and showed that an exactly extremal ReissnerNordström domain of outer communication can indeed form in gravitational collapse.  \nIn this paper, we continue our investigation of extremal black hole formation by showing that extremal Reissner-Nordström does arise as a critical solution in gravitational collapse for the Einstein-Maxwell-Vlasov model, giving an example of a new phenomenon that we call extremal critical collapse .  \n̸  \nTheorem 1. There exist extremal black holes on the threshold between collapsing and dispersing smooth configurations of charged matter. More precisely, for any mass M > 0 , fundamental charge e = 0 , and particle mass 0 ≤ m ≤ m 0 , where 0 < m 0 ≪ 1 depends only on M and e , there exists a smooth one-parameter family of smooth, spherically symmetric, one-ended asymptotically flat Cauchy data { Ψ λ } λ ∈ [0 , 1] for the EinsteinMaxwell-Vlasov system for particles of fundamental charge e and mass m , such that the maximal globally hyperbolic development of Ψ λ , denoted by D λ , has the following properties.  \n- 1. D 0 is isometric to Minkowski space and there exists λ ∗ ∈ (0 , 1) such that for λ < λ ∗ , D λ is future causally geodesically complete and disperses towards Minkowski space. In particular, D λ does not contain a black hole or naked singularity. If λ < λ ∗ is sufficiently close to λ ∗ , then for sufficiently large advanced times and sufficiently small retarded times, D λ is isometric to an appropriate causal diamond in a superextremal Reissner-Nordström solution.\n- 2. D λ ∗ contains a nonempty black hole region BH . = M\\ J -( I + ) and for sufficiently large advanced times, the domain of outer communication, including the event horizon H + . = ∂ ( BH ) , is isometric to that of an extremal Reissner-Nordström solution of mass M . The spacetime contains no trapped surfaces.\n- 3. For λ > λ ∗ , D λ contains a nonempty black hole region BH and for sufficiently large advanced times, the domain of outer communication, including the event horizon H + , is isometric to that of a subextremal Reissner-Nordström solution. The spacetime contains an open set of trapped surfaces.  \nIn addition, for every λ ∈ [0 , 1] , D λ is past causally geodesically complete, possesses complete null infinities I + and I -, and is isometric to Minkowski space near the center { r = 0 } for all time.  \nFigure 1: Penrose diagrams of the one-parameter family {D λ } from Theorem 1 in the case of massive particles. The dark gray region depicts the physical space support of the Vlasov matter beam. The region of spacetime to the left of the beam is exactly Minkowski space and the region to the right of the beam is exactly Reissner-Nordström with the parameter ratio as depicted. In every case, the beam 'bounces' before it hits the center { r = 0 } due to the repulsive effects of angular momentum and the electromagnetic field. When λ < λ ∗ , the beam bounces before a black hole is formed. We note already that the beams actually have more structure than is depicted here in these 'zoomed out' pictures. See already Fig. 11.  \n<!-- image -->  \nIn the proof of Theorem 1, we construct one-parameter families of charged Vlasov beams coming in from past timelike infinity (if m > 0 , cf. Fig. 1) or from past null infinity (if m = 0 , cf. Fig. 2). In the dispersive case λ < λ ∗ , the area-radius r of the beam grows linearly in time as the matter expands towards the future. Moreover, the macroscopic observables of the Vlasov matter (the particle current N and energy momentum tensor T ) decay at the sharp t -3 rate in the massive case and at the sharp t -2 rate in the massless case (with faster decay for certain null components), see already (5.20)-(5.24) in Proposition 5.5. In fact, this same dispersive behavior happens in the past for every λ ∈ [0 , 1] .  \nAs a direct consequence of Theorem 1, we obtain  \nCorollary 1. The very 'black hole-ness' of an extremal black hole arising in gravitational collapse can be unstable: There exist one-ended asymptotically flat Cauchy data for the Einstein-Maxwell-Vlasov system, leading to the formation of an extremal black hole, such that an arbitrarily small smooth perturbation of the data leads to a future causally geodesically complete, dispersive spacetime.  \nThis is in stark contrast to the subextremal case, where formation of trapped surfaces behind the event horizon-and hence stable geodesic incompleteness [Pen65]-is expected. Despite this inherent instability of the critical solution, we expect extremal critical collapse itself to be a stable phenomenon: We conjecture that there exists a telelologically determined 'hypersurface' B crit in moduli space which consists of asymptotically extremal black holes, contains D λ ∗ , and locally delimits the boundary in moduli space between future complete and collapsing spacetimes. This 'codimension-one' property is expected to hold for other variants of critical collapse and will be discussed in detail in Section 1.4.  \nWe further expect extremal critical collapse to be a more general phenomenon: we conjecture it to occur in the spherically symmetric Einstein-Maxwell-charged scalar field model and also for the Einstein vacuum equations, where extremal Kerr is the model critical solution. In this paper, we also prove (see already Theorem 4 in Section 4.3) that extremal critical collapse already occurs in the simpler-but singularbouncing charged null dust model , which was first introduced by Ori in [Ori91]. The proof of Theorem 1, which will be outlined in Section 5.1, can be viewed as a global-in-time desingularization of these extremal critical collapse families in dust.  \nBesides the Einstein-Maxwell-Vlasov and bouncing charged null dust models, it turns out that the thin  \nFigure 2: Penrose diagrams of the one-parameter family {D λ } from Theorem 1 in the massless case. In the ingoing (resp., outgoing) phases, the massless beams are entirely contained in slabs of finite advanced (resp., retarded) time. Therefore, for sufficiently early advanced times and sufficiently late retarded times, the solutions are vacuum and isometric to Minkowski space.  \n<!-- image -->  \ncharged shell model [Isr66; DI67] also exhibits extremal critical collapse: Prószyński observed in [Pró83] that if a thin charged shell is injected into Minkowski space (so the interior of the shell is always flat), the parameters can be continuously varied so that the exterior of the shell goes from forming a subextremal Reissner-Nordström black hole, to forming an extremal Reissner-Nordström black hole, to forming no black hole or naked singularity at all: the shell 'bounces' off to future timelike infinity. Because the thin shell model is quite singular (the energy-momentum tensor is merely a distribution and the metric can fail to be C 1 across the shell), it seems to have been discounted as a serious matter model. We refer to the discussion in [KU22, Section 1.4.1] in reference to the thin charged shell counterexample to the third law by Farrugia and Hajicek [FH79]. Theorem 1 can be viewed as a vindication of [Pró83], since our smooth Einstein-Maxwellmassive Vlasov solutions exhibit all of the qualitative features of Prószyński's dust shells. In particular, Fig. 10 below is strikingly similar to Fig. 3 in [Pró83].", '1.1 The Einstein-Maxwell-Vlasov system': 'In this paper, we consider the Einstein-Maxwell-Vlasov system, which models a distribution of collisionless, self-gravitating charged particles with mass m ≥ 0 and fundamental charge e ∈ R \\{ 0 } . The model consists of a quadruple ( M 4 , g, F, f ) , where ( M 4 , g ) is a spacetime, F is a closed 2-form representing the electromagnetic field strength, and f = f ( x, p ) , called the distribution function of the Vlasov matter, is a smooth nonnegative function defined on the mass shell  \nP m . = { ( x, p ) ∈ T M : p is future-directed causal and g ( p, p ) = -m 2 } .  \nThe equations of motion are  \nR µν -1 2 Rg µν = 2 ( T EM µν + T µν ) , (1.2)  \n∇ µ F µν = -e N ν , (1.3)  \nXf = 0 , (1.4)  \nwhere T EM µν . = F µ α F να -1 4 g µν F αβ F αβ is the energy-momentum tensor of the electromagnetic field, N and T are the number current and energy-momentum tensor of the Vlasov matter, defined by  \nN µ [ f ]( x ) . = ∫ P m x p µ f ( x, p ) dµ m x ( p ) , T µν [ f ]( x ) . = ∫ P m x p µ p ν f ( x, p ) dµ m x ( p ) , (1.5)  \nand X ∈ Γ( TT M ) is the electromagnetic geodesic spray vector field, defined relative to canonical coordinates ( x µ , p µ ) on T M by  \nX . = p µ ∂ ∂x µ -( Γ µ αβ p α p β -e F µ α p α ) ∂ ∂p µ . (1.6)  \nFor the definition of the family of measures dµ m x on P m and a proof of the consistency of the system (1.2)(1.4), we refer to Section 3.1.  \nThe integral curves of the vector field X consist of curves of the form s ↦→ ( γ ( s ) , p ( s )) ∈ T M , where p = dγ/ds and p satisfies the Lorentz force equation  \nDp µ ds = e F µ ν p ν .  \nWe refer to such curves γ as electromagnetic geodesics . The vector field X is tangent to P m for any m ≥ 0 , and the Vlasov equation (1.4) implies that f is conserved along electromagnetic geodesics. Since f ≥ 0 , N is a future-directed causal vector field on M and the model satisfies the dominant energy condition.  \nWhen m > 0 , the system (1.2)-(1.4) is locally well-posed outside of symmetry, which can be seen as a special case of results in [BC73] or by applying the general methods of [Rin13]. Well-posedness when m = 0 is conditional and is a delicate issue that we will return to in Section 3.3. We emphasize at this point that Theorem 1 produces examples of extremal critical collapse for any sufficiently small positive particle mass, where well-posedness is unconditional and valid outside of spherical symmetry.', '1.2 The problem of critical collapse': "We would like to place Theorem 1 into the larger picture of critical collapse , the general study of the black hole formation threshold. In particular, we conjecture that our examples in Theorem 1 have a suitable codimension-one property as is expected to hold for other, so far only numerically observed, critical phenomena in gravitational collapse.  \nIn order to discuss the general concept of critical collapse, it is very helpful to have a notion of 'phase space' or moduli space for initial data (or maximal Cauchy developments) for the Einstein equations. Consider, formally, the set M of one-ended asymptotically flat Cauchy data for the Einstein equations with a fixed matter model (or vacuum) and perhaps with an additional symmetry assumption. We will be intentionally vague about what regularity elements of M have, what decay conditions to impose, or what topology to endow M with. We will also not discuss gauge conditions, which could be viewed as taking specific quotients of M . These questions are related to several fundamental issues in general relativity, see for instance [Chr94; Chr99b; Chr02; DS18; LO19; Keh22; RSR23; Keh23; KM24; Sin24]. 1 Indeed, it seems likely that there is no single 'correct' definition-it is doubtful that a single moduli space will capture every interesting phenomenon.  \nNevertheless, we will pretend in this section that a 'reasonable' definition of M exists. At the very least, M ought to consist of initial data possessing a well-posed initial value problem. For each element Ψ = (¯ g, ¯ k, . . . ) ∈ M (where ¯ g is a Riemannian metric on R 3 , ¯ k the induced second fundamental form, and . . . denotes possible matter fields), we have a unique maximal globally hyperbolic development D = ( M , g, . . . ) of Ψ , where M ∼ = R 4 [Fou52; CG69; Sbi16]. 2 We assume that ( M , g ) is asymptotically flat. In particular, we assume that we have a well-defined notion of future null infinity I + and past null infinity I -.  \nRemark 1.1 . In the proof of Theorem 1, we define a 'naive moduli space' M ∞ consisting of all smooth solutions of the Einstein-Maxwell-Vlasov constraint equations on R 3 , equipped with the C ∞ loc topology, and with no identifications made. See already Definition 5.29. This topology is inadequate for addressing asymptotic stability questions but since our families are electrovacuum outside a fixed large compact set anyway, they will be continuous in any 'reasonable' topology that respects asymptotic flatness.  \n̸  \nLet C ⊂ M denote the subset of initial data with future causally geodesically complete developments. We also highlight the special class D ⊂ C of initial data with dispersive developments, i.e, those solutions whose geometry asymptotically converges to Minkowski space in the far future and matter fields decay suitably. 3 Nontrivial stationary states, if they exist, lie in C \\ D since they do not decay. 4 Let B ⊂ M denote the set of initial data leading to the formation of a nonempty black hole region , i.e., BH . = M\\ J -( I + ) = ∅ . The question of critical collapse is concerned with the study of phase transitions between C , D , and B , that is, the structure of the boundaries ∂ C , ∂ D , and ∂ B , how they interact, and characterizing solutions lying on the threshold.  \nA natural way of exploring this phase transition is by studying continuous paths of initial data interpolating between future complete and black hole forming solutions.  \nDefinition 1.2. An interpolating family is a continuous one-parameter family { Ψ λ } λ ∈ [ -1 , 1] ⊂ M such that Ψ 0 ∈ C and Ψ 1 ∈ B . Given such a family, we may define the critical parameter λ ∗ and the critical solution D λ ∗ (the development of Ψ λ ∗ ) by  \nλ ∗ . = sup { λ ∈ [0 , 1] : Ψ λ ∈ C } .  \nThe prototypical critical collapse scenario consists of a spherically symmetric self-gravitating massless scalar field pulse with fixed profile and 'total energy' ∼ λ . At λ = 0 , the solution is Minkowski space and for λ very close to 0 , the solution disperses and is future complete [Chr86]. As λ approaches 1 , a trapped surface forms in evolution, signaling the formation of a black hole [Chr91]. This is precisely the scenario first studied numerically by Christodoulou in his thesis [Chr71] and then later by Choptuik in the influential work [Cho93]. Based on numerical evidence, it is believed that the critical solutions for these types of families are naked singularities that form a codimension-one 'submanifold' in moduli space. For discussion of Choptuik's results we refer to the survey [GM07].  \nRemark 1.3 . A codimension-one submanifold of naked singularities is nongeneric and therefore compatible with the weak cosmic censorship conjecture, which has been proved in this model by Christodoulou [Chr99b].  \nRemark 1.4 . A rigorous understanding of Choptuik's critical collapse scenario would in particular give a construction of naked singularities in the Einstein-scalar field system starting from smooth initial data, in contrast to Christodoulou's examples in [Chr94]. It already follows from work of Christodoulou [Chr91] that a critical solution cannot be a black hole in this model and from work of Luk and Oh that a critical solution cannot 'scatter in BV norm' [LO15]. This leaves the possibility of either a first singularity along the center not hidden behind an event horizon 5 or a solution in C \\ D which 'blows up at infinity.' Ruling out this latter case is an interesting open problem.  \nWhen massive fields are introduced, such as in the spherically symmetric Einstein-massive Klein-Gordon or Einstein-massive Vlasov systems, then static 'star-like' critical solutions can be observed numerically [BCG97; RRS98; OC02; AR06; AAR21]. These static solutions are nonsingular and lie in C \\ D . It is interesting to note that while Einstein-Klein-Gordon also displays Choptuik-like naked singularity critical solutions, there is no numerical evidence for the existence of naked singularities in the Einstein-Vlasov system. We again refer to [GM07] for references and would also like to point out the new development [Bau+23] on numerical critical collapse in vacuum.", '1.3 Extremal critical collapse': "So far, all numerically observed critical solutions are believed to be either naked singularities or complete and nondispersive. It follows at once from Penrose's incompleteness theorem [Pen65] and Cauchy stability that a critical solution cannot contain a trapped surface. While a generic black hole is expected to contain trapped surfaces, 6 members of the extremal Kerr-Newman family do not. In view of this, we raise the question of whether extremal black holes can arise on the black hole formation threshold:  \nDefinition 1.5. An interpolating family { Ψ λ } λ ∈ [0 , 1] exhibits extremal critical collapse if the critical solution D λ ∗ asymptotically settles down to an extremal black hole.  \nOur main result, Theorem 1, proves that the Einstein-Maxwell-Vlasov system exhibits extremal critical collapse, with critical solution D λ ∗ exactly isometric to extremal Reissner-Nordström in the domain of outer communication at late advanced times. As shown by Prószyński [Pró83] and the present authors in Theorem 4, the fundamentally singular thin charged shell and charged null dust models, respectively, exhibit extremal critical collapse, also with extremal Reissner-Nordström as the critical solution. We expect this phenomenon to also occur in the spherically symmetric Einstein-Maxwell-charged scalar field system and even for the Einstein vacuum equations, where the critical solution is expected to be based on the extremal Kerr solution. Note that we only require the asymptotic geometry of the critical solution to be an extremal black hole in Definition 1.5, which is a much weaker condition than being exactly extremal as in Theorem 1.  \nRemark 1.6 . Because black holes in the spherically symmetric Einstein-scalar field model always contain trapped surfaces [Chr91], this model does not exhibit extremal critical collapse. In particular, since the presence of a trapped surface in this model already implies completeness of null infinity and the existence of a black hole [Daf05b], B is open in the spherically symmetric Einstein-scalar field model .  \nRemark 1.7 . It is not possible for a Kerr solution with nonzero angular momentum (i.e., not Schwarzschild) to appear as the asymptotic state in axisymmetric vacuum gravitational collapse. This is because the Komar angular momentum (16 π ) -1 ∫ S ⋆dZ ♭ , where Z is the axial Killing vector field, is independent of the sphere S , which is nullhomologous. Similarly, it is not possible for a Kerr-Newman solution with nonzero charge (i.e., not Kerr) to appear as the asymptotic state in gravitational collapse for the Einstein-Maxwell system. This is because the charge (4 π ) -1 ∫ S ⋆F is independent of the sphere S , which is nullhomologous. The presence of charged matter is essential in Theorem 1.  \nRemark 1.8 (Stationary solutions and the extremal limit) . In the 1960s and '70s, it was suggested that astrophysical black holes could form through quasistationary accretion processes. In a landmark work, Bardeen and Wagoner [Bar70; BW71] numerically studied axisymmetric stationary states of the Einsteindust system (modeling accretion disks) and found that a 'black hole limit' was only possible in the 'extremal limit' of the dust configuration. 7 In this limit, the exterior metric of the disk converges, in a certain sense, to the metric of the domain of outer communication of extremal Kerr.  \nHowever, the event horizon of a stationary black hole is necessarily a Killing horizon and therefore an exactly stationary black hole solution cannot admit a one-ended asymptotically flat Cauchy hypersurface. It follows that a sequence of one-ended stationary states cannot actually smoothly converge to a black hole spacetime up to and including the event horizon, and that the black hole threshold cannot be directly probed by studying limits of stationary states-black hole formation is a fundamentally dynamical process.  \nNevertheless, there is a substantial body of numerical and heuristic literature exploring 'extremal black hole limits' of stationary solutions in dust models [NM95; Mei06; Mei+08; MH11; KLM11] and using Einstein-Yang-Mills-Higgs magnetic monopoles [LW99; LW00]; see also references therein. In particular, we refer the reader to [MH11] for a cogent explanation of the exact nature of the convergence of these stationary states to extremal Reissner-Nordström/Kerr exteriors and throats. It would be interesting to see if perturbing these 'near-extremal' non-black hole stationary states can provide another route to extremal critical collapse (and also perhaps to new examples of third law violating solutions), but this seems to be a difficult and fully dynamical problem.  \nRemark 1.9 (Overcharging and overspinning) . Extremal critical collapse should not be confused with the attempt to overcharge or overspin a black hole, i.e., the attempt to destroy the event horizon and create a 'superextremal naked singularity' by throwing charged or spinning matter into a (near-)extremal black hole. The fear of forming such a naked singularity provided some impetus for the original formulation of the third law in [BCH73] 8 and many arguments for and against have appeared in the literature, see [Wal74; Hub99;  \n8 With this in mind, the formulation of the third law in [BCH73] can be thought of as simply outright forbidding the formation of extremal black holes. The formulation in Israel's work [Isr86] is more refined and specifically refers to subextremal black holes 'becoming' extremal in a dynamical process. In any case, both formulations are false as shown in [KU22] and again in the present paper.  \nJS09; SW17] and references therein. Overcharging has been definitively disproved in spherical symmetry for the class of 'weakly tame' matter models [Daf05b; Kom13], which includes the Einstein-Maxwell-Vlasov system considered in this paper. We expect overcharging and overspinning to be definitively disproved with a positive resolution of the black hole stability problem for extremal black holes, to be discussed in Section 1.4 below.", '1.4 Stability of extremal critical collapse': "Before discussing the stability of our interpolating families in Theorem 1, we must first address the expected notion of stability for the domain of outer communication of the extremal Reissner-Nordström solution.  \nFirstly, since the asymptotic parameter ratio of the black hole is inherently unstable, we can at most expect a positive codimension stability statement for extremal Reissner-Nordström. This should be compared with the codimension-three nonlinear stability theorem of the Schwarzschild solution by Dafermos, Holzegel, Rodnianski, and Taylor [DHRT]: Only a codimension-three 'submanifold' of moduli space can be expected to asymptote to Schwarzschild, which has codimension three in the Kerr family (parametrized by the mass and specific angular momentum vector). In the case of Reissner-Nordström, the set of extremal solutions has codimension one in the full family. Indeed, any fixed parameter ratio subfamily of the Reissner-Nordström family has codimension one. See already Remark 1.12.  \nSecondly, and far less trivially, the stability problem for extremal black holes is complicated by the absence of the celebrated redshift effect , which acts as a stabilizing mechanism for the event horizon of subextremal black holes. The event horizon of extremal Reissner-Nordström (and axisymmetric extremal black holes in general) suffers from a linear instability known as the Aretakis instability [Are11a; Are11b; Are15; Ape22], which causes ingoing translation invariant null derivatives of solutions to the linear wave equation to (generically) either not decay, or to blow up polynomially along the event horizon as v → ∞ . Weissenbacher has recently shown that a similar instability (non-decay of the first derivative of the energymomentum tensor) occurs for the linear massless Vlasov equation on extremal Reissner-Nordström [Wei23].  \nHowever, the Aretakis instability is weak and does not preclude asymptotic stability and decay away from the event horizon . Including the horizon, we expect a degenerate type of stability, with decay in directions tangent to it, and possible non-decay and growth transverse to it (so-called horizon hair ). This behavior has been shown rigorously for a semilinear model problem on a fixed background [Ang16; AAG20] and numerically for the coupled spherically symmetric nonlinear Einstein-Maxwell-(massless and neutral) scalar field system [MRT13].  \nTo further complicate matters, the massive and massless Vlasov equations behave fundamentally differently and we state two separate conjectures. In these statements, we consider characteristic data posed on a bifurcate null hypersurface C out ∪ C in , where C out is complete and C in penetrates the event horizon in the case of trivial data. Solutions of the linear massless Vlasov equation decay exponentially on subextremal Reissner-Nordström black holes [Big23; Wei23] and Velozo Ruiz has proved nonlinear asymptotic stability of Schwarzschild for the spherically symmetric Einstein-massless Vlasov system [Vel23]. Based on this, [MRT13; Ang16; AAG20], and [DHRT, Conjecture IV.2], we make the  \nConjecture 1. The extremal Reissner-Nordström solution is nonlinearly asymptotically stable to spherically symmetric perturbations in the Einstein-Maxwell-massless Vlasov model in the following sense: Given sufficiently small characteristic data posed on a bifurcate null hypersurface C out ∪ C in and lying on a 'codimensionone submanifold' M stab (which contains the trivial solution) of the moduli space of such initial data, the maximal Cauchy development contains a black hole which asymptotically settles down to the domain of outer communication of an extremal Reissner-Nordström solution, away from the event horizon H + . Moreover, along the horizon, the solution decays towards extremal Reissner-Nordström in tangential directions, with possibly growing 'Vlasov hair' transverse to the horizon.  \nRemark 1.10 . There exist nontrivial spherically symmetric static solutions of the Einstein-massless Vlasov system containing a black hole which are isometric to a Schwarzschild solution in a neighborhood of the event horizon [And21]. However, these are not small perturbations of Schwarzschild (as the structure of trapping for null geodesics has to be significantly modified) and their existence is therefore consistent with [Vel23] and Conjecture 1.  \nFigure 3: A cartoon depiction of the conjectured structure of a neighborhood of moduli space near an interpolating family { Ψ λ } from Theorem 1. We have suppressed infinitely many dimensions and emphasize the codimension-one property of the critical 'submanifold' B crit which consists of asymptotically extremal black holes in accordance with Conjectures 1 and 2. The interpolating family { Ψ ' λ } is a small perturbation of { Ψ λ } which also crosses B crit and exhibits extremal critical collapse. Locally, B is foliated by 'hypersurfaces' B ( r ) consisting of black hole spacetimes with asymptotic parameter ratio r close to 1 .  \n<!-- image -->  \nThe massive Vlasov equation admits many nontrivial stationary states on black hole backgrounds, which is an obstruction to decay and we do not expect a general asymptotic stability statement to hold, even in the subextremal case. In fact, it has been shown that there exist spherically symmetric static solutions of Einstein-massive Vlasov bifurcating off of Schwarzschild [Rei94; Jab21]. We refer to [Vel23] for a characterization of the 'largest' region of phase space on which one can expect decay for the massive Vlasov energy-momentum tensor on a Schwarzschild background. However, one might still hope for orbital stability of the exterior, with a non-decaying Vlasov matter atmosphere , and that the horizon itself decays to that of extremal Reissner-Nordström:  \nConjecture 2. The extremal Reissner-Nordström solution is nonlinearly orbitally stable to spherically symmetric perturbations in the Einstein-Maxwell-massive Vlasov model in the following sense: Given sufficiently small characteristic data posed on a bifurcate null hypersurface C out ∪ C in and lying on a 'codimension-one submanifold' M stab of the moduli space of such initial data, the maximal Cauchy development contains a black hole which remains close to an extremal Reissner-Nordström solution in the domain of outer communication and asymptotically settles down to extremal Reissner-Nordström tangentially along the horizon, with possibly growing 'Vlasov hair' transverse to the horizon.  \nRemark 1.11 . We emphasize that this type of nonlinear orbital stability for massive Vlasov has not yet been proven even in the subextremal case, where we do not expect horizon hair to occur.  \nWith the conjectured description of the stability properties of the exterior of the critical solution at hand, we are now ready to state our conjecture for the global stability of the extremal critical collapse families in Theorem 1. Refer to Fig. 3 for a schematic depiction of this conjecture.  \nConjecture 3. Extremal critical collapse is stable in the following sense: Consider the moduli space M of the spherically symmetric Einstein-Maxwell-Vlasov system for particles of mass m . Let { Ψ λ } be one of the interpolating families given by Theorem 1. Then there exists a 'codimension-one submanifold' B crit of M such that Ψ 0 ∈ B crit ⊂ B , which has the following properties:  \n- 1. B crit is critical in the sense that B and C locally lie on opposite sides of B crit .\n- 2. If m = 0 and Ψ ∈ B crit , the domain of outer communication of the maximal Cauchy development of Ψ asymptotically settles down to an extremal Reissner-Nordström black hole as in Conjecture 1.\n- 3. If m > 0 and Ψ ∈ B crit , the domain of outer communication of the maximal Cauchy development of Ψ remains close to an extremal Reissner-Nordström black hole and the event horizon asymptotically settles down to an extremal Reissner-Nordström event horizon as in Conjecture 2.  \nTherefore, any nearby interpolating family { Ψ ' λ } also intersects B crit and exhibits extremal critical collapse.  \nRemark 1.12 . We further conjecture that given r ∈ [1 -ε, 1] for some ε > 0 , there exists a one-parameter family of disjoint 'codimension-one submanifolds' B ( r ) ⊂ B , varying 'continuously' in r , such that B (1) = B crit and if Ψ ∈ B ( r ) , then the maximal Cauchy development of Ψ contains a black hole which asymptotes to a Reissner-Nordström black hole with parameter ratio r = e f /M f , where M f is the final renormalized Hawking mass and e f is the final charge. One can then interpret equation (5.136) below as saying that the families { Ψ λ } in Theorem 1 intersect the foliation { B ( r ) } transversally , as depicted in Fig. 3.  \nWhile one should think that B crit in Conjecture 3 corresponds to M stab in Conjectures 1 and 2, Part 1 of Conjecture 3 is also a highly nontrivial statement about the interiors of the black holes arising from B crit . In particular, by the incompleteness theorem, it would imply that there are no trapped surfaces in the maximal developments of any member of B crit ; see [GL19, Remark 1.8] and the following remark.  \nRemark 1.13 . Using arguments from [LO19, Appendix A], one can show the following statement in the spherically symmetric Einstein-Maxwell-(neutral and massless) scalar field model: If the maximal Cauchy development of a partial Cauchy hypersurface 9 with ∂ u r < 0 contains a black hole with asymptotically extremal parameter ratio, then the development does not contain trapped symmetry spheres. The argument uses crucially the constancy of charge and absence of T uv in this model.", '1.5 Extremal critical collapse of a charged scalar field and in vacuum': "It is natural to conjecture the analog of Theorem 1 for a massless charged scalar field in spherical symmetry:  \nConjecture 4. Extremal critical collapse occurs in the spherically symmetric Einstein-Maxwell-charged scalar field model and there exist critical solutions which are isometric to extremal Reissner-Nordström in the domain of outer communication after sufficiently large advanced time.  \nIn [KU22], the present authors showed that a black hole with an extremal Reissner-Nordström domain of outer communication and containing no trapped surfaces can arise from regular one-ended Cauchy data in the spherically symmetric charged scalar field model (see Corollary 3 of [KU22]). The proof is based on a characteristic gluing argument, in which we glue a late ingoing cone in the interior of extremal ReissnerNordström to an ingoing cone in Minkowski space. The desired properties of the spacetime are obtained softly by Cauchy stability arguments. In particular, the method is inadequate to address whether the solution constructed in [KU22, Corollary 3] is critical.  \nIt is also natural to conjecture the analog of Theorem 1 for the Einstein vacuum equations,  \nRic( g ) = 0 , (1.7)  \nwhere the role of extremal Reissner-Nordström is played by the rotating extremal Kerr solution. 10  \nConjecture 5. Extremal critical collapse occurs in vacuum gravitational collapse and there exist critical solutions which are isometric to extremal Kerr in the domain of outer communication after sufficiently large advanced time.  \nIn [KU23], the present authors constructed examples of vacuum gravitational collapse which are isometric to Kerr black holes with prescribed mass M and specific angular momentum a , where M and a are any Kerr parameters satisfying 0 ≤ | a | /M ≤ a 0 for some small positive constant a 0 . Like in [KU22], the proof is by characteristic gluing and uses the perturbative and obstruction-free gluing results of AretakisCzimek-Rodnianski [ACR21] and Czimek-Rodnianski [CR22] as a black box. In particular, this provided a fundamentally new proof of Christodoulou's seminal theorem on black hole formation in vacuum [Chr09]. Our proof does not work for large values of a and whether extremal Kerr black holes can form in gravitational collapse remains open.  \nRemark 1.14 . In [KU23], the Cauchy data (¯ g, ¯ k ) are constructed with regularity H 7 / 2 -loc × H 5 / 2 -loc , which is well above the threshold for classical existence and uniqueness for the Einstein vacuum equations [HKM76;  \nFigure 4: Penrose diagram of a counterexample to the third law of black hole thermodynamics in the Einstein-Maxwell-Vlasov model from Theorem 2. The broken curve A ' is the outermost apparent horizon of the spacetime. This view is zoomed in on the Vlasov beam that charges up the subextremal black hole to extremality. We refer to Fig. 15 in Section 6.1 for diagrams of the entire spacetime.  \n<!-- image -->  \nPR07; Chr13]. This limited regularity is because the characteristic gluing results [ACR21; CR22] which we use as a black box are limited to C 2 regularity of transverse derivatives in the non-bifurcate case. Using the more recent spacelike gluing results of Mao-Oh-Tao [MOT23], it is possible to construct suitable Cauchy data in H s loc × H s -1 loc for any s .  \nIf extremal critical collapse involving the Kerr solution does occur, then one may also ask about stability as in Section 1.4. In this case, the question hangs on the stability properties of extremal Kerr, which are more delicate than for extremal Reissner-Nordström. While extremal Kerr is mode-stable [TdC20], axisymmetric scalar perturbations have been shown to exhibit the same non-decay and growth hierarchy as general scalar perturbations of extremal Reissner-Nordström [Are12; Are15]. In light of the newly discovered azimuthal instabilities of extremal Kerr by Gajic [Gaj23], in which growth of scalar perturbations already occurs at first order of differentiability, the full (in)stability picture of extremal Kerr may be one of spectacular complexity!", '1.6 The third law of black hole thermodynamics and event horizon jumping at extremality': "The techniques used to prove Theorem 1 can also be immediately used to disprove the third law in the Einstein-Maxwell-Vlasov model, which complements our previous disproof in the Einstein-Maxwell-charged scalar field model [KU22]. The present method has the advantage of constructing counterexamples which are past causally geodesically complete, like the spacetimes in Theorem 1.  \nTheorem 2. There exist smooth solutions of the Einstein-Maxwell-Vlasov system for either massless or massive particles that violate the third law of black hole thermodynamics: a subextremal Reissner-Nordström apparent horizon can evolve into an extremal Reissner-Nordström event horizon in finite advanced time due to the incidence of charged Vlasov matter.  \nMore precisely, there exist smooth, spherically symmetric, one-ended asymptotically flat Cauchy data for the Einstein-Maxwell-Vlasov system for either massive or massless particles such that the maximal globally hyperbolic development D has the following properties.  \n- 1. D contains a nonempty black hole region and for sufficiently large advanced times, the domain of outer communication, including the event horizon H + , is isometric to that of an extremal ReissnerNordström solution.\n- 2. D contains a causal diamond which is isometric to a causal diamond in a subextremal ReissnerNordström black hole, including an appropriate portion of the subextremal apparent horizon. This subextremal region contains an open set of trapped surfaces.  \n- 3. The outermost apparent horizon A ' of D has at least two connected components. One component of A ' coincides in part with the subextremal apparent horizon and the last component (with respect to v ) coincides with the extremal event horizon.\n- 4. D is past causally goedesically complete, possesses complete null infinities I + and I -, and is isometric to Minkowski space near the center { r = 0 } for all time.  \nRefer to Fig. 4 for a Penrose diagram of one of these solutions. Note the disconnectedness of the outermost apparent horizon A ' , which is necessary in third law violating spacetimes-see the discussion in Section 1.4.3 of [KU22]. It is striking that the Vlasov beams we construct in the proof of Theorem 2 do not even touch the subextremal apparent horizon, which should be compared with the hypothetical situation depicted in Fig. 1 of [Isr86]. As with Theorem 1, Theorem 2 is proved by desingularizing suitable bouncing charged null dust spacetimes which we construct in Section 4.4.  \nIt is now very natural to ask if some critical behavior can be seen in the examples from Theorem 2. They are clearly not candidates for critical collapse because they contain trapped surfaces. Nevertheless, by tuning the final charge to mass ratio of the outermost beam in Theorem 2 (subextremal to superextremal as in Theorem 1), we construct one-parameter families of solutions satisfying the following  \nTheorem 3. There exist smooth one-parameter families of smooth, spherically symmetric, one-ended asymptotically flat Cauchy data { Ψ λ } λ ∈ [ -1 , 1] for the Einstein-Maxwell-Vlasov system for either massive or massless particles with the following properties. Let D λ be a choice of maximal globally hyperbolic development 11 of Ψ λ for which the double null gauge ( u, v ) is continuously synchronized as a function of λ . (See already Assumption 6.1 and Remark 6.2 for the definition of continuous synchronization.) Then the following holds:  \n̸  \n- 1. For λ = 0 , D λ contains a black hole whose domain of outer communication is isometric to that of a subextremal Reissner-Nordström black hole with mass M λ and charge | e λ | < M λ for sufficiently large advanced times.\n- 2. D 0 contains a black hole whose domain of outer communication is isometric to that of an extremal Reissner-Nordström black hole with mass M 0 and charge | e 0 | = M 0 for sufficiently large advanced times.\n- 3. The location of the event horizon is discontinuous as a function of λ : Let u λ, H + denote the retarded time coordinate of the event horizon H + λ of D λ with respect to the continuously synchronized gauge ( u, v ) . Then λ ↦→ u λ, H + is continuous from the left but discontinuous from the right, and  \nlim λ → 0 + u λ, H + > lim λ → 0 -u λ, H + . (1.8)  \n4. The functions λ ↦→ M λ and λ ↦→ e λ are continuous from the left but discontinuous from the right, and  \nlim λ → 0 + M λ < lim λ → 0 -M λ , lim λ → 0 + | e λ | < lim λ → 0 -| e λ | , (1.9)  \nlim λ → 0 + | e λ | M λ < lim λ → 0 -| e λ | M λ = 1 , lim λ → 0 + r λ, H + < lim λ → 0 -r λ, H + , where r λ, H + . = M λ + √ M 2 λ -e 2 λ .  \n(1.10)  \nIn addition, for every λ ∈ [ -1 , 1] , D λ is past causally geodesically complete, possesses complete null infinities I + and I -, and is isometric to Minkowski space near the center { r = 0 } for all time.  \nFrom the perspective of the dynamical extremal black hole D 0 , an arbitrarily small perturbation to D λ with λ > 0 causes the event horizon to jump by a definite amount in u (i.e., not o (1) in λ ) and the parameter ratio to drop by a definite amount. The proof of Theorem 3 relies crucially on the absence of trapped surfaces in a double null neighborhood of the horizon in the solutions of Theorem 2, cf. Fig. 4. In the asymptotically subextremal case, trapped surfaces are expected to asymptote towards future timelike infinity i + . In this  \ncase, we prove in Proposition 6.4 below that the location of the event horizon is continuous as a function of initial data, under very general assumptions in spherical symmetry. Therefore, (1.8) is a characteristic feature of extremal black hole formation.  \nWe expect this 'local critical behavior' to be stable in the sense of Section 1.4 and to play a key role in the general stability problem for extremal black holes.  \nRemark 1.15 . By a suitable modification of the characteristic gluing techniques in [KU22], Theorem 3 can be proved for the spherically symmetric Einstein-Maxwell-charged scalar field model, but past completeness of the solutions does not follow immediately from our methods. It is also natural to conjecture analogs for Theorem 3 in (electro)vacuum; see in particular [DHRT, Section IV.2]. 12", '1.7 Outline of the paper': "Section 2. We introduce basic definitions and properties of general Einstein-matter systems and electromagnetic fields in spherical symmetry.  \nSection 3. We first recall the Einstein-Maxwell-Vlasov system in Section 3.1 and derive its spherically symmetric formulation in double null coordinates in Section 3.2. We also prove local well-posedness in Section 3.3 (the main iteration argument being deferred to Appendix A) and the generalized extension principle in Section 3.4. Finally, we define time-symmetric seed data sets and their developments in Section 3.5 which will play an important role in the paper.  \nSection 4. Before turning to the proof of our main result Theorem 1 in Section 5, we show in Section 4 that a singular toy model-Ori's bouncing charged null dust model-exhibits extremal critical collapse. We first recall the definition of the model in Section 4.1. We then introduce a radial parametrization of bouncing charged null dust spacetimes in Section 4.2 in which we teleologically prescribe a regular, spacelike, totally geodesic bounce hypersurface. These spacetimes consist of an explicit ingoing charged Vaidya metric pasted along the radially parametrized bounce hypersurface to an outgoing charged Vaidya metric through a physically motivated surgery procedure. In Sections 4.3 and 4.4, we use the radial parametrization to construct new examples of bouncing charged null dust spacetimes. In Section 4.3, we show that Ori's model exhibits extremal critical collapse (Theorem 4) and in Section 4.4, we show that the third law of black hole thermodynamics is false in Ori's model (Theorem 5). We then discuss the fundamental flaws of Ori's model in Section 4.5: the ill-posedness across the bounce hypersurface, the singular nature of the solutions, and the ill-posedness near the center. In Section 4.6, we conclude Section 4 with the formal radial charged null dust system in double null gauge which will be important for the setup of our initial data in Section 5.  \nSection 5. This section is devoted to the proof of our main result, Theorem 1. The proof relies crucially on a very specific teleological choice of Cauchy data which aims at desingularizing the dust examples of extremal critical collapse in Theorem 4, globally in time. In Section 5.1, we give a detailed guide to the proof of Theorem 1. In Section 5.2, we define the hierarchy of beam parameters and state the key ingredient in the proof of Theorem 1: the existence and global structure of outgoing charged Vlasov beams (Proposition 5.5). These beams arise from data posed on a Cauchy hypersurface which is analogous to the 'bounce hypersurface' associated to Ori's model in Section 4. In Section 5.3, we solve the constraint equations and prove estimates for the solution along the initial data hypersurface. Section 5.4 is devoted to estimates for the 'near region' establishing the bouncing character of our Vlasov beams. To overcome certain difficulties associated with low momenta, our construction features an 'auxiliary beam' which is treated in Section 5.5. Section 5.6 is concerned with the 'far region' and in Section 5.7 we prove the sharp dispersive estimates in the case of massive particles. In Section 5.8 we conclude the proof of Proposition 5.5. This proposition is then used to prove Theorem 1 in Section 5.9. Finally, in Section 5.10 we show that in a certain hydrodynamic limit of our parameters, the family of solutions constructed in Theorem 1 converge in a weak* sense to the family constructed in the charged null dust model in Section 4. This result rigorously justifies Ori's bouncing charged null dust construction from [Ori91].  \nSection 6. In this final section, we disprove the third law of black hole thermodynamics for the EinsteinMaxwell-Vlasov model (Theorem 2) in Section 6.1. Section 6.2 is concerned with the phenomenon of event horizon jumping at extremality. We first show in Proposition 6.4 that for a general class of (so-called weakly tame ) spherically symmetric Einstein-matter systems the retarded time coordinate of the event horizon is lower semicontinuous as a function of initial data. Secondly, we show by example (Theorem 3) that event horizon jumping can occur in the Einstein-Maxwell-Vlasov system for extremal horizons, which proves the sharpness of semicontinuity in Proposition 6.4.", 'Acknowledgments': 'The authors would like to express their gratitude to Mihalis Dafermos for many stimulating discussions. They would also like to thank Carsten Gundlach, István Kádár, Georgios Moschidis, Amos Ori, Frans Pretorius, Harvey Reall, Igor Rodnianski, Andrew Strominger, and Robert Wald for their interest, helpful discussions, and comments. C.K. acknowledges support by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation. R.U. thanks the University of Cambridge and ETH Zürich for hospitality as this work was being carried out.', '2.1 Double null gauge': "Let ( M , g ) be a smooth, connected, time-oriented, four-dimensional Lorentzian manifold. We say that ( M , g ) is spherically symmetric with (possibly empty) center of symmetry Γ ⊂ M if M\\ Γ splits diffeomorphically as ˚ Q× S 2 with metric  \ng = g Q + r 2 γ,  \nwhere ( Q , g Q ) , Q = ˚ Q∪ Γ , is a (1+1)-dimensional Lorentzian spacetime with boundary Γ (possibly empty), γ . = dϑ 2 + sin 2 ϑdφ 2 is the round metric on the unit sphere, and r is a nonnegative function on Q which can be geometrically interpreted as the area-radius of the orbits of the isometric SO(3) action on ( M , g ) . In a mild abuse of notation, we denote by Γ both the center of symmetry in M and its projection to Q . Moreover, if Γ is non-empty, we assume that the SO(3) action fixes Γ and that Γ consists of one timelike geodesic along which r = 0 . We further assume that ( Q , g Q ) admits a global double-null coordinate system ( u, v ) such that the metric g takes the form  \ng = -Ω 2 dudv + r 2 γ (2.1)  \nfor a positive function Ω 2 . = -2 g Q ( ∂ u , ∂ v ) on Q and such that ∂ u and ∂ v are future-directed. Our conventions are so that u = t -r and v = t + r give a double null coordinate system on (3 + 1) -dimensional Minkowski space, with r = 1 2 ( v -u ) and Ω 2 ≡ 1 . We will also use the notation / g . = r 2 γ . The constant u and v curves are null in ( Q , g Q ) and correspond to null hypersurfaces 'upstairs' in the full spacetime ( M , g ) . We further assume that along the center Γ , the coordinate v is outgoing and u is ingoing, i.e., ∂ v r | Γ > 0 , ∂ u r | Γ < 0 . We will often refer interchangeably to ( M , g ) and the reduced spacetime ( Q , r, Ω 2 ) .  \nRecall the Hawking mass m : M→ R , which is defined by  \nm . = r 2 (1 -g ( ∇ r, ∇ r ))  \nand can be viewed as a function on Q according to  \nm = r 2 ( 1 + 4 ∂ u r∂ v r Ω 2 ) . (2.2)  \nWe will frequently use the formula  \nΩ 2 = 4( -∂ u r ) ∂ v r 1 -2 m r (2.3)  \nto estimate Ω 2 when 1 -2 m r > 0 .  \nThe isometric action of SO(3) on ( M , g ) extends to the tangent bundle T M as follows: Let ϱ : SO(3) → Diff( M ) be the representation of SO(3) given by the spherically symmetric ansatz, so that the group action is given by  \nR · x . = ϱ ( R )( x )  \nfor R ∈ SO(3) and x ∈ M . For ( x, p ) ∈ T M , we define  \nR · ( x, p ) . = ( ϱ ( R )( x ) , ϱ ( R ) ∗ p ) , (2.4)  \nwhere of course ϱ ( R ) ∗ p lies in T ϱ ( R )( x ) M .  \nFinally, we note that the double null coordinates ( u, v ) above are not uniquely defined and for any strictly increasing smooth functions U, V : R → R , we obtain new global double null coordinates (˜ u, ˜ v ) = ( U ( u ) , V ( v )) such that g = -˜ Ω 2 d ˜ ud ˜ v + / g , where ˜ Ω 2 (˜ u, ˜ v ) = ( U ' V ' ) -1 Ω 2 ( U -1 (˜ u ) , V -1 (˜ v )) and r (˜ u, ˜ v ) = r ( U -1 (˜ u ) , V -1 (˜ v )) .", '2.2 Canonical coordinates on the tangent bundle': "Given local coordinates ( ϑ 1 , ϑ 2 ) on a (proper open subset of) S 2 , the quadruple ( u, v, ϑ 1 , ϑ 2 ) defines a local coordinate system on the spherically symmetric spacetime ( M , g ) . Given p ∈ T x M , we may expand  \np = p u ∂ u | x + p v ∂ v | x + p 1 ∂ ϑ 1 | x + p 2 ∂ ϑ 2 | x .  \nThe octuple ( u, v, ϑ 1 , ϑ 2 , p u , p v , p 1 , p 2 ) defines a local coordinate system on T M , and is called a canonical coordinate system on T M dual to ( u, v, ϑ 1 , ϑ 2 ) . One is to think of p as the 'momentum coordinate' and x as the 'position coordinate.' The tangent bundle of Q trivializes globally as Q× R 2 , with coordinates p u and p v on the second factor. We let  \nπ : T Q → Q  \ndenote the canonical projection.  \nOn a spherically symmetric spacetime, we define the angular momentum function by  \nℓ : T M→ [0 , ∞ ) ( x, p ) ↦→ √ r 2 / g AB p A p B ,  \nwhere summation over A,B ∈ { 1 , 2 } is implied. This function is independent of the angular coordinate system chosen and is itself spherically symmetric as a function on T M .  \nGiven a double null gauge ( u, v ) , it will be convenient to define a 'coordinate time' function  \nτ . = 1 2 ( v + u ) .  \nAssociated to this time function is the τ -momentum coordinate  \np τ . = 1 2 ( p v + p u ) .", '2.3 The Einstein equations and helpful identities in double null gauge': "A tensor field on a spherically symmetric spacetime is said to be spherically symmetric if it is itself invariant under the SO(3) -action of the spacetime. If ( M , g ) is a spherically symmetric spacetime satisfying the Einstein equations (1.1), then the energy-momentum tensor T is a spherically symmetric, symmetric (0 , 2) -tensor field. We may decompose  \nT = T uu du 2 + T uv ( du ⊗ dv + dv ⊗ du ) + T vv dv 2 + S / g,  \nwhere  \nS . = 1 2 / g AB T AB = 1 2 tr g T + 2 Ω 2 T uv .  \nIt will be convenient to work with the contravariant energy momentum tensor, which takes the form  \nT ♯♯ = T uu ∂ u ⊗ ∂ u + T uv ( ∂ u ⊗ ∂ v + ∂ v ⊗ ∂ u ) + T vv ∂ v ⊗ ∂ v + S / g -1 ,  \nwhere  \nT uu = 1 4 Ω 4 T vv , T uv = 1 4 Ω 4 T uv , T vv = 1 4 Ω 4 T uu .  \nThe Christoffel symbols involving null coordinates are given by  \nΓ u uu = ∂ u log Ω 2 , Γ v vv = ∂ v log Ω 2 , Γ u AB = 2 ∂ v r Ω 2 r / g AB , Γ v AB = 2 ∂ u r Ω 2 r / g AB , Γ A Bu = ∂ u r δ A B , Γ A Bv = ∂ v r δ A B ,  \nr r  \nand the totally spatial Christoffel symbols Γ A BC are the same as for γ in the coordinates ( ϑ 1 , ϑ 2 ) .  \nFor a spherically symmetric metric g written in double null gauge (2.1), the Einstein equations (1.1) separate into the wave equations for the geometry,  \n∂ u ∂ v r = -Ω 2 2 r 2 m + 1 4 r Ω 4 T uv , (2.5)  \n∂ u ∂ v log Ω 2 = Ω 2 m r 3 -1 2 Ω 4 T uv -Ω 2 S , (2.6)  \n∂ u ( ∂ u r Ω 2 ) = -1 4 r Ω 2 T vv , (2.7)  \n∂ v ( ∂ v r Ω 2 ) = -1 4 r Ω 2 T uu . (2.8)  \nThe Hawking mass (2.2) satisfies the equations  \n∂ u m = 1 2 r 2 Ω 2 ( T uv ∂ u r -T vv ∂ v r ) , (2.9)  \n∂ v m = 1 2 r 2 Ω 2 ( T uv ∂ v r -T uu ∂ u r ) . (2.10)  \nIf X is a spherically symmetric vector field, then  \nX  \n=  \nX  \n∂  \nu  \n+  \nX  \n∂  \nv  \nand X satisfies div g X = 0 if and only if  \n∂ u ( r 2 Ω 2 X u ) + ∂ v ( r 2 Ω 2 X v ) = 0 . (2.11)  \nThe contracted Bianchi identity,  \ndiv g T = 0 ,  \nwhich follows from the Einstein equations (1.1), implies the following pair of identities:  \n∂ u ( r 2 Ω 4 T uu ) + ∂ v ( r 2 Ω 4 T uv ) = ∂ v log Ω 2 r 2 Ω 4 T uv -4 r∂ v r Ω 2 S , (2.12)  \n∂ v ( r 2 Ω 4 T vv ) + ∂ u ( r 2 Ω 4 T uv ) = ∂ u log Ω 2 r 2 Ω 4 T uv -4 r∂ u r Ω 2 S . (2.13)  \nIf α is a spherically symmetric two-form which annihilates TS 2 , then it may be written as  \nα = -Ω 2 f 2 r 2 du ∧ dv,  \nwhere f : Q → R is a smooth function. We then have  \n∇ µ α uµ = ∂ u f r 2 , ∇ µ α vµ = -∂ v f r 2 . (2.14)  \nu  \nv  \nand Raychaudhuri's equations", '2.4 Spherically symmetric electromagnetic fields': "We will additionally assume that our spherically symmetric spacetime ( M , g ) carries a spherically symmetric electromagnetic field with no magnetic charge. The electromagnetic field is represented by a closed two-form  \nF , which takes the Coulomb form  \nF = -Ω 2 Q 2 r 2 du ∧ dv, (2.15)  \nfor a function Q : Q → R . The number Q ( u, v ) is the total electric charge enclosed in the ( u, v ) -symmetry sphere S u,v ⊂ M , which can be seen from the gauge-invariant formula  \nQ ( u, v ) = 1 4 π ∫ S u,v ⋆F, (2.16)  \nwhere ⋆ is the Hodge star operator and we orient M by du ∧ dv ∧ dϑ ∧ dφ .  \nThe electromagnetic energy momentum tensor is defined by  \nT EM µν . = F µ α F να -1 4 g µν F αβ F αβ , (2.17)  \nand relative to a double null gauge is given by  \nT EM = Ω 2 Q 2 4 r 4 ( du ⊗ dv + dv ⊗ du ) + Q 2 2 r 4 / g  \nin spherical symmetry. If F satisfies Maxwell's equation  \n∇ α F µα = J µ  \nfor a charge current J , then the divergence of the electromagnetic energy momentum tensor satisfies  \n∇ µ T EM µν = -F να J α . (2.18)  \nIn spherical symmetry, Maxwell's equations read (see (2.14))  \n∂ u Q = -1 2 r 2 Ω 2 J v , ∂ v Q = 1 2 r 2 Ω 2 J u .  \nFinally, we will utilize the renormalized Hawking mass  \nϖ . = m + Q 2 2 r (2.19)  \nto account for the contribution of the electromagnetic field to the Hawking mass m .", '2.5 The Lorentz force': 'We next briefly recall the Lorentz force law for the motion of a charged particle. Let ( M , g, F ) be a charged spacetime, where F is a closed 2-form representing the electromagnetic field. If γ is the worldline of a particle of mass m > 0 and charge e ∈ R , then it satisfies the Lorentz force equation  \nm Du µ dτ = e F µ ν u ν ,  \nwhere τ is proper time and u . = dγ/dτ (so that g ( u, u ) = -1 ). Defining the momentum of γ by p . = m u and rescaling proper time to s = m -1 τ (so that p = dγ/ds ), we can rewrite the Lorentz force equation as  \nDp µ ds = e F µ ν p ν . (2.20)  \nThis equation, which we call the electromagnetic geodesic equation , makes sense for null curves as well, and can be taken as the equation of motion for all charged particles [Ori91].  \nRemark 2.1 . Because the Lorentz force equation (2.20) is not quadratic in p , s is not an affine parameter, which has fundamental repercussions for the dynamics of the electromagnetic geodesic flow. In particular, trajectories of the Lorentz force with parallel, but not equal, initial velocities will in general be distinct, even up to reparametrization.  \nThe electromagnetic geodesic flow has two fundamental conserved quantities which will feature prominently in this work.  \nLemma 2.2. Let γ : I → M be a causal electromagnetic geodesic in a spherically symmetric charged spacetime, where I ⊂ R is an interval. Let p = dγ/ds . Then the rest mass  \nm [ γ ] . = √ -g γ ( p, p )  \nand the angular momentum  \nℓ [ γ ] . = ℓ ( γ, p )  \nare conserved quantities along γ .  \nProof. To show that m is constant, we compute  \nd ds g ( p, p ) = 2 g ( Dp ds , p ) = 2 e F ( p, p ) = 0 ,  \nwhere the final equality follows from the antisymmetry of F .  \nSince ℓ is independent of the coordinates chosen on S 2 , we may assume that ( ϑ 1 , ϑ 2 ) are normal coordinates on S 2 at the point ( γ 1 ( s 0 ) , γ 2 ( s 0 )) . To show that ℓ is constant, we then compute  \nd ds r 4 γ AB p A p B ∣ ∣ ∣ ∣ s = s 0 = 4 r 3 dr ds γ AB ˙ γ A ˙ γ B -2 r 4 γ AB ( -2Γ A Cu ˙ γ u -2Γ A Cv ˙ γ v ) ˙ γ B ˙ γ C = 0 ,  \nwhere we used the formulas for the Christoffel symbols in spherical symmetry from Section 2.3.  \nFor an electromagnetic geodesic γ ( s ) with angular momentum ℓ = ℓ [ γ ] and mass m = m [ γ ] , the Lorentz force law can be written as  \nd ds p u = -∂ u log Ω 2 ( p u ) 2 -2 ∂ v r r Ω 2 ℓ 2 r 2 -e Q r 2 p u , (2.21)  \nd ds p v = -∂ v log Ω 2 ( p v ) 2 -2 ∂ u r r Ω 2 ℓ 2 r 2 + e Q r 2 p v , (2.22)  \nΩ 2 p u p v = ℓ 2 r 2 + m 2 , (2.23)  \nwhere p u . = dγ u /ds , p v . = dγ v /ds , and the third equation, known as the mass shell relation , is directly equivalent to the definition of mass and angular momentum. In this work, it will not be necessary to follow the angular position of the electromagnetic geodesics in the (3 + 1) -dimensional spacetime. It is very convenient to rewrite (2.21) and (2.22) as  \nd ds (Ω 2 p u ) = ( ∂ v log Ω 2 -2 ∂ v r r ) ℓ 2 r 2 -e Q r 2 (Ω 2 p u ) , (2.24)  \nd ds (Ω 2 p v ) = ( ∂ u log Ω 2 -2 ∂ u r r ) ℓ 2 r 2 + e Q r 2 (Ω 2 p v ) . (2.25)', '3 The Einstein-Maxwell-Vlasov system': 'In this section, we introduce the main matter model considered in this paper, the Einstein-Maxwell-Vlasov system. This model describes an ensemble of collisionless self-gravitating charged particles which are either massive or massless. We begin by defining the general system outside of symmetry in Section 3.1 and then specialize to the spherically symmetric case in Section 3.2. In Sections 3.3 and 3.4, we formulate the fundamental local theory for this model, local existence and a robust continuation criterion known as the generalized extension principle . Finally, in Section 3.5, we define a procedure for solving the constraint equations for the spherically symmetric Einstein-Maxwell-Vlasov system.', '3.1.1 The volume form on the mass shell': 'Let ( M 4 , g ) be a spacetime. For x ∈ M and m ≥ 0 , we define the (future) mass shell at x by  \n̸  \nP m x . = { p ∈ T x M : p is future directed and g x ( p, p ) = -m 2 , p = 0 }  \nand the associated smooth fiber bundle  \nP m . = ⋃ x ∈M P m x ,  \nwith projection maps π m : P m →M .  \nFix x ∈ M and let ( x µ ) be normal coordinates at x so that  \ng x = -( dx 0 ) 2 +( dx 1 ) 2 +( dx 2 ) 2 +( dx 3 ) 2 .  \nLet p µ be dual coordinates to x µ on T M , defined by p µ = dx µ ( p ) for p ∈ T x M . For m ≥ 0 , the restrictions of p 1 , p 2 , p 3 to P m x , denoted by ¯ p 1 , ¯ p 2 , ¯ p 3 , define a global coordinate system on P m x , with p 0 determined by  \np 0 = √ m 2 + | ¯ p 1 | 2 + | ¯ p 2 | 2 + | ¯ p 3 | 2 . (3.1)  \nDefinition 3.1. Let m ≥ 0 and x ∈ M . The canonical volume form µ m x ∈ Ω 3 ( P m x ) is defined by  \nµ m x = ( p 0 ) -1 d ¯ p 1 ∧ d ¯ p 2 ∧ d ¯ p 3 ,  \nin normal coordinates at x , where p 0 is given by (3.1).  \nOne can show that this form is independent of the choice of normal coordinates. When m > 0 , P m x is a spacelike hypersurface in T x M if it is equipped with the Lorentzian metric g x . In this case, µ m x = m -1 ω m x , where ω m x is the induced Riemannian volume form on P m x . The factor of m -1 is needed to account for the degeneration of ω m x as m → 0 , since P 0 x is a null hypersurface. For more information about the volume form on the mass shell, see [SW77; Rin13; SZ14]. The canonical volume form is uniquely characterized by the following property, which can be found in [SW77, Corollary 5.6.2].  \nLemma 3.2. The form µ m x defined above does not depend on the choice of local coordinates on M . Moreover, it is uniquely characterized by the following property. Let H ( p ) . = 1 2 g x ( p, p ) . If α is a 3-form in T x M defined along P m x such that  \ndH ∧ α = √ -det g ( x ) dp 0 ∧ dp 1 ∧ dp 2 ∧ dp 3 ,  \nthen i ∗ m α = µ m x , where i m : P m x → T x M denotes the inclusion map.  \nWe denote the integration measure associated to µ m x by dµ m x . A distribution function is a nonnegative function f ∈ C ∞ ( P m ) which decays sufficiently quickly on the fibers so that the relevant integrals are welldefined and are smooth functions of x . Given a distribution function f we may now define the number current N and the energy momentum tensor T of f by  \nN µ ( x ) . = ∫ P m x p µ f ( x, p ) dµ m x , T µν ( x ) . = ∫ P m x p µ p ν f ( x, p ) dµ m x . (3.2)  \nThese are readily verified to be tensor fields on M . Taking divergences, we have [Rin13, Appendix D]  \n∇ µ N µ = ∫ P m x X 0 ( f ) dµ m x , ∇ µ T µν = ∫ P m x p ν X 0 ( f ) dµ m x , (3.3)  \nwhere X 0 . = p µ ∂ x µ -Γ µ νρ p ν p ρ ∂ p µ ∈ Γ( TT M ) is the geodesic spray vector field.', '3.1.2 The equations': "Definition 3.3. The Einstein-Maxwell-Vlasov system for particles of mass m ∈ R ≥ 0 and fundamental charge e ∈ R \\ { 0 } consists of a charged spacetime ( M , g, F ) and a distribution function f : P m → [0 , ∞ ) satisfying the following equations:  \nR µν -1 2 Rg µν = 2 ( T EM µν + T µν ) , (3.4)  \n∇ α F µα = e N µ , (3.5)  \nXf = 0 , (3.6)  \nwhere T EM is the electromagnetic energy momentum tensor defined in (2.17), T µν and N µ are the Vlasov energy-momentum tensor and number current, respectively, defined in (3.2), and  \nX . = p µ ∂ ∂x µ -( Γ µ αβ p α p β -e F µ α p α ) ∂ ∂p µ ∈ Γ( TT M ) (3.7)  \nis the electromagnetic geodesic spray vector field.  \nThe vector field X is easily seen to be tangent to the mass shell P m , which means that the Vlasov equation (3.6) is indeed a transport equation on P m . The vector field F µ α p α ∂ p µ is itself tangent to P m and we have the integration by parts formulas  \n∫ P m x F µ α p α ∂ p µ f dµ m x = 0 , ∫ P m x F µ α p ν p α ∂ p µ f dµ m x = -F ν α N α ,  \nwhich are easily verified in normal coordinates. Combined with (3.3) and the transport equation (3.6), we obtain the fundamental conservation laws  \n∇ µ N µ = 0 , (3.8)  \n∇ µ T µν = e N α F να . (3.9)  \nWe now see that the Einstein-Maxwell-Vlasov system is consistent: (3.8) implies that Maxwell's equation (3.5) is consistent with antisymmetry of F and (3.9) implies (using also (2.18)) the contracted Bianchi identity  \n∇ µ ( T EM µν + T µν ) = 0 . (3.10)  \nfor the total energy-momentum tensor of the system.", '3.1.3 Relation to the relativistic Maxwell-Vlasov system': "The system (1.2)-(1.4) includes gravity and thus generalizes the special relativistic Maxwell-Vlasov system which is typically written in the form 13 (cf. [Gla96])  \n∂ t f K + ˆ v · ∂ x f K + e ( E + ˆ v × B ) · ∂ v f K = 0 , ∂ t E -∇× B = -j K , ∂ t B + ∇× E = 0 , ∇· E = ρ K , ∇· B = 0 ,  \nwhere f K ( t, x, v ) ≥ 0 is the distribution function, ( t, x, v ) ∈ R × R 3 × R 3 , E is the electric field, B is the magnetic field, ˆ v . = ( m 2 + | v | 2 ) -1 / 2 v is the 'relativistic velocity' and has modulus smaller than unity, and  \nρ K ( t, x ) . = e ∫ R 3 f ( t, x, v ) dv, j K ( t, x ) . = e ∫ R 3 ˆ vf ( t, x, v ) dv.  \nThis system is equivalent to the covariant equations (3.5) and (3.6) in Minkowski space under the identifications ( √ m 2 + | v | 2 , v ) = p , f K ( t, x, v ) = f ( t, x, √ m 2 + | v | 2 , v ) , E i = F i 0 , B i = 1 2 ε i jk F jk , ρ K = e N 0 , and ( j K ) i = e N i .", '3.2 Spherically symmetric definitions and equations': "Let ( Q , r, Ω 2 ) be the (1+1) -dimensional reduced spacetime associated to a spherically symmetric spacetime. For m ≥ 0 , we define the reduced mass shell by  \nP m red . = { ( u, v, p u , p v ) ∈ T Q : Ω 2 ( u, v ) p u p v ≥ m 2 , p τ > 0 } , (3.11)  \nwhere the second condition in the definition forbids Ω 2 p u p v = 0 in the m = 0 case. The angular momentum function from Section 2.2 can be defined on the reduced mass shell by  \nℓ : P m red → R ≥ 0 ( u, v, p u , p v ) ↦→ r √ Ω 2 p u p v -m 2 .  \nNote that ℓ > 0 on P 0 red . The definition of ℓ can be rewritten as the fundamental mass shell relation  \nΩ 2 p u p v = ℓ 2 r 2 + m 2 . (3.12)  \nDefinition 3.4. A spherically symmetric distribution function of massive (if m > 0 ) or massless (if m = 0 ) particles is a smooth function  \nf : P m red → R ≥ 0 .  \nWe say that f has locally compact support in p if for every compact set K ⊂ Q there exists a compact set K ' ⊂ R 2 such that spt( f ) ∩ P m red | K ⊂ K × K ' . We say that f has locally positive angular momentum if for every K ⊂ Q compact there exists a constant c K > 0 such that ℓ ≥ c K on spt( f ) ∩ P m red | K .  \nIn order to define appropriate moments of a distribution function f on a spherically symmetric spacetime ( Q , r, Ω 2 ) , we require that f decays in the momentum variables p u and p v . For σ > 0 , k ≥ 0 an integer, and K ⊂ Q compact, we define the norm  \n∥ f ∥ C k σ ( P m red | K ) . = ∑ 0 ≤ i 1 + i 2 ≤ k sup P m red | K ⟨ p τ ⟩ σ + i 2 | ∂ i 1 x ∂ i 2 p f | , (3.13)  \nwhere ∂ i 1 x ∂ i 2 p f ranges over all expressions involving i 1 derivatives in the ( u, v ) -variables and i 2 derivatives in the ( p u , p v ) -variables. If the norm (3.13) is finite for all compact sets K ⊂ Q , we say that f ∈ C ∞ σ, loc ( P m red ) . If f has locally compact support in p , then it lies in C ∞ σ, loc ( P m red ) .  \nRemark 3.5 . Our well posedness theory for the Einstein-Maxwell-Vlasov system requires that p -derivatives of f decay faster, which is the reason for the i 2 weight in (3.13).  \nGiven a spherically symmetric spacetime ( Q , r, Ω 2 ) with distribution function f , we define the Vlasov number current by  \nN u ( u, v ) . = π Ω 2 ∫ Ω 2 p u p v ≥ m 2 p u f ( u, v, p u , p v ) dp u dp v , (3.14)  \nN v ( u, v ) . = π Ω 2 ∫ Ω 2 p u p v ≥ m 2 p v f ( u, v, p u , p v ) dp u dp v (3.15)  \nand the Vlasov energy momentum tensor by  \nT uu ( u, v ) . = π Ω 2 ∫ Ω 2 p u p v ≥ m 2 ( p u ) 2 f ( u, v, p u , p v ) dp u dp v , (3.16)  \nT uv ( u, v ) . = π Ω 2 ∫ Ω 2 p u p v ≥ m 2 p u p v f ( u, v, p u , p v ) dp u dp v , (3.17)  \nT vv ( u, v ) . = π Ω 2 ∫ Ω 2 p u p v ≥ m 2 ( p v ) 2 f ( u, v, p u , p v ) dp u dp v , (3.18)  \nS ( u, v ) . = π 2 Ω 2 ∫ Ω 2 p u p v ≥ m 2 (Ω 2 p u p v -m 2 ) f ( u, v, p u , p v ) dp u dp v ≤ Ω 2 2 T uv ( u, v ) . (3.19)  \nIf f ∈ C ∞ σ, loc ( P m red ) with σ > 4 , then these moments are well defined smooth functions of u and v .  \nDefinition 3.6. The spherically symmetric Einstein-Maxwell-Vlasov model for particles of mass m ∈ R ≥ 0 and fundamental charge e ∈ R \\{ 0 } consists of a smooth spherically symmetric charged spacetime ( Q , r, Ω 2 , Q ) and a smooth distribution function f ∈ C ∞ σ, loc ( P m red ) for a decay rate σ > 4 fixed. When m = 0 , we require that f also has locally positive angular momentum. To emphasize that the distribution functions we consider in this paper have these regularity properties in p , we say that such a solution has admissible momentum .  \nThe system satisfies the wave equations  \n∂ u ∂ v r = -Ω 2 2 r 2 ( m -Q 2 2 r ) + 1 4 r Ω 4 T uv , (3.20)  \n∂ u ∂ v log Ω 2 = Ω 2 m r 3 -Ω 2 Q 2 r 4 -1 2 Ω 4 T uv -Ω 2 S, (3.21)  \n∂ u ( ∂ u r Ω 2 ) = -1 4 r Ω 2 T vv , (3.22)  \n∂ v ( ∂ v r Ω 2 ) = -1 4 r Ω 2 T uu , (3.23)  \n∂ u Q = -1 2 e r 2 Ω 2 N v , (3.24)  \n∂ v Q = + 1 2 e r 2 Ω 2 N u , (3.25)  \nwhere N u , N v , T uu , T uv , T vv , and S are defined by equations (3.14)-(3.19). Finally, f satisfies the spherically symmetric Vlasov equation  \nXf = 0 , (3.26)  \nwhere X ∈ Γ( TP m red ) is the reduced electromagnetic geodesic spray  \nX . = p u ∂ u + p v ∂ v -( ∂ u log Ω 2 ( p u ) 2 + 2 ∂ v r r Ω 2 (Ω 2 p u p v -m 2 ) + e Q r 2 p u ) ∂ p u -( ∂ v log Ω 2 ( p v ) 2 + 2 ∂ u r r Ω 2 (Ω 2 p u p v -m 2 ) -e Q r 2 p v ) ∂ p v . (3.27)  \nRemark 3.7 . Since f ≥ 0 for a solution of the Einstein-Maxwell-Vlasov system, the components N u and N v of the number current are nonnegative. It follows from the Maxwell equations (3.24) and (3.25) that e Q is decreasing in u and increasing in v , unconditionally. This monotonicity property is a fundamental feature of the spherically symmetric Einstein-Maxwell-Vlasov system and will be exploited several times in this paper.  \nRemark 3.8 . Both the electromagnetic energy-momentum tensor T EM and the Vlasov energy-momentum tensor T of the Einstein-Maxwell-Vlasov system satisfy the dominant energy condition.  \nRemark 3.9 . As an abuse of notation, we have denoted the spray (3.7) on T M and the spray (3.27) on T Q by the same letter X . It will always be clear from the context which vector field we are referring to. They are related by the pushforward along the natural projection map P m → P m red .  \nFor a solution of the Einstein-Maxwell-Vlasov system, the Hawking mass m satisfies the equations  \n∂ u m = 1 2 r 2 Ω 2 ( T uv ∂ u r -T vv ∂ v r ) + Q 2 2 r 2 ∂ u r, (3.28)  \n∂ v m = 1 2 r 2 Ω 2 ( T uv ∂ v r -T uu ∂ u r ) + Q 2 2 r 2 ∂ v r, (3.29)  \nwhich can be derived from (2.9) and (2.10). The modified Hawking mass ϖ can then be seen to satisfy  \n∂ u ϖ = 1 2 r 2 Ω 2 ( T uv ∂ u r -T vv ∂ v r ) -1 2 e r Ω 2 QN v , (3.30)  \n∂ v ϖ = 1 2 r 2 Ω 2 ( T uv ∂ v r -T uu ∂ u r ) + 1 2 e r Ω 2 QN u . (3.31)  \nthe Raychaudhuri equations  \nand the Maxwell equations  \nThe particle current N satisfies the conservation law  \n∂ u ( r 2 Ω 2 N u ) + ∂ v ( r 2 Ω 2 N v ) = 0 (3.32)  \nby (2.11) and (3.8). Alternatively, it can be directly derived from the spherically symmetric Vlasov equation, which we will do the proof of Proposition 3.14 in Appendix A. Finally, for the Einstein-Maxwell-Vlasov system, the Bianchi identities (2.12) and (2.13) read  \n∂ u ( r 2 Ω 4 T uu ) + ∂ v ( r 2 Ω 4 T uv ) = ∂ v log Ω 2 r 2 Ω 4 T uv -4 r∂ v r Ω 2 S -e Ω 4 QN u , (3.33)  \n∂ v ( r 2 Ω 4 T vv ) + ∂ u ( r 2 Ω 4 T uv ) = ∂ u log Ω 2 r 2 Ω 4 T uv -4 r∂ u r Ω 2 S + e Ω 4 QN v . (3.34)  \nAgain, this follows either from (3.10) or directly from the spherically symmetric equations.", '3.2.1 The spherically symmetric reduction': 'Proposition 3.10. Let ( Q , r, Ω 2 , Q, f sph ) be a solution of the spherically symmetric Einstein-MaxwellVlasov system as defined by Definition 3.6. Then ( M , g, F, f ) solves the Einstein-Maxwell-Vlasov system if we lift the solution according to  \nM  \n.  \n=  \nQ×  \nS  \n2  \n,  \ng . = -Ω 2 2 ( du ⊗ dv + dv ⊗ du ) + r 2 γ, (3.35)  \nF . = -Q 2 r 2 du ∧ dv, (3.36)  \nf ( u, v, ϑ 1 , ϑ 2 , p u , p v , p 1 , p 2 ) . = f sph ( u, v, p u , p v ) , (3.37)  \nwhere ( ϑ 1 , ϑ 2 ) is a local coordinate system on S 2 .  \nNote that f in (3.37) is SO(3) -invariant as a function on T M as defined in Section 2.1.  \nProof. As the equations (3.20)-(3.25) are equivalent to the Einstein equations and Maxwell equations, we must only check that f , defined by (3.37), satisfies Xf = 0 , where X is given by (3.7), and that the spherically symmetric formulas (3.14)-(3.19) appropriately reconstruct the (3 + 1) -dimensional number current and energy-momentum tensor.  \nLet γ ( s ) be an electromagnetic geodesic. Then Xf = 0 at ( γ ( s 0 ) , ˙ γ ( s 0 )) ∈ P m is equivalent to  \nd ds ∣ ∣ ∣ ∣ s = s 0 f ( γ ( s ) , ˙ γ ( s )) = 0 . (3.38)  \nUsing the chain rule  \nd ds ∣ ∣ ∣ ∣ s = s 0 f sph ( γ u ( s ) , γ v ( s ) , p u ( s ) , p v ( s )) = p u ∂ u f sph + p v ∂ v f sph + dp u ds ∂ p u f sph + dp v ds ∂ p v f sph ,  \nthe spherically symmetric Lorentz force equations (2.21) and (2.22), the mass shell relation (2.23), and the spherically symmetric Vlasov equation (3.26), we see that (3.38) holds.  \nTo compute the energy-momentum tensor in spherical symmetry, we use Lemma 3.2. Let x ∈ M and take ( ϑ 1 , ϑ 2 ) to be local coordinates on S 2 which are normal at the spherical component of x , so that γ AB = δ AB . We have  \nTherefore, if we define  \nthen  \ndH = -Ω 2 2 p v dp u -Ω 2 2 p u dp v + r 2 p 1 dp 1 + r 2 p 2 dp 2 .  \nα . = -r 2 ( p v ) -1 dp v ∧ dp 1 ∧ dp 2 ,  \ndH ∧ α = 1 2 Ω 2 r 2 dp u ∧ dp v ∧ dp 1 ∧ dp 2 .  \nTherefore, by Lemma 3.2,  \ndµ m x = r 2 ( p v ) -1 dp v dp 1 dp 2  \nas measures on (0 , ∞ ) × R 2 ( p 1 ,p 2 ) . The remaining momentum variable p u is obtained from p v , p 1 , and p 2 via the mass shell relation (3.12). If we set tan β = p 2 /p 1 and use that ℓ 2 = r 4 (( p 1 ) 2 +( p 2 ) 2 ) , then  \ndµ m x = r -2 ( p v ) -1 dp v ℓ dℓ dβ.  \nFor any weight function w = w ( p u , p v ) , we therefore have  \n∫ (0 , ∞ ) × R 2 wf dµ m x = r -2 ∫ 2 π 0 ∫ ∞ 0 ∫ ∞ 0 w ( ℓ 2 + m 2 r 2 r 2 Ω 2 p v , p v ) f sph ( u, v, ℓ 2 + m 2 r 2 r 2 Ω 2 p v , p v ) dp v p v ℓ dℓ dβ. (3.39)  \nIntegrating out β and applying the coordinate transformation ( p v , ℓ ) ↦→ ( p u , p v ) reproduces the formulas (3.14)-(3.19) for N and T .  \nRemark 3.11 . Other works on the spherically symmetric Einstein-Vlasov system in double null gauge, such as [DR16; Mos18; Mos23], represent the distribution function f differently, opting to (at least implicitly) eliminate either p u or p v in terms of ℓ using the mass shell relation (3.12). This leads to different formulas for N µ and T µν , as these will then involve an integral over ℓ , as in (3.39). To make this precise, we can define the outgoing representation 14 of the spherically symmetric f by  \nf ↗ ( u, v, p v , ℓ ) . = f ( u, v, ℓ 2 + r 2 m 2 r 2 Ω 2 p v , p v ) , (3.40)  \nand (3.39) implies, for instance,  \nT uv = 2 π r 4 Ω 2 ∫ ∞ 0 ∫ ∞ 0 ℓ 2 + r 2 m 2 p v f ↗ ( u, v, p v , ℓ ) dp v ℓ dℓ.  \nThe outgoing representation may be taken as an alternative definition of the spherically symmetric Vlasov system. We have chosen the formulation here in terms of p u and p v because of its explicit symmetry, which is key for constructing time-symmetric initial data in the proof of Theorem 1. We have also chosen to always write N µ and T µν in contravariant form, so that N u is associated with p u , etc. This causes extra factors of g uv = -1 2 Ω 2 to appear in various formulas, compared to [DR16; Mos18; Mos23].', '3.2.2 Previous work on the Einstein-Maxwell-Vlasov system': 'Besides the general local existence result of [BC73], the Einstein-Maxwell-Vlasov model does not seem to have been extensively studied. Dispersion for small data solutions of the Einstein-Maxwell-massive Vlasov model (stability of Minkowski space) in spherical symmetry was proved by Noundjeu [Nou05]. See also [NN04] for local well-posedness in Schwarzschild coordinates and [NNR04] for the existence of nontrivial solutions of the constraints. Many static solutions are known to exist for the massive system, first studied numerically by Andréasson-Eklund-Rein [AER09] and proved rigorously by Thaller [Tha19] in spherical symmetry. Thaller has also shown that stationary and axisymmetric (but not spherically symmetric) solutions exist [Tha20].', '3.3 Local well-posedness in spherical symmetry': "Electromagnetic geodesics, in contrast to ordinary geodesics, can have limit points in M . By standard ODE theory, this can only happen if p ( s ) → 0 as s → ±∞ . 15 On a fixed spherically symmetric background, one can show that an electromagnetic geodesic γ cannot have a limit point if either m [ γ ] > 0 or ℓ [ γ ] > 0 . However, even in the massless case, an electromagnetic geodesic with initially positive momentum will still have positive momentum for a short (coordinate) time. Therefore, one can show that local well-posedness in double null gauge holds in the case of massive particles or massless particles with momentum initially supported away from zero.  \nRemark 3.12 . Bigorgne has shown that the relativistic Maxwell-massless Vlasov system is classically illposed if the initial data are allowed to be supported near zero momentum [Big22]. We expect a similar result to hold for the Einstein-Maxwell-massless Vlasov system.  \nWenow state our fundamental local well-posedness result for the spherically symmetric Einstein-MaxwellVlasov system. We formulate this in terms of the characteristic initial value problem , though the techniques used apply to the Cauchy problem as well. Note that we work in function spaces that allow for noncompact support in the momentum variables, although this is not needed for the applications in this paper (but is useful in the context of cosmic censorship [DR16]). The proof of local existence is deferred to Appendix A.  \nGiven U 0 < U 1 and V 0 < V 1 , let  \nC ( U 0 , U 1 , V 0 , V 1 ) . = ( { U 0 } × [ V 0 , V 1 ]) ∪ ([ U 0 , U 1 ] ×{ V 0 } ) , R ( U 0 , U 1 , V 0 , V 1 ) . = [ U 0 , U 1 ] × [ V 0 , V 1 ] .  \nWe will consistently omit ( U 0 , U 1 , V 0 , V 1 ) from the notation for these sets when the meaning is clear. A function ϕ : C → R is said to be smooth if it is continuous and ϕ | { U 0 }× [ V 0 ,V 1 ] and ϕ | [ U 0 ,U 1 ] ×{ V 0 } are C ∞ single-variable functions. This definition extends naturally to functions f : P m red | C → R ≥ 0 .  \nDefinition 3.13. A smooth (bifurcate) characteristic initial data set for the spherically symmetric EinsteinMaxwell-Vlasov system with parameters m , e , and σ consists of smooth functions ˚ r, ˚ Ω 2 , ˚ Q : C → R with ˚ r and ˚ Ω 2 positive, and a smooth function ˚ f : P m red | C → R ≥ 0 , where P m red | C is defined using ˚ Ω 2 . Moreover, we assume that the norms  \n∥ ˚ f ∥ C k σ ( P | C ) . = ∑ 0 ≤ i 1 + i 2 ≤ k ( sup P κ | { U 0 }× [ V 0 ,V 1 ] ⟨ p τ ⟩ σ + i 2 | ∂ i 1 v ∂ i 2 p ˚ f | + sup P κ | [ U 0 ,U 1 ] ×{ V 0 } ⟨ p τ ⟩ σ + i 2 | ∂ i 1 u ∂ i 2 p ˚ f | )  \nare finite for every k ≥ 0 . In the case m = 0 , we also assume that ˚ f has locally positive angular momentum. Finally, we assume that Raychaudhuri's equations (3.22) and (3.23), together with Maxwell's equations (3.24) and (3.25) are satisfied to all orders in directions tangent to C .  \nProposition 3.14. For any m ≥ 0 , e ∈ R , σ > 4 , B > 0 , and c ℓ > 0 , there exists a constant ε loc > 0 with the following property. Let (˚ r, ˚ Ω 2 , ˚ Q, ˚ f ) be a characteristic initial data set for the spherically symmetric Einstein-Maxwell-Vlasov system on C ( U 0 , U 1 , V 0 , V 1 ) . If U 1 -U 0 < ε loc , V 1 -V 0 < ε loc ,  \n∥ log˚ r ∥ C 2 ( C ) + ∥ log ˚ Ω 2 ∥ C 2 ( C ) + ∥ ˚ Q ∥ C 1 ( C ) + ∥ ˚ f ∥ C 1 σ ( P m red | C ) ≤ B,  \nand in the case m = 0 , ℓ ≥ c ℓ on spt( ˚ f ) , then there exists a unique smooth solution ( r, Ω 2 , Q, f ) of the spherically symmetric Einstein-Maxwell-Vlasov system on R ( U 0 , U 1 , V 0 , V 1 ) which extends the initial data. If ˚ f has locally compact support in p , then so does f . Moreover, the norms  \n∥ log r ∥ C k ( R ) , ∥ log Ω 2 ∥ C k ( R ) , ∥ Q ∥ C k ( R ) , ∥ f ∥ C k σ ( P m red | R )  \nare finite for any k and can be bounded in terms of appropriate higher order initial data norms.  \nThe proof of the proposition is given in Appendix A.2.", '3.4 The generalized extension principle': "Recall that a spherically symmetric Einstein-matter model is said to satisfy the generalized extension principle if any 'first singularity' either emanates from a point on the spacetime boundary with r = 0 , or its causal past has infinite spacetime volume. This property has been shown to hold for the Einstein-massless scalar field system by Christodoulou [Chr93], for the Einstein-massive Vlasov system by Dafermos and Rendall [DR16], and for the Einstein-Maxwell-charged Klein-Gordon system by Kommemi [Kom13]. In this paper, we extend the generalized extension principle of Dafermos-Rendall to the Einstein-Maxwell-Vlasov system:  \nProposition 3.15 (The generalized extension principle) . Let ( Q , r, Ω 2 , Q, f ) be a smooth solution of the spherically symmetric Einstein-Maxwell-Vlasov system with admissible momentum, which is defined on an open set Q ⊂ R 2 u,v . If Q contains the set R ' . = R ( U 0 , U 1 , V 0 , V 1 ) \\{ ( U 1 , V 1 ) } and the following two conditions are satisfied:", "1. R ' has finite Lorentzian volume, i.e.,": "∫∫ R ' Ω 2 dudv < ∞ , (3.41)  \n- 2. the area-radius is bounded above and below, i.e.,  \nsup R ' | log r | < ∞ , (3.42)  \nthen the solution extends smoothly, with admissible momentum, to a neighborhood of ( U 1 , V 1 ) .  \nTherefore, since this system satisfies the dominant energy condition (Remark 3.8), the Einstein-MaxwellVlasov system is strongly tame in Kommemi's terminology [Kom13], under the admissible momentum assumption. This is an important 'validation' of the Einstein-Maxwell-Vlasov model over the charged null dust model and means the model enjoys Kommemi's a priori boundary characterization [Kom13], which will be used in Section 6.2.1 below.  \nProposition 3.15 is also used crucially in the proof of Theorem 1 because it provides a continuation criterion at zeroth order. This allows us to avoid commutation when treating the singular 'main beam' in the construction of bouncing charged Vlasov beams.  \nWe now give the proof of Proposition 3.15, assuming the 'fundamental local spacetime estimate' to be stated and proved in Section 3.4.2 below. The proof of the local estimate is based on a streamlining of the ideas already present in [DR16] together with the monotonicity of charge inherent to the Einstein-MaxwellVlasov system and a quantitative lower bound on the 'coordinate time momentum' p u + p v obtained from the mass shell relation.  \nProof of Proposition 3.15. By Lemma 3.16 below, (3.41) and (3.42) imply that  \nB . = ∥ log r ∥ C 2 ( R ' ) + ∥ log Ω 2 ∥ C 2 ( R ' ) + ∥ Q ∥ C 1 ( R ' ) + ∥ f ∥ C 1 σ ( P m red | R ' ) < ∞ .  \nLet U ' 1 > U 1 and V ' 1 > V 1 be such that the segments [ U 1 , U ' 1 ] ×{ V 0 } and { U 0 } × [ V 1 , V ' 1 ] lie inside of Q and let c ℓ > 0 be a lower bound for ℓ on spt( f ) ∩ P m red | C ( U 0 ,U ' 1 ,V 0 ,V ' 1 ) if m = 0 . Let ε loc > 0 be the local existence time for the spherically symmetric Einstein-Maxwell-Vlasov system with parameters ( m , e , σ, 2 B,c ℓ ) given by Proposition 3.14. Fix ( ˜ U, ˜ V ) ∈ R ' with U 1 -˜ U < ε loc and V 1 -˜ V < ε loc .  \nObserve that if U 2 > U 1 is sufficiently close to U 1 and V 2 > V 1 is sufficiently close to V 1 , then  \nB . = ∥ log r ∥ C 2 ( ˜ C ) + ∥ log Ω 2 ∥ C 2 ( ˜ C ) + ∥ Q ∥ C 1 ( ˜ C ) + ∥ f ∥ C 1 σ ( P m red | ˜ C ) ≤ 2 B  \nand ℓ ≥ c ℓ on spt( f ) ∩ P m red | ˜ C if m = 0 , where ˜ C . = C ( ˜ U,U 2 , ˜ V , V 2 ) . Indeed, this is clear for log r , log Ω 2 , and Q by smoothness of these functions on Q . For f , we can also easily show this using the mean value theorem and the finiteness of ∥ f ∥ C 2 σ ( P m red | K ) on compact sets K ⊂ Q . For ℓ , this follows immediately from conservation of angular momentum and the domain of dependence property if U 2 ≤ U ' 1 and V 2 ≤ V ' 1 .  \nTherefore, by Proposition 3.14, the solution extends smoothly, with admissible momentum, to the rectangle R ( ˜ U,U 2 , ˜ V , V 2 ) , which contains ( U 1 , V 1 ) . This completes the proof.", '3.4.1 Horizontal lifts and the commuted Vlasov equation': 'Local well-posedness for ( r, Ω 2 , Q, f ) takes place in the space C 2 × C 2 × C 1 × C 1 , since the Christoffel symbols and electromagnetic field need to be Lipschitz regular to obtain a unique classical solution of the Vlasov equation (3.26). In order to estimate ∂ 2 u Ω 2 and ∂ 2 v Ω 2 , one has to commute the wave equation for Ω 2 , (3.21), with ∂ u and ∂ v . This commuted equation contains terms such as ∂ u T uv , which can only be estimated by first estimating ∂ u f and ∂ v f . On the other hand, naively commuting the spherically symmetric Vlasov equation, (3.26), with spatial derivatives introduces highest order nonlinear error terms such as ∂ 2 u Ω 2 ∂ p u f . 16 Therefore, it would appear that the system does not close at this level of regularity.  \nHowever, as was observed by Dafermos and Rendall in [DR05a] in the case of Einstein-Vlasov (see also the erratum of [RR92]), the horizontal lifts  \nˆ ∂ u f . = ∂ u f -p u ∂ u log Ω 2 ∂ p u f,  \nˆ ∂ v f . = ∂ v f -p v ∂ v log Ω 2 ∂ p v f  \nof ∂ u f and ∂ v f with respect to the Levi-Civita connection satisfy a better system of equations without these highest order errors. In the case of Einstein-Maxwell-Vlasov, we directly commute (3.26) with { ˆ ∂ u , ˆ ∂ v , ∂ u , ∂ v } to obtain  \nX ( ˆ ∂ u f ) = p u ∂ u log Ω 2 ˆ ∂ u f + ( p u ∂ u log Ω 2 ∂ p u ζ u -∂ u ζ u -∂ u log Ω 2 ζ u ) ∂ p u f + ( ∂ u ∂ v log Ω 2 ( p u ) 2 -∂ u ζ v + p u ∂ u log Ω 2 ∂ p u ζ v ) ∂ p v f,  \n(3.43)  \nX ( ˆ ∂ v f ) = p v ∂ v log Ω 2 ˆ ∂ v f + ( ∂ u ∂ v log Ω 2 ( p v ) 2 -∂ v ζ u + p v ∂ v log Ω 2 ∂ p v ζ u ) ∂ p u f + ( p v ∂ v log Ω 2 ∂ p v ζ v -∂ v ζ v -∂ v log Ω 2 ζ v ) ∂ p v f, (3.44)  \nX ( ∂ p u f ) = -ˆ ∂ u f +(3 p u ∂ u log Ω 2 -∂ p u ζ u ) ∂ p u f -∂ p u ζ v ∂ p v f, (3.45)  \nX ( ∂ p v f ) = -ˆ ∂ v f -∂ p v ζ u ∂ p u f +(3 p v ∂ v log Ω 2 -∂ p v ζ v ) ∂ p v f, (3.46)  \nζ u . = 2 ∂ v r r Ω 2 (Ω 2 p u p v -m 2 ) + e Q r 2 p u , ζ v . = 2 ∂ u r r Ω 2 (Ω 2 p u p v -m 2 ) -e Q r 2 p v .  \nUpon using the wave equation (3.21), we see that the right-hand sides of (3.43)-(3.46) do not contain second derivatives of Ω 2 .', '3.4.2 The fundamental local spacetime estimate': "Lemma 3.16. For any m ≥ 0 , e ∈ R , σ > 4 , and C 0 > 0 , there exists a constant C ∗ < ∞ with the following property. Let ( r, Ω 2 , Q, f ) be a solution of the spherically symmetric Einstein-Maxwell-Vlasov system with admissible momentum for particles of charge e , mass m , and momentum decay rate σ defined on R ' . = R ( U 0 , U 1 , V 0 , V 1 ) \\ { ( U, V ) } . Assume U 1 -U 0 ≤ C 0 , V 1 -V 0 ≤ C 0 , the initial data estimates  \n∥ log r ∥ C 2 ( C ) + ∥ log Ω 2 ∥ C 2 ( C ) + ∥ Q ∥ C 1 ( C ) + ∥ f ∥ C 1 σ ( P m red | C ) ≤ C 0 , (3.47)  \nthe global estimates  \n∫∫ R ' Ω 2 dudv ≤ C 0 , (3.48)  \nsup R ' | log r | ≤ C 0 , (3.49)  \ninf spt( f ) ∩ P m red | C ℓ ≥ C -1 0 . (3.50)  \nThen we have the estimate  \n∥ log r ∥ C 2 ( R ' ) + ∥ log Ω 2 ∥ C 2 ( R ' ) + ∥ Q ∥ C 1 ( R ' ) + ∥ f ∥ C 1 σ ( P m red | R ' ) ≤ C ∗ .  \nProof. In this proof, we use the notation A ≲ 1 to mean that A ≤ C , where C is a constant depending only on m , e , and σ , and C 0 . When writing area integrals, we will also make no distinction between R and R ' , though the integrands are strictly speaking only assumed to be defined on R ' .  \nFrom (3.47) and the monotonicity properties of Maxwell's equations (3.24) and (3.25), it follows that  \nsup R ' | Q | ≲ 1 . (3.51)  \nRewriting (3.20), we obtain  \nwhere  \nand in the case m = 0 , assume also that  \nr 2 Ω 4 T uv = 2 ∂ u ∂ v r 2 +Ω 2 ( 1 -Q 2 r 2 ) . (3.52)  \nIntegrating over R ' and using (3.47), (3.48), and (3.49) yields  \n∫∫ R Ω 4 T uv dudv ≲ ∫∫ R r 2 Ω 4 T uv dudv ≲ ∫∫ R ∂ u ∂ v r 2 dudv + ∫∫ R Ω 2 dudv ≲ 1 . (3.53)  \nRewriting (3.52) slightly, we obtain  \n∂ u ( r∂ v r ) = -Ω 2 4 ( 1 -Q 2 r 2 ) + 1 4 r 2 Ω 4 T uv . (3.54)  \nIntegrating this in u and using (3.47), (3.49), and (3.51), we have  \nsup [ U 0 ,U 1 ] ×{ v } | r∂ v r | ≲ 1 + ∫ U 1 U 0 Ω 2 ( u, v ) du + ∫ U 1 U 0 Ω 4 T uv ( u, v ) du  \nfor any v ∈ [0 , V ] . Integrating this estimate in v and using (3.49), (3.48), and (3.53) yields  \n∫ V 1 V 0 sup [ U 0 ,U 1 ] ×{ v } | ∂ v r | dv ≲ ∫ V 1 V 0 sup [ U 0 ,U 1 ] ×{ v } | r∂ v r | dv ≲ 1 . (3.55)  \nBy Raychaudhuri's equation (3.22), ∂ u r changes signs at most once on each ingoing null cone. Therefore, by the fundamental theorem of calculus and (3.49),  \nsup v ∈ [ V 0 ,V 1 ] ∫ U 1 U 0 | ∂ u r | ( u, v ) du ≤ 2 ( sup R ' r -inf R ' r ) ≲ 1 . (3.56)  \nCombining (3.55) and (3.56) yields  \n∫∫ R | ∂ u r∂ v r | dudv ≤ ( ∫ V 1 V 0 sup [ U 0 ,U 1 ] ×{ v } | ∂ v r | dv )( sup v ∈ [ V 0 ,V 1 ] ∫ U 1 U 0 | ∂ u r | ( u, v ) du ) ≲ 1 . (3.57)  \nUsing the definition of the Hawking mass (2.2), (3.49), and (3.48), we readily infer  \n∫∫ R Ω 2 | m | dudv ≲ 1 (3.58)  \nBy the wave equation (3.21), the fundamental theorem of calculus, and the estimates (3.19), (3.48), (3.49), (3.51), (3.53), and (3.58), we have  \nsup R ' | log Ω 2 | ≲ 1 + ∣ ∣ ∣ ∣ ∫∫ R ∂ u ∂ v log Ω 2 dudv ∣ ∣ ∣ ∣ ≲ 1 + ∫∫ R ( Ω 2 +Ω 2 | m | +Ω 4 T uv ) dudv ≲ 1 .  \nWe now prove estimates for the electromagnetic geodesic flow. Let γ : [0 , S ] →R ' be an electromagnetic geodesic such that ( γ (0) , p (0)) ∈ spt( f ) ∩ P m red | C . We aim to prove that  \n∣ ∣ ∣ ∣ log ( Ω 2 p u ( s ) Ω 2 p u (0) )∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ log ( Ω 2 p v ( s ) Ω 2 p v (0) )∣ ∣ ∣ ∣ ≲ 1 (3.59)  \nfor s ∈ [0 , S ] , where the implied constant does not depend on γ .  \nIt suffices to prove this estimate for p u , as the proof of the estimate for p v differs only in notation. Following [Mos18] (see also [Daf06]), we write an integral formula for log(Ω 2 p u ) , which can then be estimated using our previous area estimates. Rewriting the mass shell relation (3.12) as  \nℓ 2 r 2 = ( ℓ 2 ℓ 2 + m 2 r 2 ) Ω 2 p u p v ,  \nwe deduce from the Lorentz force equation (2.24) that  \nd ds log(Ω 2 p u ) = ( ∂ v log Ω 2 -2 ∂ v r r )( ℓ 2 ℓ 2 + m 2 r 2 ) p v -e Q r 2 .  \nIntegrating in s and changing variables yields  \nlog ( Ω 2 p u ( s ) Ω 2 p u (0) ) = ∫ γ ([0 ,s ]) ( ∂ v log Ω 2 -2 ∂ v r r )( ℓ 2 ℓ 2 + m 2 r 2 ) dv -∫ s 0 e Q r 2 ∣ ∣ ∣ ∣ γ ( s ' ) ds ' .  \nWe use the fundamental theorem of calculus on the first integral to obtain  \n∫ γ ([0 ,s ]) ( ∂ v log Ω 2 -2 ∂ v r r )( ℓ 2 ℓ 2 + m 2 r 2 ) dv = ∫ γ v ( s ) γ v (0) ∫ γ u ( s v ) 0 ∂ u [( ∂ v log Ω 2 -2 ∂ v r r )( ℓ 2 ℓ 2 + m 2 r 2 )] dudv + ∫ { 0 }× [0 ,γ v ( s )] ( ∂ v log Ω 2 -2 ∂ v r r )( ℓ 2 ℓ 2 + m 2 r 2 ) dv,  \nwhere s v ∈ [0 , S ] is defined by γ v ( s v ) = v . Using now the wave equations (3.20) and (3.21), we arrive at  \nlog ( Ω 2 p u ( s ) Ω 2 p u (0) ) = ∫ γ v ( s ) γ v (0) ∫ γ u ( s v ) 0 ( 3Ω 2 m r 3 -3Ω 2 Q 2 2 r 4 -Ω 2 2 r 2 -Ω 4 T uv -Ω 2 S )( ℓ 2 ℓ 2 + m 2 r 2 ) dudv + ∫ γ v ( s ) γ v (0) ∫ γ u ( s v ) 0 ( 2 ∂ u r∂ v r -∂ u r∂ v log Ω 2 ) 2 m 2 ℓ 2 ( ℓ 2 + m 2 r 2 ) 2 dudv + ∫ { 0 }× [0 ,γ v ( s )] ( ∂ v log Ω 2 -2 ∂ v r r ) dv -∫ s 0 e Q r 2 ∣ ∣ ∣ ∣ γ ( s ' ) ds ' . (3.60)  \nWe now bound each of the four terms in (3.60). Using (3.48), (3.49), (3.51), (3.53), and (3.58), the first double integral in (3.60) is readily seen to be uniformly bounded. To estimate the second double integral, we integrate the wave equation (3.21) in u to obtain  \nsup [0 ,U ] ×{ v } | ∂ v log Ω 2 | ≲ 1 + ∫ U 0 ( Ω 2 | m | +Ω 2 +Ω 4 T uv ) ( u, v ) du  \nfor any v ∈ [0 , V ] . Integrating this estimate in v and using the previous area estimates yields  \n∫ V 0 sup [0 ,U ] ×{ v } | ∂ v log Ω 2 | dv ≲ 1  \nwhich when combined with (3.56) gives  \n∫∫ R | ∂ u r || ∂ v log Ω 2 | dudv ≲ 1 . (3.61)  \nCombined with (3.57), we now readily see that the second double integral in (3.60) is uniformly bounded. The integral along initial data is clearly also bounded by assumption.  \nUsing (3.49) and (3.51), we estimate  \n∫ s 0 e | Q | r 2 ds ≲ S. (3.62)  \nTo estimate S , define the function τ ( s ) . = τ | γ ( s ) , which is strictly increasing and satisfies 0 ≤ τ ( s ) ≲ 1 for every s ∈ [0 , S ] . Using the mass shell relation (3.12), we have  \n√ 4 Ω 2 ( ℓ 2 r 2 + m 2 ) ≤ p τ = dτ ds . (3.63)  \nSince either m > 0 or (3.50) holds, it follows that dτ/ds is uniformly bounded away from zero, which implies S ≲ 1 for any electromagnetic geodesic in the support of f . Combined with (3.62), this uniformly bounds the final term in (3.60) and completes the proof of (3.59).  \nSince f is constant along ( γ ( s ) , p ( s )) , we therefore have ⟨ p τ ( s ) ⟩ σ f ( γ ( s ) , p ( s )) ≲ ⟨ p τ (0) ⟩ σ f ( γ (0) , p (0)) , which implies  \n∥ f ∥ C 0 σ ( P m red | R ' ) +sup R ' ( N u + N v + T uu + T uv + T vv + S ) ≲ 1 .  \nBy (3.24) and (3.25),  \nsup R ' ( | ∂ u Q | + | ∂ v Q | ) ≲ 1 .  \nBy integrating (3.54) and also using that ∂ u ( r∂ v r ) = ∂ v ( r∂ u r ) , we now readily estimate  \nsup R ' ( | ∂ u r | + | ∂ v r | ) ≲ 1 .  \nAs this bounds the Hawking mass pointwise, we can now estimate  \nsup R ' ( | ∂ u log Ω 2 | + | ∂ v log Ω 2 | + | ∂ u ∂ v log Ω 2 | ) ≲ 1  \nusing (3.21). It then follows immediately from (3.20), (3.22), and (3.23) that  \nsup R ' ( | ∂ 2 u r | + | ∂ u ∂ v r | + | ∂ 2 v r | ) ≲ 1 .  \nAlong an electromagnetic geodesic γ lying in the support of f , we have that  \n| X ( ˆ ∂ u f ) | ≲ p τ | ˆ ∂ u f | +( p τ ) 2 | X ( ∂ p u f ) | +( p τ ) 2 | X ( ∂ p v f ) | , | X ( ˆ ∂ v f ) | ≲ p τ | ˆ ∂ v f | +( p τ ) 2 | X ( ∂ p u f ) | +( p τ ) 2 | X ( ∂ p v f ) | , | X ( ∂ p u f ) | ≲ | ˆ ∂ u f | + p τ | X ( ∂ p u f ) | + p τ | X ( ∂ p v f ) | , | X ( ∂ p v f ) | ≲ | ˆ ∂ v f | + p τ | X ( ∂ p u f ) | + p τ | X ( ∂ p v f ) |  \nby (3.43)-(3.46) and all of the estimates obtained so far. It follows that, defining  \nF ( s ) . = ( ( p τ ) σ ˆ ∂ u f, ( p τ ) σ ˆ ∂ v f, ( p τ ) σ +1 ∂ p u f, ( p τ ) σ +1 ∂ p v f ) ( s )  \nalong γ ( s ) , we have  \n∣ ∣ ∣ ∣ d ds F ∣ ∣ ∣ ∣ ≲ p τ |F| .  \nBy Grönwall's inequality and (3.63), it follows that |F| ≲ 1 along γ . Recovering ∂ u f and ∂ v f from ˆ ∂ u f and ˆ ∂ v f , we now readily bound  \n∥ f ∥ C 1 σ ( P m red | R ' ) +sup R ' ( | ∂ u T uv | + | ∂ v T uv | + | ∂ u S | + | ∂ v S | ) ≲ 1 .  \nCommuting the wave equation (3.21) with ∂ u and ∂ v , we obtain the final estimates  \nsup R ' ( | ∂ 2 u log Ω 2 | + | ∂ 2 v log Ω 2 | ) ≲ 1 ,  \nwhich completes the proof.", '3.4.3 Local existence in characteristic slabs': "The spacetime estimate Lemma 3.16 can also be used to improve Proposition 3.14 to local existence in a full double null neighborhood of a bifurcate characteristic hypersurface of arbitrary length:  \nProposition 3.17. For any m ≥ 0 , e ∈ R , σ > 4 , B > 0 , and c ℓ > 0 , there exists a constant ε slab > 0 with the following property. Let (˚ r, ˚ Ω 2 , ˚ Q, ˚ f ) be a characteristic initial data set for the spherically symmetric Einstein-Maxwell-Vlasov system on C ( U 0 , U 1 , V 0 , V 1 ) with admissible momentum. If  \n∥ log˚ r ∥ C 2 ( C ) + ∥ log ˚ Ω 2 ∥ C 2 ( C ) + ∥ ˚ Q ∥ C 1 ( C ) + ∥ ˚ f ∥ C 1 σ ( P m red | C ) ≤ B  \nand either m > 0 or m = 0 and ℓ ≥ c ℓ on spt( ˚ f ) , then there exists a unique smooth solution ( r, Ω 2 , Q, f ) of the spherically symmetric Einstein-Maxwell-Vlasov system with admissible momentum on  \nR ( U 0 , U 0 +min { ε slab , U 1 -U 0 } , V 0 , V 1 ) ∪ R ( U 0 , U 1 , V 0 , V 0 +min { ε slab , V 1 -V 0 } )  \nwhich extends the initial data.  \nProof. We prove existence in the slab which is thin in the u -direction, the proof in the other slab being identical. Let C 0 = 10 B and let C ∗ be the constant obtained from the fundamental local spacetime estimate Lemma 3.16 with this choice. Let A ⊂ [ V 0 , V 1 ] denote the set of ˜ V such that the solution exists on R ( U 0 , U ' 1 , V 0 , ˜ V ) , where U ' 1 . = U 0 +min { ε slab , U 1 -U 0 } and satisfies the estimates  \nsup R ( U 0 ,U ' 1 ,V 0 , ˜ V ) | log r | + sup R ( U 0 ,U ' 1 ,V 0 , ˜ V ) | log Ω 2 | ≤ C 0 . (3.64)  \nWe will show that if ε slab = min { ε loc ( C ∗ ) , B ( U 1 -U 0 ) -1 C -1 ∗ } , then A is nonempty, closed, and open. Nonemptyness follows from Proposition 3.14 and closedness by continuity of the bootstrap assumptions. Let now ˜ V ∈ A . To improve the bootstrap assumptions, we note that | ∂ u ∂ v log r | ≤ C ∗ on R ( U 0 , U ' 1 , V 0 , ˜ V ) by Lemma 3.16, whence | log r | ≤ ε slab ( U 1 -U 0 ) C ∗ + 3 B ≤ 1 2 C 0 by the fundamental theorem of calculus. A similar argument applies for log Ω 2 . Therefore, by applying Proposition 3.14 again, a simple continuity argument shows that ˜ V + η ∈ A for η > 0 sufficiently small.", '3.5 Time-symmetric seed data and their normalized developments': "In the proof of Theorem 1, we will pose data for the Einstein-Maxwell-Vlasov system on a mixed spacelikenull hypersurface, with the Vlasov field f supported initially on the spacelike hypersurface and away from the center. The initial data is given by a compactly supported distribution function ˚ f on the spacelike hypersurface, a numerical parameter that fixes the location of the initial outgoing null cone, together with the mass and charge of the particles. As we will only consider time-symmetric initial configurations, these data are sufficient to uniquely determine the solution.  \nDefinition 3.18. A time-symmetric seed data set S . = ( ˚ f, r 2 , m , e ) for the spherically symmetric EinsteinMaxwell-Vlasov system consists of a real numbers r 2 ∈ R > 0 , m ∈ R ≥ 0 , and e ∈ R , together with a compactly supported nonnegative function ˚ f ∈ C ∞ ( (0 , ∞ ) r × (0 , ∞ ) p u × (0 , ∞ ) p v ) which is symmetric in the second and third variables, ˚ f ( · , p u , p v ) = ˚ f ( · , p v , p u ) , and satisfies spt ( ˚ f ( · , p u , p v ) ) ⊂ (0 , r 2 ] for every p u , p v ∈ (0 , ∞ ) .  \nGiven a seed data set S = ( ˚ f, r 2 , m , e ) and r ∈ [0 , r 2 ] , we define  \n˚ ϱ ( r ) . = π ∫ ∞ 0 ∫ ∞ 0 ˚ f ( v, p u , p v ) dp u dp v , (3.65)  \n˚ N u ( r ) . = ˚ N v ( r ) . = π ∫ ∞ 0 ∫ ∞ 0 p u ˚ f ( r, p u , p v ) dp u dp v , (3.66)  \n˚ T uu ( r ) . = ˚ T vv ( r ) . = π ∫ ∞ 0 ∫ ∞ 0 ( p u ) 2 ˚ f ( r, p u , p v ) dp u dp v , (3.67)  \n˚ T uv ( r ) . = π ∫ ∞ 0 ∫ ∞ 0 p u p v ˚ f ( r, p u , p v ) dp u dp v . (3.68)  \nRemark 3.19 . These formulas are missing a factor of Ω 2 compared to (3.14)-(3.19). This is because Ω 2 is not explicitly known on the initial data hypersurface and is accounted for by extra factors of Ω 2 in the constraint system (3.69)-(3.70) below.  \nFigure 5: Penrose diagram of a normalized development U of a time symmetric seed S . The spacelike hypersurface { τ = 0 } is totally geodesic, i.e., time symmetric, and the outgoing cone { u = -r 2 } has f = 0 . To the left of the support of f , the spacetime is vacuum: both the Hawking mass m and charge Q vanish identically. For the significance of the cone { v = r 0 } , see already Remark 3.24.  \n<!-- image -->  \nLet the functions ˚ m = ˚ m ( r ) and ˚ Q = ˚ Q ( r ) be the unique solutions of the first order system  \nd dr ˚ m = r 2 4 ( 1 -2˚ m r ) -2 ( ˚ T uu +2 ˚ T uv + ˚ T vv ) + ˚ Q 2 2 r 2 , (3.69)  \nd dr ˚ Q = 1 2 e r 2 ( 1 -2˚ m r ) -2 ( ˚ N u + ˚ N v ) (3.70)  \nwith initial conditions ˚ m (0) = 0 and ˚ Q (0) = 0 . If  \nsup r ∈ [0 ,r 2 ] 2˚ m r < 1 ,  \nthen ˚ m and ˚ Q exist on the entire interval [0 , r 2 ] and we say that S is untrapped . We also define  \n˚ Ω 2 . = ( 1 -2˚ m r ) -1 , ˚ ϖ . = ˚ m + ˚ Q 2 2 r .  \nFinally, we say that S is consistent with particles of mass m if ˚ Ω 2 ( r ) p u p v ≥ m 2 for every ( r, p u , p v ) ∈ spt ˚ f . Remark 3.20 . We have not attempted to formulate the most general notion of seed data for the spherically symmetric Einstein-Maxwell-Vlasov Cauchy problem here as it is not needed for our purposes.  \nAssociated with time-symmetric seed data as in Definition 3.18, we will introduce normalized developments of such data in the following. For r 2 > 0 , let  \nC r 2 . = { τ ≥ 0 } ∩ { v ≥ u } ∩ { u ≥ -r 2 } ⊂ R 2 u,v  \nand let U r 2 denote the collection of connected relatively open subsets U ⊂ { v ≥ u } ⊂ R 2 u,v for which there exists a (possibly empty) achronal curve ζ ⊂ C r 2 , extending from the center { u = v } and reaching the cone { u = -r 2 } , such that U = C r 2 ∩ { u + v < ζ u + ζ v } and { τ = 0 } ∩ { 0 ≤ v ≤ r 2 } ⊂ U . See Fig. 5.  \nWe also define the cones  \nC u 0 . = U ∩ { u = u 0 } , C v 0 . = U ∩ { v = v 0 } .  \nDefinition 3.21. Let S = ( ˚ f, r 2 , m , e ) be an untrapped time-symmetric seed data set which is consistent with particles of mass m . A normalized development of S consists of a domain U ∈ U r 2 and a spherically symmetric solution ( r, Ω 2 , Q, f ) of the Einstein-Maxwell-Vlasov system for particles of mass m and fundamental charge e defined over U \\ { u = v } such that the following holds.  \n- 1. For every ( v, p u , p v ) ∈ (0 , r 2 ] × (0 , ∞ ) × (0 , ∞ ) ,  \nr ( -v, v ) = v, (3.71)  \n∂ v r ( -v, v ) = 1 2 , (3.72)  \n∂ u r ( -v, v ) = -∂ v r ( -v, v ) , (3.73)  \nΩ 2 ( -v, v ) = ˚ Ω 2 ( v ) , (3.74)  \n∂ v Ω 2 ( -v, v ) = 1 2 ( d dr ˚ Ω 2 ) ( v ) , (3.75)  \n∂ u log Ω 2 ( -v, v ) = -∂ v log Ω 2 ( -v, v ) , (3.76)  \nf ( -v, v, p u , p v ) = ˚ f ( v, p u , p v ) . (3.77)  \n- 2. Along the initial outgoing null cone C -r 2 ,  \nr ( -r 2 , v ) = 1 2 r 2 + 1 2 v, (3.78)  \nΩ 2 = ˚ Ω 2 ( r 2 ) , (3.79)  \nQ = ˚ Q ( r 2 ) , (3.80)  \nf = 0 . (3.81)  \n- 3. The functions r , Ω 2 , Q , and m extend smoothly to the center Γ . = U ∩ { u = v } and satisfy there the boundary conditions  \nr = m = Q = 0 , (3.82)  \n∂ u r < 0 , ∂ v r > 0 . (3.83)  \n- 4. Let γ : [0 , S ) →U\\ Γ be a future-directed electromagnetic geodesic such that r ( γ ( s )) → 0 as s → S . 17 Then ( γ u ( s ) , γ v ( s ) , p u ( s ) , p v ( s )) attains a limit on Γ , say ( u ∗ , v ∗ , p u ∗ , p v ∗ ) , and there exists a unique electromagnetic geodesic γ ' : ( S, S + ε ) →U\\ Γ for some ε > 0 such that ( γ ' u ( s ) , γ ' v ( s ) , p ' u ( s ) , p ' v ( s )) → ( u ∗ , v ∗ , p u ∗ , p v ∗ ) as s → S . We then require that  \nlim s ↗ S f ( γ u ( s ) , γ v ( s ) , p u ( s ) , p v ( s )) = lim s ↘ S f ( γ ' u ( s ) , γ ' v ( s ) , p ' u ( s ) , p ' v ( s )) .  \nWe use the adjective 'normalized' to emphasize the choice of a development with double null gauge anchored to the data as in points 1. and 2. above.  \nRemark 3.22 . The 'time-symmetric' aspect of the development is captured by the first equalities in (3.67) and (3.68), and the equations (3.73) and (3.76). One can moreover easily verify, using (3.73), (3.76), and the formulas for the Christoffel symbols in Section 2.3, that { τ = 0 } ∩ U is a totally geodesic spacelike hypersurface with respect to the (3 + 1) -dimensional metric (2.1).  \nFor a normalized development of seed data, we clearly have  \nN u = ˚ Ω 2 ˚ N u , N v = ˚ Ω 2 ˚ N v , T uu = ˚ Ω 2 ˚ T uu , T vv = ˚ Ω 2 ˚ T vv , T uv = ˚ Ω 2 ˚ T uv , S = ˚ Ω 4 2 ˚ T uv -˚ Ω 2 m 2 ˚ ϱ, Q = ˚ Q, ϖ = ˚ ϖ  \nalong { τ = 0 } ∩ U .  \nProposition 3.23. Let S be an untrapped time-symmetric seed data set which is consistent with particles of mass m . Then there exists a δ > 0 and a unique normalized development ( r, Ω 2 , Q, f ) of S defined on { 0 ≤ τ < δ } ∩ C r 2 .  \nProof. Using essentially the same methods as the proof of Proposition 3.14 in Appendix A, we obtain a unique local smooth solution ( r, Ω 2 , Q, f ) to the system of equations (3.20), (3.21), (3.24), and (3.26), with initial data given by (3.71)-(3.81). It remains to show that the constraints (3.22), (3.23), and (3.25) hold.  \nBy the same calculation as in the proof of Proposition 3.14, equation (3.26) implies the conservation law (3.32) for N . Let v ∈ (0 , r 2 ) . By integration of (3.24),  \nQ ( u, v ) = ˚ Q ( v ) -∫ u -v 1 2 e r 2 Ω 2 N v du '  \nfor u ≥ -v . Differentiating in v , using (3.70), (3.32), and the fundamental theorem of calculus yields (3.25) at ( u, v ) .  \nTo prove that (3.22) and (3.23) hold, we argue as in the proof of Proposition 3.14. Therefore, it suffices to show that (3.23) holds on initial data (the corresponding argument for (3.22) being the same). By (3.78) and (3.81), (3.23) clearly holds on the initial outgoing cone. By taking the absolute v -derivative of ∂ v r ( -v, v ) = 1 2 , we obtain ∂ 2 v r ( -v, v ) = ∂ u ∂ v r ( -v, v ) . Therefore, using (3.20), (3.69), and (3.75), we readily compute  \n∂ 2 v r -∂ v r∂ v log Ω 2 + 1 4 r Ω 4 T vv = ∂ u ∂ v r -1 4Ω 2 d dr ( 1 -2˚ m r ) -1 + 1 4 r Ω 4 T vv = -Ω 2 m 2 r 2 + Ω 2 Q 2 4 r 3 + 1 4 r Ω 4 T uv -1 4Ω 2 ( -2Ω 4 ˚ m r 2 + r Ω 6 2 ( T uu +2 T uv + T vv ) + Ω 4 Q 2 r 3 ) + 1 4 r Ω 4 T vv = 0 ,  \nwhere every function is being evaluated at ( -v, v ) . This is equivalent to (3.23) and completes the proof.  \nRemark 3.24 . Let r 0 ∈ (0 , r 2 ) be such that ˚ f ( r, p u , p v ) = 0 if r ∈ (0 , r 0 ] . Since ˚ f is assumed to be compactly supported, such an r 0 necessarily exists. Then if ( U , r, Ω 2 , Q, f ) is a normalized development of S , the portion of the triangle { v ≤ r 0 } inside of U is identically Minkowskian in the sense that  \nr = 1 2 ( v -u ) , Ω 2 = 1 , Q = f = 0  \non U ∩ { v ≤ r 0 } . In fact, we may therefore assume that any normalized development of S contains the full corner C r 2 ∩ { v ≤ r 0 } .  \nRemark 3.25 . One can verify that a normalized development as in Definition 3.21 defines a solution of the constraint equations associated to the (3 + 1) -dimensional Einstein-Maxwell-Vlasov system after applying the correspondence of Proposition 3.10. In particular, the lift of τ = 0 will be totally geodesic in the (3 + 1) -dimensional spacetime.  \nRemark 3.26 . One can 'maximalize' Proposition 3.23 to show the existence of a maximal globally hyperbolic development of S , but this requires treating the local existence and uniqueness problem for the spherically symmetric Einstein-Maxwell-Vlasov system at the center of symmetry, which we do not address in this paper. 18 Indeed, since our charged Vlasov beams spacetimes will always be vacuum near r = 0 , existence and uniqueness near the center will be completely trivial in our specific construction and is established in the following lemma.  \nLemma 3.27. Let u 0 ≤ v 0 < v 1 , r 0 ≥ 0 , λ 0 > 0 , and α : [ u 0 , v 0 ] → R > 0 and β : [ v 0 , v 1 ] : R > 0 be smooth functions satisfying the relations  \nα ( u 0 ) = β ( v 0 ) , α ( v 0 ) = 4 λ 2 0 , r 0 = 1 4 λ 0 ∫ v 0 u 0 α ( u ' ) du ' .  \nThen there exists a unique smooth solution ( r, Ω 2 , Q, f ) of the spherically symmetric Einstein-MaxwellVlasov system on  \n[ u 0 , v 0 ] × [ v 0 , v 1 ] ∪ ( { v ≥ u } ∩ { u ≥ v 0 } ∩ { v ≤ v 1 } )  \nwith Q and f identically vanishing, satisfying the boundary conditions of Definition 3.21 along { u = v } , together with  \nr ( u 0 , v 0 ) = r 0 , ∂ v r ( u 0 , v 0 ) = λ 0 , Ω 2 | [ u 0 ,v 0 ] ×{ v 0 } = α, Ω 2 | { u 0 }× [ v 0 ,v 1 ] = β.  \nThe solution is given by the explicit formulas  \nr ( u, v ) = r 0 + λ 0 β ( v 0 ) ∫ v v 0 β ( v ' ) dv ' -1 4 λ 0 ∫ u u 0 α ( u ' ) du ' , Ω 2 ( u, v ) = α ( u ) β ( v ) β ( v 0 )  \nfor ( u, v ) ∈ [ u 0 , v 0 ] × [ v 0 , v 1 ] and  \nr ( u, v ) = λ 0 β ( v 0 ) ∫ v u β ( v ' ) dv ' , Ω 2 ( u, v ) = 4 λ 0 β ( v 0 ) 2 β ( u ) β ( v )  \nfor ( u, v ) ∈ { v ≥ u } ∩ { u ≥ v 0 } ∩ { v ≤ v 1 } .  \nRemark 3.28 . From the last formula, it follows that  \n∂ u Ω 2 ( u, u ) = ∂ v Ω 2 ( u, u ) . (3.84)", '4 A singular toy model: bouncing charged null dust': "In this section, we introduce Ori's bouncing charged null dust model [Ori91]. We then show that Ori's model exhibits extremal critical collapse and can be used to construct counterexamples to the third law of black hole thermodynamics. Later, in Section 5, we will then show that these dust solutions can be (in an appropriate sense) globally desingularized by passing to smooth solutions of the Einstein-Maxwell-Vlasov system. The constructions in this section are crucial to motivate the choice of initial data in the proof of Theorem 1 in Section 5.", "4.1 Ori's bouncing charged null dust model": "We begin by recalling the general notion of charged null dust from [Ori91]:  \nDefinition 4.1. The Einstein-Maxwell-charged null dust model for particles of fundamental charge e ∈ R \\ { 0 } consists of a charged spacetime ( M , g, F ) , a future-directed null vector field k representing the momentum of the dust particles, and a nonnegative function ρ which describes the energy density of the dust. The equations of motion are  \nR µν -1 2 Rg µν = 2 ( T EM µν + T µν ) , (4.1)  \n∇ α F µα = e ρk µ , (4.2)  \nk ν ∇ ν k µ = e F µ ν k ν , (4.3)  \n∇ µ ( ρk µ ) = 0 , (4.4)  \nwhere T EM µν was defined in (2.17) and T µν . = ρk µ k ν is the energy-momentum tensor of a pressureless perfect fluid. By the forced Euler equation (4.3), the integral curves of k are electromagnetic null geodesics.  \nAny two functions ϖ in , Q in ∈ C ∞ ( R ) determine a spherically symmetric solution to the system (4.1)-(4.4) by the formulas  \ng in [ ϖ in , Q in ] . = -D ( V, r ) dV 2 +2 dV dr + r 2 γ, (4.5)  \nF . = -Q in r 2 dV ∧ dr, (4.6)  \nk . = e ˙ Q in ( ˙ ϖ in -Q in ˙ Q in r ) ( -∂ r ) , ρ . = ( ˙ Q in ) 2 e 2 r 2 ( ˙ ϖ in -Q in ˙ Q in r ) -1 , (4.7)  \nwhere · denotes differentiation with respect to V and  \nD ( V, r ) . = 1 -2 ϖ in ( V ) r + Q in ( V ) 2 r 2 .  \nThe metric (4.5) is known as the ingoing charged Vaidya metric [PS68; BV70] and describes a 'time dependent' Reissner-Nordström spacetime in ingoing Eddington-Finkelstein-type coordinates ( V, r, ϑ, φ ) . The spacetime is time oriented by -∂ r . The metric (4.5) and Maxwell field (4.6) are spherically symmetric and may therefore be considered as a spherically symmetric charged spacetime in the framework of Section 2. One easily sees that D = 1 -2 m r , Q = Q in , and ϖ = ϖ in .  \nWe will always make the assumption ˙ ϖ in ≥ 0 so that T µν = ρk µ k ν satisfies the weak energy condition for r sufficiently large. We also assume that e > 0 and impose the condition ˙ Q in ≥ 0 on the seed function Q in , which just means that positively charged particles increase the charge of the spacetime. (If e < 0 , we would instead assume ˙ Q in ≤ 0 and the discussion would otherwise remain unchanged.)  \nWe define a function r b = r b ( V ) , called the bounce radius , by  \nr b . = Q in ˙ Q in ˙ ϖ in  \nwhenever ˙ ϖ in ( V ) > 0 . The reason for this terminology will become clear shortly. By inspection of (4.7), we observe the following: For r > r b ( V ) , ( g in , F, k, ρ ) defines a solution of the Einstein-Maxwell-charged null dust system, k is future-directed null, and ρ ≥ 0 . If r b ( V ) > 0 and r ↘ r b ( V ) , then k and T dust vanish. If also ˙ Q in ( V ) > 0 , then ρ blows up at r = r b ( V ) , but ρk is nonzero and bounded. Finally, for r < r b ( V ) , k is past-directed null and ρ < 0 , so T dust violates the weak energy condition.  \nPhysically, the ingoing Vaidya metric and (4.7) describe an ingoing congruence of radial charged massless dust particles which interact with the electromagnetic field that they generate. One can interpret the vanishing of k as the dust being 'stopped' by the resulting repulsive Lorentz force. Integral curves of k are ingoing radial electromagnetic null geodesics γ ( s ) with limit points on the bounce hypersurface Σ b . = { r = r b } as s → ∞ . The charged null dust system is actually ill-posed across Σ b since the transport equation (4.3) breaks down there. Because of this, Ori argued in [Ori91] that the ingoing charged Vaidya metric (4.5) (and the associated formulas in (4.7)) should only be viewed as physical to the past of Σ b and must be modified if we wish to continue the solution beyond Σ b .  \nRemark 4.2 . The divergence of ρ along Σ b does not seem to have been explicitly mentioned by Ori, but it is one of the fundamentally singular features of charged null dust. One can also see that ρ can blow up if ˙ Q in /Q in blows up as a function of V , which occurs if the dust is injected into Minkowski space.  \nRemark 4.3 . Before Ori's paper [Ori91], the 'standard interpretation' [SI80; LZ91] of the ingoing Vaidya metric (4.5) did not actually involve Maxwell's equation and the fluid equation was simply taken to be the standard geodesic equation. The set { r < r b } was included in the ingoing solution and the dust was thought to violate the weak energy condition in this region. We refer to [Ori91] for discussion.  \nIn order to continue the dust solution across Σ b , we must make some further (nontrivial!) assumptions on the seed functions ϖ in and Q in . In order to not trivially violate causality, we must demand that Σ b is spacelike , so that the 'other side' { r < r b } of Σ b does not intersect the past of Σ b . This is equivalent to  \nD -2˙ r b < 0 on Σ b . (4.8)  \nWe further assume that Σ b does not contain trapped symmetry spheres, which is equivalent to  \nD > 0 on Σ b . (4.9)  \nBy examining the behavior of almost-radial electromagnetic null geodesics in Reissner-Nordström, Ori proposed the following bouncing continuation of the solution through Σ b : it should be as an outgoing charged Vaidya metric. This metric takes the form  \ng out [ ϖ out , Q out ] . = -D ( U, r ) dU 2 -2 dUdr + r 2 γ, (4.10)  \nFigure 6: Penrose diagram of Ori's bouncing charged null dust model. The geometry of the beams is described by the ingoing and outgoing Vaidya metrics, g in and g out , which are related by the gluing map G . The spacetime to the left and right of the bouncing beam is described by the Reissner-Nordström solution, with parameters ( ϖ 1 , Q 1 ) and ( ϖ 2 , Q 2 ) with ϖ 1 < ϖ 2 and Q 1 < Q 2 . The endpoints of Σ b correspond to symmetry spheres in these Reissner-Nordström spacetimes with radii r 1 < r 2 . In this diagram, the V coordinate is normalized according to the ingoing solution. We have depicted here the case of a totally geodesic bounce hypersurface Σ b and the outgoing beam is exactly the time-reflection of the ingoing beam.  \n<!-- image -->  \nwhere  \nD ( U, r ) . = 1 -2 ϖ out ( U ) r + Q out ( U ) 2 r 2 ,  \nfor free functions ϖ out and Q out . The coordinates ( U, r, ϑ, φ ) are now outgoing Eddington-Finkelstein-like. Ori defined a procedure for gluing an outgoing Vaidya metric to the ingoing Vaidya metric along Σ b by demanding continuity of the second fundamental form of Σ b from both sides. One sets  \n( ϖ out , Q out )( U ) = ( ϖ in , Q in ) · G -1 ( U ) ,  \nwhere the gluing map G = G ( V ) is determined by  \nd G dV = D ( V, r b ( V )) -2˙ r b ( V ) D ( V, r b ( V )) , (4.11)  \nup to specification of the (unimportant) initial condition. Notice that G is strictly monotone decreasing on account of (4.8) and (4.9). It turns out that this continuation preserves the weak energy condition through Σ b . We formalize this choice of extension of the Vaidya metric with the following  \nDefinition 4.4. Let ϖ in and Q in be nondecreasing charged Vaidya seed functions such that spt( ˙ ϖ in ) = spt( ˙ Q in ) = [ V 1 , V 2 ] and r b is well-defined and positive on [ V 1 , V 2 ] . Assume also the conditions (4.8) and (4.9). Ori's bouncing charged null dust model consists of the ingoing charged Vaidya metric g in [ ϖ in , Q in ] on M in . = { V ∈ spt( ˙ ϖ in ) , r ≥ r b ( V ) } × S 2 with spacelike, untrapped bounce hypersurface Σ in b . = { V ∈ spt( ˙ ϖ in ) , r = r b ( V ) }× S 2 glued to the outgoing charged Vaidya metric g out [ ϖ in ·G -1 , Q in ·G -1 ] on M out . = { U ∈ spt( ˙ ϖ out ) , r ≥ r b · G -1 ( U ) } × S 2 with spacelike, untrapped bounce hypersurface Σ out b . = { U ∈ spt( ˙ ϖ out ) , r = r b ( G -1 ( U )) } × S 2 along the map G × id r : Σ in b → Σ out b defined by (4.11). Outside the support of the dust, Ori's bouncing charged null dust model extends by attaching two Reissner-Nordström solutions with parameters ( ϖ 1 , Q 1 ) . = ( ϖ in , Q in )( V 1 ) and ( ϖ 2 , Q 2 ) . = ( ϖ in , Q in )( V 2 ) as depicted in Fig. 6.  \nThe model can be generalized to allow for multiple beams of dust by iterating the above definition in the obvious manner.", '4.2 The radial parametrization of bouncing charged null dust spacetimes': "It is not immediately clear that interesting seed functions ϖ in and Q in satisfying the requirements of Definition 4.4 exist. Therefore, it is helpful to directly prescribe the geometry of Σ b and the dust along it in the following sense. Given a spacetime as in Fig. 6, we can parametrize Σ b by the area-radius function r . Then the renormalized Hawking mass ϖ and charge Q , which are gauge-invariant quantities, can be viewed as functions of r on Σ b , and we wish to prescribe these functions. We will also prescribe Σ b to be totally geodesic . While not essential, this condition greatly simplifies Proposition 4.7 below and will later play a key role in our Vlasov construction in Section 5.  \nDefinition 4.5. Let P denote the set of points ( r 1 , r 2 , ϖ 1 , ϖ 2 , Q 1 , Q 2 ) ∈ R 6 ≥ 0 subject to the conditions  \n0 < r 1 < r 2 , Q 1 < Q 2 , ϖ 1 < ϖ 2 (4.12)  \n2 r 1 ϖ 1 ≥ Q 2 1 , (4.13)  \n2 r 1 ( ϖ 2 -ϖ 1 ) < Q 2 2 -Q 2 1 < 2 r 2 ( ϖ 2 -ϖ 1 ) , (4.14)  \nmin r ∈ [ r 1 ,r 2 ] ( r 1 r 2 -2 ϖ 1 r 1 r + Q 2 1 r -Q 2 2 r + Q 2 2 r 1 ) > 0 . (4.15)  \nElements of P will typically be denoted by the letter α and are called admissible parameters . Let V denote the set of triples ( α, ˇ ϖ, ˇ Q ) ∈ P × C ∞ ( [0 , ∞ ) ) × C ∞ ( [0 , ∞ ) ) such that the functions ˇ ϖ = ˇ ϖ ( r ) and ˇ Q = ˇ Q ( r ) are monotone increasing and satisfy  \nspt( ˇ ϖ ' ) = spt( ˇ Q ' ) = [ r 1 , r 2 ] , (4.16)  \nd dr ˇ Q 2 ( r ) = 2 r d dr ˇ ϖ ( r ) , (4.17)  \nˇ ϖ ( r 1 ) = ϖ 1 , ˇ Q ( r 1 ) = Q 1 , ˇ ϖ ( r 2 ) = ϖ 2 , ˇ Q ( r 2 ) = Q 2 , (4.18)  \nwhere ' denotes differentiation with respect to r .  \nRemark 4.6 . In the proof of Theorem 1 we will employ the regular center parameter space P Γ , consisting of those α ∈ P with ϖ 1 = Q 1 = 0 .  \nProposition 4.7 (Radial parametrization of bouncing charged null dust) . Let ( α, ˇ ϖ, ˇ Q ) ∈ V , and define strictly monotone functions V , U : [ r 1 , r 2 ] → R by  \nV ( r ) = -U ( r ) = ∫ r r 1 D ( r ' ) -1 dr ' ,  \nwhere D ( r ) . = 1 -2 ˇ ϖ ( r ) r + ˇ Q 2 ( r ) r 2 . Then:  \n- 1. The seed functions ( ϖ in , Q in ) . = (ˇ ϖ, ˇ Q ) · V -1 and ( ϖ out , Q out ) . = (ˇ ϖ, ˇ Q ) · U -1 define a bouncing charged null dust spacetime as in Definition 4.4 with gluing map G ( V ) = -V and bounce radius r b ( V ) = V -1 ( V ) .\n- 2. The bounce hypersurface Σ b is spacelike and untrapped. With the setup as in Fig. 6, the left edge of Σ b has area-radius r 1 and Reissner-Nordström parameters ( ϖ 1 , Q 1 ) and the right edge has area-radius r 2 and Reissner-Nordström parameters ( ϖ 2 , Q 2 ) . The Hawking mass m is nonnegative on Σ b .\n- 3. The bounce hypersurface Σ b is totally geodesic with respect to g in and g out .  \nProof. We must check that ϖ in . = ˇ ϖ · V -1 and Q in . = ˇ Q · V -1 satisfy the assumptions of Definition 4.4. Using the chain rule and (4.17), we compute  \nr b ( V ) = ˇ Q ( V -1 ( V )) ˇ Q ' ( V -1 ( V )) ˇ ϖ ' ( V -1 ( V )) = V -1 ( V ) .  \nDifferentiating, we obtain  \n˙ r b ( V ) = D ( V -1 ( V )) = D ( V, r b ( V )) , (4.19)  \nwhich implies that d G /dV = -1 . To prove (4.9), we show that D ( r ) > 0 for r ∈ [ r 1 , r 2 ] . Integrating (4.17) in r and integrating by parts yields  \nˇ ϖ ( r ) = ϖ 1 + 1 2 ∫ r r 1 1 r ' d dr ' ˇ Q 2 ( r ' ) dr ' = ϖ 1 + ˇ Q 2 ( r ) 2 r -Q 2 1 2 r 1 + 1 2 ∫ r r 1 ˇ Q 2 ( r ' ) r ' 2 dr ' . (4.20)  \nUsing condition (4.15) and ˇ Q ≤ Q 2 , we then find  \nD ( r ) = 1 -2 ˇ ϖ ( r ) r + ˇ Q 2 ( r ) r 2 = 1 -2 ϖ 1 r + Q 2 1 r 1 r -1 r ∫ r r 1 ˇ Q 2 ( r ' ) r ' 2 dr ' > 1 r 1 r 2 ( r 1 r 2 -2 ϖ 1 rr 1 + Q 2 1 r -Q 2 2 r + Q 2 2 r 1 ) > 0 (4.21)  \nfor r ∈ [ r 1 , r 2 ] . This proves (4.9) and since D -2˙ r b = -D , also proves (4.8). Condition (4.13) implies that the Hawking mass is nonnegative at r 1 . Finally, that Σ b is a totally geodesic hypersurface is shown by directly computing its second fundamental form and using (4.19).  \nThe definition of V involves many more conditions than just (4.8) and (4.9) alone, but it turns out that these are relatively easy to satisfy. In particular, we have:  \nProposition 4.8. The natural projection map V →P admits a smooth section ς : P → V . In other words, given any smooth family of parameters in P we may associate a smooth family of bouncing charged null dust spacetimes attaining those parameters, with totally geodesic bounce hypersurfaces.  \nRemark 4.9 . In the remainder of the paper, we fix the choice of section to be the one constructed in the proof below.  \nProof. Define a smooth, surjective function ψ : R 2 → (0 , 1) by  \nψ ( x, ξ ) = 1 1 + exp [ -( x -ξ ) e ( x -ξ ) 2 ] .  \nis strictly monotone increasing and surjective.  \nNote that for each fixed ξ ∈ R the function x ↦→ ψ ( x, ξ ) Moreover, for x ∈ R we have ψ ( x, ξ ) → 0 as ξ →∞ and ψ ( x, ξ ) → 1 as ξ →-∞ . We now define the function ˇ Q : ( r 1 , r 2 ) × R ×P → R by  \nˇ Q ( r, ξ, α ) . = Q 1 +( Q 2 -Q 1 ) ψ ( log ( r -r 1 r 2 -r ) , ξ ) .  \n(4.22)  \nBy construction of ψ , the function ˇ Q extends smoothly to [0 , ∞ ) × R × P by setting ˇ Q ( r, ξ, α ) = Q 1 for 0 ≤ r ≤ r 1 and ˇ Q ( r, ξ, α ) = Q 2 for r ≥ r 2 .  \nWith our family of candidate ˇ Q 's at hand, we aim to satisfy the constraint ˇ ϖ ( r 2 ) = ϖ 2 , where ˇ ϖ ( r ) is defined by (4.20). Consider the smooth map Π : ( ξ, α ) ∈ R ×P → R defined by  \nΠ( ξ, α ) . = ϖ 1 + Q 2 2 2 r 2 -Q 2 1 2 r 1 + ∫ r 2 r 1 ˇ Q 2 ( r ' , ξ, α ) 2 r ' 2 dr ' .  \nSince ψ satisfies ∂ψ ∂ξ < 0 on R 2 , we have that ∂ Π ∂ξ < 0 on R ×P . Moreover, using the pointwise limits of ψ , a direct computation gives  \nlim ξ →∞ Π( α, ξ ) = ϖ 1 + Q 2 2 2 r 2 -Q 2 1 2 r 2 , lim ξ →-∞ Π( α, ξ ) = ϖ 1 + Q 2 2 2 r 1 -Q 2 1 2 r 1 .  \nBy condition (4.14), this implies that  \nlim ξ →∞ Π( α, ξ ) < ϖ 2 < lim ξ →-∞ Π( α, ξ ) .  \nThus, the intermediate value theorem and the fact that ∂ Π ∂ξ < 0 show that there exists a unique ξ ( α ) such that Π( α, ξ ( α )) = ϖ 2 . Moreover, a direct consequence of the implicit function theorem is that the assignment P ∋ α ↦→ ξ ( α ) ∈ R is smooth. The above construction shows that the functions ˇ Q ( r, ξ ( α ) , α ) and ˇ ϖ satisfy all required properties.  \nFigure 7: Penrose diagrams of extremal critical collapse in Ori's bouncing charged null dust model. Compare with Fig. 2. In Theorem 4, λ ∗ = 1 .  \n<!-- image -->  \nThe set P is defined by simple polynomial relations and includes many interesting examples as we will see in the next two sections.", "4.3 Extremal critical collapse in Ori's model": "The first application of Propositions 4.7 and 4.8 is the construction of one-parameter families of bouncing charged null dust spacetimes exhibiting extremal critical collapse . We first show that the regular center parameter space P Γ contains elements with arbitrary final Reissner-Nordström parameters:  \nLemma 4.10. Let ϖ 2 , Q 2 > 0 . Then there exist 0 < r 1 < r 2 such that ( r 1 , r 2 , 0 , ϖ 2 , 0 , Q 2 ) ∈ P Γ . If ϖ 2 ≥ Q 2 , then r 2 can moreover be chosen so that r 2 < ϖ 2 -√ ϖ 2 2 -Q 2 2 .  \nProof. Let  \nr 1 . = Q 2 ( Q 2 2 ϖ 2 -ε ) , r 2 . = Q 2 ( Q 2 2 ϖ 2 + ε ) ,  \nwhere ϵ > 0 is a small parameter to be determined. With this choice, (4.14) is clearly satisfied. Let p ( r ) . = r 1 r 2 -Q 2 2 r + Q 2 2 r 1 and observe that  \nlim ε → 0 p ( Q 2 2 2 ϖ 2 ) = Q 6 2 8 ϖ 3 2 > 0 .  \nIt follows that (4.15) is satisfied for ε sufficiently small. If x ≥ 1 , then (2 x ) -1 < x -√ x 2 -1 , so taking ε perhaps smaller ensures that r 2 < ϖ 2 -√ ϖ 2 2 -Q 2 2 .  \nUsing this, we can show that Ori's model exhibits extremal critical collapse. Compare the following theorem with Theorem 1 and refer to Fig. 7 for Penrose diagrams.  \nTheorem 4. For any M > 0 and fundamental charge e ∈ R \\{ 0 } , there exist a small parameter δ > 0 and a smooth two-parameter family of regular center parameters { α λ,M ' } ⊂ P Γ for λ ∈ (0 , 2] and M ' ∈ [ M -δ, M + δ ] such that the two-parameter family of bouncing charged null dust spacetimes {D λ,M ' } , obtained by applying Proposition 4.7 to ς ( α λ,M ' ) , has the following properties:  \n- 1. For 0 < λ < 1 , D λ,M ' is isometric to Minkowski space for all sufficiently late retarded times u and hence future causally geodesically complete. In particular, it does not contain a black hole or naked singularity, and for λ < 1 sufficiently close to 1 , sufficiently large advanced times v ≥ v 0 and sufficiently small  \nretarded times u ≤ u 0 , the spacetime is isometric to an appropriate causal diamond in a superextremal Reissner-Nordström solution. Moreover, D λ,M ' converges smoothly to Minkowski space as λ → 0 .  \n- 2. λ = 1 is critical: D 1 ,M ' contains a nonempty black hole region BH and for sufficiently large advanced times v ≥ v 0 , the domain of outer communication, including the event horizon H + , is isometric to that of an extremal Reissner-Nordström solution of mass M ' . The spacetime contains no trapped surfaces.\n- 3. For 1 < λ ≤ 2 , D λ,M ' contains a nonempty black hole region BH and for sufficiently large advanced times v ≥ v 0 , the domain of outer communication, including the event horizon H + , is isometric to that of a subextremal Reissner-Nordström solution. The spacetime contains an open set of trapped surfaces.  \nIn addition, for every λ ∈ [0 , 2] , D λ,M ' is isometric to Minkowski space for sufficiently early advanced time and near the center { r = 0 } for all time, and possesses complete null infinities I + and I -.  \nProof. Using Lemma 4.10, choose 0 < r 1 < r 2 < r -(4 M, 2 M ) such that ( r 1 , r 2 , 0 , 4 M, 0 , 2 M ) ∈ P Γ . We consider  \nα λ,M ' . = ( r 1 , r 2 , 0 , λ 2 M ' , 0 , λM ' ) (4.23)  \nand note that α λ,M ' lies in P Γ for | λ -2 | sufficiently small and | M -M ' | ≤ δ sufficiently small by the openness of the conditions defining P Γ . Moreover, from the scaling properties of (4.14) and the monotonicity of (4.15), we observe that α λ,M ' ∈ P Γ for all 0 < λ ≤ 2 and | M -M ' | ≤ δ .  \nAfter applying Proposition 4.7 to ς ( α λ,M ' ) for λ > 0 , it remains only to show that D λ,M ' extends smoothly to Minkowski space as λ → 0 . Indeed, a direct inspection of the proof of Proposition 4.8 shows that ξ ( α λ,M ' ) is independent of λ , so that the function r ↦→ ˇ Q ( r, ξ ( α λ,M ' ) , α λ,M ' ) defined in (4.22) converges smoothly to the function ˇ Q ≡ 0 as λ → 0 . Therefore, ˇ ϖ also converges smoothly to the zero function and hence D λ,M ' converges smoothly to Minkowski space as λ → 0 .  \nThe construction in the proof of Theorem 1 can be thought of as a global-in-time desingularization of this family of dust solutions. In fact, we will make essential use of the one-parameter family { ς ( α λ,M ' ) } when constructing initial data for the Einstein-Maxwell-Vlasov system.", "4.4 A counterexample to the third law of black hole thermodynamics in Ori's model": "Using the radial parametrization, we can now give a very simple disproof of the third law:  \nTheorem 5. There exist bouncing charged null dust spacetimes that violate the third law of black hole thermodynamics: a subextremal Reissner-Nordström apparent horizon can evolve into an extremal ReissnerNordström event horizon in finite advanced time due to the incidence of charged null dust.  \nProof. Apply Propositions 4.7 and 4.8 to ( r 1 , r 2 , ϖ 1 , ϖ 2 , Q 1 , Q 2 ) ∈ P satisfying r 2 < ϖ 2 , Q 1 < ϖ 1 , and Q 2 = ϖ 2 . For example, one may take (0 . 85 , 0 . 88 , 0 . 56 , 1 , 0 . 5 , 1) ∈ P . See Fig. 8.  \nRemark 4.11 . Since the energy-momentum tensor remains bounded in Ori's model and the weak energy condition is satisfied, this is indeed a counterexample to Israel's formulation of the third law [Isr86].  \nThe counterexample in Theorem 5 explicitly displays the disconnectedness of the outermost apparent horizon which is also present in our charged scalar field counterexamples to the third law [KU22]. Note that the bouncing dust beam does not cross the subextremal apparent horizon, as is required by (4.9).  \nRemark 4.12 . In the example depicted in Fig. 8, the parameters ˇ ϖ and ˇ Q satisfy ˇ ϖ ( r ) < ˇ Q ( r ) for r ∈ ( r 2 -ε, r 2 ) and some ε > 0 . Indeed, the ODE (4.17) implies  \nˇ Q ' ( r ) = r ˇ Q ( r ) ˇ ϖ ' ( r ) < ˇ ϖ ' ( r )  \nnear r 2 , where we have used r 2 < Q 2 . The possibility (in fact, apparent inevitability) of the Vaidya parameters being superextremal right before extremality is reached seems to have been overlooked in the literature [SI80; FGS17]. 19  \nFigure 8: Penrose diagram of a counterexample to the third law of black hole thermodynamics in Ori's charged null dust model, Theorem 5. Note that the bounce Σ b lies behind the extremal event horizon H + since r 2 < ϖ 2 and ϖ 2 is the area-radius of the extremal horizon. The broken curve A ' is the outermost apparent horizon of the spacetime. The disconnectedness of A ' is necessary in third law violating spacetimes-see the discussion in Section 1.4.3 of [KU22]. A crucial feature of this counterexample is that Σ b lies strictly between the (initially) subextremal apparent horizon and the (eventually) extremal event horizon. Compare with Fig. 4.  \n<!-- image -->  \nRemark 4.13 . If one applies the old 'standard interpretation' of the ingoing Vaidya metric from [SI80; LZ91] to the seed functions ϖ in ( V ) and Q in ( V ) constructed in the proof of Theorem 5, one sees that the beam will hit the subextremal apparent horizon with a negative energy density, which is consistent with [SI80].  \nUsing methods from the proof of Theorem 1, the dust spacetimes in Theorem 5 can be 'desingularized' to smooth Einstein-Maxwell-Vlasov solutions. The desingularized solutions can also be chosen to have the property that the matter remains strictly between the subextremal apparent horizon and the event horizon and we refer back to Section 1.6.", '4.5 Issues with the bouncing charged null dust model': 'While Proposition 4.7 allows us to construct these interesting examples, the bouncing charged null dust model is unsatisfactory and we should seek to replace it for several reasons:  \n- 1. The model does not arise as a well-posed initial value problem for a system of PDEs. Pasting the ingoing and outgoing Vaidya solutions together is a deliberate surgery procedure that only works for seed functions ϖ in and Q in satisfying several nontrivial and nongeneric conditions.\n- 2. The solutions are generally not smooth along Σ b , nor along any cone where Q = 0 (recall Remark 4.2). The fluid density ρ is unbounded along Σ b and the number current N = ρk is discontinuous across Σ b .\n- 3. Null dust is ill-posed once the dust reaches the center of symmetry [Mos17].  \nNevertheless, we will show in the course of this paper that the bouncing charged null dust model can be well-approximated (near Σ b ), in a precise manner, by smooth solutions of the Einstein-Maxwell-Vlasov system. See already Section 5.10.', '4.6 The formal radial charged null dust system in double null gauge': "In order to precisely phrase the manner in which Einstein-Maxwell-Vlasov approximates bouncing charged null dust, as well as to motivate the choice of Vlasov initial data, we now reformulate Ori's model in double null gauge. Following Moschidis [Mos17; Mos20], we reformulate the system by treating N and T as the fundamental variables. By eliminating the fluid variables k and ρ , we can view the ingoing and outgoing phases as two separate well-posed initial value problems, with data posed along the bounce hypersurface. This helpfully suppresses the issue of blowup of ρ on Σ b .  \nDefinition 4.14. The spherically symmetric formal outgoing charged null dust model for particles of fundamental charge e ∈ R \\ { 0 } consists of a smooth spherically symmetric charged spacetime ( Q , r, Ω 2 , Q ) and two nonnegative smooth functions N v and T vv on Q .  \nThe system satisfies the wave equations  \n∂ u ∂ v r = -Ω 2 2 r 2 ( m -Q 2 2 r ) , (4.24)  \n∂ u ∂ v log Ω 2 = Ω 2 m r 3 -Ω 2 Q 2 r 4 , (4.25)  \n∂ u ( ∂ u r Ω 2 ) = -1 4 r Ω 2 T vv , (4.26)  \n∂ v ( ∂ v r Ω 2 ) = 0 , (4.27)  \n∂ u Q = -1 2 e r 2 Ω 2 N v , (4.28)  \n∂ v Q = 0 . (4.29)  \nThe number current satisfies the conservation law  \n∂ v ( r 2 Ω 2 N v ) = 0 (4.30)  \nand the energy-momentum tensor satisfies the Bianchi equation  \n∂ v ( r 2 Ω 4 T vv ) = + e Ω 4 QN v . (4.31)  \nIn the outgoing model, we may think of N u , T uu , and T uv to just be defined as identically zero. From (4.24) and (4.26)-(4.29) one easily derives  \n∂ u m = -1 2 r 2 Ω 2 T vv ∂ v r + Q 2 2 r 2 ∂ u r, ∂ v m = Q 2 2 r 2 ∂ v r, (4.32)  \n∂ u ϖ = -1 2 r 2 Ω 2 T vv ∂ v r -1 2 e r Ω 2 QN v , ∂ v ϖ = 0 . (4.33)  \nFurthermore, if we set k v . = T vv /N v , then  \nk v ∂ v k v + ∂ v log Ω 2 ( k v ) 2 = + e Q r 2 k v , (4.34)  \nwhich is the spherically symmetric version of (4.3) for the vector field k . = k v ∂ v . The energy density of the fluid is defined by ρ . = ( N v ) 2 /T vv whenever the denominator is nonvanishing.  \nDefinition 4.15. The spherically symmetric formal ingoing charged null dust model for particles of fundamental charge e ∈ R \\ { 0 } consists of a smooth spherically symmetric charged spacetime ( Q , r, Ω 2 , Q ) and two nonnegative smooth functions N u and T uu on Q . The system satisfies the same equations as the ingoing system with u ↔ v and the opposite sign in front of N u .  \nIn the ingoing case, k u . = T uu /N u and ρ . = ( N u ) 2 /T uu .  \nRemark 4.16 . By (4.34), these formal systems define solutions of the Einstein-Maxwell-charged null dust system (see Definition 4.1) whenever k and ρ are well-defined.  \nRemark 4.17 . Inspection of (4.31) reveals that T vv can reach zero in finite backwards time. If one were to continue the solution further, T vv could become negative, which shows that the formal system actually reproduces the old 'standard interpretation' of the charged Vaidya metric discussed in [Ori91]. As we will see, because the dominant energy condition holds in the Einstein-Maxwell-Vlasov model, only dust solutions with T uu , T vv ≥ 0 will arise as limiting spacetimes, confirming Ori's heuristic picture discussed in [Ori91].  \nthe Raychaudhuri equations  \nand the Maxwell equations", '4.6.1 The Cauchy problem for outgoing formal charged null dust': "Mirroring the treatment of time-symmetric 20 seed data for the Einstein-Maxwell-Vlasov system in Section 3.5, we make the following definition:  \nDefinition 4.18. A time-symmetric seed data set S d . = ( ˚ N v , ˚ T vv , r 2 , e ) for the spherically symmetric formal outgoing charged null dust system consists of real numbers r 2 ∈ R > 0 and e ∈ R \\ { 0 } , together with nonnegative compactly supported smooth functions ˚ N v and ˚ T vv with support contained in (0 , r 2 ] .  \nIn the dust case, we define ˚ m and ˚ Q on [0 , r 2 ] with ˚ m (0) = ˚ Q (0) = 0 by solving  \nd dr ˚ m = r 2 4 ( 1 -2˚ m r ) -2 ˚ T vv + ˚ Q 2 2 r 2 , (4.35)  \nd dr ˚ Q = 1 2 e r 2 ( 1 -2˚ m r ) -2 ˚ N v , (4.36)  \nprovided 2˚ m < r on [0 , r 2 ] . The remaining definitions from the Vlasov case, in particular Definition 3.21, can be carried over to dust with the obvious modification that N v = T vv = 0 along Γ . 21  \nProposition 4.19. Let S d be an untrapped time-symmetric seed data set for outgoing dust. Then there exists a unique global smooth solution of the formal outgoing charged null dust system ( r, Ω 2 , Q, N v , T vv ) on C r 2 attaining the seed data.  \nRemark 4.20 . Let r 1 . = inf(spt ˚ N v ∪ spt ˚ T vv ) . Then ( r, Ω 2 , Q ) is isometric to Minkowski space for u ≥ -r 1 .  \nProof. This can be proved by applying a suitable coordinate transformation to a suitable outgoing charged Vaidya metric. However, it is instructive to give a direct proof using the evolution equations.  \nWe pose initial data  \n˚ r ( r ) = r, ˚ Ω 2 ( r ) = ( 1 -2˚ m r ) -1 , ˚ Q ( r ) = ∫ r 0 1 2 e r 2 ( 1 -2˚ m r ) -2 ˚ N v dr ' ,  \nand for derivatives according to Definition 3.21, for the equations (4.24), (4.25), and (4.29). By a standard iteration argument, this determines the functions ( r, Ω 2 , Q ) uniquely. The existence of a global development is strictly easier than the corresponding proof in Proposition 5.5 once the rest of the system has been derived and is omitted. We now define  \nN v . = -2 e r 2 Ω 2 ∂ u Q, T vv . = -4 r Ω 2 ∂ u ( ∂ u r Ω 2 )  \nand aim to show that the rest of the equations in Definition 4.14 are satisfied.  \nTo prove (4.30), simply rearrange the definition of N v and use (4.29). Note that the definition of N v is consistent with ˚ N v = ˚ Ω -2 ˚ N v by (4.36).  \nUsing (4.24), (4.25), and (4.29), a tedious calculation yields  \n∂ u ( r∂ 2 v r -r∂ v r∂ v log Ω 2 ) = 0 . (4.37)  \nArguing as in Proposition 3.23, we see that (4.27) holds on initial data and is therefore propagated by (4.37). This proves the evolution equation ∂ v m = ∂ v rQ 2 / (2 r 2 ) and by using (4.24) once more, we see that  \n∂ u m = -2 r∂ v r∂ u ( ∂ u r Ω 2 ) + Q 2 2 r 2 ∂ u r.  \nComparing this with (4.35) and the definition of T vv yields ˚ T vv = ˚ Ω -2 ˚ T vv , as desired. Finally, (4.31) is proved by directly differentiating the definition of T vv and using (4.24), (4.25), and (4.28).  \nΣ b totally geodesic  \n<!-- image -->  \nFigure 9: An outgoing charged null dust beam obtained by applying Proposition 4.19 to the seed S d ,α for parameters α = ( r 1 , r 2 , 0 , ϖ 2 , 0 , Q 2 ) . The electrovacuum boundary C ⋆ can be attached to a ReissnerNordström spacetime with parameters ϖ 2 and Q 2 .", '4.6.2 Outgoing charged Vaidya as formal outgoing dust': "We now want to represent the outgoing portion of a regular center bouncing charged null dust beam given by Proposition 4.7 in terms of the outgoing formal system. Let α ∈ P Γ , ς ( α ) = ( α, ˇ ϖ, ˇ Q ) be given by Proposition 4.8, and consider the time-symmetric dust seed data S d ,α . = ( ˚ N v d , 0 , r 2 , e ) , where  \n˚ N v d . = 2 e r 2 ( 1 -2 ˇ ϖ r + ˇ Q 2 r 2 ) 2 ˇ Q ' . (4.38)  \nFor this choice of seed, the constraints (4.35)-(4.36) read  \nd dr ˚ m = ˚ Q 2 2 r 2 , (4.39)  \nd dr ˚ Q = ( 1 -2˚ m r ) -2 ( 1 -2 ˇ m r ) 2 ˇ Q ' . (4.40)  \nTherefore, by (4.17), ˚ m = ˇ m and ˚ Q = ˇ Q , where ˇ m . = ˇ ϖ -ˇ Q 2 / (2 r ) . It follows that the outgoing formal dust solution ( r d , Ω 2 d , Q d , N v d , T vv d ) provided by Proposition 4.19 on C r 2 is indeed the same as the outgoing charged Vaidya metric provided by the radial parametrization method, Proposition 4.7. See Fig. 9.", '5 The construction of bouncing charged Vlasov beams and the proof of the main theorem': 'In this section, we prove Theorem 1 by constructing bouncing charged Vlasov beams as in Fig. 1 and Fig. 2 with prescribed final parameters. This is achieved by a very specific choice of time-symmetric Vlasov seed data and global estimates for the resulting developments. We give a detailed outline of the proof in Section 5.1 and the proof itself occupies Sections 5.2 to 5.9. Finally, in Section 5.10, we show that these bouncing charged Vlasov beams weak* converge to the bouncing charged null dust spacetimes of Proposition 4.7 in a hydrodynamic limit of the beam parameters.', '5.1.1 The heuristic picture': "The essential idea in the proof of Theorem 1 is to 'approximate' the bouncing radial charged null dust solutions from Theorem 4 and Fig. 7 by smooth families of smooth Einstein-Maxwell-Vlasov solutions. Indeed, at least formally, dust can be viewed as Vlasov matter f ( x, p ) concentrated on a single momentum p = k ( x ) at each spacetime point x . One is faced with having to perform a global-in-time desingularization of families of dust solutions which are singular in both the space and momentum variables.  \nAssuming that this can be done, the heuristic picture is that of a focusing beam of Vlasov matter coming in from infinity with particles of mass m = 0 or m ≪ 1 (so that the particles look almost massless for very large time scales) and very small angular momentum 0 < ℓ ≪ 1 , which are decelerated by the electromagnetic field that they generate. Then, along some 'approximate bounce hypersurface,' the congruence smoothly 'turns around' and becomes outgoing, escaping to infinity if a black hole has not yet formed. Along the way, the particles do not hit the center of symmetry. By appropriately varying the beam parameters, we can construct families of spacetimes as depicted in Fig. 1 or Fig. 2.  \nAs should be apparent from the treatment of the Cauchy problem for the Einstein-Maxwell-Vlasov system in Section 3.5 and for charged null dust in Section 4.6, we want to pose Cauchy data on (what will be) the approximate bounce hypersurface for the desingularized Vlasov solutions. We will choose the initial data for f to be supported on small angular momenta ℓ ∼ ε and so that the charge ˚ Q and Hawking mass ˚ m profiles closely approximate the initial data for dust as described in Section 4.6.2. The Vlasov beam which is intended to approximate charged null dust is called the main beam .  \nAs we will see, desingularizing bouncing charged null dust requires an ansatz for ˚ f which necessarily degenerates in ε . Closing estimates in the region of spacetime where Q ≲ ε is then a fundamental issue because the repulsive effect of the electromagnetic field is relatively weak there. We overcome this issue by adding an auxiliary beam to the construction, which stabilizes the main beam by adding a small amount of charge on the order of η ≫ ε . This beam is not dust-like, consists of particles with angular momentum ∼ 1 , and is repelled away from the center by the centrifugal force.  \nThe goal will be to construct a smooth family of Vlasov seeds λ ↦→ S λ for λ ∈ [ -1 , 2] such that S -1 is trivial (i.e., evolves into Minkowski), S 2 forms a subextremal Reissner-Nordström black hole with charge to mass ratio ≈ 1 2 , and λ ∗ ≈ 1 is the critical parameter for which an extremal Reissner-Nordström black hole with mass M forms. For λ ∈ [0 , 2] , the Vlasov development D λ closely approximates the dust developments from Theorem 4 (in a sense to be made precise in Section 5.10 below) and λ ∈ [ -1 , 0] smoothly 'turns on' the auxiliary beam. At the very end of the proof, λ is simply rescaled to have range [0 , 1] .  \nIn fact, our methods allow us to desingularize any bouncing charged null dust beam given by Proposition 4.8. Adding dependence on λ is then essentially only a notational hurdle. We now highlight specific aspects of the construction in more detail.", '5.1.2 Time symmetry and reduction to the outgoing case': "The starting point of the construction of bouncing charged Vlasov beams is the prescription of Cauchy data on an approximate bounce hypersurface Σ b , using the radial parametrization of bouncing charged null dust as a guide. We can now see the utility of the time-symmetric ansatz in Section 3.5: it reduces the problem to constructing an outgoing beam, which is then reflected and glued to maximally extended ReissnerNordström to construct a time symmetric spacetime as depicted in Fig. 10 below. These 'maximal timesymmetric spacetimes' are constructed in Section 5.9.1. The globally hyperbolic developments in Theorem 1 are obtained by taking appropriate subsets and identifying suitable Cauchy hypersurfaces.  \nThe problem now reduces to constructing the region bounded to the past by C ⋆ , Σ b , and the center in Fig. 10. In this region, the solution is always dispersive . Therefore, we can actually treat the subextremal, extremal, and superextremal cases at once. Detection of whether a black hole forms in the doubled spacetime takes place on the level of initial data and we heavily exploit the global structure of the Reissner-Nordström family itself in this process. Note that while we prescribe data in the black hole interior when λ ≤ 0 , there clearly exist Cauchy surfaces lying entirely in the domain of outer communication. In fact, the solutions are always past complete and disperse to the past. See already Proposition 5.30.  \nFigure 10: Penrose diagrams of the 'maximal time-symmetric doubled spacetimes' used in the proof of Theorem 1 when m > 0 . When λ ≤ 0 , these spacetimes are evidently not globally hyperbolic, but one can easily observe that the globally hyperbolic spacetimes depicted in Fig. 1 when λ ≤ 0 are simply the above spacetimes restricted to the past of CH + ∪ { i + } ∪ I + . The exterior region is isometric to a subset of the maximally extended Reissner-Nordström solution with parameters depending on λ .  \n<!-- image -->", '5.1.3 The choice of seed data': 'We now describe our desingularization procedure for bouncing charged null dust on the level of initial data. Consider the outgoing portion of a charged null dust beam ( r d , Ω 2 d , Q d , N v d , T vv d ) as in Section 4.6.2, with Cauchy data posed along the bounce hypersurface Σ b . The geometry of the outgoing dust beam is entirely driven by the choice of renormalized number current ˚ N v d in (4.38). Importantly, the energy-momentum tensor of dust vanishes identically along Σ b .  \nSince radial charged null dust has ℓ = 0 , we wish to approximate dust with Vlasov matter consisting of particles with angular momentum ℓ ∼ ε , where 0 < ε ≪ 1 is a small parameter to be chosen. We want to choose the initial distribution function ˚ f so that  \n˚ N u + ˚ N v = ˚ N v d = 1 e r 2 ( 1 -2 ˇ ϖ r + ˇ Q 2 r 2 ) 2 d ˇ Q dr , (5.1)  \n˚ Q ≈ ˇ Q, ˚ ϖ ≈ ˇ ϖ, ˚ T uu , ˚ T uv , ˚ T vv ≈ 0 (5.2)  \non Σ b , as ε → 0 . These conditions are satisfied if we choose  \n˚ f α,ε main ( r, p u , p v ) . = c e r 2 ε ( 1 -2 ˇ ϖ r + ˇ Q 2 r 2 ) 2 d ˇ Q dr δ ε ( p u ) δ ε ( p v ) (5.3)  \nfor r ∈ [ r 1 , r 2 ] , where δ ε are approximations of the identity with support [ ε, 2 ε ] and c is a normalization constant that depends on the precise choice of the family δ ε . In order for the mass shell inequality Ω 2 p u p v ≥ m 2 to hold on the support of ˚ f α,ε main , (5.3) forces us to choose m ∈ [0 , m 0 ] with 0 < m 0 ≪ ε .  \nFigure 11: Penrose diagram of outgoing charged Vlasov beams (evolution of the seed data S α,η,ε ). Note that the beams do not intersect when m = 0 . When m > 0 , one can show that they do, but it is not necessary to do so for our purposes here.  \n<!-- image -->  \nRemark 5.1 . In the full bouncing null dust model, N is discontinuous across Σ b . Indeed, to the past of Σ b , N points in the u -direction and has a nonzero limit along Σ b , but to the future points in the v -direction and also has a nonzero limit. In the Vlasov case, time symmetry demands N be smooth across, and orthogonal to, Σ b . By comparing (3.70) with (4.36), we see that ˚ N u + ˚ N v in Vlasov takes the role of ˚ N v in dust.  \nObserve directly from (5.3) that ˚ f α,ε main behaves pointwise like ε -3 and therefore pointwise estimates for N and T in evolution must utilize precise estimates of the electromagnetic flow to cancel factors of ε . Closing estimates independently of ε is the main challenge of this scheme and we directly exploit the null structure of the spherically symmetric Einstein-Maxwell-Vlasov system in the proof. The main mechanisms ensuring boundedness of N and T in the main beam are:  \n- 1. The angular momentum ℓ is conserved, so that ℓ ∼ ε throughout the main beam.\n- 2. If γ is an electromagnetic geodesic arising from the support of ˚ f α,ε main , then p v should rapidly increase due to electromagnetic repulsion. Dually, p u should rapidly decrease, which ought to suppress the ingoing moments N u , T uu , and T uv . This should be compared with the vanishing of N u , T uu , and T uv in outgoing null dust. We say that the main beam bounces due to electromagnetic repulsion .  \nAs is apparent from (2.22), the magnitude of the repulsive effect is proportional to Q . If we were to evolve the seed ˚ f α,ε main on its own, the inner edge of the beam would experience less electromagnetic repulsion since Q is potentially quite small in the inner region.  \nIn order to reinforce the repulsive effect of the electric field in the main beam and get a consistent hierarchy of powers of ε , we introduce an auxiliary beam on the inside of the main beam which bounces due to the centrifugal force associated to electromagnetic geodesics with large angular momentum. The initial data for the auxiliary beam is chosen to be  \n˚ f r 1 ,η aux ( r, p u , p v ) . = η φ ( r, p u , p v ) , (5.4)  \nwhere η ≫ ε is a constant determining the amplitude, φ is a cutoff function supported on the set [ 1 3 r 1 , 2 3 r 1 ] × [Λ -1 , Λ+1] × [Λ -1 , Λ+1] , and Λ is a fixed large constant that determines the strength of the centrifugal force felt by the auxiliary beam. The auxiliary beam ensures that the main beam always interacts with an electric field of amplitude ≳ η , which acts as a crucial stabilizing mechanism.', '5.1.4 The near and far regions and the hierarchy of scales': "The total seed for an outgoing Vlasov beam is taken to be S α,η,ε . = ( ˚ f α,η,ε tot , r 2 , m , e ) , where  \n˚ f α,η,ε tot . = ˚ f r 1 ,η aux + ˚ f α,ε main , (5.5)  \nthe fundamental charge e > 0 is fixed, the mass m lies in the interval [0 , m 0 ] , and η, ε , and m 0 need to be chosen appropriately small.  \nTo study the evolution of S α,η,ε , depicted in Fig. 11, we distinguish between the near region { v ≤ ˘ v } and the far region R ˘ v, ∞ far = { v ≥ ˘ v } , where ˘ v is a large advanced time to be determined. Roughly, the ingoing cone { v = ˘ v } is chosen so that the geometry is very close to Minkowskian and the Vlasov field is 'strongly outgoing' and supported far away from the center, i.e.,  \np u p v ≲ r -2 ≪ 1 (5.6)  \n̸  \nfor every p u and p v such that f ( · , ˘ v, p u , p v ) = 0 . The near region is further divided into the main and auxiliary regions, corresponding to the physical space support of the main and auxiliary beams and denoted by R ˘ v main and R ˘ v aux , respectively. 22  \nThe beam parameters η , ε, m 0 and the auxiliary parameter ˘ v satisfy the hierarchy  \n0 < m 0 ≪ ε ≪ η ≪ ˘ v -1 ≪ 1 . (5.7)  \nTo prove the sharp rate of dispersion when m > 0 , we augment this hierarchy with  \n0 < v -1 # ≪ m ,  \nwhere v # is a very large time after which the additional dispersion associated to massive particles kicks in.  \nThe proof of Theorem 1 proceeds by showing that if (5.7) holds, then the solution exists, with certain properties, in the regions R ˘ v main , R ˘ v aux , and R ˘ v, ∞ far , in that order. The sharp decay rates of N and T are then shown a posteriori by re-analyzing the electromagnetic geodesic flow in the far region.  \nRemark 5.2 . The final Reissner-Nordström parameters of the total Vlasov seed (5.5) depend on the approximation parameters η and ε , but are O ( η ) -close to ( ϖ 2 , Q 2 ) . Therefore, in order to reach any fixed set of parameters, the background dust seed has to be appropriately modulated. See already Section 5.9.5.", '5.1.5 Outline of the main estimates': "The main beam in the near region: In this region, the main goal is proving smallness (in ε ) of N u , T uu , and T uv , which are identically zero in the background dust solution. We define the phase space volume function V : Q → R ≥ 0 by  \n̸  \nV ( u, v ) . = Ω 2 ( u, v ) |{ ( p u , p v ) : f ( u, v, p u , p v ) = 0 }| , (5.8)  \nwhere | · | is the Lebesgue measure on R 2 p u ,p v . The function V is invariant under gauge transformations of u and v . Using the mass shell relation (3.12) and the change of variables formula, we find  \nV ( u, v ) = 2 r 2 ∫ ∞ 0 ∫ { p v : f ( u,v,p u ,p v ) =0 } dp v p v ℓ dℓ, (5.9)  \n̸  \nwhere we view p u as a function of p v and ℓ . Because of the addition of ˚ f r 1 ,η aux to the seed data and the good monotonicity properties of (3.24) and (3.25), it holds that Q ≳ η in R ˘ v main . Under relatively mild bootstrap assumptions, any electromagnetic geodesic γ in the main beam is accelerated outwards at a rate ≳ η , i.e.,  \np v ≳ ε + η min { τ, 1 } , p u ≲ ε 2 r 2 ( ε + η min { τ, 1 } ) ,  \nwhere τ . = 1 2 ( u + v ) is a 'coordinate time.' We also show that if γ 1 and γ 2 are two electromagnetic geodesics in the main beam which reach the same point ( u, v ) ∈ R ˘ v main , then  \n| p v 1 -p v 2 | ≲ ε η 2  \nat ( u, v ) . Using these estimates, conservation of angular momentum, and the hierarchy (5.7), we show that  \nV ( u, v ) ≲ η ε 3 ε + η min { τ, 1 } ,  \nwhere the notation A ≲ η B means A ≤ CB , where C is a constant depending on η . Then, simply using the transport nature of the Vlasov equation, we obtain the estimates  \nT uu ( u, v ) ≲ ε 1 / 2 , T uv ( u, v ) ≲ ε 1 / 2 , ∫ v -u N u ( u, v ' ) dv ' ≲ ε 1 / 2 ,  \nwhich capture the fundamental characteristic of outgoing null dust. These estimates allow us to control the geometry at C 1 order, which is more than enough to use the generalized extension principle, Proposition 3.15, to extend the solution. For details, see Section 5.4. When λ ∈ [ -1 , 0] and the main beam has not yet been turned on, constructing the solution in this region is trivial since the solution is electrovacuum.  \nThe auxiliary beam in the near region: Since the auxiliary beam is genuinely weak ( ˚ f r 1 ,η aux ≲ η pointwise), the bootstrap argument in R ˘ v aux is a standard Cauchy stability argument, perturbing off of Minkowski space. We use explicit knowledge of the impact parameter and asymptotics of null geodesics with angular momentum ∼ Λ on Minkowski space and treat the charge as an error term in this region. For details, see Section 5.5.  \nExistence in the far region: The argument in this region is a refinement of Dafermos' proof of the stability of Minkowski space for the spherically symmetric Einstein-massless Vlasov system [Daf06] (see also [Tay15, Chapter 4]). Because of the singular nature of f main in powers of ε , it seems difficult to obtain uniform in ε pointwise decay estimates for T uv by the usual method of estimating decay of the phase space volume of the support of f at late times. Fortunately, we are able to exploit the a priori energy estimates  \n∫ r 2 Ω 2 T uv ∂ u r du ' ≲ 1 , ∫ r 2 Ω 2 T uv ∂ v r dv ' ≲ 1 (5.10)  \ncoming from the monotonicity of the Hawking mass when ∂ v r > 0 and ∂ u r < 0 (see [Daf05b]). It is important to note that these energy estimates are independent of initial data and are a fundamental feature of the spherically symmetric Einstein equations. Under the bootstrap assumption that the electromagnetic geodesics making up the support of f are 'outgoing' for v ≥ ˘ v , the energy estimates (5.10) imply decay for the unweighted fluxes of T uv . This shows that the geometry remains close to Minkowski in C 1 and recovering the bootstrap assumption on the support of f follows from good monotonicity properties of the electromagnetic geodesic flow when close to Minkowski. We also note that this approach using energy estimates allows us to treat the cases m = 0 and m > 0 simultaneously. For details, see Section 5.6.  \nDispersion in the far region: Once the solution has been shown to exist globally, we prove sharp (in coordinate time τ ) pointwise decay statements for N and T (see [RR92; Nou05; Tay15]). As the decay rates differ when m = 0 or m > 0 , these two cases are treated separately.  \nThe massless case. It follows immediately from the mass shell relation (3.12) that p u ≲ r -2 in the far region. Since this is integrable, the beams are confined to null slabs and can even be shown to be disjoint as depicted in Fig. 11. Since each p u contributes a factor of r -2 and our solutions have bounded angular momentum, we obtain the sharp dispersive hierarchy  \nN v + T vv ≤ C (1 + τ ) -2 , N u + T uv ≤ C (1 + τ ) -4 , T uu ≤ C (1 + τ ) -6 ,  \nwhere the constant C depends on α, η , and ε . For details, see Section 5.7.  \nThe massive case. When m > 0 , p u does not decay asymptotically. After a very late time v # ≫ m -1 , p u ∼ η m 2 , which drives additional decay of the phase space volume. We prove this by a change of variables  \nargument, turning volume in p u at later times v ≥ v # into physical space volume of the support of f at time v = v # . This leads to the sharp isotropic decay rate  \nM≤ C (1 + τ ) -3  \nfor any moment M of f , where C depends on α, η, ε , and a lower bound for m . For details, see Section 5.8.", '5.2.1 The beam parameters, fixed constants, and conventions': "First, we fix once and for all the fundamental charge e ∈ R \\ { 0 } . Without loss of generality, we may take e > 0 , as all of the arguments and definitions in the remainder of Section 5 require only minor cosmetic modifications to handle the case e < 0 . Next, we fix an even function φ ∈ C ∞ c ( R ) satisfying spt φ = [ -1 , 1] , φ ≥ 0 , and  \n∫ 1 -1 φdx = 1 .  \nLet θ ∈ C ∞ ( R ) be a nondecreasing function such that θ ( λ ) = 0 for λ ≤ -1 and θ ( λ ) = 1 for λ ≥ 0 . Let ζ ∈ C ∞ ( R ) be a nondecreasing function such that ζ ( λ ) = 0 for λ ≤ 0 , ζ ( λ ) = λ for λ ≥ 1 2 , and ζ ' ( λ ) > 0 for λ ∈ (0 , 1 2 ] . Finally, we fix a large 23 number Λ ≥ 1 , such that  \nmin p 1 ,p 2 ∈ [Λ -1 , Λ+1] p 1 p 2 ( p 1 + p 2 ) 2 ≥ 81 400 . (5.11)  \nWe emphasize that:  \nThe quintuple ( e , φ, θ, ζ, Λ) is fixed for the remainder of the paper.  \nRecall the set P Γ of regular center admissible parameters of the form α = ( r 1 , r 2 , 0 , ϖ 2 , 0 , Q 2 ) which was defined in Section 4.2. Let η, ε , and m 0 be positive real numbers. In the course of the proofs below, the particle mass m will be restricted to satisfy 0 ≤ m ≤ m 0 .  \nDefinition 5.3. Let α = ( r 1 , r 2 , 0 , ϖ 2 , 0 , Q 2 ) ∈ P Γ , η > 0 , and ε > 0 . The time-symmetric outgoing charged Vlasov beam seed S α,η,ε is given by ( ˚ f r 1 ,η aux + ˚ f α,ε main , r 2 , m , e ) , where  \n˚ f r 1 ,η aux ( r, p u , p v ) . = η φ ( 6 r 1 r -3 ) φ ( p u -Λ) φ ( p v -Λ) , (5.12)  \nand  \n˚ f α,ε main ( r, p u , p v ) . = 8 3 π e ε 3 r 2 ( 1 -2 ˇ ϖ ( r ) r + ˇ Q 2 ( r ) r 2 ) 2 ˇ Q ' ( r ) φ ( 2 p u ε -3 ) φ ( 2 p v ε -3 ) , (5.13)  \nwhere ˇ ϖ and ˇ Q are taken from ς ( α ) , where the map ς was defined in Proposition 4.8 (cf. Remark 4.9).  \nDefinition 5.4. Let M > 0 , let 0 < r 1 < r 2 , and let { α λ,M ' } (with | M -M ' | ≤ δ ) be as in (4.23) in the proof of Theorem 4. For λ ∈ [ -1 , 2] , η > 0 , ε > 0 , m 0 , we define  \nS λ,M ' ,η,ε . = S α ζ ( λ ) ,M ' ,θ ( λ ) η,ε (5.14)  \nfor particles of mass 0 ≤ m ≤ m 0 . For λ ≤ 0 , the ˇ ϖ and ˇ Q components of ς ( α ζ ( λ ) ,M ' ) are interpreted as identically zero, in correspondence with the proof of Theorem 4.  \nThroughout Section 5, the notation A ≲ B means that there exists a constant C > 0 , which only depends on e , φ , θ , ζ , Λ , M , r 1 , and r 2 such that A ≤ CB . The notation A ≳ B is defined similarly and A ∼ B means A ≲ B and A ≳ B . Moreover, we use the convention that all small (large) constants in 'sufficiently small (large)' may depend on e , φ , θ , ζ , Λ , M , r 1 , and r 2 . In Section 5.7, we will also use the notation  \nA ≲ η B (resp., A ≲ η,ε B ), in which we allow the constants to also depend on η (resp., η and ε ). The relations A ∼ η B and A ∼ η,ε B are defined in the obvious way.  \nFor the evolution problem, we will introduce a large parameter ˘ v to separate C r 2 into the 'near' and 'far' regions. We will always assume that the parameter hierarchy  \n0 < m 0 ≪ ε ≪ η ≪ ˘ v -1 ≪ 1 (5.15)  \nholds, by which we mean that any given statement holds for ˘ v sufficiently large, η sufficiently small depending on ˘ v , ε sufficiently small depending on ˘ v and η , and m 0 sufficiently small depending on ˘ v , η , and ε . To prove dispersion in the massive case, we introduce an even larger parameter v # satisfying  \n0 < v -1 # ≪ m ≤ m 0 ,  \nso that v # is chosen sufficiently large depending on ˘ v, η, ε , and m .", '5.2.2 The global structure of outgoing charged Vlasov beams': "Proposition 5.5. Fix a fundamental charge e > 0 , cutoff functions φ, θ , and ζ as in Section 5.2.1, a number Λ ≥ 1 satisfying (5.11) , M > 0 , and 0 < r 1 < r 2 < r -(4 M, 2 M ) . Let δ > 0 be as in the statement of Theorem 4 and define S λ,M ' ,η,ε as in Definition 5.4 for λ ∈ [ -1 , 2] , | M ' -M | ≤ δ , η > 0 , ε > 0 , and for particles of mass 0 ≤ m ≤ m 0 , where m 0 > 0 .  \nIf η is sufficiently small, ε is sufficiently small depending on η , and m 0 is sufficiently small depending on η and ε , then for any λ ∈ [ -1 , 2] and | M ' -M | ≤ δ , the seed S λ,M ' ,η,ε is untrapped and consistent with particles of mass m . There exists a unique maximal normalized development ( U , r, Ω 2 , Q, f ) of S λ,M ' ,η,ε for particles of charge e and mass m with the following properties. 24 If m > 0 :  \n- 1. The development is global in the normalized double null gauge, i.e., U = C r 2 .\n- 2. The (3 + 1) -dimensional spacetime obtained by lifting ( U , r, Ω 2 ) is future causally geodesically complete and satisfies globally the estimates  \n∂ v r ∼ 1 , | ∂ u r | ≲ 1 , Ω 2 ∼ 1 , (1 + u 2 ) | ∂ u Ω 2 | +(1 + v 2 ) | ∂ v Ω 2 | ≲ 1 . (5.16)  \n- 3. Define the final Reissner-Nordström parameters ˜ M and ˜ e of S λ,M ' ,η,ε to be the constant values of ϖ and Q , respectively, on the cone C -r 2 . Then ˜ M and ˜ e are smooth functions of ( λ, M ' , η, ε, m ) , satisfy the estimate  \n| ˜ M -ζ ( λ ) 2 M ' | + | ˜ e -ζ ( λ ) M ' | ≲ η, (5.17)  \nand extend smoothly to η = ε = 0 , where they equal ζ ( λ ) 2 M ' and ζ ( λ ) M ' , respectively. The spacetime ( U , r, Ω 2 ) contains antitrapped surfaces (symmetry spheres where ∂ u r ≥ 0 ) if and only if ˜ e ≤ ˜ M and r 2 < r -, where r ± . = ˜ M ± √ ˜ M 2 -˜ e 2 . In this case, we nevertheless have ∂ u r ∼ -1 for v sufficiently large and the antitrapped surfaces are restricted to lie in the slab { 2 r --r 2 ≤ v ≤ 2 r + -r 2 } .  \n- 4. The Vlasov distribution function f is quantitatively supported away from the center,  \ninf π (spt f ) r ≥ 1 6 r 1 , (5.18)  \nand the beam asymptotes to future timelike infinity i + in the sense that  \nπ (spt f ) ⊂ { C 1 v ≤ u ≤ C 2 v } , (5.19)  \nwhere C 1 and C 2 are positive constants that may additionally depend on η, ε, m , and λ . The connected component of U \\ π (spt f ) containing the center is isometric to Minkowski space. The connected component of U \\ π (spt f ) containing future null infinity I + is isometric to an appropriate neighborhood of future null infinity in the Reissner-Nordström solution with parameters M and e .  \n- 5. The Vlasov matter disperses in the sense that the macroscopic observables decay pointwise:  \nM≤ C (1 + τ ) -3 , (5.20)  \nwhere M∈{ N u , N v , T uu , T uv , T vv , S } and the constant C may additionally depend on η, ε, m , and λ .  \nThe same conclusions hold if m = 0 , but points 4. and 5. are improved to:  \n- 4. ' The estimate (5.18) still holds but (5.19) is improved to  \nπ (spt f ) ⊂ {-r 2 ≤ u ≤ 1 3 r 1 } , (5.21)  \ni.e., the beam is confined to a null slab. The spacetime is isometric to Minkowski space for u ≥ r 1 .  \n- 5. ' The Vlasov matter disperses in the sense that the macroscopic observables decay pointwise:  \nN v + T vv ≤ C (1 + τ ) -2 , (5.22)  \nN u + T uv ≤ C (1 + τ ) -4 , (5.23)  \nT uu ≤ C (1 + τ ) -6 , (5.24)  \nwhere the constant C may additionally depend on η, ε , and λ .  \nRemark 5.6 . An analogous version of Proposition 5.5 may be proved for any α = ( r 1 , r 2 , 0 , ϖ 2 , 0 , Q 2 ) ∈ P Γ or α = ( r 1 , r 2 , 0 , 0 , 0 , 0) by evolving the seed data S α,η,ε given by Definition 5.3. In that case, (5.17) becomes  \n| ˜ M -ϖ 2 | + | ˜ e -Q 2 | ≲ η. (5.25)  \nRemark 5.7 . The decay rate τ -3 in (5.20) is sharp for massive particles [RR92; Nou05]. The hierarchy of decay rates in (5.22)-(5.24) is sharp for massless particles [Tay15].", '5.3 Estimates on the initial data': "For the remainder of Section 5, we assume the notation and hypotheses of Proposition 5.5.  \nLemma 5.8. For η , ε , and m 0 sufficiently small and any λ ∈ [ -1 , 2] and | M ' -M | ≤ δ , S λ,M ' ,η,ε is untrapped and consistent with particles of mass 0 ≤ m ≤ m 0 . Let ( U , r, Ω 2 , Q, f ) be a development of S λ,M ' ,η,ε , such as the one obtained from Proposition 3.23. Then the following holds:  \n- 1. The set U can be assumed to contain the corner C r 2 ∩ { v ≤ 1 3 r 1 } . The solution is equal to Minkowski space in this region in the sense that  \nr = 1 2 ( v -u ) , 2  \nΩ = 1 ,  \nQ = f = 0  \non C r 2 ∩ { v ≤ 1 3 r 1 } .  \n- 2. Estimates on the initial data for the auxiliary beam:  \n˚ f r 1 ,θ ( λ ) η aux ≲ θ ( λ ) η 1 [ 1 3 r 1 , 2 3 r 1 ] × [Λ -1 , Λ+1] × [Λ -1 , Λ+1] , (5.26)  \nsup v ∈ [ 1 3 r 1 ,r 1 ] | (Ω 2 -1 , ∂ u log Ω 2 , ∂ v log Ω 2 , Q, m )( -v, v ) | ≲ θ ( λ ) η, (5.27)  \nQ ( -2 3 r 1 , 2 3 r 1 ) ≳ θ ( λ ) η, (5.28)  \nℓ ( -v, v, p u , p v ) ≈ 1 for every ( v, p u , p v ) ∈ spt( ˚ f r 1 ,θ ( λ ) η aux ) . (5.29)", '3. Estimates on the initial data for the main beam:': "˚ f α ζ ( λ ) ,M ' ,θ ( λ ) η,ε main ≲ ε -3 1 [ r 1 ,r 2 ] × [ ε, 2 ε ] × [ ε, 2 ε ] · { 1 if λ > 0 0 if λ ≤ 0 , (5.30)  \nsup v ∈ [ 2 3 r 1 ,r 2 ] | (Ω 2 , ∂ u log Ω 2 , ∂ v log Ω 2 , Q, m )( -v, v ) -( ˇ Ω 2 , ˇ ω, ˇ ω, ˇ Q, ˇ m )( v ) | ≲ θ ( λ ) η, (5.31)  \ninf v ∈ [ 2 3 r 1 ] Q ( -v, v ) ≳ θ ( λ ) η, (5.32)  \nℓ ( -v, v, p u , p v ) ≈ ε for every ( v, p u , p v ) ∈ spt( ˚ f α ζ ( λ ) ,M ' ,θ ( λ ) η,ε main ) , (5.33)  \nwhere  \nˇ m ( v ) . = ˇ ϖ ( v ) -ˇ Q 2 ( v ) 2 v , ˇ Ω 2 ( v ) . = ( 1 -2 ˇ m ( v ) v ) -1 , ˇ ω . = -ˇ ω . = 1 2 d dv log ˇ Ω 2 ( v ) . (5.34)", '4. Estimates on the initial outgoing cone C -r 2 :': "ϖ ( -r 2 , v ) = ϖ ( -r 2 , r 2 ) , (5.35)  \nQ ( -r 2 , v ) = Q ( -r 2 , r 2 ) , (5.36)  \n0 ≤ m ( -r 2 , v ) ≤ 10 M, (5.37)  \nr ( -r 2 , v ) = 1 2 v + 1 2 r 2 (5.38)  \nfor v ≥ r 2 .  \nProof. Consistency with particles of mass m ≤ m 0 follows immediately from Definition 5.3 and the estimates (5.27) and (5.31) by taking m 0 sufficiently small. We therefore focus on proving the estimates and as a byproduct infer the untrapped property of S λ,M ' ,η,ε .  \nPart 1. This is a restatement of Remark 3.24.  \nPart 2. The estimate (5.26) follows immediately from the definition (5.12). Inserting the ansatz (5.26) into (3.66)-(3.68), we find  \n˚ N u ( r ) = ˚ N v ( r ) = πθ ( λ ) η Λ φ ( 6 r 1 r -3 ) , ˚ T uu ( r ) = ˚ T vv ( r ) = πθ ( λ ) η ( Λ 2 + ∫ 1 -1 x 2 φ ( x ) dx ) φ ( 6 r 1 r -3 ) , ˚ T uv ( r ) = πθ ( λ ) η Λ 2 φ ( 6 r 1 r -3 )  \nfor r ∈ [ 1 3 r 1 , 2 3 r 1 ] . For η sufficiently small, it then follows readily from the system (3.69) and (3.70) that ˚ Q and ˚ m are nonnegative, nondecreasing functions, and  \n0 ≤ ˚ Q ( r ) + ˚ m ( r ) ≲ θ ( λ ) η  \nfor r ∈ [ 1 3 r 1 , 2 3 r 1 ] and  \n˚ Q ( 2 3 r 1 ) ≳ θ ( λ ) η.  \nUsing the definition of ˚ Ω 2 , we infer | ˚ Ω 2 -1 | ≲ θ ( λ ) η , and to estimate | ∂ v log Ω 2 ( -v, v ) | , we observe that  \n| ∂ v log Ω 2 ( -v, v ) | = ∣ ∣ ∣ ∣ d dv log ˚ Ω 2 ( v ) ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ˚ Ω -2 ( 2˚ m ( v ) v 2 -2 v d dv ˚ m ( v ) )∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ 2˚ m ˚ Ω 2 v 2 -2 ˚ Ω 2 v ( ˚ Ω 4 v 2 4 ( ˚ T uu +2 ˚ T uv + ˚ T vv ) + ˚ Q 2 2 v 2 )∣ ∣ ∣ ∣ ∣ ≲ θ ( λ ) η.  \nThe estimate for ∂ u log Ω 2 ( -v, v ) follows from (3.76). This establishes (5.27) and (5.28). Finally, (5.29) follows from the mass shell relation and (5.27), provided m 0 is chosen sufficiently small.  \nPart 3. The estimate (5.30) follows immediately from the definition (5.13). Inserting the ansatz (5.13) into (3.66)-(3.68), we find  \n˚ N u ( r ) = ˚ N v ( r ) = 1 e r 2 ( 1 -2 ˇ ϖ ( r ) r + ˇ Q 2 ( r ) r 2 ) 2 ˇ Q ' ( r ) , (5.39)  \n˚ T uu ( r ) = ˚ T vv ( r ) = ε 6 e r 2 ( 9 + ∫ 1 -1 x 2 φ ( x ) dx )( 1 -2 ˇ ϖ ( r ) r + ˇ Q 2 ( r ) r 2 ) 2 ˇ Q ' ( r ) , (5.40)  \n˚ T uv ( r ) = 3 ε 4 e r 2 ( 1 -2 ˇ ϖ ( r ) r + ˇ Q 2 ( r ) r 2 ) 2 ˇ Q ' ( r ) , (5.41)  \nwhere ˇ ϖ and ˇ Q are obtained from ς ( α λ,M ' ) . Inserting (5.39)-(5.41) into (3.69) and (3.70) yields  \nd ˚ m = ˚ Q 2 2 +Err  \ndr 2 r ,  \nd dr ˚ Q = ( 1 -2˚ m r ) -2 ( 1 -2 ˇ m r ) 2 ˇ Q ' ,  \n| Err | ≲ ( 1 -2˚ m r ) -2 ε  \nwhere  \nand ˚ m ( r 1 ) , ˚ Q ( r 1 ) ≲ θ ( λ ) η . Therefore, by (4.17), (4.21), and a simple Grönwall and bootstrap argument, ˚ m and ˚ Q exist on [ r 1 , r 2 ] and satisfy  \nsup r ∈ [ r 1 ,r 2 ] | (˚ m -ˇ m, ˚ Q -ˇ Q )( r ) | ≲ θ ( λ ) η.  \nThis implies the same estimate for | ˚ Ω 2 -ˇ Ω 2 | by definition. To estimate the other quantities, we may now argue as in the proof of Part 2.  \nPart 4. Equations (5.35), (5.36), and (5.38) follow immediately from the definitions. Inequality (5.37) follows from (5.31) provided η is chosen sufficiently small.", '5.4 The main beam in the near region': "For τ 0 > 0 , let  \nR τ 0 main . = { 0 ≤ τ ≤ τ 0 } ∩ {-2 3 r 1 ≤ u ≤ -r 2 } ⊂ C r 2 . (5.42)  \nLemma 5.9. For any ˘ v , η , ε , and m 0 satisfying (5.15) , the following holds. Any normalized development ( U , r, Ω 2 , Q, f ) of S λ,M ' ,η,ε , such as the one obtained from Proposition 3.23, can be uniquely extended to R 1 2 ˘ v -1 3 r 1 main . Moreover, the solution satisfies the estimates  \n0 ≤ m ≤ 10 M, 0 ≤ Q ≤ 6 M, (5.43)  \nr ∼ v, Ω 2 ∼ 1 , (5.44)  \n∂ v r ∼ 1 , | ∂ u r | ≲ 1 , (5.45)  \n| ∂ u Ω 2 | ≲ 1 , | ∂ v Ω 2 | ≲ v -3 (5.46)  \non R 1 2 ˘ v -1 3 r 1 main and (5.47)  \n1 3 ≤ ∂ v r ≤ 2 3 , ∂ u r ∼ -1  \non R 1 2 ˘ v -1 3 r 1 main ∩ C ˘ v . Finally, the support of the distribution function satisfies  \nπ (spt f ) ∩ R 1 2 ˘ v -1 3 r 1 main ⊂ {-5 6 r 1 ≤ u ≤ -r 2 } (5.48)  \nFigure 12: Penrose diagram of the bootstrap region R τ 0 main used in the proof of Lemma 5.9.  \n<!-- image -->  \n̸  \nand if u ∈ [ -r 2 , -5 6 r 1 ] , p u , and p v are such that f ( u, ˘ v, p u , p v ) = 0 , then  \np u p v ≲ ˘ v -2 , ℓ 2 p v ≲ 1 . (5.49)  \nWhen λ ≤ 0 , (5.43) reads instead  \n0 ≤ m ≲ θ ( λ ) η, 0 ≤ Q ≲ θ ( λ ) η.  \nThe proof of Lemma 5.9 will be given on Page 61. We will make use of a bootstrap argument in the regions R τ 0 main , where τ 0 ranges over [0 , 1 2 ˘ v -1 3 r 1 ] . For the basic geometric setup of the lemma and its proof, refer to Fig. 12. As the proof is much simpler when λ ≤ 0 (the main beam is absent), we focus only on the case λ > 0 , in which case θ ( λ ) η = η .  \nWe first make some definitions that will be used to define the bootstrap assumptions. Let C 1 > 0 be a constant such that  \nC -1 1 ≤ ( 1 -2 ˇ m ( v ) v ) -1 ≤ C 1  \nfor v ∈ [ 2 3 r 1 , r 2 ] , where ˇ m is given by (5.34). (Recall that ˇ m ( v ) = 0 for v ≤ r 1 .) We then define  \nC 2 . = 8 C 1 ( 3 r 2 2 r 1 -1 )( 5 M + 18 M 2 r 1 ) ,  \nC 3 . = 2 max v ∈ [ 2 3 r 1 ,r 2 ] | ˇ ω ( v ) | +100 C 1 e C 2 ( 5 M + 27 M 2 r 1 )∫ ∞ 2 3 r 1 dv ( 2 3 (1 -1 6 e -C 2 ) r 1 + 1 6 e -C 2 v ) 3 ,  \nThe constants C 1 , C 2 , and C 3 do not depend on η , ε , or m 0 .  \nThe quantitative bootstrap assumptions for the proof of Lemma 5.9 are  \n1 6 e -C 2 ≤ ∂ v r ≤ 3 2 e C 2 , (5.50)  \n1 8 C -1 1 ≤ Ω -2 ∂ v r ≤ C 1 , (5.51)  \n| ∂ u log Ω 2 | ≤ C 3 , (5.52)  \nϖ ≤ 5 M, (5.53)  \nN v ≤ Ae Bτ , (5.54)  \non R τ 0 main where A ≥ 1 and B ≥ 1 are constants to be determined which may depend on ˘ v and η , but not on ε . We now derive some consequences of the bootstrap assumptions for the geometry of the solution.  \nLemma 5.10. If (5.15) holds, τ 0 ∈ [0 , 1 2 ˘ v -1 3 r 1 ] , R τ 0 main ⊂ U , and the bootstrap assumptions (5.50) -(5.53) hold on R τ 0 main , then  \nη ≲ Q ≤ 6 M, (5.55)  \non R τ 0 main .  \nWe will frequently use that (5.56) implies  \n2 3 (1 -1 6 e -C 2 ) r 1 + 1 6 e -C 2 v ≤ r ≤ r 2 + 3 2 e C 2 v, (5.56)  \nΩ 2 ≈ 1 , (5.57)  \n| ∂ u r | ≲ 1 (5.58)  \nr  \n∼  \nv  \non R τ 0 main without further comment.  \nProof. For η and ε sufficiently small, η ≲ Q ≤ 6 M on { τ = 0 } ∩ {-r 2 ≤ u ≤ -2 3 r 1 } and { τ ≥ 0 } ∩ { v = r 2 } by Lemma 5.8. Since N v ≥ 0 by definition, Maxwell's equation (3.24) implies the upper bound in (5.55). The lower bound also follows from Maxwell's equation (3.25) and N u ≥ 0 . The inequality (5.56) follows from integrating the bootstrap assumption (5.50). The inequality (5.57) follows directly by multiplying the bootstrap assumptions (5.50) and (5.51). To estimate ∂ u r , we rewrite the definition of the Hawking mass (2.2) and the renormalized Hawking mass (2.19) as  \n∂ u r = -1 4 ( 1 -2 ϖ r + Q 2 r 2 ) Ω 2 ∂ v r . (5.59)  \nNow (5.58) follows immediately from (5.51), (5.53), and (5.56).  \nWe now use the basic geometric control obtained in Lemma 5.10 to obtain crucial control of the electromagnetic geodesic flow. It is convenient to first introduce some notation. Let Γ f denote the set of maximally extended electromagnetic geodesics γ : I →U , where I is an interval, such that ( γ, p )( I ) ⊂ spt f , where p = dγ/ds . If γ passes through the point ( u, v ) , we denote by s u,v the parameter value such that γ ( s u,v ) = ( u, v ) . Let Γ f ( u, v ) denote the subset of Γ f consisting of curves passing through ( u, v ) . Note that every curve in Γ f intersects C r 2 ∩ { τ = 0 } .  \nLemma 5.11. If (5.15) holds, A is sufficiently large depending only on α , τ 0 ∈ [0 , 1 2 ˘ v -1 3 r 1 ] , R τ 0 main ⊂ U , the bootstrap assumptions (5.50) -(5.54) hold on R τ 0 main , and ( u, v ) ∈ R τ 0 main , then  \nV ( u, v ) ≲ ε 3 η ( ε + η min { τ, 1 } ) ( 1 + A B e Bτ ) , (5.60)  \nwhere V is the phase space volume function defined by (5.9) . Furthermore, if γ ∈ Γ f ( u, v ) , then  \n0 < u -u 0 ≲ η -1 ε, (5.61)  \nε + η min { τ, 1 } ≲ p v ( s u,v ) ≲ ε +min { τ, 1 } . (5.62)  \nwhere u 0 is the retarded time coordinate of the intersection of γ with { τ = 0 } .  \nProof. Let γ ∈ Γ f ( u, v ) . We will use the Lorentz force written in the form of equation (2.25) to estimate p v . Since ( u, v ) ∈ R τ 0 main , γ intersects { τ = 0 } in spt( ˚ f α,ε main ) and therefore has angular momentum ℓ ∼ ε . By the bootstrap assumptions and Lemma 5.10, it holds that  \n∣ ∣ ∣ ∣ ( ∂ u log Ω 2 -2 ∂ u r r ) ℓ 2 r 2 ∣ ∣ ∣ ∣ ≲ ε 2 v 2 (5.63)  \nalong the entire length of γ . Let ( u 0 , v 0 ) be the coordinate of the intersection of γ with { τ = 0 } . Using (5.55)-(5.57) and the fact that p v ( s u 0 ,v 0 ) ∈ [ ε, 2 ε ] , we have  \ne Q r 2 (Ω 2 p v ) ∣ ∣ ∣ ∣ s =0 ≳ ηε.  \nIf ε is sufficiently small (independent of γ , but depending on η ), these estimates show that  \nd ds (Ω 2 p v ) ≳ ηε > 0  \nalong γ . Using (5.57), we see that  \np v ≳ ε (5.64)  \nalong γ .  \nIt is now convenient to parametrize γ by the advanced time coordinate v of the spacetime. The Lorentz force equation then becomes  \ndγ u dv = p u p v = ℓ 2 r -2 + m 2 Ω 2 ( p v ) 2 , (5.65)  \nd dv (Ω 2 p v ) = 1 p v ( ∂ u log Ω 2 -2 ∂ u r r ) ℓ 2 r 2 + e Q r 2 Ω 2 , (5.66)  \nwhere however p v is still given by dγ v /ds and we have used the mass shell relation in (5.65). By (5.63) and (5.64),  \n∣ ∣ ∣ ∣ 1 p v ( ∂ u log Ω 2 -2 ∂ u r r ) ℓ 2 r 2 ∣ ∣ ∣ ∣ ≲ ε v 2  \nalong γ . This implies, using v 0 ≥ r 1 and hence ∫ ∞ v 0 v '-2 dv ' ≲ 1 , that  \n∣ ∣ ∣ ∣ ∣ Ω 2 p v ∣ ∣ ( γ u ( v ) ,v ) -∫ v v 0 e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( γ u ( v ' ) ,v ' ) dv ' ∣ ∣ ∣ ∣ ∣ ≲ ε. (5.67)  \nUsing Lemma 5.10, we readily deduce that  \n∫ v v 0 e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( γ u ( v ' ) ,v ' ) dv ' ≲ 1 (5.68)  \nand  \n∫ v v 0 e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( γ u ( v ' ) ,v ' ) dv ' ≳ η ( 1 v 0 -1 v ) ≳ η min { 1 , v -v 0 } .  \nCombining this with (5.64) and (5.67), we deduce  \nε + η min { 1 , v -v 0 } ≲ p v ( v ) ≲ 1 (5.69)  \nalong γ .  \nWe are now able to prove (5.61). Since r ≲ v ≲ ˘ v , r 2 m 2 ≲ ε 2 ≲ ℓ 2 for m 0 sufficiently small while respecting the hierarchy (5.15). Therefore, using also Lemma 5.10 and (5.69), we find  \ndγ u dv ≲ ε 2 v 2 ( ε + η min { 1 , v -v 0 } ) 2 . (5.70)  \nIf v ∈ [ v 0 , v 0 +1] , we compute  \n∫ v v 0 ε 2 v ' 2 ( ε + η ( v ' -v 0 )) 2 dv ' ≲ ε ( v -v 0 ) ε + η ( v -v 0 ) ≲ η -1 ε  \nand if v ∈ [ v 0 +1 , ∞ ) , we compute  \n∫ v v 0 +1 ε 2 v ' 2 ( ε + η ) 2 dv ' ≲ η -2 ε 2 ≲ η -1 ε,  \nfor ε ≤ η . Therefore, integrating (5.70), we find  \nu -u 0 = γ u ( v ) -γ u ( v 0 ) = ∫ v v 0 dγ u dv dv ' ≲ η -1 ε, (5.71)  \nwhich proves (5.61).  \nNow u 0 = -v 0 , so (5.71) implies u + v 0 ≲ η -1 ε . Therefore, we have  \n2 τ = v + u = ( v -v 0 ) + ( u + v 0 ) ≲ η -1 ε + v -v 0  \nand  \nε + ητ ≲ ε + η ( v -v 0 ) ≲ p v ( v )  \nfor τ ≲ 1 . This, together with (5.67) and (5.68), proves (5.62).  \nTo prove (5.60), we use the approximate representation formula (5.67) for p v and the change of variables formula (5.9). Using the bootstrap assumptions, Lemma 5.10, the mean value theorem, Maxwell's equation (3.24), and the estimate (5.61), we have  \n∣ ∣ ∣ ∣ ∣ Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( γ u ( v ' ) ,v ' ) -Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( u,v ' ) ∣ ∣ ∣ ∣ ∣ ≲ ( 1 + sup [ γ u ( v ' ) ,u ] ×{ v ' } | ∂ u Q | ) ( γ u ( v ' ) -u ) ≲ Aη -1 εe B ( u + v ' ) / 2  \nfor every v ' ∈ [ v 0 , v ] and A sufficiently large depending only on α . Using this and (5.67), we find  \n∣ ∣ ∣ ∣ ∣ Ω 2 p v ∣ ∣ ( γ u ( v ) ,v ) -∫ v v 0 e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( u,v ' ) dv ' ∣ ∣ ∣ ∣ ∣ ≲ ∣ ∣ ∣ ∣ ∣ Ω 2 p v ∣ ∣ ( γ u ( v ) ,v ) -∫ v v 0 e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( γ u ( v ' ) ,v ' ) dv ' ∣ ∣ ∣ ∣ ∣ + ∫ v v 0 ∣ ∣ ∣ ∣ ∣ Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( γ u ( v ' ) ,v ' ) -Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( u,v ' ) ∣ ∣ ∣ ∣ ∣ dv ' ≲ ε + ∫ v v 0 Aη -1 εe B ( u + v ' ) / 2 dv ' ≤ ε ( 1 + A Bη e B ( u + v ) / 2 ) . (5.72)  \nNext, we estimate  \n0 ≤ ∫ v 0 -u e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( u,v ' ) dv ' ≲ v 0 + u = u -u 0 ≲ η -1 ε. (5.73)  \nCombining (5.72) and (5.73) yields  \n∣ ∣ ∣ ∣ ∣ Ω 2 p v ∣ ∣ ( u,v ) -∫ v -u e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( u,v ' ) dv ' ∣ ∣ ∣ ∣ ∣ ≲ ε η ( 1 + A B e B ( u + v ) / 2 ) . (5.74)  \nTherefore, if γ 1 , γ 2 ∈ Γ f ( u, v ) and we parametrize both by advanced time, and denote the v -momentum of γ i by p v i for i = 1 , 2 , we find  \n| p v 1 ( v ) -p v 2 ( v ) | ≤ ∣ ∣ ∣ ∣ ∣ p v 1 ( v ) -1 Ω 2 ( u, v ) ∫ v -u e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( u,v ' ) dv ' ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ p v 2 ( v ) -1 Ω 2 ( u, v ) ∫ v -u e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( u,v ' ) dv ' ∣ ∣ ∣ ∣ ∣ ≲ ε η ( 1 + A B e B ( u + v ) / 2 ) .  \nInserting this estimate, (5.62), and ℓ ∼ ε in (5.9) yields (5.60), as desired.  \nProof of Lemma 5.9. The proof is a bootstrap argument based on the bootstrap assumptions (5.50)-(5.54) and continuation criterion given by the extension principle Proposition 3.15. Let  \nA . = { τ 0 ∈ [0 , 1 2 ˘ v -1 3 r 1 ] : the solution extends uniquely to R τ 0 main and (5.50)-(5.54) hold on R τ 0 main } .  \nThe set A is nonempty by Proposition 3.23 if A is chosen sufficiently large and η , ε , and m 0 are sufficiently small. It is also manifestly connected and closed by continuity of the bootstrap assumptions. We now show that if A and B are sufficiently large depending on η and (5.15) holds, then A is also open.  \nLet τ 0 ∈ A . First, we use Lemmas 5.10 and 5.11 to estimate N v and improve (5.54). Since f is transported along electromagnetic geodesics, we have f ( u, v, p u , p v ) ≲ ε -3 for ( u, v ) ∈ R τ 0 main . Using (5.60) and (5.62), we infer directly from the definition of N v that  \nN v ( u, v ) ≲ ( ε +min { τ, 1 } ) ε -3 V ( u, v ) ≲ ε +min { τ, 1 } η ( ε + η min { τ, 1 } ) ( 1 + A B ) e Bτ ≲ η -2 ( 1 + A B ) e Bτ .  \nLetting C ∗ = C ∗ ( e , φ, Λ , α ) denote the implicit constant in this inequality, and choosing A = 4 C ∗ η -2 and B ≥ A , we see that  \nN v ( u, v ) ≤ 1 2 Ae Bτ ,  \nwhich improves (5.54).  \nTo continue, we now estimate N u , T uu , and T uv in the same fashion, making use now of the strong decay of p u . If γ ∈ Γ f ( u, v ) , then  \np u ( s u,v ) = ℓ 2 r -2 + m 2 Ω 2 p v ( s u,v ) ≲ ε 2 ε + η min { τ, 1 }  \nby (5.62). Using (5.60), we therefore find  \nN u ≲ ε 2 η 1 ( ε + η min { 1 , τ } ) 2 Ae Bτ , (5.75)  \nT uv ≲ ε 2 η ε +min { τ, 1 } ε + η min { 1 , τ } Ae Bτ , (5.76)  \nT uu ≲ ε 4 η 1 ( ε + η min { 1 , τ } ) 3 Ae Bτ . (5.77)  \nUsing the hierarchy (5.15), these estimates imply  \nr 2 T uv + r 2 T uu + ∫ v -u r 2 N u ( u, v ' ) dv ' ≲ ε 1 / 2 (5.78)  \nfor any ( u, v ) ∈ R τ 0 main . With these final estimates in hand, we may begin to improve the remaining bootstrap assumptions (5.50)-(5.53). We will then carry out the rest of the continuity argument and prove all of the stated conclusions of the lemma.  \nImproving (5.50): The wave equation (3.20) can be rewritten as  \n∂ u ∂ v r = -1 2 r 2 Ω 2 ∂ v r ( ϖ -Q 2 r ) ∂ v r + r 4 Ω 4 T uv .  \nUsing an integrating factor, we find  \n∂ u [ exp (∫ u -r 2 1 2 r Ω 2 ∂ v r ( ϖ -Q 2 r ) du ' ) ∂ v r ] = r 4 Ω 4 T uv exp (∫ u -r 2 1 2 r Ω 2 ∂ v r ( ϖ -Q 2 r ) du ' ) , (5.79)  \nwhere the integral is taken over fixed v . The bootstrap assumptions imply  \n∫ -2 3 r 1 -r 2 1 2 r Ω 2 ∂ v r ∣ ∣ ∣ ∣ ϖ -Q 2 r ∣ ∣ ∣ ∣ du ' ≤ C 2 .  \nFor ε sufficiently small, the right-hand side of (5.79) is pointwise ≤ 1 10 on R τ 0 main , so integrating this equation yields  \n2 5 e -C 2 ≤ ∂ v r ( u, v ) ≤ 3 5 e C 2  \nfor any ( u, v ) ∈ R τ 0 main , which improves (5.50).  \nImproving (5.51): Raychaudhuri's equation (3.23) can be rewritten as  \n∂ v log ( ∂ v r Ω 2 ) = -r 4 ∂ v rT uu . (5.80)  \nTo improve the upper bound in (5.51), note that the right-hand side of (5.80) is nonpositive by (5.50) and hence  \n∂ v r Ω 2 ( u, v ) ≤ ∂ v r Ω 2 ( -v, v ) = 1 2 ( 1 -2 ˇ m ( v ) v ) ≤ C 1 2 .  \nTo improve the lower bound, note that the right-hand side of (5.80) is bounded by log 2 in absolute value for ε sufficiently small and hence  \n∂ v r Ω 2 ( u, v ) ≥ 1 2 ∂ v r Ω 2 ( -v, v ) ≥ 1 4 C 1 ,  \nwhich improves (5.51).  \nImproving (5.52): The wave equation (3.21) can be rewritten as  \n∂ v ∂ u log Ω 2 = Ω 2 r 3 ( ϖ -3 Q 2 2 r ) -1 2 Ω 4 T uv -Ω 2 S. (5.81)  \nIntegrating this equation in v and using the bootstrap assumptions, (5.31), and (5.78) yields  \n| ∂ u log Ω 2 | ≤ 3 4 C 3  \nfor η and ε sufficiently small, which improves (5.52).  \nImproving (5.53): Integrating the evolution equation for the renormalized Hawking mass (3.31) and using (5.78), we have  \n| ϖ ( u, v ) -ϖ ( -v, v ) | ≲ ε 1 / 2 ,  \nwhich improves (5.53).  \nWe have thus improved the constants in all of the bootstrap assumptions (5.50)-(5.54). Using the local existence theory Proposition 3.14 and generalized extension principle Proposition 3.15, there exists a τ ' 0 > τ 0 such that U ⊂ R τ ' 0 main . Choosing τ ' 0 > τ 0 perhaps smaller, the bootstrap assumptions (5.50)-(5.54) extend to R τ ' 0 main by continuity. Therefore, A is open and the bootstrap argument is complete.  \nWe now prove the remaining conclusions of the lemma. First, m ( u, v ) ≥ 0 for every ( u, v ) ∈ R 1 2 ˘ v -1 3 r 1 main because either ∂ u r ( u, v ) < 0 (and by Raychaudhuri (3.22) also for any u ' > u ) or ∂ u r ( u, v ) ≥ 0 and then m ( u, v ) ≥ 0 directly from the definition (2.2). In the first case, the evolution equation (3.29) implies m is nondecreasing along the outgoing cone terminating at ( u, v ) . Since this cone either intersects { τ = 0 } , where m ≥ 0 , or a sphere where ∂ u r ≤ 0 (and hence m ≥ 0 ), we conclude m ( u, v ) ≥ 0 . Now integrating the wave equation (3.21) in u and using (5.78), we see that | ∂ v log Ω 2 | ≲ v -3 . Together with the bootstrap assumptions and Lemma 5.10, this proves (5.43)-(5.46). Next, (5.47) follows from integrating the wave equation (3.20) in u along C ˘ v and taking ˘ v ∼ r sufficiently large and similarly in (5.59). The inclusion (5.48) follows immediately from the u -deflection estimate (5.61) for electromagnetic geodesics in Γ f and the hierarchy (5.15). Finally, to prove (5.49) we use the mass shell relation, (5.62), and the parameter hierarchy to estimate  \nr 2 p u p v = ℓ 2 + r 2 m 2 Ω 2 ( p v ) 2 ≲ ε 2 η 2 ≲ 1 , ℓ 2 p v ≲ ε 2 η ≲ 1 ,  \nwhich completes the proof.  \n<!-- image -->  \nFigure 13: Penrose diagram of the bootstrap region R v 0 aux used in the proof of Lemma 5.12.  \n<!-- image -->", '5.5 The auxiliary beam in the near region': "For v 0 > 0 , let  \nR v 0 aux . = { v ≥ u } ∩ { τ ≥ 0 } ∩ {-2 3 r 1 ≤ u ≤ 1 3 r 1 } ∩ { 1 3 r 1 ≤ v ≤ v 0 } , (5.82)  \n˜ R v 0 aux . = { v ≥ u } ∩ { τ ≥ 0 } ∩ { u ≥ -2 3 r 1 } ∩ { v ≤ v 0 } . (5.83)  \nLemma 5.12. For any ˘ v , η , ε , and m 0 satisfying (5.15) , the following holds. The development of S λ,M ' ,η,ε obtained in Lemma 5.9 can be uniquely extended to ˜ R ˘ v aux . The spacetime is vacuum for u ≥ 1 3 r 1 and v ≤ ˘ v . Moreover, the solution satisfies the estimates  \n0 ≤ m ≲ θ ( λ ) η, 0 ≤ Q ≲ θ ( λ ) η, Ω 2 ∼ 1 , ∂ v r ∼ -∂ u r ∼ 1 , (1 + u 2 ) | ∂ u Ω 2 | +(1 + v 2 ) | ∂ v Ω | 2 ≲ 1  \non ˜ R ˘ v aux and  \n1 4 ≤ ∂ v r ≤ 3 4  \non ˜ R ˘ v aux ∩ C ˘ v . Finally, the support of the distribution function satisfies  \nπ (spt f ) ∩ ˜ R ˘ v aux ⊂ {-2 r 1 ≤ u ≤ 1 r 1 } ,  \ninf spt( f ) ∩ ˜ R ˘ v aux r ≥ 1 6 r 1  \n3 6 ,  \n̸  \nand if u ∈ [ -2 3 r 1 , 1 3 r 1 ] , p u , and p v are such that f ( u, ˘ v, p u , p v ) = 0 , then  \np u p v ≲ ˘ v -2 , ℓ 2 p v ≲ 1 . (5.84)  \nThe proof of Lemma 5.12 will be given on Page 66. We will make use of a bootstrap argument in the regions R v 0 aux , where v 0 ranges over [ 1 3 r 1 , ˘ v ] . The triangle { v ≥ u }∩{ u ≥ 1 3 r 1 }∩{ v ≤ ˘ v } is Minkowskian and can simply be attached at the very end of the argument, cf. Lemma 3.27. For the basic geometric setup of the lemma and its proof, refer to Fig. 13.  \nFor ( u, v ) ∈ R ˘ v aux , let  \nˆ r ( u, v ) = r 1 6 -u 2 + 1 2 ∫ v 1 3 r 1 β ( v ' ) dv ' , ˆ Ω 2 ( u, v ) = β ( v ) ,  \nwhere  \nβ ( v ) . = { 1 if v < 2 3 r 1 Ω 2 ( -2 3 r 1 , v ) if v ≥ 2 3 r 1 .  \nIt is easily verified that (ˆ r, ˆ Ω 2 ) is a solution of the spherically symmetric Einstein vacuum equations and matches smoothly in v with ( r, Ω 2 ) from R 1 2 ˘ v -1 3 r 1 main along C -2 3 r 1 .  \nThe first bootstrap assumption is  \n| Q | + | ϖ | + | ∂ v r -∂ v ˆ r | + | ∂ u r -∂ u ˆ r | + | Ω 2 -ˆ Ω 2 | + | ∂ v Ω 2 -∂ v ˆ Ω 2 | + | ∂ u Ω 2 | ≤ Aθ ( λ ) ηe Bτ , (5.85)  \nwhere A ≥ 1 and B ≥ 1 are constants to be determined that may depend on ˘ v but not η . We also make the following assumption on the electromagnetic geodesic flow. For ( v ' 0 , p u 0 , p v 0 ) ∈ spt( ˚ f r 1 ,θ ( λ ) η aux ) , let γ be an electromagnetic geodesic of mass m for ( r, Ω 2 , Q ) starting at ( -v ' 0 , v ' 0 , p u 0 , p v 0 ) , and let ˆ γ be a null geodesic for (ˆ r, ˆ Ω 2 ) starting at ( -v ' 0 , v ' 0 , p u 0 , p v 0 ) . Then, assuming both γ and ˆ γ remain within R v 0 aux , we assume that  \n| Ω 2 p u -ˆ Ω 2 ˆ p u | + | Ω 2 p v -ˆ Ω 2 ˆ p v | ≤ Aθ ( λ ) ηe B ( γ v -v ' 0 ) . (5.86)  \nFirst, we note the following immediate consequences of the first bootstrap assumption:  \nLemma 5.13. If (5.15) holds, v 0 ∈ [ 1 3 r 1 , ˘ v ] , R v 0 aux ⊂ U , the bootstrap assumption (5.85) holds on R v 0 aux , and η is sufficiently small depending on A and B , then on R v f aux it holds that  \n0 ≤ ϖ ≲ θ ( λ ) η, 0 ≤ Q ≲ θ ( λ ) η, | log ∂ v r | + | log( -∂ u r ) | + | log Ω 2 | ≲ 1 , | ∂ v Ω 2 | + | ∂ u Ω 2 | ≲ 1 .  \nNext, we use the second bootstrap assumption to obtain  \nLemma 5.14. If (5.15) holds, v 0 ∈ [ 1 3 r 1 , ˘ v ] , R v 0 aux ⊂ U , the bootstrap assumptions (5.85) and (5.86) hold on R v 0 aux , B is sufficiently large, and η is sufficiently small depending on A and B , then the following holds. Let γ : [0 , S ] →R v 0 aux be a future-directed electromagnetic geodesic starting in spt( ˚ f r 1 ,θ ( λ ) η aux ) , then  \nr ≥ 1 6 r 1 , u ≤ 1 3 r 1 , r 2 p u p v ≲ 1 , ℓ 2 p v ≲ 1  \nalong γ .  \nProof. Upon making the coordinate transformation  \n˜ u = u, ˜ v = 1 3 r 1 + ∫ v 1 3 r 1 β ( v ' ) dv ' , (5.87)  \nthe metric (ˆ r, ˆ Ω 2 ) is brought into the standard Minkowski form ( 1 2 (˜ v -˜ u ) , 1) . If ˜ t . = 1 2 (˜ v + ˜ u ) , ˆ γ is a null geodesic in R ˘ v aux with respect to (ˆ r, ˆ Ω 2 ) intersecting { τ = 0 } with momentum ( p u 0 , p v 0 ) = ( p ˜ u 0 , p ˜ v 0 ) at an area-radius of ˆ r 0 , then it is easy to check that  \nˆ r 2 =   ˜ t +sign( p v 0 -p u 0 ) √ ˆ r 2 0 -ˆ ℓ 2 ˆ E 2   2 + ˆ ℓ 2 ˆ E 2 , (5.88)  \np ˜ u =    ˆ E + √ ˆ E 2 -ˆ ℓ 2 / ˆ r 2 if ˜ t < -sign( p v 0 -p u 0 ) √ ˆ r 2 0 -ˆ ℓ 2 / ˆ E 2 ˆ E -√ ˆ E 2 -ˆ ℓ 2 / ˆ r 2 if ˜ t ≥ -sign( p v 0 -p u 0 ) √ ˆ r 2 0 -ˆ ℓ 2 / ˆ E 2 , (5.89)  \np ˜ v =    ˆ E -√ ˆ E 2 -ˆ ℓ 2 / ˆ r 2 if ˜ t < -sign( p v 0 -p u 0 ) √ ˆ r 2 0 -ˆ ℓ 2 / ˆ E 2 ˆ E + √ ˆ E 2 -ˆ ℓ 2 / ˆ r 2 if ˜ t ≥ -sign( p v 0 -p u 0 ) √ ˆ r 2 0 -ˆ ℓ 2 / ˆ E 2 (5.90)  \nalong ˆ γ , where ˆ ℓ 2 . = ˆ r 2 p ˜ u p ˜ v and ˆ E . = 1 2 ( p ˜ v + p ˜ u ) are conserved quantities. If ( p u 0 , p v 0 ) ∈ [Λ -1 , Λ+1] 2 and Λ satisfies (5.11), then  \n81 100 ˆ r 2 0 ≤ ˆ ℓ 2 ˆ E 2 ≤ ˆ r 2 0 .  \nFrom (5.88) it is apparent that  \nmin ˆ γ r ≥ ˆ ℓ ˆ E ≥ 9 10 ˆ r 0 , (5.91)  \nsup ˆ γ ˜ u = -sign( p v 0 -p u 0 ) √ ˆ r 2 0 -ˆ ℓ 2 ˆ E 2 ≤ 45 100 ˆ r 0 , (5.92)  \nand by inspection of (5.89) and (5.90) that  \nˆ r 2 p ˜ u p ˜ v ≲ 1 , ˆ ℓ 2 p ˜ v ≲ 1 (5.93)  \nalong ˆ γ .  \nLet γ and ˆ γ be as defined before (5.86). Parametrize γ and ˆ γ by v as in the proof of Lemma 5.11. Then  \n∣ ∣ ∣ ∣ d dv ( γ u -ˆ γ u ) ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ p u p v -ˆ p u ˆ p v ∣ ∣ ∣ ∣ ≲ Aθ ( λ ) ηe B ( v -v ' 0 )  \nby (5.86) and the observation that p v ≳ 1 and p ˜ v ≳ 1 , so that  \n| γ u -ˆ γ u | ≤ θ ( λ ) ηe B ( v -v ' 0 ) (5.94)  \nfor B chosen sufficiently large. Therefore, the conclusions of the lemma follow from the estimates (5.91)(5.93) and the fact that ˆ r 0 ∈ [ 1 3 r 1 , 2 3 r 1 ] after undoing the coordinate transformation (5.87) and applying the bootstrap assumptions.  \nProof of Lemma 5.12. The proof is a bootstrap argument based on the bootstrap assumptions (5.85) and (5.86), the continuation criterion Proposition 3.15, and Lemma 3.27. Define the bootstrap set  \nA . = { v 0 ∈ [ 1 3 r 1 , ˘ v ] : the solution extends uniquely to R v 0 aux and (5.85), (5.86) hold on R v 0 aux } .  \nThe set A is nonempty by Propositions 3.17 and 3.23 if A is chosen sufficiently large and is manifestly closed and connected. We now show that if the parameters satisfy (5.15), then A is also open.  \nLet v 0 ∈ A . Taking m 0 sufficiently small and using the formula (5.9), the initial data estimate Lemma 5.8, and Lemmas 5.13 and 5.14, we immediately find  \nN u + N v + T uu + T uv + T vv + S ≲ r -2 θ ( λ ) η 1 { r ≥ 1 6 r 1 } (5.95)  \non R v 0 aux . By (5.27) and the observation that f = 0 along C -2 3 r 1 ∩ R v 0 aux , we have  \n| Q | + | ϖ | + | ∂ v r -∂ v ˆ r | + | Ω -2 ∂ v r -ˆ Ω -2 ∂ v ˆ r | + | Ω 2 -ˆ Ω 2 | + | ∂ v Ω 2 -∂ v ˆ Ω 2 | + | ∂ u Ω 2 | ≲ θ ( λ ) η (5.96)  \nalong { τ = 0 } and C -2 3 r 1 in R v 0 aux . Then, using (5.95) and the Einstein-Maxwell-Vlasov system, we see that (5.96) holds on R v 0 aux . This improves (5.85) for an appropriate choice of A (independent of η ).  \nWe now improve (5.86). Using the Lorentz force equations (2.24) and (2.25), we have  \nd dv (Ω 2 p u -ˆ Ω 2 ˆ p u ) = ( ∂ v log Ω 2 -2 ∂ v r r ) ℓ 2 r 2 p v ∣ ∣ ∣ ∣ γ -( ∂ v log ˆ Ω 2 -2 ∂ v ˆ r ˆ r ) ˆ ℓ 2 ˆ r 2 ˆ p v ∣ ∣ ∣ ∣ ∣ ˆ γ -e Ω 2 Q r 2 p u p v , d dv (Ω 2 p v -ˆ Ω 2 ˆ p v ) = ( ∂ u log Ω 2 -2 ∂ u r r ) ℓ 2 r 2 p v ∣ ∣ ∣ ∣ γ + 2 ∂ u ˆ r ˆ r ˆ ℓ 2 ˆ r 2 ˆ p v ∣ ∣ ∣ ∣ ∣ ˆ γ + e Ω 2 Q r 2 .  \nwhere  \nFigure 14: Penrose diagram of the bootstrap region R ˘ v,v f far used in the proof of Lemma 5.15.  \n<!-- image -->  \nUsing the parameter hierarchy (5.15), the bootstrap assumptions (5.85) and (5.86), Lemmas 5.13 and 5.14, and the bound (5.94), we can estimate  \n∣ ∣ ∣ ∣ d dv (Ω 2 p u -ˆ Ω 2 ˆ p u ) ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ d dv (Ω 2 p u -ˆ Ω 2 ˆ p u ) ∣ ∣ ∣ ∣ ≲ θ ( λ ) η + Aθ ( λ ) ηe B ( v -v 0 ) .  \nIntegrating and choosing the constants A and B sufficiently large in terms of the implied constants and ˘ v improves (5.86) and shows that the solution extends to R ˘ v aux .  \nThe solution is at once extended to ˜ R ˘ v aux by Lemma 3.27. The rest of the conclusions of the lemma follow immediately from Lemmas 3.27, 5.9, 5.13, and 5.14 and (5.95).", '5.6 The far region': "Lemma 5.15. For any ˘ v , η , ε , and m 0 satisfying (5.15) , there exists a constant C ν > 0 such that the following holds. The development of S λ,M ' ,η,ε obtained in Lemma 5.12 can be uniquely extended globally to C r 2 . Moreover, the solution satisfies the estimates  \n0 ≤ m ≤ 10 M, 0 ≤ Q ≤ 6 M, (5.97)  \nΩ 2 ∼ 1 , ∂ v r ∼ -∂ u r ∼ 1 , (5.98)  \n(1 + u 2 ) | ∂ u Ω 2 | ≲ 1 , v 2 | ∂ v Ω 2 | ≲ 1 (5.99)  \non C r 2 ∩ { v ≥ ˘ v } , and the distribution function satisfies  \nπ (spt f ) ∩ { v ≥ ˘ v } ⊂ { 6 C ν u ≤ v } .  \nWe will make use of a bootstrap argument in the regions  \nR ˘ v,v f far . = { v ≥ u } ∩ { τ ≥ 0 } ∩ { u ≥ -r 2 } ∩ { ˘ v ≤ v ≤ v f } ,  \nwhere v f ≥ ˘ v . Refer to Fig. 14. The bootstrap assumptions are  \n-C ν ≤ ∂ u r ≤ -C -1 ν , (5.100)  \n1 5 ≤ ∂ v r ≤ 1 , (5.101)  \nπ (spt f ) ∩ R ˘ v,v f far ⊂ W , (5.102)  \nW . = { 6 C ν u ≤ v } ∩ { v ≥ ˘ v }  \nand the constant 10 ≤ C ν ≲ 1 is chosen so that -1 2 C ν ≤ ∂ u r ≤ -2 C -1 ν on C ˘ v . Such a constant exists by Lemmas 5.9 and 5.12.  \nLemma 5.16. If (5.15) holds, v f ≥ ˘ v , R ˘ v,v f far ⊂ U , and the bootstrap assumptions (5.100) -(5.102) hold on R ˘ v,v f far , then  \n0 ≤ m ≤ 10 M, 0 ≤ Q ≤ 6 M, (5.103)  \nΩ 2 ∼ 1 , (5.104)  \n∂ v log Ω 2 ≲ v -2 , (5.105)  \n1 2 ≤ 1 -2 m r ≤ 1 (5.106)  \nr ∼ v, (5.107)  \n∂ v log Ω 2 < 2 ∂ v r r (5.108)  \nm = Q = 0 (5.109)  \non R ˘ v,v f far \\ W .  \nProof. Proof of (5.103) and (5.109): The bootstrap assumptions (5.100) and (5.101) imply ∂ u r < 0 and ∂ v r > 0 . Therefore, (5.103) follows from the monotonicity properties of the Einstein-Maxwell-Vlasov system, Lemmas 5.8, 5.9, and 5.12, and the boundary condition (3.82).  \nProof of (5.107): By the bootstrap assumption (5.100),  \nr ( u, v ) -r ( -r 2 , v ) ≥ -C ν ( u + r 2 ) ≥ -1 6 v -C ν r 2  \nfor ( u, v ) ∈ W . By (5.38), the lower bound in (5.107) easily follows if ˘ v is taken sufficiently large. Since ∂ u r < 0 , r ( u, v ) ≤ r ( -r 2 , v ) ≲ v for v ≥ ˘ v and ˘ v sufficiently large, which proves the upper bound in (5.107). Proof of (5.106): This is immediate for ˘ v chosen sufficiently large in light of (5.103) and the fact that  \ninf W r ≳ ˘ v, (5.110)  \nwhich follows from (5.107).  \nProof of (5.104): This follows from (2.3) by combining the bootstrap assumptions (5.100) and (5.101) with (5.106).  \nProof of (5.105): Let ( u, v ) ∈ W . We will show that  \n∫ u -r 2 T uv ( u ' , v ) du ' ≲ v -2 , (5.111)  \nwhich together with (3.21), (5.103), (5.104), and (5.107), readily implies (5.105). To prove (5.111), we observe that by the bootstrap assumptions (5.100) and (5.101) and the evolution equation (3.28), ∂ u m ≤ 0 . Using that m ≥ 0 in R ˘ v,v f aux , we therefore infer  \n∫ u -r 2 [ 1 2 r 2 Ω 2 ( T uv ( -∂ u r ) + T vv ∂ v r ) + Q 2 2 r 2 ( -∂ u r ) ]∣ ∣ ∣ ∣ ( u ' ,v ) du ' = m ( -r 2 , v ) -m ( u, v ) ≤ 2 ϖ 2 .  \nSince all three integrands are nonnegative, this energy estimate, taken together with (5.100), (5.104) and (5.107) imply  \nv 2 ∫ u -r 2 T uv ( u ' , v ) du ' ≲ ∫ u -r 2 r 2 Ω 2 T uv ∂ v r ∣ ∣ ( u ' ,v ) du ' ≲ 1  \nfor ( u, v ) ∈ W , which proves (5.111).  \nProof of (5.108): This follows from the fact that ∂ v r/r ≳ v -1 in W by (5.101) and (5.107).  \non R ˘ v,v f far and  \non W . Furthermore,  \nProof of Lemma 5.15. The proof is a bootstrap argument based on the bootstrap assumptions (5.100)(5.102). Let  \nA . = { v f ∈ [˘ v, ∞ ) : R ˘ v,v f aux ⊂ U and (5.100)-(5.102) hold on R ˘ v,v 0 aux } .  \nThe set A is nonempty by Proposition 3.23, Lemma 5.9, and Lemma 5.12. It is also manifestly closed by continuity of the bootstrap assumptions. We now show that if (5.15) holds, then A is also open. Let v f ∈ A . ˘ v,v '  \nImproving (5.101): Let ( u, v ) ∈ R f aux . Integrating the wave equation (3.20) in u starting at u = -r 2 and using the estimates of Lemma 5.16 yields  \n| ∂ v r ( u, v ) -1 2 | ≤ ∫ u -r 2 ( Ω 2 2 r 2 ( m + Q 2 2 r ) + 1 4 r Ω 4 T uv )∣ ∣ ∣ ∣ ( u ' ,v ) du ' ≲ v -1 ≤ ˘ v -1 ,  \nwhich improves (5.101) for ˘ v sufficiently large.  \nImproving (5.100): Using (3.29), ∂ v m ≥ 0 , and (5.110), we obtain (similarly to (5.111))  \n∫ v 2 v 1 rT uv | ( u,v ' ) dv ' ≲ ˘ v -1 (5.112)  \nfor any ( u, v 1 ) , ( u, v 2 ) ∈ R ˘ v,v f aux . Let ( u, v ) ∈ R ˘ v,v f aux . We integrate the wave equation (3.20) in v starting at C ˘ v if u ≤ ˘ v and at ( u, u ) ∈ Γ if u > ˘ v . In the former case,  \n| ∂ u r ( u, v ) -∂ u r ( u, ˘ v ) | ≤ ∫ v ˘ v ( Ω 2 2 r 2 ( m + Q 2 2 r ) + 1 4 r Ω 2 T uv )∣ ∣ ∣ ∣ ( u,v ' ) dv ' ≲ ˘ v -1 ,  \nwhich improves (5.100) for ˘ v sufficiently large by the definition of C ν . In the latter case, the boundary condition (3.82) implies ∂ u r ( u, u ) = -∂ v r ( u, u ) , so  \n| ∂ u r ( u, v ) + ∂ v r ( u, u ) | ≤ ∫ v u ( Ω 2 2 r 2 ( m + Q 2 2 r ) + 1 4 r Ω 2 T uv )∣ ∣ ∣ ∣ ( u,v ' ) dv ' ≲ ˘ v -1 ,  \nwhich by (5.101) improves (5.100) for ˘ v sufficiently large by the definition of C ν .  \nImproving (5.102): Let γ : [0 , S ) →R ˘ v,v f aux be an electromagnetic geodesic in the support of f starting at C ˘ v at s = 0 . By (2.24) and (5.108),  \nd ds (Ω 2 p u ) ≤ 0 ,  \nso by (5.104),  \np u ( s ) ≲ p u (0) . (5.113)  \nUsing (2.22), the signs of ∂ u r and Q , and parametrizing γ by v yields  \nd dv log p v ≥ -∂ v log Ω 2 . (5.114)  \nBy (5.105), it is easy to see that  \nand therefore  \n∫ v ˘ v | ∂ v log Ω 2 ( γ u ( v ' ) , v ' ) | dv ' ≲ ˘ v -1  \nexp ( -∫ v ˘ v ∂ v log Ω 2 ( γ u ( v ' ) , v ' ) dv ' ) ≥ 1 2  \nfor ˘ v sufficiently large. It follows from (5.114) that  \np v ( s ) ≳ p v (0) . (5.115)  \nCombining (5.113) and (5.115) yields  \np u p v ( s ) ≤ 1 7 C ν ,  \nfor ˘ v sufficiently large by (5.49) and (5.84). It is then easy to show that γ ( s ) stays in { 7 C ν u ≤ v } for every s ∈ [0 , S ) , which quantitatively improves (5.102). The rest of the existence and uniqueness proof now follows a standard continuity argument using Proposition 3.17 and Lemma 3.27.  \nTo estimate | ∂ u Ω 2 | , note that | ∂ u Ω 2 | ≲ (1 + u 2 ) -1 along C ˘ v ∪ Γ by Lemma 5.12 and (3.84). Observe that  \ninf C u ∩W∩R ˘ v, ∞ far r ∼ inf C u ∩W∩R ˘ v, ∞ far v = max { ˘ v, 6 C ν u } ≳ 1 + u,  \nso the energy estimate (5.112) can be improved to  \n∫ v 2 v 1 T uv ( u, v ' ) dv ' ≲ (1 + u ) -2  \nfor any u ≤ v 1 ≤ v 2 . Therefore the desired estimate can be propagated to the interior by integrating the wave equation (3.21) in v . Together with the bootstrap assumptions and Lemma 5.16, this completes the proof of the estimates (5.97)-(5.99).", '5.7 The dispersive estimate in the massive case': "Let m > 0 and consider the solution ( r, Ω 2 , Q, f ) given by Lemma 5.15, defined globally on C r 2 . We augment the hierarchy (5.15) with a large parameter v # satisfying  \n0 < v -1 # ≪ m ≤ m 0 (5.116)  \nand aim to prove the following  \nLemma 5.17. For any ˘ v, η, ε, m 0 , m , and v # satisfying (5.15) and (5.116) , we have the decay  \nM≤ Cv -3  \nfor v ≥ v # and any M∈{ N u , N v , T uu , T uv , T vv , S } , where C may depend on η, ε , m , and v # .  \nThe proof is based on a bootstrap argument for the dispersion of ingoing momentum p u along spt f as v →∞ , which leads to cubic decay of the phase space volume V , which was defined in (5.8). Using the mass shell relation (3.12) and the change of variables formula, we have  \nV ( u, v ) = 2 r 2 ∫ ∞ 0 ∫ { p u : f ( u,v,p u ,p v ) =0 } dp u p u ℓ dℓ, (5.117)  \n̸  \nwhere we view p v as a function of p u and ℓ . Compare with (5.9).  \nLemma 5.18. If (5.15) and (5.116) hold and ( u, v, p u , p v ) ∈ spt f , then p v ≲ 1 if v ≥ ˘ v and p u ∼ η m 2 if v ≥ v # .  \nRemark 5.19 . The estimate p v ≲ 1 also holds in the massless case. The non-decay of p u for massive particles drives the decay rate v -3 , but only at very late times.  \nProof. Let γ ∈ Γ f and v ≥ ˘ v . Parametrizing γ by v and using (5.49), (5.66), (5.84), and the estimates in Lemma 5.15, we infer  \n∣ ∣ ∣ ∣ d dv (Ω 2 p v ) ∣ ∣ ∣ ∣ ≲ v -2 ,  \nwhich is integrable and hence shows that p v ≲ 1 . By (5.49) and (5.84), we have p v ≳ η 1 . We then obtain  \np u = ℓ 2 r -2 + m 2 Ω 2 p v ∼ η ℓ 2 r 2 + m 2 ∼ m 2  \nfor r ≳ v # sufficiently large.  \nUsing this lemma and (5.65), we immediately infer:  \n<!-- image -->  \nLemma 5.20. If (5.15) and (5.116) hold, ( u, v, p u , p v ) ∈ spt f , and v ≥ v # , then u ∼ η m 2 v . If γ 1 , γ 2 ∈ Γ f ( u, v ) , then we have | γ u 1 ( s 1 v # ) -γ u 2 ( s 2 v # ) | ( s v # ) ≲ η m 2 ( v -v # ) , where s i v # is the parameter time for which γ v i ( s i # ) = v # .  \nLet ( u 0 , v 0 ) ∈ C r 2 and let γ ∈ Γ f ( u 0 , v 0 ) have ingoing momentum p u 0 and angular momentum ℓ at ( u 0 , v 0 ) . We parametrize γ by v going backwards in time and denote this by  \nγ u ( v ) . = γ u ( v ; u 0 , v 0 , p u 0 , ℓ ) , p u ( v ) . = p u ( v ; u 0 , v 0 , p u 0 , ℓ ) .  \nWe readily derive the equations  \nd dv γ u = (Ω 2 p u ) 2 Ω 2 ( ℓ 2 r -2 + m 2 ) , (5.118)  \nd dv (Ω 2 p u ) = ( ∂ v log Ω 2 -2 ∂ v r r ) ℓ 2 Ω 2 p u r 2 m 2 + ℓ 2 -e Q r 2 m 2 + ℓ 2 (Ω 2 p u ) 2 . (5.119)  \nNext, we define the variational quantities  \nu ( v ; u 0 , v 0 , p u 0 , ℓ ) . = ∂ ∂p u 0 γ u ( v ; u 0 , v 0 , p u 0 , ℓ ) , p ( v ; u 0 , v 0 , p u 0 , ℓ ) . = ∂ ∂p u 0 (Ω 2 p u )( v ; u 0 , v 0 , p u 0 , ℓ ) ,  \nwhere we emphasize that the derivative in p u 0 is taken with ℓ fixed. From (5.118) and (5.119) we obtain  \nd dv u = 2 p u ℓ 2 r -2 + m 2 p -( p u ) 2 ( ℓ 2 r -2 + m 2 ) 2 [ ∂ u Ω 2 ( ℓ 2 r 2 + m 2 ) -2Ω 2 ℓ 2 ∂ u r r 3 ] u , (5.120) d dv p = [( ∂ v log Ω 2 -2 ∂ v r r ) ℓ 2 r 2 m 2 + ℓ 2 -2 e Q Ω 2 p u r 2 m 2 + ℓ 2 ] p + [( ∂ u ∂ v log Ω 2 -2 ∂ u ∂ v r r +2 ∂ v r∂ u r r 2 ) ℓ 2 Ω 2 p u r 2 m 2 + ℓ 2 -( ∂ v log Ω 2 -2 ∂ v r r ) 2 m 2 r∂ u rℓ 2 Ω 2 p u ( r 2 m 2 + ℓ 2 ) 2 -e (Ω 2 p u ) 2 r 2 m 2 + ℓ 2 ∂ u Q + 2 e Q m 2 r∂ u r (Ω 2 p u ) 2 ( r 2 m 2 + ℓ 2 ) 2 ] u . (5.121)  \nNote that u ( v 0 ) = 0 and p ( v 0 ) = Ω 2 ( u 0 , v 0 ) ∼ 1 .  \nLemma 5.21. If (5.15) and (5.116) hold, and v ≥ v # , then  \nV ( u, v ) ≲ η,ε v # v 3 . (5.122)  \nProof. We claim that there exists a constant C ∗ , depending on η and ε , such that  \nC -1 ∗ ≤ u ( v ) v -v 0 ≤ C ∗ (5.123)  \nfor any v # ≤ v ≤ v 0 . To see how this proves (5.122), let Φ u 0 ,v 0 ,ℓ ( p u 0 ) . = γ u ( v # ; u 0 , v 0 , p u 0 , ℓ ) and observe that Φ ' u 0 ,v 0 ,ℓ ( p u 0 ) = u ( v # ) < 0 . Changing variables in the p u integral in (5.117) to γ u ( v # ) and using Lemma 5.20 to estimate the u -dispersion along C v # , we find  \nV ( u 0 , v 0 ) ≲ η,ε 1 r 2 m 2 C ∗ | v # -v 0 | min { m 2 v # , m 2 ( v 0 -v # ) } ≲ C ∗ v # v 3 0 . (5.124)  \nWe prove (5.123) by a bootstrap argument as follows. Let v f ≥ v # and assume (5.123) holds for all ( v, u 0 , v 0 , p u 0 , ℓ ) with v # ≤ v ≤ v 0 ≤ v f . The assumption is clearly satisfied for some choice of C ∗ for v f sufficiently close to v # on account of u ( v 0 ) = 0 .  \nWe now show that for m sufficiently small and v # sufficiently large, we can improve the constant in (5.123). Using (5.122), we estimate  \nN v ≲ η,ε C ∗ v # v 3 , T uv ≲ η,ε C ∗ m 2 v # v 3  \nfor v ∈ [ v # , v f ] . Using this, as well as (3.21), (3.24), Lemma 5.15, and p u ∼ η m 2 in (5.121) yields  \n∣ ∣ ∣ ∣ d dv p ∣ ∣ ∣ ∣ ≲ η,ε ( 1 m 2 v 3 + 1 v 2 ) | p | + ( 1 v 4 + m 2 v 3 + C ∗ m 2 v # v 3 ) | u | .  \nWe use the bootstrap assumption (5.123) and (5.116) to infer  \n∣ ∣ ∣ ∣ d dv ( p -p ( v 0 )) ∣ ∣ ∣ ∣ ≲ η,ε 1 v 2 | p -p ( v 0 ) | + 1 v 2 + C 2 ∗ m 2 v # v 3 | v -v 0 | .  \nand then use Grönwall's inequality to obtain  \n| p ( v ) -p ( v 0 ) | ≲ η,ε v -1 # + C 2 ∗ m 2 v # v 2 | v -v 0 | .  \nUsing p ( v 0 ) ∼ 1 , we therefore have  \n∫ v 0 v p ( v ' ) dv ' ∼ η,ε v 0 -v + O η,ε ( C 2 ∗ m 2 | v -v 0 | ) ∼ v 0 -v (5.125)  \nfor m sufficiently small. Finally, we use (5.99) and Lemma 5.20 to estimate  \n∫ v 0 v | ∂ u Ω 2 | ( γ u ( v ' ) , v ' ) ( ℓ 2 r 2 + m 2 ) dv ' ≲ η ∫ γ u ( v 0 ) γ u ( v ) 1 1 + u ' 2 du ' ≲ 1 γ u ( v # ) ≲ 1 m 2 v # . (5.126)  \nIntegrating (5.120) and using (5.125) and (5.126) improves the constant in (5.123) for m sufficiently small and v # sufficiently large, which completes the proof.  \nProof of Lemma 5.17.  \nImmediate from (5.122).", '5.8 Proof of Proposition 5.5': "Proof. Part 1. This follows immediately from combining Lemmas 5.9, 5.12, and 5.15.  \nPart 2. The estimates in (5.16) follow directly from the estimates in Lemmas 5.9, 5.12, and 5.15. We will now prove causal geodesic completeness of the spacetime. Let γ be a future-directed causal geodesic in the (3 + 1) -dimensional spacetime. Then the projection of γ to the reduced spacetime, still denoted γ , satisfies  \nd ds (Ω 2 p u ) = ( ∂ v log Ω 2 -2 ∂ v r r ) ℓ 2 r 2 , (5.127)  \nd ds (Ω 2 p v ) = ( ∂ u log Ω 2 -2 ∂ u r r ) ℓ 2 r 2 , (5.128)  \nΩ 2 p u p v = ℓ 2 r 2 + m 2 , (5.129)  \nwhere s is an affine parameter and γ is continued through the center according to Part 4 of Definition 3.21. We will show for any future-directed causal geodesic γ : [0 , S ) →C r 2 , p τ is uniformly bounded in any compact interval of coordinate time 0 ≤ τ ≤ τ 0 along γ . This implies that γ can be extended to [0 , ∞ ) by the normal neighborhood property of the geodesic flow in Lorentzian manifolds [ONe83].  \nBy (5.127)-(5.129) and (5.16),  \n∣ ∣ ∣ ∣ d ds (Ω 2 p τ ) ∣ ∣ ∣ ∣ ≲ ( 1 + ∣ ∣ ∣ ∣ ∂ v r + ∂ u r r ∣ ∣ ∣ ∣ ) (Ω 2 p τ ) 2 . (5.130)  \nWhen r ≥ 1 6 r 1 , the term in the absolute value on the right-hand side of this estimate is clearly bounded by (5.16). When r ≤ 1 6 r 1 , the spacetime is Minkowski and the formulas in Lemma 3.27 can be used to show that  \n∂ v r + ∂ u r ≲ v -u ≲ r. (5.131)  \nTherefore, passing to a τ parametrization of γ , (5.130) implies  \n∣ ∣ ∣ ∣ d dτ (Ω 2 p τ ) ∣ ∣ ∣ ∣ ≲ Ω 2 p τ ,  \nand the proof is completed by an application of Grönwall's lemma. 25  \nPart 3. The estimate (5.17) for the final parameters follows from (5.31). The remaining claims in this part follow from the proof of Lemma 5.8, (5.59) on C -2 3 r 1 , and Lemma 5.15.  \nPart 4. The estimate (5.18), the upper bound in (5.19), and the claim about the neighborhood of the center follow from Lemma 5.15. The lower bound in (5.19) follows from Lemma 5.18, which implies that γ u grows linearly in v at very late times for any electromagnetic geodesic γ in the support of f . Since a neighborhood of I + is then electrovacuum, it is isometric to an appropriate Reissner-Nordström solution by Birkhoff's theorem.  \nPart 5. This follows immediately from Lemma 5.17.  \nPart 4 ' . We now take m = 0 and seek to prove the upper bound in (5.21), the rest following immediately from Birkhoff's theorem. Let γ ( s ) be an electromagnetic geodesic lying in the support of f with γ v (0) = ˘ v . When m = 0 , the mass shell relation, together with (5.49), (5.84), and (5.115) gives the estimate  \np u p v ( s ) = r 2 (0) r 2 ( s ) Ω 2 (0) Ω 2 ( s ) p u (0) p v ( s ) p v (0) p v ( s ) ≲ v -2 ,  \nwhich has integral ≲ ˘ v -1 . Therefore, the upper bound in (5.21) follows from (5.65) and taking ˘ v sufficiently large.  \nPart 5 ' . When m = 0 , the mass shell relation implies p u ≲ v -2 for any electromagnetic geodesic lying in the support of f . The estimates (5.22)-(5.24) follow from this and the formula (5.9).", '5.9.1 The maximal time-symmetric doubled spacetime': "Let α ∈ P Γ or be of the form ( r 1 , r 2 , 0 , 0 , 0 , 0) . Let η and ε be beam parameters for which the conclusion of Proposition 5.5 holds, recalling also Remark 5.6. Let ˜ M and ˜ e denote the final Reissner-Nordström parameters of S = S α,η,ε . We say that S is subextremal if ˜ e < ˜ M , extremal if ˜ e = ˜ M , and superextremal if ˜ e > ˜ M . 26 If S is not superextremal, we may define  \nr ± = ˜ M ± √ ˜ M 2 -˜ e 2 ,  \nwhich is the formula for the area radii of the outer and inner horizons in Reissner-Nordström.  \nThe following lemma is an easy consequence of the well-known structure of the maximally extended Reissner-Nordström solution:  \nLemma 5.22. For any ˜ M, ˜ e > 0 and 0 < r 2 < r -(if ˜ e ≤ ˜ M ), there exists a relatively open set  \nE ˜ M, ˜ e,r 2 ⊂ { u ≤ -r 2 } ∩ { v ≥ r 2 } ⊂ R 2 u,v  \nand an analytic spherically symmetric solution ( r, Ω 2 , Q ) of the Einstein-Maxwell equations on E ˜ M, ˜ e,r 2 with the following properties: The renormalized Hawking mass ϖ = ˜ M globally, the charge Q = ˜ e globally, r ( -r 2 , r 2 ) = r 2 , ∂ v r ( -r 2 , r 2 ) = -∂ u r ( -r 2 , r 2 ) = 1 2 , and Ω 2 is constant along the cones { u = -r 2 }∩{ v ≥ r 2 } and { v = r 2 } ∩ { u ≤ -r 2 } . Moreover, we may choose ( E ˜ M, ˜ e,r 2 , r, Ω 2 , Q ) to be maximal with these properties, and it will then be unique.  \n̸  \nRemark 5.23 . The (3 + 1) -dimensional lift of ( E ˜ M, ˜ e,r 2 , r, Ω 2 , Q ) is isometric to a subset of the maximally extended Reissner-Nordström solution with parameters ˜ M and ˜ e . The hypersurface ( { τ = 0 }∩E ˜ M, ˜ e,r 2 ) × S 2 is then totally geodesic.  \nRemark 5.24 . If ˜ M < ˜ e , then E ˜ M, ˜ e,r 2 = { u ≤ -r 2 } ∩ { v ≥ r 2 } . In the case ˜ e ≤ ˜ M , E ˜ M, ˜ e,r 2 is a strict subset of { u ≤ -r 2 } ∩ { v ≥ r 2 } with this choice of gauge.  \nDefinition 5.25. Let ( C r 2 , r, Ω 2 , Q, f ) be the maximal normalized development of S = S α,η,ε for particles of mass m with final Reissner-Nordström parameters ( ˜ M, ˜ e ) for which the conclusion of Proposition 5.5 holds, recalling also Remark 5.6. Let  \n˜ C r 2 . = { τ ≤ 0 } ∩ { v ≥ u } ∩ { v ≤ r 2 }  \nbe the time-reflection of C r 2 . For ( u, v ) ∈ ˜ C r 2 , define  \n˜ r ( u, v ) = r ( -v, -u ) , ˜ Ω 2 ( u, v ) = Ω 2 ( -v, -u ) , ˜ Q ( u, v ) = Q ( -v, -u ) , ˜ f ( u, v, p u , p v ) = f ( -v, -u, p v , p u ) .  \nLet M . = C r 2 ∪ ˜ C r 2 ∪E ˜ M, ˜ e,r 2 and define ( r, Ω 2 , Q, f ) on M by simply gluing the corresponding functions across the boundaries of the sets C r 2 , ˜ C r 2 , and E ˜ M, ˜ e,r 2 . (We therefore now drop the tilde notation on the solution in C r 2 , except for in the proof of Lemma 5.26 below.) The tuple ( M , r, Ω 2 , Q, f ) is called the maximal time-symmetric doubled spacetime associated to S .  \nLemma 5.26. ( M , r, Ω 2 , Q, f ) is a smooth solution of the spherically symmetric Einstein-Maxwell-Vlasov system. The hypersurface { τ = 0 }∩ M is a totally geodesic hypersurface once lifted to the (3+1) -dimensional spacetime by Proposition 3.10.  \nProof. This is immediate except perhaps across { τ = 0 } ∩ { v ≤ r 2 } . We only have to show that the solution is C 2 × C 2 × C 1 × C 1 regular across this interface by the regularity theory of Proposition 3.14. By Definition 3.18 and Definition 3.21, the solution is clearly C 1 × C 1 × C 0 × C 0 regular. Using Raychaudhuri's equations (3.22), (3.23) and Maxwell's equations (3.24), (3.25), it is easy to see that r is C 2 and Q is C 1 across { τ = 0 } . To check second derivatives of Ω 2 , differentiate ( ∂ u + ∂ v )Ω 2 ( -v, v ) = 0 in v to obtain ∂ 2 u Ω 2 ( -v, v ) = ∂ 2 v Ω 2 ( -v, v ) , which together with the wave equation (3.21) implies C 2 matching. The p u and p v derivatives of f are also continuous by inspection and continuity of spatial derivatives can be proved as follows: On { τ = 0 } , ∂ v f can be eliminated in terms of ∂ u f and d dv ˚ f by (3.77). Then the Maxwell-Vlasov equation (3.26) can be solved for ∂ u f . Performing the same calculation for ˜ f shows that ∂ u f is continuous across { τ = 0 } . The same argument applies for ∂ v f and the proof is complete.", '5.9.2 The anchored Cauchy hypersurface': "In order to define the family of Cauchy data in Theorem 1, we need to identify an appropriate Cauchy hypersurface in each M . Let  \nv ∞ . = { 2 r --r 2 if ˜ e ≤ ˜ M ∞ if ˜ e > ˜ M (5.132)  \nand let Σ u : [ -50 ˜ M,v ∞ ) → R be the smooth function defined by  \nΣ u ( v ) = -100 ˜ M -v  \nfor v ∈ [ -50 ˜ M,r 1 ] , by solving the ODE  \nd dv Σ u ( v ) = ∂ v r ∂ u r ∣ ∣ ∣ ∣ (Σ u ( v ) ,v ) (5.133)  \nwith initial condition Σ u ( r 2 ) = -100 ˜ M -r 2 for v ∈ [ r 2 , v ∞ ) , and by applying the following easy consequence of Borel's lemma for v ∈ [ r 1 , r 2 ] :  \nLemma 5.27. Given 0 < r 1 < r 2 , ˜ M > 0 , and a sequence of real numbers a 1 , a 2 , . . . with a 1 < 0 , there exists a smooth function Σ u : [ r 1 , r 2 ] → [ -100 ˜ M -r 2 , -100 ˜ M -r 1 ] such that d dv Σ u ( v ) < 0 for v ∈ [ r 1 , r 2 ] , Σ u has Taylor coefficients ( -100 ˜ M -r 1 , -1 , 0 , 0 , . . . ) at r 1 , and Taylor coefficients ( -100 ˜ M -r 2 , a 1 , a 2 , . . . ) at r 2 . Moreover, if ˜ M and each a j are smooth functions of the parameters ϖ 2 , Q 2 , ε, η , then f can be chosen to depend smoothly on ϖ 2 , Q 2 , ε, η .  \nRemark 5.28 . For v ≥ r 2 , the curve Σ : v ↦→ (Σ u ( v ) , v ) ∈ E ˜ M, ˜ e,r 2 lies in the domain of outer communication if ˜ e ≤ ˜ M and is contained in a constant time hypersurface in Schwarzschild coordinates. Indeed, the timetranslation Killing vector field in E ˜ M, ˜ e,r 2 is given by the Kodama vector field  \nK . = 2Ω -2 ∂ v r ∂ u -2Ω -2 ∂ u r ∂ v ,  \nwhich is clearly orthogonal to Σ .", '5.9.3 Cauchy data for the Einstein-Maxwell-Vlasov system': 'Let ( M , g, F, f ) be a solution of the (3 + 1) -dimensional Einstein-Maxwell-Vlasov system as defined in Section 3.1. Let i : R 3 → ˜ Σ ⊂ M be a spacelike embedding with future-directed unit timelike normal n to ˜ Σ . As usual, we may consider the induced metric ¯ g . = i ∗ g and second fundamental form k (pulled back to R 3 along i ) of ˜ Σ . The electric field is defined by ¯ E . = i ∗ F ( · , n ) and the magnetic field is defined by ¯ B . = ( ⋆ ¯ g i ∗ F ) ♯ , where ♯ is taken relative to ¯ g . Since the domain of f is the spacetime mass shell P m which is not intrinsic to ˜ Σ , one first has to define a projection procedure to T ˜ Σ ∼ = T R 3 , after which the restriction of f to ˜ Σ can be thought of as a function ¯ f : T R 3 → [0 , ∞ ) . Similarly, the volume forms in the Vlasov energy momentum tensor T and number current N have to be written in terms of ¯ g , after which ¯ ρ T . = i ∗ T ( n, n ) , ¯ j T . = i ∗ T ( n, · ) and ¯ ρ N . = i ∗ N ( n ) can be evaluated on ˜ Σ only in terms of ¯ g and ¯ f . For details of this procedure, we refer to [Rin13, Section 13.4].  \nDefinition 5.29. A Cauchy data set for the Einstein-Maxwell-Vlasov system for particles of mass m and fundamental charge e consists of the tuple Ψ = (¯ g, ¯ k, ¯ E, ¯ B, ¯ f ) on R 3 satisfying the constraint equations 27  \nR ¯ g -| ¯ k | 2 ¯ g +(tr ¯ g ¯ k ) 2 = | ¯ E | 2 ¯ g + | ¯ B | 2 ¯ g +2¯ ρ T [ ¯ f ] , div ¯ g ¯ k -d tr ¯ g ¯ k = -( ⋆ ¯ g ( ¯ E ♭ ∧ ¯ B ♭ )) ♯ + ¯ j T [ ¯ f ] div ¯ g ¯ E = e ¯ ρ N [ ¯ f ] , div ¯ g ¯ B = 0 .  \nWe denote by M ∞ ( R 3 , m , e ) the set of solutions Ψ of the Einstein-Maxwell-Vlasov constraint system on R 3 with the C ∞ loc subspace topology.', '5.9.4 The globally hyperbolic region': 'By Proposition 5.5, Remark 5.6, their time-reversed ( u ↦→ -v , v ↦→ -u ) versions, Remark 5.28, and the structure of the Reissner-Nordström family, we have:  \nProposition 5.30. Let S and ( M , r, Ω 2 , Q, f ) be as in Definition 5.25 and let Σ u ( v ) be the function defined in Section 5.9.2. Let  \nX . = { v ≥ u } ∩ { v < v ∞ } ⊂ M  \nand let Σ = { (Σ u ( v ) , v ) : v ∈ [ -50 ˜ M,v ∞ ) } . Then the following holds:  \n- 1. The manifold M . = (( X \\ Γ) × S 2 ) ∪ Γ with metric g = -Ω 2 dudv + r 2 γ , electromagnetic field, and Vlasov field lifted according to Proposition 3.10 is a globally hyperbolic, asymptotically flat spacetime, free of antitrapped surfaces, with Cauchy hypersurface ˜ Σ . = ((Σ \\ Γ) × S 2 ) ∪ (Σ ∩ Γ) .\n- 2. ( M , g ) possesses complete null infinities I ± and is past causally geodesically complete.\n- 3. If ˜ e > ˜ M , then ( M , g ) is future causally geodesically complete.  \n- 4. If ˜ e ≤ ˜ M , there are two options. (If S α,η,ε is to be untrapped, then necessarily r 2 / ∈ [ r -, r + ] .)  \n̸  \n- (a) If r 2 < r -, then ( M , g ) is future causally geodesically incomplete. The spacetime contains a nonempty black hole region, i.e., BH . = M\\ J -( I + ) = ∅ . The Cauchy hypersurface ˜ Σ is disjoint from BH . The event horizon H + = ∂ ( BH ) is located at u = r 2 -2 r + .\n- (b) If r 2 > r + , then ( M , g ) is future causally geodesically complete.', '5.9.5 Proof of the main theorem': "Proof of Theorem 1. Fix a fundamental charge e > 0 , cutoff functions φ , θ , and ζ as in Section 5.2.1, and a number Λ ≥ 1 satisfying (5.11). Fix the extremal black hole target mass M > 0 and let 0 < r 1 < r 2 and δ > 0 be as in Theorem 4. Let η 0 > 0 be such that Proposition 5.5 applies to the multiparameter family of seed data S λ,M ' ,η,ε (which was defined in Definition 5.4) for λ ∈ [ -1 , 2] , | M ' -M | ≤ δ , 0 < η ≤ η 0 , ε > 0 sufficiently small depending on η , and particle mass 0 ≤ m ≤ m 0 , where m 0 is sufficiently small depending on η and ε .  \nLet F : [0 , ∞ ) 2 → [0 , ∞ ) 2 be defined by F ( λ, M ' ) = ( λ 2 M ' , λM ' ) , which is easily verified to be smoothly invertible on (0 , ∞ ) 2 . Define the function F η,ε ( λ, M ' ) = ( ˜ M, ˜ e ) , the final Reissner-Nordström parameters of S λ,M ' ,η,ε . By (5.17), we find  \n| F η,ε ( λ, M ' ) -F ( ζ ( λ ) , M ' ) | ≲ η (5.134)  \nfor λ ∈ [ -1 , 2] , | M ' -M | ≤ δ , and 0 < η ≤ η 0 . There exists a constant 0 < λ 0 ≪ 1 depending on η 0 such that if λ ∈ [ -1 , λ 0 ] , then | F η,ε ( λ, M ' ) | < 1 2 r 1 . Also, (5.134) implies  \nsup λ 0 ≤ λ ≤ 2 , | M ' -M |≤ δ ∣ ∣ ∣ ∣ ˜ e ˜ M -1 ζ ( λ ) ∣ ∣ ∣ ∣ ≲ λ 0 η. (5.135)  \nFrom smooth convergence of F η,ε to F as η → 0 , we obtain  \nsup λ 0 ≤ λ ≤ 2 , | M ' -M |≤ δ ∣ ∣ ∣ ∣ d dλ ( ˜ e ˜ M ) + ζ ' ( λ ) ζ ( λ ) 2 ∣ ∣ ∣ ∣ ≲ λ 0 η. (5.136)  \nIt follows that the charge to mass ratio ˜ e/ ˜ M is strictly decreasing as a function of λ , for λ ≥ λ 0 .  \nOn any fixed neighborhood of (1 , M ) ∈ R 2 , F -1 · F η,ε converges uniformly to the identity map by (5.134). Therefore, by a simple degree argument 28 we find an assignment ( η, ε ) → ( λ ( η, ε ) , M ' ( η, ε )) such that F η,ε ( λ ( η, ε ) , M ' ( η, ε )) = ( M,M ) . We claim that for η sufficiently small, the family of seed data λ ↦→ S λ . = S λ,M ' ( η,ε ) ,η,ε gives rise to the desired family of spacetimes and Cauchy data:  \n- 1. For λ ∈ [ -1 , λ 0 ] , the final Reissner-Nordström parameters of S λ are < 1 2 r 1 by definition of λ 0 and hence the globally hyperbolic spacetime D λ associated to S λ by Proposition 5.30 is future causally geodesically complete and dispersive. At λ = -1 , the seed data is trivial and hence the development is isometric to Minkowski space.\n- 2. For λ ∈ [ λ 0 , λ ( η, ε )) , the final charge to mass ratio ˜ e/ ˜ M is strictly decreasing towards 1 by (5.136). Therefore, D λ is future causally geodesically complete and dispersive. A neighborhood of spatial infinity i 0 is isometric to a superextremal Reissner-Nordström solution.\n- 3. λ = λ ( η, ε ) is, by construction, critical, with parameter ratio ˜ e/ ˜ M = 1 . D λ contains a nonempty black hole region and for sufficiently large advanced time, the domain of outer communication and event horizon are isometric to an appropriate portion of an extremal Reissner-Nordström black hole.\n- 4. For λ ∈ ( λ ( η, ε ) , 2] , ˜ e/ ˜ M decreases away from 1 by (5.136). By definition of r 2 , r -> r 2 for η sufficiently small and hence D λ contains a nonempty black hole region and for sufficiently large advanced time, the domain of outer communication and event horizon are isometric to an appropriate portion of a subextremal Reissner-Nordström black hole. By (5.135), the charge to mass ratio at λ = 2 can be made arbitrarily close to 1 2 .  \nTo complete the proof, we assign a smooth family of Cauchy data to D λ . Let i λ : [0 , ∞ ) → Σ λ be the arc length parametrization (with respect to the metric g λ ) of the Cauchy surface associated to D λ by Proposition 5.30. Then we may define the embedding ˜ i λ = i λ × id S 2 / ∼ : R 3 → ˜ Σ λ ⊂ M λ (where the central sphere is collapsed to a point). The natural map λ ↦→ Ψ λ , where Ψ λ is the Cauchy data induced on ˜ Σ λ by pullback along ˜ i λ , is smooth. This completes the proof of the theorem.", '5.10 Weak* convergence to dust': "In this section, we show that the spacetimes constructed in Proposition 5.30 weak* converge in an appropriate sense to the bouncing charged null dust spacetimes given by Proposition 4.7. First, we show convergence of an outgoing Vlasov beam to the underlying outgoing formal dust beam:  \nProposition 5.31. Let α ∈ P Γ , let { η i } and { ε j } be decreasing sequences of positive numbers tending to zero, let ( r, Ω 2 , Q, f ) be the solution of the Einstein-Maxwell-Vlasov system associated to ( α, η i , ε j ) by Proposition 5.5 for j ≫ i and arbitrary allowed mass, and let ( r d , Ω 2 d , Q d , N v d , T vv d ) be the outgoing formal dust solution from Section 4.6.2 on C r 2 . Then the following holds for any relatively compact and relatively open set U ⊂ C r 2 :  \n- 1. We have the following strong convergence:  \nlim i →∞ j ≫ i ( ∥ r -r d ∥ C 1 ( U ) + ∥ Ω 2 -Ω 2 d ∥ C 1 ( U ) + ∥ Q -Q d ∥ C 0 ( U ) + ∥ T uu ∥ C 0 ( U ) + ∥ T uv ∥ C 0 ( U ) ) = 0 , (5.137)  \nwhere the limit is to be understood as taking i → ∞ while keeping j sufficiently large for each i such that the conclusion of Proposition 5.5 applies for S α,η i ,ε j .  \n- 2. If U ' ⊂ U is disjoint from a neighborhood of { τ = 0 } , then  \nlim i →∞ j ≫ i ∥ N u ∥ C 0 ( U ' ) = 0 . (5.138)  \n- 3. We have the following weak* convergence: For any φ ∈ C 1 c ( R 2 ) with spt φ ∩ C r 2 ⊂ U ,  \nlim i →∞ j ≫ i ∫ U ( N u , N v , T vv ) φdudv = ∫ U (0 , N v d , T vv d ) φdudv. (5.139)  \n- 4. We have the following weak* convergence: For any φ ∈ L 1 ( U ) ,  \nlim i →∞ lim j →∞ ∫ U ( N u , N v , T vv ) φdudv = ∫ U (0 , N v d , T vv d ) φdudv. (5.140)  \nProof. It is clear from the estimates used in the proof of Lemma 5.12 that the Vlasov solutions converge strongly to Minkowski space in the regions R ˘ v aux . It therefore suffices to prove the proposition only for the case U = R 1 2 ˘ v -1 3 r 1 main , where we use the estimates of Lemma 5.9.  \nPart 1. Using Lemma 5.8, (3.25), and (5.78), we already obtain  \n| Q ( u, v ) -ˇ Q ( u ) | ≲ η  \nfor any ( u, v ) ∈ U . = R 1 2 ˘ v -1 3 r 1 main and note that ˇ Q ( u ) = Q d ( u, v ) by (4.29). Using this estimate, Lemma 5.8, (5.78), and Grönwall's inequality on the differences of (3.20), (3.21) and (4.24), (4.25), we readily infer  \n| r -r d | + | Ω 2 -Ω 2 d | ≲ η  \non U , which completes the proof of (5.137).  \nPart 2. This follows immediately from (5.75) since τ is bounded below on U ' .  \nPart 3. Since φ is bounded,  \n∣ ∣ ∣ ∣ ∫ U N u φdudv ∣ ∣ ∣ ∣ ≲ ∫ -2 3 r 1 -r 2 ∫ ˘ v -u N u dvdu ≲ ε 1 / 2  \nby (5.78). Next, let ˜ φ . = φ/ ( -e r 2 Ω 2 ) , use Maxwell's equation (3.24), and integrate by parts:  \n∫ U N v φdudv = ∫ U ∂ u Q ˜ φdudv = -∫ U Q∂ u ˜ φdudv + ∫ ∂U Q ˜ φ (5.141)  \nBy Part 1, Q∂ u ˜ φ → Q d ∂ u ˜ φ d and Q ˜ φ → Q d ˜ φ d uniformly as i → ∞ and j ≪ i , where ˜ φ d . = φ/ ( -e r 2 d Ω 2 d ) . Therefore, passing to the limit, we have  \nlim i →∞ j ≫ i ∫ U N v φdudv = -∫ U Q d ∂ u ˜ φ d dudv + ∫ U Q d ˜ φ d = ∫ U ∂ u Q d ˜ φ d = ∫ U N v d dudv,  \nwhere we have again integrated by parts and used (4.28). A similar argument applies for the convergence of T vv using the Raychaudhuri equations (3.22) and (4.26). This completes the proof of (5.139).  \nPart 4. We first fix i and take j →∞ . By (5.75), N u ≲ i 1 , for every j ≪ i , where we use the notation A ≲ i B to denote A ≤ CB , where C may depend on i . By the Banach-Alaoglu theorem, there exists a subsequence j n and an L ∞ ( U ) function h such that N u ∗ ⇀h in L ∞ ( U ) . However, by (5.138), it is clear that h = 0 almost everywhere. Since the subsequential limit is unique, N u ∗ ⇀ 0 as j → ∞ . This completes the proof of (5.140) for N u .  \nLet ˚ m V ,i and ˚ Q V ,i be the values of ˚ m and ˚ Q at r = 2 3 r 1 for the Vlasov seed S α,η i ,ε j . Note that these numbers do not depend on j . Let ˚ m i and ˚ Q i be the solutions of the system (4.35) and (4.36) on [ 2 3 r 1 , r 2 ] with initial conditions ˚ m V ,i and ˚ Q V ,i at r = 2 3 r 1 , current ˚ N v given by (4.38), and identically vanishing energy-momentum tensor ˚ T vv . Following the proof of Proposition 4.19, we obtain a unique smooth solution ( r d ,i , Ω 2 d ,i , Q d ,i , N v d ,i , T vv d ,i ) of the formal outgoing charged null dust system on U which attains the initial data just described.  \nBy repeating the proof of Part 1 of the present proposition, we see that  \nlim j →∞ ( ∥ r -r d ,i ∥ C 1 ( U ) + ∥ Ω 2 -Ω 2 d ,i ∥ C 1 ( U ) + ∥ Q -Q d ,i ∥ C 0 ( U ) ) = 0  \nfor fixed i . Arguing as in Part 2, we then see that  \nlim j →∞ ∫ U ( N v , T vv ) φdudv = ∫ U ( N v d ,i , T vv d ,i ) φdudv (5.142)  \nfor any φ ∈ C 1 c ( R 2 ) . Since N v , T vv ≲ i 1 , a standard triangle inequality argument shows that φ can be replaced by any L 1 function in (5.142). Now it follows by construction that  \n| N v d ,i -N v d | + | T vv d ,i -T vv d | ≲ η i  \non U , so we can safely take i →∞  \nin (5.142), which completes the proof of (5.140).  \nIn order to globalize this, we must first define bouncing charged null dust in the formal system in double null gauge. So consider again the outgoing solution ( r d , Ω 2 d , Q d , N v d , T vv d ) on C r 2 with seed data ( ˚ N v d , 0 , r 2 , e ) given by (4.38). As in Definition 5.25, we extend r d , Ω 2 d , and Q d to ˜ C r 2 by reflection, and set  \nN u d ( u, v ) = N v d ( -v, -u ) , T uu d ( u, v ) = T vv d ( -v, -u )  \nfor ( u, v ) ∈ ˜ C r 2 \\ { τ = 0 } . We extend N v d and T vv d to zero in ˜ C r 2 \\ { τ = 0 } and similarly extend N u d and T uu d to zero in C r 2 . Using Lemma 5.22, we attach a maximal piece of Reissner-Nordström with parameters ϖ 2 and Q 2 to C r 2 ∪ ˜ C r 2 . Let v ∞ be as defined in (5.132).  \nDefinition 5.32. The globally hyperbolic bouncing formal charged null dust spacetime associated to a set of parameters α ∈ P Γ is the tuple ( X d , r d , Ω 2 d , Q d , N u d , N v d , T uu d , T vv d ) , where X d . = { v ≥ u } ∩ { v < v ∞ } .  \nFor τ ≥ 0 , ( r d , Ω 2 d , N v d , T vv d ) solves the outgoing formal dust system and for τ < 0 , ( r d , Ω 2 d , N u d , T uu d ) solves the ingoing formal dust system. The functions r d and Ω 2 d are C 1 across { τ = 0 } and Q d , T uu d , and T vv d are C 0 across { τ = 0 } . The currents N u d and N v d are discontinuous across { τ = 0 } (since we extended by zero), but of course N u d = N v d at { τ = 0 } .  \nUsing this definition and Proposition 5.31, we immediately obtain the following  \nTheorem 6. Let α ∈ P Γ , let { η i } and { ε j } be decreasing sequences of positive numbers tending to zero, let ( X , r, Ω 2 , Q, f ) be the globally hyperbolic solution of the Einstein-Maxwell-Vlasov system associated to ( α, η i , ε j ) by Proposition 5.30 for j ≪ i and arbitrary allowed mass, and let ( X d , r d , Ω 2 d , Q d , N u d , N v d , T uu d , T vv d ) be the globally hyperbolic bouncing formal charged null dust spacetime associated to α by Definition 5.32. Then the following holds for any relatively compact open set U ⊂ X d :", '1. We have the following strong convergence:': 'lim i →∞ j ≫ i ( ∥ r -r d ∥ C 1 ( U ) + ∥ Ω 2 -Ω 2 d ∥ C 1 ( U ) + ∥ Q -Q d ∥ C 0 ( U ) + ∥ T uv ∥ C 0 ( U ) ) = 0 .  \n- 2. We have the following weak* convergence: For any φ ∈ C 1 c ( U ) ,  \nlim i →∞ j ≫ i ∫ U ( N u , N v , T uu , T vv ) φdudv = lim i →∞ j ≪ i ∫ U ( N u d , N v d , T uu d , T vv d ) φdudv.  \n- 3. We have the following weak* convergence: For any φ ∈ L 1 ( R 2 ) with spt φ ⊂ U ,  \nlim i →∞ lim j →∞ ∫ U ( N u , N v , T uu , T vv ) φdudv = lim i →∞ j ≪ i ∫ U ( N u d , N v d , T uu d , T vv d ) φdudv.  \nRemark 5.33 . The globally hyperbolic region X depends on η i and ε j , but it always holds that U ⊂ X for i sufficiently large and j ≫ i .', '6 The third law and event horizon jumping at extremality': 'Using the technology of bouncing Vlasov beams developed in the proof of Theorem 1, we are now able to quickly prove Theorems 2 and 3. As complete proofs would require more lengthy setup, we only sketch the proofs of these results. We refer the reader back to Section 1.6 for the theorem statements and discussion.', '6.1 Counterexamples to the third law': "Refer to Fig. 15 for global Penrose diagrams.  \nProof of Theorem 2. Let α ∈ P be third law violating dust parameters as in Theorem 5. We desingularize this dust beam as in the proof of Theorem 1, noting that the charge on the inner edge of the beam is bounded below and hence no auxiliary beam is required. In order to achieve extremality we must modulate α slightly as in Theorem 1, but all required inequalities for this construction are strict, so this can be done. By this procedure we obtain the time-symmetric solution D ext depicted in Fig. 16 below.  \nLet ( ϖ 1 , Q 1 ) be the initial Reissner-Nordström parameters of α , which will also be the initial parameters of D ext . Using the same methods as Theorem 1 and Lemma 4.10, we can construct a solution D sub of Einstein-Maxwell-Vlasov collapsing to a subextremal Reissner-Nordström black hole with parameters ϖ 1 and Q 1 . In the case of massless particles, the desired third law violating spacetime is then obtained by deleting an appropriate double null rectangle from D sub and gluing in an appropriate piece of D ext . In the case of massive particles, the beams from D ext and D sub will possibly interact in the early past, but as is clear from the proof of Proposition 5.5, this happens in the dispersive region and the proof of Lemma 5.15 can be repeated to show global existence and causal geodesic completeness in the past.  \nFigure 15: Penrose diagrams of counterexamples to the third law of black hole thermodynamics in the Einstein-Maxwell-Vlasov model. The disconnected thick black curve denotes the outermost apparent horizon A ' , which jumps outward as the black hole becomes extremal. This behavior is necessary in third law violating spacetimes which obey the weak energy condition, see [KU22, Proposition 1.1] and [Isr86].  \n<!-- image -->", '6.2.1 General results for spherically symmetric event horizons': "We now consider general weakly tame spherically symmetric Einstein-matter systems, i.e., those satisfying the dominant energy condition and the weak extension principle. We refer to [Daf05b; Kom13] for the precise definition of the weak extension principle, but note that it is a strictly weaker condition than the generalized extension principle as formulated in Proposition 3.15 and therefore holds for the Einstein-Maxwell-KleinGordon, Einstein-Higgs, and Einstein-Maxwell-Vlasov systems [Chr93; Daf05a; Kom13; DR16].  \nLet Ψ = (Σ , ¯ g, ¯ k, . . . ) be a spherically symmetric, asymptotically flat Cauchy data set on R 3 with a regular center in the given matter model. We will assume that Ψ contains no spherically symmetric antitrapped surfaces, i.e., that ∂ u r < 0 on Σ . This assumption is physically motivated by the observation that if the maximal Cauchy development D = ( Q , r, Ω 2 , . . . ) does not contain a white hole, then there are no antitrapped surfaces in the spacetime. Furthermore, by Raychaudhuri's equation (2.7) and the dominant energy condition, ∂ u r < 0 is propagated to the future of the Cauchy hypersurface Σ .  \nUnder these assumptions, a very general a priori characterization of the boundary of ( Q , r, Ω 2 ) is available due to the work of Dafermos [Daf05b] and we refer to Kommemi [Kom13] for a detailed account. We will utilize the following two facts:  \n̸  \nFact 1. If ( Q , r, Ω 2 ) contains a trapped or marginally trapped surface, i.e., ∂ v r ( u 0 , v 0 ) ≤ 0 for some ( u 0 , v 0 ) ∈ Q , then the black hole region is nonempty ( BH . = Q\\ J -( I + ) = ∅ ), future null infinity is complete in the sense of Christodoulou [Chr99a], and ( u 0 , v 0 ) ∈ BH .  \nFact 2. The Hawking mass m extends to a (not necessarily continuous) nonincreasing and nonnegative function on future null infinity I + , called the Bondi mass . If the Hawking mass of Σ is bounded, 29 then the Bondi mass is also bounded and the final Bondi mass M f . = inf I + m is finite. Then the 'event horizon Penrose inequality' sup H + r ≤ 2 M f holds and by the no antitrapped surfaces condition we also obtain  \nsup BH r ≤ 2 M f . (6.1)  \nWe wish to consider sequences of initial data and their developments. In order to compare them, we have to ensure that the double null gauges are synchronized in an appropriate sense. We consider only  \nFigure 16: Penrose diagram of the time symmetric Einstein-Maxwell-Vlasov solution D ext interpolating between subextremality and extremality. This diagram is valid for both massive and massless particles.  \n<!-- image -->  \ndevelopments D = ( Q , r, Ω 2 , . . . ) for which the center Γ is a subset of { u = v } ⊂ R 2 u,v and if i : [0 , ∞ ) → Σ denotes the embedding map of the Cauchy hypersurface into Q , we demand that i (0) ∈ Γ . Clearly, these conditions can always be enforced by an appropriate transformation of the double null gauge.  \nAssumption 6.1. Let { Ψ j } be a sequence of spherically symmetric asymptotically flat Cauchy data for a weakly tame Einstein-matter system. Let D j = ( Q j , r j , Ω 2 j , . . . ) denote the maximal development of Ψ j with Cauchy hypersurface Σ j ⊂ Q j and embedding map i j : [0 , ∞ ) → Σ j normalized as above. We assume that Ψ j converges to another data set Ψ ∞ and the developments converge in the following sense:  \n- 1. Gauge condition: Let D ∞ denote the maximal development of Ψ ∞ with Cauchy hypersurface and embedding map i ∞ : [0 , ∞ ) → Σ ∞ . We assume that D j and D ∞ have continuously synchronized gauges in the sense that ( u, v ) · i j : [0 , ∞ ) → R 2 converges uniformly on compact sets as j →∞ .\n- 2. Cauchy stability of the area-radius: If U ⊂ Q ∞ is a relatively compact open set, then U ⊂ Q j for j sufficiently large and r j → r ∞ in C 1 ( U ) .  \nRemark 6.2 . The notion of continuous synchronization also makes sense for continuous one-parameter families of Cauchy data λ ↦→ Ψ λ . In this case we require the maps ( u, v ) · i λ ( x ) to be jointly continuous in λ and x ∈ [0 , ∞ ) .  \nRemark 6.3 . The initial data and developments given by Proposition 5.30 are continuously synchronized as functions of the beam parameters ( α, η, ε, m ) .  \nProposition 6.4. Let Ψ j → Ψ ∞ be a convergent sequence of one-ended asymptotically flat Cauchy data for a weakly tame spherically symmetric Einstein-matter system, containing no spherically symmetric antitrapped surfaces, and satisfying Assumption 6.1. Assume that the sequence has uniformly bounded Bondi mass and that the development D j of Ψ j contains a black hole for each j ∈ N . Let r j, H + denote the limiting area-radius of the event horizon H + j of D j and u j, H + its retarded time coordinate (also for j = ∞ if D ∞ contains a black hole). Then the following holds:  \n- 1. If future timelike infinity i + ∞ is a limit point of the center Γ in D ∞ (in particular, D ∞ does not contain a black hole), then  \nlim j →∞ u j, H + = sup Q ∞ u. (6.2)  \n- 2. If D ∞ contains a black hole, then\n- (a) The retarded time of the event horizon is lower semicontinuous:  \nlim inf j →∞ u j, H + ≥ u ∞ , H + . (6.3)  \n- (b) Assume further that there are trapped surfaces asymptoting to i + ∞ in the following sense: Let ( u ∞ ,i + , v ∞ ,i + ) denote the coordinates 30 of i + ∞ and suppose there exist sequences u 1 > u 2 > · · · → u ∞ ,i + and v 1 < v 2 < · · · → v ∞ ,i + such that ∂ v r ∞ ( u i , v i ) < 0 for every i ≥ 1 . Then (6.3) is upgraded to  \nlim j →∞ u j, H + = u ∞ , H + (6.4)  \nand it additionally holds that  \nlim inf j →∞ r j, H + ≥ r ∞ , H + . (6.5)  \nProof. Part 1: The inequality ≤ in (6.2) follows directly from Cauchy stability, which implies the stronger statement  \nlim sup j →∞ sup Q j u ≤ sup Q ∞ u.  \nWe now prove the inequality ≥ . Let u 0 < sup Q ∞ u and let M be an upper bound for the Bondi mass of the sequence {D j } . Since the cone C u 0 in Q ∞ reaches future null infinity, we can choose v 0 such that r ∞ ( u 0 , v 0 ) > 2 M . By Cauchy stability, r j ( u 0 , v 0 ) > 2 M for j sufficiently large, so ( u 0 , v 0 ) / ∈ BH j by (6.1). This implies u j, H + ≥ u 0 which completes the proof.  \nPart 2 (a): The argument to prove (6.3) is the same as the proof of Part 1, since C u 0 reaches future null infinity in Q ∞ for any u 0 < u ∞ , H + .  \nPart 2 (b): By (6.3) we must now show that for u 0 > u ∞ , H + , u j, H + ≤ u 0 for j sufficiently large. Let ( u i , v i ) be a trapped sphere for D ∞ as in the statement, with u 0 > u i > u ∞ , H + . By Cauchy stability, ( u i , v i ) is then trapped in D j for j sufficiently large and therefore u i > u j, H + , which completes the proof of (6.4). Using this, we now have by Cauchy stability and monotonicity of r j along H + j that  \nlim inf j →∞ r j, H + ≥ lim j →∞ r j ( u j, H + , v 0 ) = r ∞ ( u ∞ , H + , v 0 )  \nfor any v 0 < v ∞ ,i + . Letting v 0 → v ∞ ,i + completes the proof.  \nRemark 6.5 . It is natural to ask if the 'reverse' of (6.5), i.e.,  \nlim sup j →∞ r j, H + ≤ r ∞ , H + , (6.6)  \nholds at this level of generality (even assuming trapped surfaces asymptoting towards i + ∞ ). It turns out that (6.6) is false without additional assumptions. On the one hand, assuming additionally a strict inequality in (6.3), a minor modification of the arguments used to show (6.3) can be used to show (6.6). 31 On the other hand, without the assumption of a strict inequality in (6.3), using the ingoing (uncharged) Vaidya metric, one can construct a counterexample to (6.6) which moreover satisfies the asymptotic trapped surface assumption of Part 2 (b). One is to imagine inflating the event horizon of a Schwarzschild black hole by injecting a fixed dust packet at later and later advanced times v ∼ j . In the limit j → ∞ , the dust disappears, and lim sup r j, H + > r ∞ , H + . Curiously, since black holes in the Vaidya model always have trapped surfaces behind the horizon, Part 2 (b) of the proposition implies that u j, H + is actually continuous in this process. This is because injecting a fixed amount of matter at later and later times causes the horizon to move outwards less and less (in u ), causing it to converge back to the original Schwarzschild horizon as j → ∞ . Therefore, in order for (6.6) to hold, one must assume that Ψ j converges to Ψ ∞ in a norm that sufficiently respects the asymptotically flat structure. We emphasize at this point that the conclusions of Proposition 6.4 hold only under an assumption of local Cauchy stability -no asymptotic stability or weighted convergence is required.", '6.2.2 Proof of event horizon jumping in the Einstein-Maxwell-Vlasov model': 'Proof of Theorem 3. This is proved by following the proof of Theorem 2 and varying the final parameters of D ext as in the proof of Theorem 1.  \nThis theorem shows that it is not always possible to have equality in (6.3) when D ∞ is extremal: the event horizon can and does jump as a function of the initial data.', 'A The characteristic initial value problem for spherically symmetric nonlinear wave-transport systems': "In this appendix, we prove local well-posedness for the spherically symmetric Einstein-Maxwell-Vlasov system in small characteristic rectangles away from the center. In fact, we consider the general system of equations  \n∂ u ∂ v Ψ = F (Ψ , ∂ Ψ , Q, M [ f ] , M uv [ f ]) , (A.1)  \n∂ u Q = K (Ψ , M v [ f ]) , (A.2)  \nX ( p u , p v , Ψ , ∂ Ψ , Q ) f = 0 , (A.3)  \nwhere Ψ : R 2 u,v → R N is a vector-valued function taking the role of the 'wave-type' variables r and Ω 2 , Q : R 2 u,v → R is the charge, f : T R 2 u,v → R ≥ 0 is the distribution function,  \nM [ f ] . = ∫ ∞ 0 ∫ ∞ 0 f dp u dp v , M v [ f ] . = ∫ ∞ 0 ∫ ∞ 0 p v f dp u dp v , M uv [ f ] . = ∫ ∞ 0 ∫ ∞ 0 p u p v f dp u dp v  \nare moments of f , F = ( F 1 , . . . , F N ) and K are smooth functions of their variables, and X is a vector field on R 2 u,v of the form  \nX ( p u , p v , Ψ , ∂ Ψ , Q ) = p u ∂ u + p v ∂ v + ξ u ( p u , p v , Ψ , ∂ Ψ , Q ) ∂ p u + ξ v ( p u , p v , Ψ , ∂ Ψ , Q ) ∂ p v ,  \nwhere ξ u and ξ v are smooth functions of their variables. Letting a ∈ { u, v } , we can write X using Einstein notation as  \nX = p a ∂ a + ξ a ∂ p a .  \nWe assume that there exist functions G k : R ≥ 0 → R ≥ 0 for k ≥ 0 such that  \n| D i 1 Ψ ,∂ Ψ ,Q ∂ i 2 p ξ u ( p u , p v , Ψ , ∂ Ψ , Q ) | + | D i 1 Ψ ,∂ Ψ ,Q ∂ i 2 p ξ v ( p u , p v , Ψ , ∂ Ψ , Q ) | ≤ G k ( M ) ⟨ p τ ⟩ 2 -i 2 (A.4)  \nif | Ψ | + | ∂ Ψ | + | Q | ≤ M and i 1 + i 2 = k , where ⟨ s ⟩ . = √ 1 + s 2 and p τ . = 1 2 ( p u + p v ) . Here D i 1 Ψ ,∂ Ψ ,Q ∂ i 2 p denotes any expression involving i 1 derivatives in the (Ψ , ∂ Ψ , Q ) -variables and i 2 derivatives in the ( p u , p v ) -variables. We also assume that there exists a constant m ≥ 0 such that  \nξ u p v + ξ v p u = 0 (A.5)  \nwhenever p u p v = m 2 . These structural assumptions are verified for a renormalized version of the spherically symmetric Einstein-Maxwell-Vlasov system.  \nGiven U 0 < U 1 and V 0 < V 1 , let  \nC ( U 0 , U 1 , V 0 , V 1 ) . = ( { U 0 } × [ V 0 , V 1 ]) ∪ ([ U 0 , U 1 ] ×{ V 0 } ) , R ( U 0 , U 1 , V 0 , V 1 ) . = [ U 0 , U 1 ] × [ V 0 , V 1 ] .  \nWe will consistently omit ( U 0 , U 1 , V 0 , V 1 ) from the notation for these sets. We also define  \nP m . = { ( u, v, p u , p v ) ∈ T R 2 : p u ≥ 0 , p v ≥ 0 , p τ > 0 , p u p v ≥ m 2 } , H κ . = { ( u, v, p u , p v ) ∈ T R 2 : p u ≥ 0 , p v ≥ 0 , p τ > κ }  \nand set P . = P 0 . A function ϕ : C → R is said to be smooth if it is continuous and ϕ | { U 0 }× [ V 0 ,V 1 ] and ϕ | [ U 0 ,U 1 ] ×{ V 0 } are C ∞ single-variable functions. This definition extends naturally to functions f : P m | C → R . A smooth characteristic initial data set for the system (A.1)-(A.3) consists of a triple ( ˚ Ψ , ˚ Q, ˚ f ) and numbers κ > 0 , σ > 4 , where ˚ Ψ : C → R N , ˚ Q : { U 0 }× [ V 0 , V 1 ] → R , and ˚ f : P m | C → R ≥ 0 are smooth. We additionally assume that spt( ˚ f ) ⊂ H κ | C for some κ > 0 (which is only an extra assumption when m = 0 ) and that  \n∥ ˚ f ∥ C k σ ( P | C ) . = ∑ 0 ≤ i 1 + i 2 ≤ k ( sup P m | { U 0 }× [ V 0 ,V 1 ] ⟨ p τ ⟩ σ + i 2 | ∂ i 1 v ∂ i 2 p ˚ f | + sup P m | [ U 0 ,U 1 ] ×{ V 0 } ⟨ p τ ⟩ σ + i 2 | ∂ i 1 u ∂ i 2 p ˚ f | ) (A.6)  \nis finite for every k ≥ 0 . For f : P m | R → R ≥ 0 and k ≥ 0 , we define the norms  \n∥ f ∥ C k σ ( P | R ) . = ∑ 0 ≤ i 1 + i 2 ≤ k sup P m | R ⟨ p τ ⟩ σ + i 2 | ∂ i 1 x ∂ i 2 p f | ,  \nwhere ∂ i 1 x denotes i 1 derivatives in the ( u, v ) -variables.  \nProposition A.1. For any B > 0 , κ > 0 , and σ > 4 there exists a constant ε > 0 (depending also on F , K , and X ) with the following property. Let ( ˚ Ψ , ˚ Q, ˚ f ) be a smooth characteristic initial data set for the system (A.1) -(A.3) on C ( U 0 , U 1 , V 0 , V 1 ) . If U 1 -U 0 < ε , V 1 -V 0 < ε , and  \n∥ ˚ Ψ ∥ C 2 ( C ) + ∥ ˚ Q ∥ C 1 ( C ) + ∥ ˚ f ∥ C 1 σ ( P | C ) ≤ B, (A.7)  \nthen there exists a unique smooth solution (Ψ , Q, f ) of (A.1) -(A.3) on R ( U 0 , U 1 , V 0 , V 1 ) which extends the initial data. Moreover, the distribution function f is supported in H κ/ 2 and for any k ≥ 0 , the norm  \n∥ Ψ ∥ C k ( R ) + ∥ Q ∥ C k ( R ) + ∥ f ∥ C k σ ( P | R )  \nis finite and can be bounded in terms of initial data norms.  \nRemark A.2 . While we assume that the initial data ( ˚ Ψ , ˚ Q, ˚ f ) are smooth (and that ˚ f satisfies the nontrivial bound (A.6) at any order), the existence time ε in the proposition depends only on the estimate (A.7).  \nRemark A.3 . If we assume that ˚ f has compact support in the momentum variables, then (A.6) is automatic  \nby smoothness.", 'A.1 Proof of Proposition A.1': "In this section, we assume the hypotheses and setup of Proposition A.1. We also set U 0 = V 0 = 0 and define τ . = 1 2 ( v + u ) . Therefore, 0 ≤ τ ≤ ε on R .  \nWe will construct the solution (Ψ , Q, f ) as the limit of an iteration scheme.  \nLemma A.4. There exist sequences of constants { ˜ C k } and { C k } such that the following holds. For any ε sufficiently small and every n ≥ 1 , there exist functions (Ψ n , Q n , f n ) ∈ C ∞ ( R ) × C ∞ ( R ) × C ∞ ( P m | R ) solving the iterative system  \n∂ u ∂ v Ψ n = F (Ψ n -1 , ∂ Ψ n -1 , Q n -1 , M [ f n -1 ] , M uv [ f n -1 ]) , (A.8)  \n∂ u Q n = K (Ψ n -1 , M v [ f n -1 ]) , (A.9)  \nX ( p u , p v , Ψ n -1 , ∂ Ψ n -1 , Q n -1 ) f n = 0 , (A.10)  \nwhere we set (Ψ 0 , Q 0 , f 0 ) to be identically zero, with initial conditions  \nΨ n | C = ˚ Ψ , Q n | C = ˚ Q, f n | P | C = ˚ f. (A.11)  \nMoreover, spt( f n ) ⊂ H κ/ 2 and these functions satisfy the bounds  \n∥ Ψ ∥ C k ( R ) ≤ ˜ C k e C k τ , (A.12)  \n∥ Q n ∥ C k ( R ) ≤ ˜ C k +1 e C k +1 τ , (A.13)  \n∥ f n ∥ C k σ ( P | R ) ≤ ˜ C k +1 e C k +1 τ . (A.14)  \nIt is convenient to set  \nF n . = F (Ψ n , ∂ Ψ n , Q n , M [ f n ] , M uv [ f n ]) , K n . = K (Ψ n , M v [ f n ]) , X n . = X ( p u , p v , Ψ n , ∂ Ψ n , Q n )  \nfor n ≥ 0 . We first require a preliminary lemma about integral curves of the vector field X n .  \nLemma A.5. For n ≥ 0 , let Γ m ,κ n denote the set of maximal integral curves ˜ γ = ( γ, p ) : I → T R (where I is a closed interval containing 0 ) of the vector field X n subject to the condition that ˜ γ (0) ∈ H κ ∩ P m . Assume that Ψ n satisfies (A.12) for k = 1 and Q n satisfies (A.13) for k = 0 . Then for ε sufficiently small (depending in particular on κ ) and any ˜ γ ∈ Γ n , γ is a future-directed causal curve in R connecting C with the future boundary of R , ˜ γ ( s ) ∈ P m ∩ H κ/ 2 for every s ∈ I , and  \n1 2 p τ (0) ≤ p τ ( s ) ≤ 2 p τ (0) (A.15)  \nfor every s ∈ I .  \nProof. Let ˜ γ = ( γ, p ) ∈ Γ n and set τ 0 . = γ τ (0) . By definition,  \ndγ a ds = p a , dp a ds = ξ a n  \nfor a ∈ { u, v } , where ξ a n . = ξ a ( p u , p v , Ψ n , ∂ Ψ n , Q n ) . Observe that X n is tangent to the the boundary of P m , { ( u, v, p u , p v ) ∈ T R : p u p v = m 2 } , by (A.5), so ˜ γ remains within P m . Reparametrizing ˜ γ by τ gives  \nd dτ ( p τ ) 2 = 2( ξ u n + ξ v n ) .  \nUsing (A.4), the assumptions on Ψ n and Q n , and Grönwall's inequality, we have  \np τ ( τ ) 2 ≤ e O ( ε ) ( p τ ( τ 0 ) 2 + O ( ε ) ) ≤ 2 p τ ( τ 0 ) 2  \nfor τ in the domain of ˜ γ and for ε sufficiently small. This proves the second inequality in (A.15). To prove the first inequality, we observe that by the estimate we have just proved,  \n| ( p τ ( τ )) 2 -( p τ ( τ 0 )) 2 | ≲ ε sup ˜ γ | ξ u n + ξ v n | ≲ εp τ ( τ 0 ) 2  \nChoosing ε perhaps even smaller proves (A.15) and completes the proof of the lemma.  \nProof of Lemma A.4. The proof is by induction on n and induction on k for each fixed n . As the existence and estimates for the base case n = 0 are trivial, we assume the existence of (Ψ n -1 , Q n -1 , f n -1 ) satisfying (A.12)-(A.14), where the constants are still to be determined. We will choose the constants to satisfy ˜ C k ≤ C k ≤ ˜ C k +1 , which we use without comment in the sequel. By (A.12)-(A.14) for (Ψ n -1 , Q n -1 , f n -1 ) and iterating the chain rule, it is easy to see that  \n| ∂ k M n -1 | ≤ C ( ˜ C k +1 ) e C k +1 τ ( k ≥ 0) , (A.16)  \n| F n -1 | ≤ C ( C 1 ) , (A.17)  \n| ∂ k F n -1 | ≤ C ( ˜ C k +1 ) e C k +1 τ ( k ≥ 1) , (A.18)  \n| K n -1 | ≤ C ( C 1 ) , (A.19)  \n| ∂ k K n -1 | ≤ C ( ˜ C k +1 ) e C k +1 τ ( k ≥ 1) , (A.20)  \nwhere M n -1 ∈ { M [ f n -1 ] , M v [ f n -1 ] , M uv [ f n -1 ] } . We also define the number  \nB ' . = | F ( ˚ Ψ , ∂ ˚ Ψ , ˚ Q,M [ ˚ f ] , M uv [ ˚ f ]) | (0 , 0) + | K ( ˚ Ψ , M v [ ˚ f ]) | (0 , 0) .  \nStep 1. The function Ψ n is defined by the explicit formula  \nΨ n ( u, v ) . = ∫ u 0 ∫ v 0 F n -1 ( u ' , v ' ) dv ' du ' + ˚ Ψ( u, 0) + ˚ Ψ(0 , v ) -˚ Ψ(0 , 0) . (A.21)  \nIt follows by inspection of this representation formula and (A.17) that  \n∥ Ψ n ∥ C 1 ( R ) ≤ 10 B (A.22)  \nif ε is sufficiently small depending on C 1 . We estimate k -th order derivatives ( k ≥ 2) of the form ∂ k u Ψ n , ∂ k -2 x ∂ u ∂ v Ψ n , and ∂ k v Ψ n separately. For the first type, we simply differentiate the representation formula (A.21) k times to obtain  \n| ∂ k u Ψ n | ≤ (data) + ∫ v 0 | ∂ k -1 u F n -1 | dv ' ≤ C + C ( ˜ C k ) C k e C k τ ≤ ˜ C k e C k τ  \nfor appropriate choices of ˜ C k and C k . For mixed derivatives, we differentiate the wave equation to obtain  \n| ∂ k -2 x ∂ u ∂ v Ψ n -1 | ≤ sup { 0 }× [0 ,V 0 ] | ∂ k -2 x F n -1 | + ∫ u 0 | ∂ u ∂ k -2 F n -1 | du ' ≤ C ( C k -1 ) + C ( ˜ C k ) C k e C k τ ≤ ˜ C k e C k τ  \nfor appropriate choices of ˜ C k and C k . The estimate for ∂ k v Ψ n is similar to ∂ k u Ψ n . For later use, we derive a slightly improved estimate for ∂ 2 x Ψ n . Using the mean value theorem and (A.18), we have  \n| F n -1 -B ' | ≤ C ( C 2 ) τ.  \nWe also have  \n| ∂ 2 u Ψ n -(data) | + | ∂ 2 v Ψ n -(data) | ≤ C ( C 2 ) τ.  \nTherefore, for ε sufficiently small depending on C 2 , combined with (A.22), we infer  \n∥ Ψ n ∥ C 2 ( R ) ≤ 20( B + B ' ) . (A.23)  \nThis completes Step 1.  \nStep 2. The function Q n is defined by the explicit formula  \nQ n ( u, v ) = ∫ u 0 K n -1 ( u ' , v ) du ' + ˚ Q (0 , v ) .  \nIt follows by inspection of this representation formula and (A.19) that  \n∥ Q ∥ C 0 ( R ) ≤ 10 B  \nfor ε sufficiently small depending on C 1 . For k ≥ 1 , we estimate  \n| ∂ k v Q n | ≤ (data) + ∫ u 0 | ∂ k v K n -1 | du ' ≤ C + C ( ˜ C k +1 ) C k +1 e C k +1 τ ≤ ˜ C k +1 e C k +1 τ ,  \n| ∂ k -1 ∂ u Q n | ≤ sup { 0 }× [0 ,V 0 ] | ∂ k -1 K n -1 | + ∫ u 0 | ∂ k K n -1 | du ' ≤ C ( C k ) + C ( ˜ C k +1 ) C k +1 e C k +1 τ ≤ ˜ C k +1 e C k +1 τ  \nfor appropriate chices of ˜ C k +1 and C k +1 . Arguing as in Step 1, for ε sufficiently small depending on C 2 , we also infer  \n∥ Q n ∥ C 1 ( R ) ≤ 20( B + B ' ) . (A.24)", 'This completes Step 2.': "Step 3. For f n we do not have an explicit representation formula and must instead infer its existence from general properties of flows of vector fields. A slight technical issue is that the 'initial data hypersurface' P | C is not smooth because of the corner in C . Therefore, in order to prove the existence of a smooth f n , we will construct it as a smooth limit of smooth solutions to the X n -1 transport equation, corresponding to initial data where we smooth out the corner in C . To carry out this idea, we first extend ˚ f to C ∞ σ ( P m | R ) according to  \nE ˚ f ( u, v, p u , p v ) . = ˚ f ( u, 0 , p u , p v ) + ˚ f (0 , v, p u , p v ) -˚ f (0 , 0 , p u , p v )  \nand set, for j ≥ 1 ,  \nS . = { ( u, v ) ∈ R : uv = 2 -j }  \nj , R j . = { ( u, v ) ∈ R : uv ≥ 2 -j . }  \n̸  \nFor any j ≥ j 0 sufficiently large that S j = ∅ , there exists a unique function f n,j ∈ C ∞ ( P m | R j ) such that  \nX n -1 f n,j = 0 (A.25)  \nin R j , with initial data f n,j = E ˚ f on P m | S j . The existence follows immediately from the flowout theorem (see [Lee13, Theorem 9.20]), the fact that X n -1 is transverse to P m | S j , and Lemma A.5. It also follows from Lemma A.5 that spt( f n,j ) ⊂ H κ/ 2 . We will use the fact that p τ ≥ κ/ 2 for any ˜ γ ∈ Γ m ,κ n -1 often and without further comment in the sequel.  \nWe now claim that we can choose ˜ C k and C k such that  \n∥ f n,j ∥ C k σ ( P | R j ) ≤ ˜ C k +1 e C k +1 τ (A.26)  \nfor every n ≥ 0 , j ≥ j 0 , and k ≥ 0 . Let F n,j,k denote the vector with ( k +3 k ) components of the form  \n( p τ ) σ + i 2 ∂ i 1 x ∂ i 2 p f n,j , (A.27)  \nwhere i 1 + i 2 = k . We will show inductively that ˜ C k +1 and C k +1 can be chosen so that  \nsup P m | R j |F n,j,k | ≤ ˜ C k +1 e C k +1 τ , (A.28)  \nwhich would imply (A.26).  \nOrders k = 0 and k = 1 are slightly anomalous in our scheme and we handle them first. We require the estimate  \n⟨ p τ ⟩ -2 | ∂ ≤ 1 x ξ a n -1 | + ⟨ p τ ⟩ -1 | ∂ p ξ a n -1 | ≲ 1 , (A.29)  \nwhich follows from (A.4), (A.23), and (A.24). Using (A.25), we compute  \nX n -1 (( p τ ) σ f n,j ) = σ 2 ( p τ ) σ -1 ( ξ u n -1 + ξ v n -1 ) f n,j .  \nFrom (A.29) we then infer  \n| X n -1 (( p τ ) σ f n,j ) | ≲ ( p τ ) σ +1 f n,j .  \nLet ˜ γ ∈ Γ m ,κ n -1 be parametrized by coordinate time τ . Then along ˜ γ we have  \n∣ ∣ ∣ ∣ d dτ F n,j, 0 ∣ ∣ ∣ ∣ ≲ |F n,j, 0 | ,  \nwhence by Grönwall's inequality  \n|F n,j, 0 | ≲ ∥E ˚ f ∥ C 0 σ ( P | R ) ≲ ∥ ˚ f ∥ C 0 σ ( P | C ) .  \nNext, we compute  \n| X n -1 (( p τ ) σ ∂ x f n,j ) | ≲ ( p τ ) σ +1 | ∂ x f n,j | +( p τ ) σ | [ X n -1 , ∂ x ] f n,j | .  \nThe commutator is estimated using (A.29) by  \n| [ X n -1 , ∂ x ] f n,j | ≲ | ∂ x ξ a n -1 || ∂ p f n,j | ≲ ( p τ ) 2 | ∂ p f n,j | .  \nThe p derivative of f n,j satisfies  \n| X n -1 (( p τ ) σ +1 ∂ p f n,j ) | ≲ ( p τ ) σ +2 | ∂ p f n,j | +( p τ ) σ +1 | [ X n -1 , ∂ p ] f n,j | ,  \nwhere the commutator is now estimated by  \n| [ X n -1 , ∂ p ] f n,j | ≲ | ∂ x f n,j | + | ∂ p ξ a n -1 || ∂ x f n,j | ≲ | ∂ x f n,j | + p τ | ∂ p f n,j | .  \nPutting these estimates together, we find that  \n∣ ∣ ∣ ∣ d dτ F n,j, 1 ∣ ∣ ∣ ∣ ≲ |F n,j, 1 | ,  \nwhence again by Grönwall's inequality we conclude a uniform bound  \n|F n,j, 1 | ≲ ∥ ˚ f ∥ C 1 σ ( P | C ) . (A.30)  \nHaving now established cases k = 0 and k = 1 of (A.28), we now assume (A.28) up to order k -1 . Let φ i 1 ,i 2 be a component of F n,j,k . We adopt the convention that if either i 1 or i 2 are negative, then φ i 1 ,i 2 is interpreted as identically zero. Using (A.4) and (A.29), we estimate  \n| X n -1 ( φ i 1 ,i 2 ) | ≲ ( p τ ) σ + i 2 -1 | ξ u n -1 + ξ v n -1 || ∂ i 1 x ∂ i 2 p f n,j | +( p τ ) σ + i 2 | [ X n -1 , ∂ k ] f n,j | ≲ p τ | φ i 1 ,i 2 | +( p τ ) σ + i 2 | [ p a ∂ a , ∂ i 1 x ∂ i 2 p ] f n,j | +( p τ ) σ + i 2 | [ ξ a n -1 ∂ p a , ∂ i 1 x ∂ i 2 p ] f n,j | . (A.31)  \nThe first commutator, [ p a ∂ a , ∂ i 1 x ∂ i 2 p ] f n,j , vanishes unless i 1 ≥ 1 and therefore consists of terms of the form ∂ i 1 +1 x ∂ i 2 -1 p f n,j , which implies  \n( p τ ) σ + i 2 | [ p a ∂ a , ∂ i 1 x ∂ i 2 p ] f n,j | ≲ p τ | φ i 1 +1 ,i 2 -1 | ≲ p τ |F n,j,k | . (A.32)  \nThe second commutator can be estimated by  \n| [ ξ a n -1 ∂ p a , ∂ i 1 x ∂ i 2 p ] f | ≲ ∑ 1 ≤ j 1 + j 2 j 1 ≤ i 1 ,j 2 ≤ i 2 | ∂ j 1 x ∂ j 2 p ξ a n -1 || ∂ i 1 -j 1 x ∂ i 2 +1 -j 2 p f n,j | .  \nBy inspection, ∂ j 1 x ∂ j 2 p ξ a n -1 is linear in ∂ j 1 +1 x Ψ n -1 , which is the worst behaved term in our inductive hierarchy. Therefore, using again (A.4), we may estimate  \n| ∂ j 1 x ∂ j 2 p ξ a n -1 | ≲ ( C ( C k ) + ˜ C j 1 +1 e C j 1 +1 τ ) ( p τ ) 2 -j 2  \nand therefore infer  \n( p τ ) σ + i 2 | [ ξ a n -1 ∂ p a , ∂ i 1 x ∂ i 2 p ] f n,j | ≲ p τ ∑ 1 ≤ j 1 + j 2 j 1 ≤ i 1 ,j 2 ≤ i 2 ( C ( C k ) + ˜ C j 1 +1 e C j 1 +1 τ ) |F n,j,k +1 -j 1 -j 2 | .  \nIf j 1 < k , then ˜ C j 1 +1 e C j 1 +1 τ ≤ C ( C k ) . If j 1 = k , then j 2 = 0 and  \n( C ( C k ) + ˜ C j 1 +1 e C j 1 +1 τ ) |F n,j,k +1 -j 1 -j 2 | = ( C ( C k ) + ˜ C k +1 e C k +1 τ ) |F n,j, 1 | ≲ C ( C k ) ˜ C k +1 e C k +1 τ , (A.33)  \nwhere we have used (A.30). Therefore, the sum in (A.1) can be estimated by  \n≲ C ( C k ) ( |F n,j,k | + ˜ C k +1 e C k +1 τ ) .  \nPutting (A.31), (A.32), (A.1), and (A.33) together, we arrive at  \n| X n -1 F n,j,k | ≲ C ( C k ) p τ ( |F n,j,k | + ˜ C k +1 e C k +1 τ ) .  \nAs before, a simple Grönwall argument now establishes (A.26) for appropriate choices of ˜ C and C .  \nk +1 k +1 Having now established the boundedness of the sequence f n,j , we may take the limit j →∞ (after perhaps passing to a subsequence). This shows the existence of a function f n ∈ C ∞ ( P m | R ) with spt( f n ) ⊂ H κ/ 2 , satisfying the estimates (A.14), and attaining ˚ f on P m | C . Finally, uniqueness of f n is immediate since it is constant along the integral curves of X n -1 .  \nProof of Proposition A.1. We prove first that the sequence iterative sequence (Ψ n , Q n , f n ) constructed in Lemma A.4 is Cauchy in C 2 × C 1 × C 1 σ . We claim that if ε is sufficiently small, then the following estimate holds for every n ≥ 2 :  \n∥ Ψ n -Ψ n -1 ∥ C 2 ( R ) + ∥ Q n -Q n -1 ∥ C 1 ( R ) + ∥ f n -f n -1 ∥ C 1 σ ( P | R ) ≤ 1 ( ∥ Ψ n -1 -Ψ n -2 ∥ C 2 ( R ) + ∥ Q n -1 -Q n -2 ∥ C 1 ( R ) + ∥ f n -1 -f n -2 ∥ C 1 σ ( P | R ) ) . (A.34)  \n2  \nUsing the mean value theorem and the boundedness of the iterative sequence, we immediately estimate  \n∥ F n -1 -F n -2 ∥ C 1 ( R ) ≲ ∥ Ψ n -1 -Ψ n -2 ∥ C 2 ( R ) + ∥ Q n -1 -Q n -2 ∥ C 1 ( R ) + ∥ f n -1 -f n -2 ∥ C 1 σ ( P | R ) . (A.35)  \nUsing the formula  \n(Ψ n -Ψ n -1 )( u, v ) = ∫ u 0 ∫ v 0 ( F n -1 -F n -2 ) dv ' du '  \nwe readily infer that for ε sufficiently small, Ψ n -Ψ n -1 and ∂ i a (Ψ n -Ψ n -1 ) for a ∈ { u, v } and i ∈ { 1 , 2 } are bounded by an arbitrarily small multiple of the right-hand side of (A.34). To estimate the mixed second partial derivative, we simply use the fundamental theorem of calculus to bound  \n∥ Ψ n -1 -Ψ n -2 ∥ C 1 ( R ) + ∥ Q n -1 -Q n -2 ∥ C 0 ( R ) + ∥M n -1 -M n -2 ∥ C 0 ( R )  \nby an arbitrarily small multiple of the right-hand side of (A.34) and then use the mean value theorem to estimate  \n| ∂ u ∂ v (Ψ n -Ψ n -1 ) | ≤ ∥ F n -1 -F n -2 ∥ C 0 ( R ) ≲ ∥ Ψ n -1 -Ψ n -2 ∥ C 1 ( R ) + ∥ Q n -1 -Q n -2 ∥ C 0 ( R ) + ∥M n -1 -M n -2 ∥ C 0 ( R ) .  \nThe argument for bounding ∥ Q n -Q n -1 ∥ C 1 ( R ) is essentially the same and is omitted. To estimate f n -f n -1 , we examine the quantity  \nF n . = ( p τ ) σ ( f n -f n -1 , ∂ x ( f n -f n -1 ) , p τ ∂ p ( f n -f n -1 ))  \n̸  \nalong integral curves of X n -1 . First, note that F n = 0 only along curves in Γ m ,κ n -1 ∪ Γ m ,κ n -2 . By Lemma A.5 (and choosing ε perhaps smaller), any value ˜ γ n -2 ( s ) for a curve ˜ γ n -2 ∈ Γ m ,κ n -2 can be realized as an initial value ˜ γ n -1 (0) for a curve ˜ γ n -1 ∈ Γ m ,κ/ 2 n -1 . Therefore, it suffices to observe that the following estimate holds along any curve ˜ γ n -1 ∈ Γ m ,κ/ 2 n -1 :  \n∣ ∣ ∣ ∣ d dτ F n ∣ ∣ ∣ ∣ ≲ |F n | + ∥ Ψ n -1 -Ψ n -2 ∥ C 2 ( R ) + ∥ Q n -1 -Q n -2 ∥ C 1 ( R ) + ∥ f n -1 -f n -2 ∥ C 1 σ ( P | R ) ,  \nwhich is obtained by simply differentiating F n and using the estimates proved in Lemma A.4. Therefore, since F n vanishes along P m | C , Grönwall's inequality implies  \n|F n | ≲ ε ( ∥ Ψ n -1 -Ψ n -2 ∥ C 2 ( R ) + ∥ Q n -1 -Q n -2 ∥ C 1 ( R ) + ∥ f n -1 -f n -2 ∥ C 1 σ ( P | R ) ) .  \nAfter choosing ε sufficiently small, the proof of (A.35) is complete.  \nTherefore, (Ψ n , Q n , f n ) converges to a solution (Ψ , Q, f ) of the system (A.1)-(A.3) in C 2 × C 1 × C 1 , which is moreover C ∞ smooth by the higher order estimates proved in Lemma A.4. Uniqueness of the solution can be proved along the same lines as the proof of the estimate (A.34) and is omitted.", 'A.2 Proof of local well posedness for the Einstein-Maxwell-Vlasov system': "In this section, we prove Proposition 3.14, local well-posedness for the Einstein-Maxwell-Vlasov system in small characteristic rectangles. The proof has 3 steps: In the first step, we solve the wave equations (3.20) and (3.21), the ingoing Maxwell equation (3.24), and the Maxwell-Vlasov equation (3.26) using Proposition A.1. In order to directly quote Proposition A.1, we in fact consider a renormalized system that fixes the location of the mass shell to the fixed family of hyperbolas ˜ p u ˜ p v = m 2 . In step 2, we show that the outgoing Maxwell equation holds as a result of conservation law (3.32), which follows from the Maxwell-Vlasov equation (3.26). Finally, in step 3, we show that Raychaudhuri's equations (3.22) and (3.23) hold, using now the Bianchi identities (3.33) and (3.34), which again follow from (3.26). Steps 2 and 3 may be thought of as propagation of constraints , as they require the relevant equations to hold on initial data.  \nProof of Proposition 3.14. Step 1. Consider the wave-transport system  \n∂ u ∂ v r = -Ω 2 4 r -∂ u r∂ v r r + πr Ω 2 4 ∫ ˜ p u ˜ p v ≥ m 2 ˜ p u ˜ p v ˜ f ( u, v, ˜ p u , ˜ p v ) d ˜ p u d ˜ p v , (A.36)  \n∂ u ∂ v log Ω 2 = Ω 2 2 r 2 + 2 ∂ u r∂ v r r 2 + π Ω 2 2 ∫ ˜ p u ˜ p v ≥ m 2 ˜ p u ˜ p v ˜ f ( u, v, ˜ p u , ˜ p v ) d ˜ p u d ˜ p v  \n-π Ω 2 2 ∫ ˜ p u ˜ p v ≥ m 2 (˜ p u ˜ p v -m 2 ) ˜ f ( u, v, ˜ p u , ˜ p v ) d ˜ p u d ˜ p v , (A.37)  \n∂ u Q = -π 2 e r 2 Ω ∫ ˜ p u ˜ p v ≥ m 2 ˜ p u ˜ f ( u, v, ˜ p u , ˜ p v ) d ˜ p u d ˜ p v , (A.38)  \n˜ X ˜ f = 0 , (A.39)  \nwhere  \n˜ X . = ˜ p u ∂ u + ˜ p v ∂ v -( ∂ u log Ω(˜ p u ) 2 -∂ v log Ω˜ p u ˜ p v + 2 ∂ v r r (˜ p u ˜ p v -m 2 ) + e Ω Q r 2 ˜ p u ) ∂ ˜ p u ,  \n-( ∂ v log Ω(˜ p v ) 2 -∂ u log Ω˜ p u ˜ p v + 2 ∂ u r r (˜ p u ˜ p v -m 2 ) -e Ω Q r 2 ˜ p v ) ∂ ˜ p v  \nwith initial data ˚ Ψ = (log˚ r, log ˚ Ω 2 ) , ˚ Q , and ˚ ˜ f ( u, v, ˜ p u , ˜ p v ) = ˚ f ( u, v, ˚ Ω -1 ˜ p u , ˚ Ω -1 ˜ p v ) . Since log ˚ Ω 2 is bounded on C and either m > 0 or ℓ ≥ c ℓ on spt( ˚ f ) , there exists a κ > 0 such that spt( ˚ ˜ f ) ⊂ H κ . Furthermore, the structural conditions (A.4) and (A.5) are easily verified for ˜ X , so Proposition A.1 produces a unique local smooth solution to (A.36)-(A.39) if ε loc is chosen sufficiently small. Making the change of variables ˜ p u ↦→ Ω p u and ˜ p v ↦→ Ω p v , defining f ( u, v, p u , p v ) = ˜ f ( u, v, Ω p u , Ω p v ) , and observing that  \nXf = Ω -1 ˜ X ˜ f = 0 ,  \nwe have obtained a unique local smooth solution ( r, Ω 2 , Q, f ) for the system (3.20), (3.21), (3.24), and (3.26) on R which extends the initial data.  \nStep 2. We first aim to prove (3.32) using only (3.26). At this point, one could apply Proposition 3.10 and (2.11) to derive (3.32), but we give now a direct proof.  \nFirst, using the definitions (3.14) and (3.15), we have  \n∂ u ( r 2 Ω 2 π N u ) + ∂ v ( r 2 Ω 2 π N v ) = ∂ u ∫ ∞ 0 ∫ ∞ m 2 / (Ω 2 p v ) r 2 Ω 4 p u f dp u dp v + ∂ v ∫ ∞ 0 ∫ ∞ m 2 / (Ω 2 p u ) r 2 Ω 4 p v f dp v dp u = ∫ Ω 2 p u p v ≥ m 2 ( 2 r∂ u r Ω 4 p u f +2 r 2 Ω 4 ∂ u log Ω 2 p u f + r 2 Ω 4 p u ∂ u f ) dp u dp v + ∫ Ω 2 p u p v ≥ m 2 ( 2 r∂ v r Ω 4 p v f +2 r 2 Ω 4 ∂ v log Ω 2 p v f + r 2 Ω 4 p v ∂ v f ) dp u dp v + ∫ ∞ 0 m 4 r 2 ( p v ) 2 ∂ u log Ω 2 f dp v + ∫ ∞ 0 m 4 r 2 ( p u ) 2 ∂ v log Ω 2 f dp u . (A.40)  \nAdding both terms involving spatial derivatives of f and using (3.26) yields  \n∫ Ω 2 p u p v ≥ m 2 r 2 Ω 4 ( p u ∂ u f + p v ∂ v f ) dp u dp v = ∫ ∞ 0 ∫ ∞ m 2 / (Ω 2 p v ) r 2 Ω 4 Ξ u ∂ p u f dp u dp v + ∫ ∞ 0 ∫ ∞ m 2 / (Ω 2 p u ) r 2 Ω 4 Ξ v ∂ p v f dp v dp u , (A.41)  \nwhere  \nΞ u . = ∂ u log Ω 2 ( p u ) 2 + 2 ∂ v r r Ω 2 (Ω 2 p u p v -m 2 ) + e Q r 2 p u , Ξ v . = ∂ v log Ω 2 ( p v ) 2 + 2 ∂ u r r Ω 2 (Ω 2 p u p v -m 2 ) -e Q r 2 p v .  \nIntegrating the first term on the right-hand side of (A.41) by parts, we find  \n∫ ∞ 0 ∫ ∞ m 2 / (Ω 2 p v ) r 2 Ω 4 Ξ u ∂ p u f dp u dp v = -∫ Ω 2 p u p v ≥ m 2 ( 2 r 2 Ω 4 ∂ u log Ω 2 p u +2 r∂ v r Ω 4 p v + e Ω 4 Q ) f dp u dp v -∫ ∞ 0 ( m 4 r 2 ( p v ) 2 ∂ u log Ω 2 + e m 2 Ω 2 p v Q ) f dp v , (A.42)  \nwhere f is evaluated at p u = m 2 / (Ω 2 p v ) , and for the second term,  \n∫ ∞ 0 ∫ ∞ m 2 / (Ω 2 p u ) r 2 Ω 4 Ξ v ∂ p v f dp v dp u = -∫ Ω 2 p u p v ≥ m 2 ( 2 r 2 Ω 4 ∂ v log Ω 2 p v +2 r∂ u r Ω 4 p u -e Ω 4 Q ) f dp u dp v -∫ ∞ 0 ( m 4 r 2 ( p u ) 2 ∂ v log Ω 2 -e m 2 Ω 2 p u Q ) f dp u , (A.43)  \nwhere f is evaluated at p v = m 2 / (Ω 2 p u ) . Combining (A.40)-(A.43) yields (3.32) after noting that  \n∫ ∞ 0 e m 2 Ω 2 p v Qf dp v = ∫ ∞ 0 e m 2 Ω 2 p u Qf dp u .  \nWe can now derive the ingoing Maxwell equation (3.25). By (3.24), we have  \nQ ( u, v ) = Q ( U 0 , v ) -∫ u U 0 1 2 e r 2 Ω 2 N v du '  \non R . We then derive  \n∂ v Q ( u, v ) = ∂ v Q ( U 0 , v ) -∫ u U 0 ∂ v ( 1 2 e r 2 Ω 2 N v ) du ' = ∂ v Q ( U 0 , v ) + ∫ u U 0 ∂ u ( 1 2 e r 2 Ω 2 N u ) du ' = 1 2 e r 2 Ω 2 N v ( u, v ) ,  \nwhere in the final equality we used the fundamental theorem of calculus and the assumption that (3.25) holds on { U 0 } × [ V 0 , V 1 ] .  \nStep 3. By a lengthy calculation which is very similar to the one performed in step 2, one may use (3.26) to derive the Bianchi identities in the form (3.33) and (3.34). Using the Maxwell equations (3.24) and (3.25), this implies the Bianchi identities in the form (2.12) and (2.13), where  \nT uu = T uu , T uv = T uv + Q 2 Ω 2 r 4 , T vv = T vv , S = S + Q 2 2 r 4 .  \nBy another lengthy calculation, using now also the wave equations (3.20) and (3.21), one can derive the pair of identities  \n∂ v ( r∂ 2 u r -r∂ u r∂ u log Ω 2 + 1 4 r 2 Ω 4 T vv ) = 0 , ∂ u ( r∂ 2 v r -r∂ v r∂ v log Ω 2 + 1 4 r 2 Ω 4 T uu ) = 0 .  \nThese identities, together with the assumption that (3.22) holds on [ U 0 , U 1 ] × { V 0 } and (3.23) holds on { U 0 } × [ V 0 , V 1 ] , prove that (3.22) and (3.23) hold throughout R .", 'References': "[AAR21] E. Ames, H. Andréasson, and O. Rinne. 'Dynamics of gravitational collapse in the axisymmetric Einstein-Vlasov system'. Classical Quantum Gravity 38.10 (2021), Paper No. 105003, 34.  \n[AH23] X. An and T. He. 'Dynamics of apparent horizon and a null comparison principle' (2023). arXiv: 2312.15666 ."}