11institutetext: Martin Fisher School of Physics, Brandeis University, Waltham
MA, USA22institutetext: Center for Quantum Mathematics and Physics (QMAP)
Department of Physics & Astronomy, University of California, Davis CA, USA

# Covariant bit threads

Matthew Headrick 2 and Veronika E. Hubeny<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We derive several new reformulations of the Hubeny-Rangamani-Takayanagi
covariant holographic entanglement entropy formula. These include: (1) a
minimax formula, which involves finding a maximal-area achronal surface on a
timelike hypersurface homologous to $D(A)$ (the boundary causal domain of the
region $A$ whose entropy we are calculating) and minimizing over the
hypersurface; (2) a max V-flow formula, in which we maximize the flux through
$D(A)$ of a divergenceless bulk 1-form $V$ subject to an upper bound on its
norm that is non-local in time; and (3) a min U-flow formula, in which we
minimize the flux over a bulk Cauchy slice of a divergenceless timelike 1-form
$U$ subject to a lower bound on its norm that is non-local in space. The two
flow formulas define convex programs and are related to each other by Lagrange
duality. For each program, the optimal configurations dynamically find the HRT
surface and the entanglement wedges of $A$ and its complement. The V-flow
formula is the covariant version of the Freedman-Headrick bit thread
reformulation of the Ryu-Takayanagi formula. We also introduce a measure-
theoretic concept of a “thread distribution”, and explain how Riemannian
flows, V-flows, and U-flows can be expressed in terms of thread distributions.

††preprint: BRX-TH-6708

##  0 Executive summary

In this paper we derive a number of new formulas that are equivalent to the
HRT covariant holographic entanglement entropy formula. These formulas can be
grouped into three classes: minimax, max V-flow, and min U-flow.

We fix a boundary spatial region $A$, and define $B:=A^{c}$ as its complement
on a boundary Cauchy slice. For simplicity, we assume that the full system
$AB$ is in a pure state. These spatial regions induce a decomposition of the
conformal boundary into the four spacetime regions $D(A)$, $D(B)$,
$J^{\pm}(\partial A)$, where $\partial A=\partial B$ is the entangling
surface. We also define $\hat{\mathcal{I}}^{\pm}$ as the future/past boundary
of the bulk spacetime (which may be a singularity or at infinite time), and
$\hat{\mathcal{I}}^{0}$ as the end-of-the-world brane (if there is one). The
HRT surface ${\gamma_{{}_{\text{HRT}}}}$ divides the bulk into four spacetime
regions: the entanglement wedge $\mathcal{W}(A)$, the complementary
entanglement wedge $\mathcal{W}(B)$, and the future and past
$J^{\pm}({\gamma_{{}_{\text{HRT}}}})$; these bulk regions meet the conformal
boundary at $D(A)$, $D(B)$, $J^{\pm}(\partial A)$ respectively Wall:2012uf ;
Headrick:2014cta .111 In the main text we are careful with the issue of the UV
regulator, so that all quantities are finite and the maximizations and
minimizations are meaningful. Specifically, we apply the entanglement wedge
cross-section regulator Dutta:2019gen . In this summary we will ignore this
issue.

##### Minimax:

$S(A)=\frac{1}{4G_{\rm
N}}\inf_{\tau}\sup_{\gamma\subset\tau}\operatorname{area}(\gamma)\,.$ (1)

The infimum is over piecewise timelike or null hypersurfaces $\tau$, which we
call time-sheets, that are homologous to $D(A)$ relative to
$\hat{\mathcal{I}}^{+}\cup\hat{\mathcal{I}}^{-}\cup\hat{\mathcal{I}}^{0}\cup
J^{+}(\partial A)\cup J^{-}(\partial A)$; in other words, there exists a bulk
spacetime region interpolating between $\tau$ and a part of the boundary that
includes all of $D(A)$ and none of $D(B)$. Note that this is a _spacetime_
homology condition, as opposed to the _spatial_ homology condition on a given
Cauchy slice familiar from the RT, HRT, and maximin formulas. As a consequence
of this homology condition, $\tau$ necessarily contains the entangling surface
$\partial A$. The supremum in (1) is over achronal codimension-2 surfaces
$\gamma$ contained in $\tau$ and containing the entangling surface $\partial
A$ (thereby excluding $\gamma$ from the bulk chronal future and past of
$\partial A$).

The minimax surface is the HRT surface (or any of them, if there is more than
one). The minimizing time-sheet, on which the minimax surface is maximal, is
highly non-unique; examples include the entanglement horizon (boundary of the
entanglement wedge) of $A$ and that of $B$.

##### Max V-flow:

$S(A)=\frac{1}{4G_{\rm N}}\sup_{V}\int_{D(A)}*V\,.$ (2)

$V$ is a 1-form in the bulk, and the objective is its flux through $D(A)$. It
is subject to a divergencelessness condition ($d*V=0$), a no-flux condition on
$\hat{\mathcal{I}}^{+}\cup\hat{\mathcal{I}}^{-}\cup\hat{\mathcal{I}}^{0}$, and
a norm bound. The norm bound can be expressed in two equivalent ways; the
first is non-local while the second is local but involves an auxiliary scalar
field:

  1. 1.

$V=0$ in the _bulk_ chronal future and past of $\partial A$, and, for every
bulk timelike curve, $\int dt\,|V_{\perp}|\leq 1$, where $t$ is the proper
time along the curve and $V_{\perp}$ is the projection of $V$ perpendicular to
the curve. Equivalently, the flux of $V$ through any codimension-1 timelike
ribbon of spatial area $a$ is bounded above by $a$.

  2. 2.

There exists a function $\phi$ in the bulk that equals $\pm 1/2$ on
$\hat{\mathcal{I}}^{\pm}\cup J^{\pm}(\partial A)$, such that the 1-forms
$d\phi\pm V$ are everywhere future-directed causal.

We can equivalently trade the 1-form $V$ for a set of “V-threads”, bulk curves
connecting $D(A)$ and $D(B)$. The first norm bound above would be interpreted
as the statement that the total number of threads crossing a window of area
$a$ being carried by an observer, over the observer’s lifetime, is bounded
above by $a$.

The V-threads are the covariant version of the bit threads introduced in
Freedman:2016zud . They may be spread out in time, but when collimated onto a
single Cauchy slice, they reduce to Riemannian threads. A crucial point is
that, in covariantizing the threads, they remain 1-dimensional, rather than
becoming extended into world-sheets, and their endpoints remain spacetime
points in $D(A)$ and $D(B)$, rather than becoming world-lines.

The maximal V-flows (or V-thread configurations) are highly non-unique but are
restricted to the entanglement wedges $\mathcal{W}(A)$ and $\mathcal{W}(B)$,
squeezing from the former to the latter via the HRT surface. This is
schematically illustrated in the left panel of figure 1.

One important difference between the V-flows and the Riemannian flows (or bit
threads) is the following. For Riemannian flows, the choice of boundary region
$A$ entered only in the objective (which is the flux through $A$), but not in
the definition of a flow. Here, however, the region $A$ enters in the norm
bound. That being said, a uniform definition of a V-flow can be given for
multiple boundary regions $A,B,\ldots$, provided they lie on a common boundary
Cauchy slice, by replacing the entangling surface $\partial A$ entering in the
norm bound (in either version) by the union of all of the entangling surfaces
$\partial A\cup\partial B\cup\cdots$ (which is equivalent to imposing the norm
bound for all the regions simultaneously). This uniform definition of a V-flow
can be used, for example, to prove subadditivity or to compute mutual
informations. On the other hand, for boundary regions _not_ lying on a common
Cauchy slice, the corresponding V-flow definitions differ essentially, and no
uniform definition can be given.

Figure 1:  Cross section of generic maximal V-flow (left) and minimal U-flow
(right). [Left]: The V-flow (indicated by green curves) goes from $D(A)$ to
$D(B)$, staying within their entanglement wedges $\mathcal{W}(A)$ and
$\mathcal{W}(B)$, squeezing through the HRT surface
${\gamma_{{}_{\text{HRT}}}}$ (red dot), and avoiding its past and future.
[Right]: The U-flow (indicated by blue curves) passes from
$\hat{\mathcal{I}}^{-}\cup J^{-}(\partial A)$ to $\hat{I}^{+}\cup
J^{+}(\partial A)$, squeezing through the HRT surface, and avoiding the
entanglement wedges. (The future/past boundaries $\hat{\mathcal{I}}^{\pm}$ may
be at infinity, as shown in the illustration, and/or singularities.)

##### Min U-flow:

$S(A)=\frac{1}{4G_{\rm N}}\inf_{U}\int_{\sigma}*U\,.$ (3)

$U$ is a future-directed causal 1-form in the bulk, and the objective is its
flux through any bulk Cauchy slice $\sigma$ containing $\partial A$. It is
subject to a divergencelessness condition ($d*U=0$), a no-flux condition on
$D(A)\cup D(B)\cup\hat{\mathcal{I}}^{0}$, and a norm bound. The norm bound,
which is a _lower_ bound on the norm, in contrast to the upper bound
constraining the V-flow, can be expressed in two equivalent ways; the first is
non-local while the second is local but involves an auxiliary scalar field:

  1. 1.

For every spacelike bulk curve connecting $D(A)$ to $D(B)$, $\int
ds\,|U_{\perp}|\geq 1$, where $s$ is the proper length along the curve and
$U_{\perp}$ is the projection of $U$ perpendicular to the curve.

  2. 2.

There exists a function $\psi$ in the bulk that equals $-1/2$ on $D(A)$ and
$+1/2$ on $D(B)$, such that the 1-forms $U\pm d\psi$ are everywhere future-
directed causal.

We can equivalently trade the 1-form $U$ for a set of timelike “U-threads”
beginning on the boundary region $\hat{\mathcal{I}}^{-}\cup J^{-}(\partial A)$
and ending on $\hat{\mathcal{I}}^{+}\cup J^{+}(\partial A)$.

The minimal U-flows (or U-thread configurations) are highly non-unique, but
are restricted to the bulk regions $J^{-}({\gamma_{{}_{\text{HRT}}}})$ and
$J^{+}({\gamma_{{}_{\text{HRT}}}})$, squeezing from the former to the latter
via the HRT surface and avoiding the entanglement wedges $\mathcal{W}(A)$ and
$\mathcal{W}(B)$. This is schematically illustrated in the right panel of
figure 1.

Possible applications of these new formulations of the HRT formula are outside
the scope of this paper, but since different ways of writing a given quantity
are often useful for different purposes, it is generally advantageous to have
as many such ways as possible. As one example, the formulas (2), (3) define
convex programs (which are actually Lagrange duals of each other), which may
make them particularly amenable to numerical computation. Some of the new
formulations may also be useful for proving general properties of holographic
entanglement entropies, such as inequalities they obey. The new formulations
may also have conceptual implications for our understanding of the
relationship between geometry and entanglement in quantum gravity.

##  1 Introduction

In this paper we develop a set of new, fully covariant geometrical
prescriptions for holographic entanglement entropy (EE). These formulas are
equivalent — but not obviously so — to the HRT and maximin formulas
Hubeny:2007xt ; Wall:2012uf . We begin by providing, in this section, a self-
contained introduction and summary of our results. We motivate the need for
such reformulations in subsection 1.1 and review the previously obtained
prescriptions in 1.2. The reader familiar with the HRT, maximin, and
Riemannian bit thread prescriptions is invited to skip to subsection 1.3,
which attempts to motivate intuitively what we might expect in covariantizing
the bit threads and what further physical insight might be gained. Subsection
1.4 then describes the key aspects of the actual results, but still focusing
on the conceptual rather than the technical side. In 1.5 we give a detailed
outline of the rest of the paper.

###  1.1 Motivation

Holographic EE offers intriguing insights into the bulk geometry of
holographic dualities. Indeed, many now suspect that entanglement crucially
underlies the emergence of spacetime, stimulating the investigation of
entanglement structure in holography. An early hint at an interesting relation
between spacetime geometry and entanglement came with the Ryu-Takayanagi (RT)
prescription Ryu:2006bv ; Ryu:2006ef , promptly uplifted to a fully covariant
formulation in general time-dependent context by Hubeny-Rangamani-Takayanagi
(HRT) Hubeny:2007xt : The EE $S(A)$ of a boundary region $A$ is given by the
proper area of a smallest-area bulk codimension-2 extremal surface
${\gamma_{{}_{\text{HRT}}}}(A)$ homologous to $A$.222 The homology condition
can be rephrased as the existence of a _homology region_ whose boundary
consists only of $A$ and ${\gamma_{{}_{\text{HRT}}}}(A)$ (the two meeting on
the boundary at the entangling surface $\partial A$).  The fact that the HRT
prescription relates a simple geometric construct, the extremal surface, to
the EE is a priori highly non-trivial, since both the EE and the holographic
mapping are individually rather intricate and complex.

Although we now have compelling evidence for the HRT conjecture (for a review,
see for example Rangamani:2016dms ), the prescription nevertheless retains
mysterious features.333 Since these puzzles are compounded rather than
dispelled in the quantum version of the holographic EE prescription
Faulkner:2013ana ; Engelhardt:2014gca , in this paper we will restrict to the
classical regime of $N\to\infty$, $\lambda\to\infty$.  On one hand, mapping a
sharply-delimited CFT region to a sharply-defined bulk object, the extremal
surface, whose location is likewise determined by entangling surface along
with the bulk geometry) naively seems to be in tension with the usual
holographic UV/IR correspondence: one might have expected the bulk construct
to be more delocalized. On the other hand, the HRT relation also has a
peculiar global (as well as a topological) aspect: amongst all the extremal
surfaces anchored on the entangling surface which are homologous to the
entangled region, we are instructed to pick the one with least area. This
allows for the extremal surface to jump to a different locus in the bulk under
smooth deformations of the entangling surface or the CFT state, while at the
phase transition itself we have a multiplicity of distinct but apparently
admissible surfaces. This discontinuity is particularly perplexing in light of
the bold conjecture Headrick:2014cta ; Wall:2012uf 444 Cf. Jafferis:2015del ;
Dong:2016eik ; Faulkner:2017vdd ; Cotler:2017erl ; Chen:2019gbt  for recent
evidence.  that the entanglement wedge555 The entanglement wedge is defined as
the bulk spacetime region spacelike-separated from the extremal surface
${\gamma_{{}_{\text{HRT}}}}(A)$ and connected to the boundary region $A$ in
question, or equivalently the bulk domain of dependence of the homology
region. (As usual, in asymptotically AdS spacetime we assume the requisite
boundary conditions for bulk evolution, so the domain of dependence extends
temporally along the boundary. We will correspondingly take the generalized
notion of global hyperbolicity and Cauchy surface.)  is the spacetime region
which is most naturally “dual to the reduced density matrix” $\rho_{A}$. It
suggests that, near such phase transitions, a tiny deformation of the reduced
density matrix could suddenly allow us to encode a huge additional spacetime
region in the bulk.666 One extreme version of this, involving a large number
of intervals in 3-d bulk, would change the entanglement wedge from covering
‘almost all’ of the compactified Poincare disk to ‘almost none’ of it. This
feature readily generalizes to higher dimensions as well.

In light of these curious features, one is compelled to reexamine the meaning
of holographic EE, or at an even more basic level, entanglement as such. We
will not attack this question directly here. Instead, we want to obtain a more
convenient characterization of holographic EE in terms of a distinct bulk
construct that, while equivalent to HRT, would offer more suggestive hints as
to its nature. Indeed, one broadly expects that different formulations tend to
demystify different features, so it is desirable to obtain as many distinct
prescriptions for holographic EE as possible.

Of course, in order for a given prescription to be even physically meaningful,
it must be fully covariant: it cannot rely on any choice of coordinates,
foliation, or other extra baggage that is not part of the physics. This
requirement then automatically enables us to apply the prescription to general
time-dependent settings. Such explorations are interesting and useful in many
contexts and indeed are presently being pursued with increasing vigor. More
importantly, a hitherto underutilized aspect of covariant formulations is that
they can naturally inspire conceptual advances, since a likely crucial but
still mostly missing piece regarding the underpinnings of the holographic
dictionary is the temporal aspect of the mapping.

###  1.2 Previous prescriptions

Let us briefly review previous reformulations of HRT. First, as already
explained in Hubeny:2007xt , the codimension-2 extremal surface can be thought
of as a surface with vanishing null expansions,777 Correspondingly, it admits
4 lightsheets Bousso:1999xy , generated by the future/past directed in/out-
going null normal congruences. Equivalently, it is a surface of vanishing
trace of the extrinsic curvature, and therefore vanishing expansion along any
normal congruence.  which turns out to be a convenient characterization for
using the Raychaudhuri equation to prove certain properties of the extremal
surface (such as its consistency with CFT causality Headrick:2014cta ) under
the usual physical assumptions, in particular the null energy condition (NEC).
In this situation, one can show that in the static context (or more generally
on a surface of time-reflection symmetry), HRT reduces to the original RT
prescription, involving a globally minimal surface on a preferred spatial
slice. Once restricted to Riemannian geometry, global minimality can be
utilized to prove important properties of the holographic EE such as strong
subadditivity (SSA) in an amazingly straightforward fashion Headrick:2007km ;
Headrick:2013zda . Unfortunately this convenience is not retained by the full
Lorentzian context, which makes the corresponding properties rather more
difficult to prove.

To remedy this, Wall Wall:2012uf  reformulated the holographic EE via a
maximin prescription, which entails taking an arbitrary bulk Cauchy slice
passing through the entangling surface, finding the globally minimal area
surface on it, and then maximizing this area over all possible slices; a
codimension-2 bulk surface which realizes this maximin procedure is called a
maximin surface, and any supporting slice on which it is globally minimal is
called a maximin slice. Wall showed, under reasonable assumptions, first that
the maximin surface coincides with the HRT surface, and second that its area
obeys SSA in the general time-dependent context. Moreover, the maximin
construction specifies the homology constraint more naturally than HRT,888 As
pointed out in Hubeny:2013gta , to maintain causality, the HRT surface must
remain spacelike-separated from the boundary region, which is ensured by
requiring the homology region is achronal. We call this the spacelike homology
constraint, and in the maximin construction it is implemented automatically by
extremizing only over minimal surfaces which lie on Cauchy slices containing
the entangling surface.  but its two-step minimization and maximization veils
the nature of entanglement quantity even further, and is perhaps conceptually
less appealing (since it contains a vast amount of intermediate extra baggage
and involves the breaking of a natural symmetry between the spatial and
temporal directions).

So far, all reformulations involved a bulk codimension-2 surface, which leaves
the conceptual meaning of holographic EE obscure and suffers from the
associated puzzles such as a possible discontinuity in the location of the
bulk surface; see Freedman:2016zud  for further discussion. However, in the
static context, these puzzles were circumvented by a completely different
prescription, which utilizes a construct dubbed _bit thread_ , a 1-dimensional
object which can be thought of as a field line of a flow connecting the
boundary region $A$ to its complement. In particular, the RT reformulation put
forward by Headrick-Freedman Freedman:2016zud  used the Riemannian max-flow
min-cut (MFMC) theorem Federer74 ; MR700642 ; MR1088184 ; MR2685608  to
express EE of a given region $A$ in terms of flows. The equivalence with RT
was explained in greater detail in Headrick:2017ucz , which further develops
the tools we will use in the present work.

The setting of Freedman:2016zud  is as for RT, namely the Riemannian geometry
of a spatial slice $\sigma$ which is a surface of time reflection symmetry in
the full Lorentzian geometry. Define a flow to be any divergenceless vector
field $v$ with unit-bounded norm:

$\nabla\cdot{}v=0\,,\qquad|v|\leq 1\ .$ (4)

We can equivalently think of this vector field in terms of oriented flow lines
(hence motivating the term threads999 Although this terminology suggests a
discrete structure, this is merely employed as a conceptual crutch; the bound
is in units of Planck area (or more accurately $4\ell_{P}^{d-1}$ in
$d+1$-dimensional spacetime), so we are typically dealing with a macroscopic
number of threads within any region of interest. ) which cannot end in the
bulk (due to the divergencelessness condition) and have bounded transverse
density (due to the norm bound). For any boundary region $A$, the MFMC theorem
states that the maximum flux of such a flow from $A$ equals the minimal area
achievable by any surface (or “cut”) $m$ homologous to $A$:

$\max_{v}\int_{A}v=\min_{\gamma\sim A}\,{\rm area}(\gamma)\ .$ (5)

Intuitively, any flow from $A$ is clearly bounded by the minimal-area
bottleneck $\gamma_{\rm min}$ the flow has to pass through, and the main
content is that a maximizing (or optimal) flow achieves this bound. The EE
$S(A)$ is then given by the flux of any such optimal flow (or maximal number
of threads) from $A$:

$S(A)=\max_{v}\int_{A}v\ .$ (6)

Note that although an optimal flow $v$ is far from unique, the bottleneck
$\gamma_{\rm min}$ generically is unique, and corresponds to the RT minimal
surface. At this locus, the flow $v$ saturates the norm bound and is normal to
$\gamma_{\rm min}$. The homology constraint is implemented automatically, with
the flow lines generating the requisite homology region between the boundary
region and the bottleneck. Moreover, the number of flow lines has the familiar
UV divergence coming from the divergent area of $m$, and in fact the flow
picture enables us to compare these divergent quantities more easily. For pure
states, we immediately see that $S(A)=S(A^{c})$, implemented by the same flow
configuration (with flipped directionality).

Despite the flow prescription (6) being equivalent to the RT prescription, it
has a number of technical and conceptual advantages. For example, showing
certain properties such as subadditivity and SSA Freedman:2016zud  is even
more immediate than for RT. In fact, the utility of the difference in
respective proof methods goes well beyond the mere confirmation of a
previously-established result. For example, it elucidated the difference
between the universally-true SSA property and the holographically-true
monogamy of mutual information (MMI) property Hubeny:2018bri ; Cui:2018dyq ,
whereas the surface-based method proves these two properties equivalently. In
the cooperative flow construction of Hubeny:2018bri  this distinction was
interpreted as the MMI being more intrinsically tied to bulk locality than
SSA.

Conceptually, the bit thread picture is evocative of a bipartite nature of the
entanglement structure. One might think of each flow line as joining an EPR
pair which straddles the entangling surface $\partial A$. However, it is
important to note that the flows depend not just on the state itself, but also
on the entangling surface. In other words, changing the region of interest
generically changes the flow, unless the regions are nested (in which case one
can find flows that simultaneously maximize both).

Given the utility of bit threads, the obvious goal is to generalize them to
the Lorentzian context, which allows for time dependence. The equivalence
Freedman:2016zud ; Headrick:2017ucz  between RT and bit thread formulations
naturally suggests applying the same techniques (convex relaxation and
Lagrange duality) to HRT. While that is indeed the route we will take in this
paper, before embarking, it will be instructive to pause to see what we might
naively expect, and its pitfalls.

###  1.3 Naive expectation

Figure 2:  Schematic sketch of two natural covariantizations of Riemannian bit
threads [Middle], indicated by purple curves on constant time slice of
Lorentzian global AdS spacetime. [Left]: Threads extend in time to form 1+1
dimensional worldsheets. [Right]: Threads remain 1-d objects but are no longer
localized on a single time slice. The blue curve in each figure is the HRT
surface. (For ease of illustration, the number of covariantized threads which
we display varies between the three pictures, but of course in the actual
constructions they would all agree.)

If we view a Riemannian bit thread as a 1-dimensional string-like object that
somehow embodies entanglement between the boundary regions at the string’s
endpoints, it then seems most natural to expect that when uplifted to the
Lorentzian context, this string will extend in time, spanning a
$(1+1)$-dimensional worldsheet, as schematically illustrated in the left panel
of figure 2. If the original endpoints characterized a Bell pair, the
intersection of the worldsheet with the boundary would now correspond to the
worldlines of these individual entangled particles. And since their
entanglement cannot be created or destroyed acausally, we might furthermore
presume such worldlines, and so plausibly the interpolating worldsheet, to be
generically timelike (or at least causal). Intersecting the worldsheet by
constant-time slices would then recover the bit threads as “snapshots” at the
given boundary time, effectively tracking the entanglement dynamics.

This worldsheet picture is further bolstered by the observation that in the
Lorentzian context, the HRT prescription entails a spacetime codimension-2
surface (uplifted from a spatial codimension-1 surface on a fixed-time slice),
so that the uplift of the threads, being in some sense the dual objects,
should be correspondingly bulk 2-dimensional sheets, locally extending in the
orthogonal directions from the HRT surface. We might then try to recast the EE
$S(A)$ as the maximal number of such worldsheets which can pass through the
boundary domain of dependence $D(A)$.

Interestingly, this naive expectation does _not_ seem to be realized. One
immediate challenge has to do with a suitable generalization of the norm bound
in (4). Since in the Riemannian context this condition restricted the threads
from getting too close together, we might expect that, similarly, the
worldsheets should have suitably bounded density. It is also natural that the
divergencelessness condition in (4) translates to the restriction that the
worldsheets cannot simply terminate in the bulk. But these two conditions
together appear to impose too global a constraint on the worldsheets, which in
particular can violate causality. For example, if the bulk spacetime happens
to contract in the future, this would seem to teleologically expel the
worldsheets in the present.101010 We thank Juan Maldacena for originally
raising this possibility.

Moreover, if each thread uplifts to a physical worldsheet, one might expect
that this object is intrinsically 2-dimensional, analogously to a fundamental
string worldsheet, with no residual information about its foliation by the
‘snapshot’ threads. In other words, observers with relative boosts should
experience the same worldsheet, but naturally associate different thread-
foliations to it. This however makes the optimal thread congruence
overconstrained, since temporally deforming the entangling surface deforms the
position of the corresponding HRT surface, where the bit threads are required
to be maximally packed and directed perpendicularly.111111 As a simple
example, boosting the boundary region in opposite directions of a
translational symmetry yields two distinct HRT surfaces that intersect in the
middle but with different normals. Said differently, the collection of HRT
surfaces anchored somewhere on the boundary ‘entangling tube’ $\partial A(t)$,
without preserving the $t$-foliation, spans a bulk codimension-0 region
instead of a timelike codimension-1 hypersurface, analogously to the set of
HRT surfaces anchored on a given boundary Cauchy slice generically spanning a
codimension-0 bulk region instead of all lying on the same bulk Cauchy
surface.

Another way of viewing the potential incompatibility of the threads forming
worldsheets under time evolution is the following. Considering for a moment
the original Riemannian context, it was observed already by Freedman:2016zud
that a given thread configuration cannot optimize on two crossing regions
simultaneously. The intuitive reason is that the threads have to be maximally
packed on the RT surface and traverse it perpendicularly, but crossing regions
have intersecting RT surfaces, and therefore incompatible perpendicular
directions at their intersection. Coming back to the Lorentzian case, the
physically pertinent geometrical object is not the entangling surface
$\partial A$ as such, but the boundary domain of dependence $D(A)$. It is then
tempting to view the case of time-evolved regions $A(t_{1})$ and $A(t_{2})$ as
likewise crossing, in the sense of their domains of dependence forming non-
nested sets with non-empty intersection. Even though the HRT surfaces as such
do not intersect, one might nevertheless worry that it won’t be possible in
general to find a single set of thread worldsheets whose $t_{1}$ and $t_{2}$
cross-sections would be simultaneously optimizing.

The basic flaw with the naive expectation of bit thread worldsheets is to
think of the EE as pertaining to a boundary space _time_ , in particular as
admitting a canonical temporal extent. On a static boundary spacetime (for
example, the Einstein static universe for an asymptotically globally AdS
spacetimes), there is indeed a natural way of fixing a given boundary region
in space and considering the evolution of its EE in time, in response to the
evolution of the state. However, in general, such a setting would be too
limiting. For example, there is no canonical way of spatially delimiting a
region in time-evolving asymptotically locally AdS spacetimes, or for non-
static observers. EE pertains to a given boundary state and region at a single
instant in time, and in these more general contexts there is no preferred
extension from one time to another. This is the reason that a codimension-2
surface does not dualize to a 2-dimensional object: we lose one dimension
because of this instantaneous nature of EE.

The conclusion (already emphasized by Freedman:2016zud ) is that the threads
are not to be viewed as physical objects, and their endpoints are not
naturally extended in time. What could then be an interpretation of such a
scenario? A holographic CFT is necessarily strongly coupled, so dynamically a
Bell pair should quickly decohere and the shared entanglement spread out.
Accordingly, we would expect entanglement to be generically delocalized, but
nevertheless one might imagine that one could momentarily localize it, perhaps
by an entanglement distillation process. Since we can think of this process as
a spacetime event (with negligible time duration), we might associate the
thread ends to such events. The number of threads which can end in $D(A)$
would then be counting the number of distillation processes we could perform
on the state specified at $A$, which would in turn characterize its
entanglement.121212 Although for general mixed states, the entanglement of
distillation does not coincide with EE, it does so for pure states. In the
present context, we think of the full geometry as encoding a pure state, so
one might hope that its entanglement could be viewed in this way.

How does this bear on the relevant geometrical constructs in the bulk which
should be associated with EE? The above speculation suggests that on the
boundary we retain string endpoints, and for the Lorentzian formulation to
correctly reduce to the Riemannian one for the static cases, the most natural
mathematical construct is then still a thread.131313 In principle, due to the
diverging conformal factor at the AdS boundary, another possible construct
connecting two boundary points is an extended flux-tube-type object; but this
does not immediately reduce to a Riemannian thread. In fact, it will turn out
that threads already allow requisite delocalization, while retaining
relatively simple description.  But since we have a Lorentzian spacetime (with
no preferred time slice in general), we would expect that these threads can
meander in both space and time, as schematically illustrated in the right
panel of figure 2. The upgraded expectation is that the EE is captured by the
maximal number of threads adhering to certain restrictions. Causality requires
that they end within $D(A)$. But what further constraints should we impose on
them? Can they be timelike somewhere? What bulk regions are they allowed to
penetrate? How do they interface with each other?

###  1.4 Preview of results

To answer these questions we will dualize HRT. Our derivation effectively
entails a double convex relaxation (in both spatial and temporal directions),
which combines features of the previously studied MFMC theorem in the
Riemannian setting, as well as the min flow-max cut theorem in the Lorentzian
setting Headrick:2017ucz . This allows us to construct a web of geometrically
distinct prescriptions, or reformulations, in terms of flows, or equivalently
in terms of threads (which we will slightly generalize from the original
picture of integral curves of a nowhere-vanishing flow field). Altogether we
will present ten new formulas for computing the holographic EE in a general
time-dependent holographic spacetime (in addition to the already-known HRT and
maximin).

To develop the mathematical framework, it will be instructive to start in a
more general context and only restrict to holography at a later point, which
will simultaneously enable us to identify the special features implemented by
holography. Hence at the outset, we will retain only the “kinematical” aspects
of the holographic context, but not specialize to keeping the “dynamical” ones
until section 6.

We will see that in optimization problems, given a function of two variables
$f(x,y)$, its maximin $\sup_{x}\,\inf_{y}\,f(x,y)$ is generically distinct
from its minimax $\inf_{y}\,\sup_{x}\,f(x,y)$ where we simply switch the order
of the two extremizations. However, the minimax universally provides an upper
bound for the maximin.141414 The reader who seeks a more intuitively obvious
mneumotic is invited to observe that the shortest giant is still taller than
the tallest dwarf. The Lagrange duality of convex optimization problems
utilizes this structure: weak duality gives the bound while strong duality
gives a sufficient condition to saturate it. The two optimizations are over
the original variables and over the Lagrange multipliers that implement the
constraints, respectively; hence the dual problem is phrased in terms of the
Lagrange multipliers instead of the original variables.

One natural class of situations where the maximin and the minimax values
coincide is when there exists a “global saddle point” $(x_{0},y_{0})$ such
that $f(x_{0},y)$ is $y$-minimized at $y_{0}$ while $f(x,y_{0})$ is
$x$-maximized at $x_{0}$. More broadly, the criterion for the minimax to equal
the maximin is specified by the minimax theorem (originally developed in the
context of game theory). In the continuum context, the crucial criterion is
for the function to be convex-concave in its respective arguments. Before
devising such a function in our context, we first consider a more localized
geometric prescription where all the action effectively takes place within a
single hypersurface of a specified class, and then optimize over such
hypersurfaces. The maximin prescription for holographic EE Wall:2012uf  is but
one example; here we can think of the action as taking place within a single
Cauchy slice, and within this slice we can use Riemannian max flow-min cut
theorem to convert it to slice flow, so that upon maximizing over all slices
we arrive at an alternative, “maximax” prescription. But instead of Cauchy
slices, one could equally start with a different class of hypersurfaces.
Since, roughly-speaking, an HRT surface area increases under spatial
deformations and decreases under temporal deformations,151515 This is just a
heuristic to build intuition; the separator is not precisely null, and in fact
one can typically find spacelike Cauchy slices (which are not maximin slices)
along which the HRT surface is not the minimal area surface. This is possible
whenever the expansion of the null normal congruence from the extremal surface
becomes negative (which is generically the case).  one could first find the
maximal-area surface within a timelike hypersurface (which we will dub “time-
sheet”) and then minimize this area over all time-sheets. This is our
“minimax” prescription.

To unify both maximin and minimax into a common phrasing, we can view the
extremal co-dimension-2 surface in question as the intersection of two
codimension-1 hypersurfaces, namely a spatial slice and a time-sheet. The
difference in the two prescriptions then boils down to merely the order of
extremizations, and hence becomes a subject of the minimax theorem. But since
the respective sets of hypersurfaces (and intersections thereof) are not
convex sets, the two quantities thus identified (which we’ll still refer to as
minimax and maximin), need not coincide, and in fact it is easy to construct
an example of spacetime where they differ. We will see that allowing partial
relaxation brings the quantities closer together, and the minimax theorem
indicates that if we can embed the problem into a fully convex-relaxed one,
then the corresponding convex maximin and convex minimax would indeed
coincide. Quite remarkably, in the actual holographic context (characterized
by the correct dynamics), we will see in section 6 that maximin and minimax in
fact do coincide even without any convex relaxation, due to the existence of a
global saddle point, as already heralded by the HRT prescription.

However, in the more general context (retaining the kinematics but not the
dynamics), to achieve the equivalence criterion of the minimax theorem, we
need to convex-relax these hypersurfaces. As in Headrick:2017ucz , we can view
a hypersurface as a level set of a scalar field, and impose conditions on the
scalar field so as to comprise a convex set. The slice is then convex-relaxed
(“smeared”) to a continuous collection of level sets, weighed by the gradient
norm. The novel feature here compared to the implementation in
Headrick:2017ucz  is that we do this not with just a single scalar field but
with two scalar fields simultaneously; in particular, we’ll associate a field
$\phi$ with a temporal smearing of a spatial slice, and a field $\psi$ with
spatially-smearing a time-sheet.

The objective function to be optimized, generalizing the area of a
codimension-2 surface, is constructed from a certain scalar pairing between
the respective gradient 1-forms, such that it has the requisite convexity
properties. The minimax theorem then states that we can exchange the order of
the extremizations without changing the optimal value. The formulation in
terms of these two scalar fields therefore provides a convenient starting
point for subsequent reformulations. However, while mathematically central to
our story and appealingly treating the spatial and temporal directions on a
similar footing, it is not the most intuitively suggestive formulation.
Instead, it turns out to be conceptually more convenient to Lagrange-dualize
on one or both scalar fields, recasting the formulation in terms of flows. We
will dub the primarily-spatial flows (dual to $\psi$) V-flows and the temporal
flows (dual to $\phi$) U-flows. In the next few paragraphs, we will preview
the respective prescriptions in greater detail.

##### V-flows:

The convex program of minimizing over the scalar field $\psi$ (which smears
out the time-sheets) at fixed $\phi$ can be dualized to obtain a concave
program, which entails maximizing the flux of a divergenceless 1-form $V$
subject to a certain norm bound. This norm bound is analogous to the familiar
one $|v|\leq 1$ in the Riemannian setting, but because of the Lorentzian
geometry, it now has a richer structure, and in particular is non-local.
Specifically, it is implemented by requiring the 1-forms $d\phi\pm V$ to be
everywhere future-directed causal.

We can re-cast this norm bound in a way that actually does not invoke $\phi$
at all, instead posing a more global norm condition, namely, we upper-bound
the integral over an arbitrary timelike curve $\mathpzc{q}$, parameterized by
proper time $t$, of the norm of the perpendicular projection of $V$:

$\int{q}dt\,|V_{\perp}|\leq 1\ .$ (7)

$S(A)$ is then calculated by maximizing the flux of a divergenceless flow $V$
from $D(A)$ subject to the requisite boundary conditions and the above norm
bound. Using time-sheets, we can interpret (7) physically as the requirement
that any observer carrying a unit-area window cannot capture more than a unit
amount of total flux of $V$ over their entire lifetime. To recover the
original bit thread formulation in the Riemannian context Freedman:2016zud ,
we can specialize to the case where all the flux is localized on a single
slice, in which case the integrated bound (7) collapses to the simple norm
bound of (4).161616 The norm bound (7) also ratifies our previous naive
expectation that there cannot exist a compatible flow that simultaneously
maximizes on temporally-separated regions with crossing domains of dependence:
even if the respective HRT surfaces do not intersect, they are necessarily
timelike-separated somewhere, which allows us to construct a curve
$\mathpzc{q}$ for which this norm bound is violated.  One might think that in
the more general Lorentzian case this is now infinitely more complicated since
we have (continuously) infinitely many observer worldlines to check, but in
fact the initial formulation in terms of $\phi$ can be viewed as providing a
single “certificate” that guarantees the bound for every worldline.

Although the original scalar field $\psi$ was introduced as a tool to smear
out a time-sheet, and correspondingly the dual $V$-flow 1-form has flow lines
which emanate from $D(A)$ and go into $D(A^{c})$, these flow lines need not
actually remain spacelike everywhere; they can have timelike (or null) pieces
along the way, subject to (7).

##### U-flows:

The U-flow case follows a very similar story as the one for V-flows, but now
pertains to dualizing on $\phi$ instead, starting from the minimax
formulation. In particular, keeping $\psi$ fixed, we can dualize the concave
program of maximizing over $\phi$ (i.e. smeared Cauchy slices) to the convex
program of minimizing the flux of a divergenceless 1-form $U$ at future
infinity, again subject to a norm bound, namely that the 1-forms $U\pm d\psi$
must be everywhere future-directed causal.171717 We can also obtain the U-flow
by directly dualizing the V-flow program; see the diagram (81) for a summary
of these relations.

Analogously to the V-flow case, we can write the norm bound as a global
condition in terms of an arbitrary spacelike curve $\mathpzc{p}$ passing
between the domain of dependence of the region $A$ and that of its complement:

$\int{p}ds\,|U_{\perp}|\geq 1\,,$ (8)

where $U_{\perp}$ is the projection of $U$ perpendicular to $\mathpzc{p}$.
Note the direct analogy with (7); however, whereas in that case the constraint
$\mathpzc{q}$ had no information about the given region but the objective
function (the boundary region on which the flux is evaluated) did, here it is
the other way around: the objective function is region-agnostic while the
region determines the constraint via requisite set of curves $\mathpzc{p}$.

The difference of viewpoint between the two sides provides us with different
toolkits that complement each other. For example, in proving holographic
entropy inequalities, one can optimize the smaller side in V-flow language to
bound the larger side from below, or conversely one can optimize the larger
side in the U-flow language and show that this provides an upper bound for the
lower side. It is worth noting however that although the two prescriptions are
closely analogous (seemingly amounting mainly to spatio-temporal flip
accompanied by minimization-maximization flip), there are still crucial
differences due to the boundary conditions, as mentioned above. An avatar of
this feature appears already in the maximin and minimax formulation using
hypersurfaces, where specifying a Cauchy slice does not fix the homology class
of codimension-2 surfaces, while specifying a time-sheet does.

##### Thread distributions:

Once we have formulated the convex-relaxed problem in terms of flows, it
follows immediately that we can re-cast them also in terms of threads. In
particular, for a flow field $V$, which is a 1-form, the flow lines (which can
be thought of either as integral curves of the dual vector field, or in terms
of the Hodge dual $*V$), can be viewed as threads joining the given boundary
subsystem to its complement.181818 One might wonder why we didn’t try to
construct threads already from $d\psi$ which is after all also a 1-form. This
is generically not possible, since $d\psi$ is not divergence-free, so that it
is not extendible into a full thread; an extreme case being when $\psi$ is
just a simple step function corresponding to a localized time-sheet.  However,
an alternative, and perhaps a more appealing, notion of a “thread” is simply
an unoriented curve in the spacetime, which is allowed to intersect other
threads. While this generalizes the notion of integral curves of a smooth
vector field (which cannot intersect by construction), one can reformulate the
requisite norm bounds in terms of a restriction on the measure on the space of
curves.

In the Riemannian case, this measure is restricted to ensure that the thread
density is $\leq 1$ everywhere, and the corresponding linear program maximizes
the total measure, subject to this constraint, of the set of threads which
join the given region and its complement. We can also dualize this program,
which amounts to minimizing a positive function $\lambda$ subject to it
integrating to a value $\geq 1$ over any thread. The minimum is attained by a
step function along the minimal surface, which again recovers the EE. We can
readily generalize this setup to multiple regions and for example obtain a
thread version of max multiflow theorem of Cui:2018dyq . Indeed, we can map
any (multi)flow to a thread distribution and vice-versa (though not via an
isomorphism due to the non-orientation of treads and non-crossing of flow
lines), so that every statement pertaining to threads has an avatar in flows.

In the full Lorentzian context, we can similarly translate V-flows and U-flows
into V-threads and U-threads, respectively. However, the since the two flows
depend on each other, the corresponding constraints on the threads are now
non-local: instead of bounding thread density at every spacetime point, we
need to bound thread density integrated over each dual thread. In this way,
the density bounds for the V- and U-threads enforce each other. The EE is
obtained by maximizing the number of V-threads, or equivalently (in the dual
picture) minimizing the number of U-threads, subject to these constraints. The
V-threads join the boundary domain of dependence of the given region with that
of its complement, but are not restricted to remain spacelike in the bulk. The
U-threads, on the other hand, are necessarily causal and join the past
boundary of the spacetime with the future one. Their knowledge of the given
region comes through the norm bound, which amounts to the U-threads forming a
sufficient barrier separating the requisite domains of dependence. The summary
of all of the prescriptions for the convex-relaxed value is in the diagram
(205).

##### Optimized flow/thread configurations in holography:

The restrictions (7) and (8), on the V- and U-flows, or the corresponding
rephrasing in terms of the threads, still allows these geometrical objects to
permeate the full spacetime without singling out any more localized regions.
However, for _optimized_ flows in the holographic context, this allowed set
collapses to become more colimated, as sketched in figure 10. In particular,
the optimized flows must all pass through the HRT surface! The V-threads are
confined to the entanglement wedge of $A$ and that of its complement, while
the U-threads are confined to the future and past of the HRT surface. These
four regions naturally partition the bulk spacetime Headrick:2014cta , and the
V-threads intersect the U-threads only along the HRT surface, thereby
naturally counting its area as the EE. Notice that this description implements
just the right amount of localization, without imposing any non-geometrical
features, and retaining maximal democracy between spatial and temporal
directions. The localization comes about collectively: each thread by itself
does not exhibit any special points along its length — rather, its physical
significance lies in what regions it connects and how it interfaces with the
other threads.

###  1.5 Outline of the paper

Having previewed our results, in the remainder of the paper we develop them in
full technical detail. We start by explaining our setup and assumptions in
section 2. Since our primary focus is on geometrical prescriptions for
calculating EE in holography, we first clarify in subsection 2.1 how we
regulate EE to obtain a meaningful quantity. In particular, in most of our
derivations we work in “regulated spacetime” consisting of the entanglement
wedge of the union of a given boundary region $A$ and its disjoint complement
$B$ (with the separating entangling surface slightly thickened so that we can
use the entanglement wedge cross section to regulate the EE). This is merely a
matter of presentational convenience; as implied by CFT causality (and
confirmed explicitly in section 7.1), the details of the spacetime in the past
or future of the HRT surface do not influence the holographic EE.

To appreciate the special features of holography, it will furthermore be
instructive to work in a more general setting — namely a globally hyperbolic
spacetime whose conformal boundary includes timelike components — which we
specify in subsection 2.2. This allows us to identify the key codimension-1
constructs, schematically illustrated in figure 4, consisting of distinct
components of the spacetime boundary, as well as slices and time-sheets, which
will pave the way for defining the maximin and minimax constructs (in section
3). In order to develop the framework to recast these prescriptions in terms
of flows (in section 4), it will be convenient to use covectors and their
Hodge duals instead of the more familiar vector fields; subsection 2.3 reviews
this formalism and constructs a scalar concave-convex “wedgedot” pairing of
two covectors that will play a crucial role throughout the paper.

Section 3 then proceeds to explain the essential points from minimax theory
and convex relaxation. To anchor the reader, we first specify the geometric
maximin and minimax constructs and variations thereon in subsection 3.1. To
understand the relation between them, we step back to review minimax theory
(in its original game theory context) in subsection 3.2. In subsection 3.3 we
apply the theory to our geometrical context and perform convex relaxation to
define a new quantity. We use the suggestive notation $S_{-}$ for maximin,
$S_{+}$ for minimax, and $S_{c}$ for the convex-relaxed quantity.191919 The
choice of the letter $S$ and acronym “EE” in our summary of results were in
anticipation of the holographic context; however strictly speaking in the
broader context our results apply to, there is no dual theory in which to
formulate an EE. The preceding paragraphs pertaining to V-flows, U-flows, and
threads all give prescriptions for $S_{c}$, whereas the hypersurface-localized
prescriptions give $S_{-}$ in slice-localized and $S_{+}$ in time-sheet-
localized contexts.  In general, the minimax theorem only ensures that
$S_{-}\leq S_{c}\leq S_{+}$, while in the more physically relevant context of
holography (discussed in section 6), all three quantities in fact do coincide,
$S_{-}=S_{c}=S_{+}$, and provide alternate prescriptions for the holographic
EE.

Having formulated $S_{c}$ in terms of a convex-concave pairing of the scalar
fields $\phi$ and $\psi$ (whose level sets implement convex-relaxation of
slices and time-sheets), we finally get to the core of the paper in section 4,
which reformulates these in terms of flows by applying Lagrange duality.
Subsection 4.1 dualizes on $\psi$ at fixed $\phi$ to obtain the V-flow
program, while subsection 4.2 dualizes on $\phi$ at fixed $\psi$ to obtain the
U-flow program. To avoid fragmenting the narrative overmuch, in both
subsections we relegate the actual dualizations, as well as proofs of lemmas
needed for the reformulations of the norm bounds etc., to subsection 4.4 which
serves as a mini-appendix to section 4. In order to hone intuition for the
effect of convex relaxation, in both subsections we devote a subsubsection to
an explicit toy example of a spacetime wherein $S_{-}\neq S_{+}$ (introduced
in subsection 3.3). Instead of a full convex relaxation, we perform a
particularly simple partial relaxation, to demonstrate how it diminishes the
gap between maximin and minimax values, bringing them closer to $S_{c}$; we
refine this with a further (but still partial) convex relaxation in appendix
B, which generalizes the previous calculations, now presenting the V-flow and
U-flow cases in parallel in a self-contained manner. In subsection 4.3 we
prove subadditivity of $S_{c}$ in these two formulations, which illustrates
that despite the close parallel between the V-flow and U-flow programs, there
is a non-trivial difference in how they implement various features of $S_{c}$.
Finally, the heart of the mathematical framework resides in subsection 4.4
where we prove the various statements asserted earlier. We start by presenting
six covector-pair lemmas, followed by five explicit dualizations, and
culminating in proving the equivalence of the norm bounds appearing in the
respective flow programs which involves a beautiful generalization of
Hamilton-Jacobi theory for non-differentiable Lagrangians, this curious
feature arising due to signature-dependence in the Lorentzian context.

Section 5 substantiates (and surpasses) the title and original motivation of
the paper; in addition to covariantizing the Riemannian bit thread formulation
of Freedman:2016zud  captured by V-threads, it also provides an alternate
covariant prescription in terms of U-threads, mimicking the V-flow and U-flow
formulations of section 4. To generalize the notion of threads viewed as flow
lines, subsection 5.1 develops the framework of thread distributions in the
Riemannian context from scratch, applying the technology of convex
optimization to measures on sets of curves and proving the analog of the MFMC
theorem. In subsection 5.2 we return to the Lorentzian context, and formulate
the V-thread and U-thread prescriptions for computing the holographic EE.

In section 6, we finally specialize to the holographic context. We first
explain in subsection 6.1 why the non-convex maximin and minimax values
coincide with the convex-relaxed one by identifying the HRT surface as a
global saddle point. This relies on certain physical assumptions about the
bulk spacetime and its boundary, and demonstrates the equivalence between
maximin and HRT prescriptions already shown in Wall:2012uf , as well as that
between minimax and HRT. In subsection 6.2 we consider the optimized flows and
show that they pass through the HRT surface. While hitherto most of our
constructions pertained to a single region $A$ on the boundary, in subsection
6.3 we indicate how to generalize the discussion to multiple regions, with a
suitably enlarged regulated spacetime.

In section 7.1, we show how we can enlarge our constructions pertaining to the
regulated spacetime, by embedding the latter in the full spacetime, without
changing any of the results. We also explain what happens to the constructions
in the limit that the regulator is removed.

We conclude in section 8 with a discussion and possible applications and
future directions.

As the paper is fairly notationally heavy, we also include for the reader’s
convenience in appendix A a table of notation.

We will assume that the reader is familiar with the content of the paper
Headrick:2017ucz , in particular the concepts of convex relaxation and
Lagrange duality and their application to geometrical theorems such as the
Riemannian max flow-min cut theorem.

##  2 Background

In this section, we explain the basic setup, assumptions, and notation we will
use in the rest of the paper. We start in subsection 2.1 by explaining how we
deal with ultraviolet divergences in holographic EEs. In subsection 2.2, we
then detail the assumptions and notation we use for the spacetime we’ll be
working in. Finally, in subsection 2.3 we explain important notation for
1-forms that we will use throughout the paper. We remind the reader that
appendix A contains a table summarizing the notation we use in the paper.

###  2.1 EWCS regulator

The purpose of this paper is to give several formulas, equivalent to the
original HRT formula Hubeny:2007xt , for the EE $S(A)$ of a spatial region $A$
in a holographic field theory. This task only really makes sense if $S(A)$ is
a finite quantity, in other words if any infrared and ultraviolet divergences
have been regulated in some way. The choice of IR regulator will not have any
bearing on our work, so we will not deal with it explicitly, but will simply
assume that any IR divergences have been regulated somehow. On the other hand,
among the various kinds of UV regulators that have been employed for
holographic EEs, there is one that will be particularly natural for our
purposes, involving the so-called entanglement wedge cross section (EWCS)
Takayanagi:2017knl ; Nguyen:2017yqw ; Dutta:2019gen , which we will now
review.202020 For systematic discussions of UV regulators for holographic EE,
see Sorce:2019zce ; Grado-White:2020wlb ; paper1 .

To set the stage, it is useful to first recall the case where the entropy is
naturally UV-finite and does not need to be regulated. This happens when the
conformal boundary has multiple connected components — i.e. the spacetime is a
multiboundary wormhole — and we are computing the entropy of an entire
connected component or a set of them. To fix some notation, we denote the bulk
by $\mathcal{M}$ and its conformal boundary by $\mathcal{N}$. $\mathcal{M}$
also has past and future boundaries $\mathcal{I}^{-}$, $\mathcal{I}^{+}$,
which may be at infinite time and/or include singularities. Let
$\\{\mathcal{N}_{i}\\}$ be the connected components of $\mathcal{N}$, and
choose a Cauchy slice $\Sigma_{i}$ for $\mathcal{N}_{i}$. Each $\Sigma_{i}$ is
boundaryless. So if we let $A$ be the union of a subset of $\\{\Sigma_{i}\\}$,
then the entangling surface $\partial A$ is empty and there is no UV
divergence in $S(A)$.

On the other hand, if $\partial A$ is non-empty, then $S(A)$ contains a UV-
divergent piece. To regulate this divergence, we essentially want to turn the
bulk spacetime into something akin to a multiboundary wormhole; this is what
the entanglement wedge and EWCS will do for us. We choose a boundary Cauchy
slice $\Sigma$ containing $A$, and a region $B\subset\Sigma\setminus A$ that
almost fills the complement but leaves a small buffer between $A$ and $B$
Dutta:2019gen . The EWCS $S(A:B)$ is defined as the area of the minimal
extremal surface in the entanglement wedge of $AB$ that is homologous to $A$
relative to the joint HRT surface ${\hat{\gamma}_{{}_{\text{HRT}}}}(AB)$. (The
reason for the hat on $\gamma$ will become clear shortly.) From a field-theory
viewpoint, the buffer between $A$ and $B$ eliminates the short-wavelength
modes shared between $A$ and its complement that cause the divergent EE. We
should also emphasize, of course, that the EWCS is an interesting quantity in
its own right, quite apart from its use as a regulator, with various
conjectured field-theory interpretations Nguyen:2017yqw ; Takayanagi:2017knl ;
Dutta:2019gen . Although we have motivated the EWCS as a regulator, the
results of this paper apply equally well to any EWCS calculation; they do not
depend on the buffer between $A$ and $B$ being small in any sense.

$A$$B$$\mathcal{M}$$\mathcal{I}^{+}$$\mathcal{I}^{-}$

Figure 3:  EWCS regulator. [Left]: On the Poincare disk, the region outside
the $AB$ entanglement wedge (shaded gray) is excised from the spacetime, and
the regulated HRT surface for $A$,
${\gamma_{{}_{\text{HRT}}}}:={\gamma_{{}_{\text{HRT}}}}(A)$ (blue curve), is
anchored on ${\hat{\gamma}_{{}_{\text{HRT}}}}(AB)$. [Right]: In the Lorentzian
AdS spacetime, the future and past of the removed regions is corresponding
excised, so that the regulated spacetime $\mathcal{M}$ is bounded by the
ingoing null congruences from ${\hat{\gamma}_{{}_{\text{HRT}}}}(AB)$ shown.

Now, the key point that makes the EWCS an appealing regulator from our
viewpoint is that computing the EWCS is equivalent to simply applying the
usual HRT formula _within the spacetime defined by the $AB$ entanglement
wedge_, with its future and past boundaries playing the role of
$\mathcal{I}^{\pm}$. This is illustrated in figure 3. This entanglement wedge
is thus essentially acting as a multiboundary wormhole; the only novelty is
that the homology must be computed relative to the joint HRT surface
${\hat{\gamma}_{{}_{\text{HRT}}}}(AB)$. (Thus here
${\hat{\gamma}_{{}_{\text{HRT}}}}(AB)$ is playing a role similar to that of an
end-of-the-world brane, for which we would also use relative homology.) This
is the point of view we will take in most of this paper: the conformal
boundary $\mathcal{N}$ will by definition consist _only_ of $D(A)$ and $D(B)$,
and the bulk $\mathcal{M}$ _only_ of their joint entanglement wedge
$\hat{\mathcal{W}}(AB)$. The rest of the original boundary and bulk spacetime
will simply be discarded. However, in section 7.1, we will return to the
original spacetime (denoted $\hat{\mathcal{M}}$, with future/past boundaries
$\hat{\mathcal{I}}^{\pm}$), and explain how our constructions can be extended
to include it. In general, we will use hatted symbols to refer to constructs
defined in the original, unregulated spacetime, and unhatted symbols to refer
to constructs in the regulated spacetime.

Of course, we may be interested in calculating the entropy of more than one
boundary region. For example, we may want to compute the mutual information
$I(A:B):=S(A)+S(B)-S(AB)$ between disjoint regions $A$, $B$ of a common
boundary Cauchy slice $\Sigma$. If $A$, $B$ are separated (not touching), then
the mutual information is finite and regulator-independent, but to compute it
using the HRT formula we must first regulate the entropies appearing in the
definition. To get a meaningful answer, it is important to use the same
regulator for all three of those entropies. So we should choose a region
$C\subset(AB)^{c}$ that leaves a buffer between $AB$ and $C$. We then proceed
as above, letting $\mathcal{M}$ be the entanglement wedge
$\hat{\mathcal{W}}(ABC)$ and deleting the rest of the original spacetime. The
new conformal boundary is $\mathcal{N}=D(A)\cup D(B)\cup D(C)$, and the
(regulated) mutual information is $S(A:BC)+S(B:AC)-S(AB:C)$. More generally,
however many boundary regions $A,B,C,\ldots$ we are interested in, we assume
that all entangling surfaces have been buffered and deleted from the boundary
Cauchy slice, and the bulk replaced with the entanglement wedge of the
remainder. We will call this entanglement wedge the “regulated spacetime”.

The authors of Grado-White:2020wlb  studied a different kind of regulator for
the HRT formula, in which the Cauchy slices are anchored on a fixed surface
(spacelike codimension-2 submanifold) at finite distance, and the HRT surface
is anchored on a fixed codimension-3 submanifold on that surface. Our
formalism, described in the next subsection, can also accommodate this choice
of regulator.

###  2.2 Spacetime setup

As explained in the previous subsection, we assume that the boundary regions
we consider — or more precisely their causal domains $D(A)$, $D(B)$, etc. —
are composed of entire connected components of the boundary $\mathcal{N}$. The
bulk $\mathcal{M}$ is either a multiboundary wormhole or the regions’ joint
entanglement wedge within the original, unregulated spacetime. In the latter
case, the future and past boundaries of the bulk are null hypersurfaces
emanating from the joint HRT surface
$\gamma^{0}:={\hat{\gamma}_{{}_{\text{HRT}}}}(AB\cdots)$. In imposing the
homology condition on the HRT (or cross-section) surface for any of the
regions (or subset of them), we work in homology relative to $\gamma^{0}$,
which implies that the surface is allowed to end on $\gamma^{0}$.

We would also like to allow for the possible presence of a timelike boundary
at finite distance: an end-of-the-world brane that we call $\mathcal{I}^{0}$.
This brane is assumed not to carry entropy, so its area does not count in
computing the HRT entropy (or EWCS), and the HRT (or cross-section) surface is
allowed to end on it. This is reflected mathematically in the fact that when
we apply the homology condition we work in homology relative to
$\mathcal{I}^{0}$.

$\mathcal{I}^{0}$ and $\gamma^{0}$ play the same role: the HRT (or cross-
section) surface is allowed to end on them, and we work in homology relative
to them. The main difference is that $\mathcal{I}^{0}$ is codimension-1
whereas $\gamma^{0}$ is codimension-2. Notationally, it will be a significant
simplification not to have to deal separately with the two cases. For this
reason, we subsume $\gamma^{0}$ into $\mathcal{I}^{0}$, by imagining that
$\gamma^{0}$ has an infinitesimal extent in the time direction and is
therefore formally codimension-1. Similarly, we may want to employ the
regulator suggested in Grado-White:2020wlb , in which the conformal boundary
$\mathcal{N}$ is replaced by a (spacelike codimension-2) surface $n$ at finite
distance, which is divided into regions $A,B,\ldots$. Everything we do in this
paper goes through for that setup. Just like for $\gamma^{0}$, in order to
avoid dealing separately with that case, imagine that $n$ has an infinitesimal
extent in the time direction, making it formally codimension-1. We emphasize
that these maneuvers are purely for the purpose of decluttering our formulas.
The diligent reader is invited to follow all the steps of our derivations with
codimension-2 boundaries $\gamma^{0}$, $n$ in place of (or in addition to)
$\mathcal{I}^{0}$, $\mathcal{N}$.

With that formality out of the way, we now lay out in a slightly more
mathematically precise way our setting, assumptions, and terminology. We fix a
compact oriented manifold-with-boundary $\bar{\mathcal{M}}$ of dimension
$D\geq 3$.212121 With certain notational adjustments, our analysis applies to
the $D=2$ case as well, e.g. in the setting of Jackiw-Teitelboim gravity
Jackiw:1984je ; Teitelboim:1983ux . For this purpose, the dilaton field $\Phi$
plays the role of the “area”. A codimension-2 “surface” is then a set of
points $x_{i}$ and its effective area is $\sum_{i}\Phi(x_{i})$; the effective
flux of a 1-form $W$ over a hypersurface $H$ is $\int_{H}\Phi\,{*W}$ (compare
to (22)); the divergenceless condition is $d(\Phi{*W})=0$ (compare to (21));
and so on. An easy way to understand the $D=2$ case is by “dimensional
oxidation” to $D=3$: convert $\mathcal{M}$ into a 3-dimensional spacetime by
fibering over it a circle of circumference $\Phi$, with all fields and
geometric objects lifted to ones translationally invariant along the circle.
The interior $\mathcal{M}$ of $\bar{\mathcal{M}}$ has a Lorentzian metric $g$
and a time orientation. The causal structure of $(\mathcal{M},g)$ extends to a
causal structure on $\bar{\mathcal{M}}$. The boundary of $\bar{\mathcal{M}}$
is the union of the following compact codimension-1 submanifolds-with-
boundary, which are disjoint except along codimension-2 common boundaries:

  * •

$\mathcal{N}$ is timelike.

  * •

$\mathcal{I}^{-}$ is spacelike and/or null and is the past boundary; every
inextendible timelike curve in $\bar{\mathcal{M}}$ begins in
$\mathcal{I}^{-}$.

  * •

$\mathcal{I}^{+}$ is spacelike and/or null and is the future boundary; every
inextendible timelike curve in $\bar{\mathcal{M}}$ ends in $\mathcal{I}^{+}$.

  * •

$\mathcal{I}^{0}$ may be empty; otherwise it is timelike and the metric $g$ on
$\mathcal{M}$ extends to the interior of $\mathcal{I}^{0}$ (in other words
points in the interior of $\mathcal{I}^{0}$ are at finite distance from points
in $\mathcal{M}$).

$\mathcal{I}^{+}$ and $\mathcal{I}^{-}$ are disjoint (though they may both
emanate from the infinitesimally time-stretched surface $\gamma^{0}$). We
define

$\mathcal{I}:=\mathcal{I}^{+}\cup\mathcal{I}^{0}\cup\mathcal{I}^{-}\,.$ (9)

Figure 4:  Schematic summary diagram of spacetime setup. The boundary of the
spacetime $\bar{\mathcal{M}}$ consists of the conformal boundary
$\mathcal{N}=D(A)\cup D(B)$ (the right and left sides respectively); the
future boundary $\mathcal{I}^{+}$ (top faces); the past boundary
$\mathcal{I}^{-}$ (bottom faces); and possibly an end-of-the-world brane
$\mathcal{I}^{0}$ (front and back faces). $\mathcal{I}^{\pm}$ are spacelike
and/or null. $\mathcal{I}^{0}$ may be empty; otherwise it is timelike. If the
bulk $\mathcal{M}$ is the entanglement wedge of $AB$ within a larger
holographic spacetime, then $\mathcal{I}^{0}$ includes the HRT surface for
$AB$, thickened slightly in the time direction to make it codimension-1. The
(blue) hypersurface $\sigma$ is a (Cauchy) slice for $\bar{\mathcal{M}}$; its
intersection with $\mathcal{I}^{0}$ is denoted $\gamma^{0}_{\sigma}$ and its
intersection with $D(A)$ is denoted $A_{\sigma}$. The (yellow) hypersurface
$\tau$ is a time-sheet homologous to $D(A)$ (relative to
$\mathcal{I}=\mathcal{I}^{+}\cup\mathcal{I}^{-}\cup\mathcal{I}^{0}$). The
intersection $\gamma$ (red) of slice $\sigma$ and timesheet $\tau$ is a
codimension-2 surface.

A Cauchy slice for $\bar{\mathcal{M}}$ (which we will abbreviate as “bulk
slice”) is a hypersurface intersected exactly once by every inextendible
timelike curve, and that does not intersect $\mathcal{I}^{\pm}$. (Note that in
our definition a slice is achronal but not necessarily acausal, i.e. it is
allowed to have null regions.) We denote by $\mathscr{S}$ the set of all bulk
slices. We assume that $\bar{\mathcal{M}}$ is globally hyperbolic in the sense
that $\mathscr{S}$ is non-empty. Given $\sigma\in\mathscr{S}$, we define the
surface

$\gamma^{0}_{\sigma}:=\mathcal{I}^{0}\cap\sigma\,.$ (10)

We also assume that every inextendible timelike curve in $\mathcal{N}$ is
inextendible in $\bar{\mathcal{M}}$. This implies that, for any
$\sigma\in\mathscr{S}$, $\sigma\cap\mathcal{N}$ is a (Cauchy) slice for
$\mathcal{N}$, and therefore that $\mathcal{N}$ is also globally hyperbolic.
See figure 4 for a summary of this setup.

We will make use of the following lemma:

######  Lemma 2.1.

Any closed achronal subset of $\bar{\mathcal{M}}$ that does not intersect
$\mathcal{I}^{\pm}$ is contained in a bulk slice.

###### Proof.

Let $s$ be a closed achronal subset of $\bar{\mathcal{M}}$ that does not
intersect $\mathcal{I}^{\pm}$. Let $\sigma$ be a bulk slice and define the
following subset of $\bar{\mathcal{M}}$:

$R:=(J^{-}(\sigma)\cup J^{-}(s))\setminus I^{+}(s)\,.$ (11)

$R$ is a closed subset of $\bar{\mathcal{M}}$ that obeys $J^{-}(R)=R$. It is
disjoint from $\mathcal{I}^{+}$ and contains an open neighborhood of
$\mathcal{I}^{-}$. Therefore its future boundary $R\setminus I^{-}(R)$ is a
bulk slice. Furthermore the future boundary contains $s$. ∎

Given that $\mathcal{N}$ is globally hyperbolic, each connected component
$\mathcal{N}_{i}$ of $\mathcal{N}$ is globally hyperbolic as well. Fix a slice
$\Sigma_{i}$ for each $\mathcal{N}_{i}$; then $\Sigma:=\cup_{i}\Sigma_{i}$ is
a slice for $\mathcal{N}$. A _boundary region_ $A$ is defined as the union of
a set of the $\Sigma_{i}$; its boundary causal domain $D(A)$ is the union of
the corresponding $\mathcal{N}_{i}$.222222 Importantly, nothing we do will
depend on the choice of boundary slices $\Sigma_{i}$. In fact, there will be
no dependence of anything on $A$ except through $D(A)$. $A$ thus essentially
just serves as a label for $D(A)$ and associated objects and quantities.
Regions $A,B,\ldots$ are disjoint. With a few exceptions (namely in
subsections 4.3 and 6.3), we will consider only two regions $A$, $B$, so
$AB=\Sigma$. Given any $\sigma\in\mathscr{S}$, its intersection with $D(A)$,

$A_{\sigma}:=D(A)\cap\sigma\,,$ (12)

is a slice for $D(A)$.

We will also work with timelike counterparts to slices that we call _time-
sheets_. More precisely, a time-sheet $\tau$ is a piecewise-timelike
hypersurface in $\bar{\mathcal{M}}$ that does not intersect
$\mathcal{N}$.232323 In subsection (6.1), we will expand the definition of
time-sheets to allow null pieces. For most of the paper, however, it is more
convenient to restrict time-sheets to be everywhere timelike. The reason is
that we will be very interested in the intersection of a given time-sheet with
a given slice; if the time-sheet is timelike then we are guaranteed that they
intersect transversely, with a codimension-2 intersection. Otherwise one has
to consider separately the case where they coincide on a null hypersurface. We
define $\Gamma_{\tau}$ as the set of surfaces of the form $\tau\cap\sigma$ for
some $\sigma\in\mathscr{S}$. We denote by $\mathscr{T}$ (or $\mathscr{T}_{A}$
when considering more than two boundary regions) the set of time-sheets
homologous to $D(A)$ relative to $\mathcal{I}$. Note that such a time-sheet
may have “seams” where two timelike pieces meet on their common future or past
boundary.

In sections 3–5, we do not impose any equations of motion or energy conditions
on the metric on $\mathcal{M}$, or any boundary conditions (such
asymptotically AdS ones) beyond those given above. Starting in section 6, we
will make further assumptions about the spacetime, related to the existence
and properties of the HRT surface; these are laid out in subsection 6.1. Then
in section 7.1 we return to the full, original spacetime.

###  2.3 1-forms

Given that we work with a fixed metric on $\mathcal{M}$, a vector $W^{\mu}$ at
a point $x\in\mathcal{M}$ can be expressed equivalently in terms of the
covector or 1-form $W=W_{\mu}dx^{\mu}$ or the $(D-1)$-form $*W$. We will
mainly employ the 1-form notation, supplemented by liberal use of the Hodge
star.

####  2.3.1 Pointwise notions

Fix a point $x\in\mathcal{M}$ and let $T^{*}:=T^{*}_{x}$ be the cotangent
space at $x$. Given covectors $W,X\in T^{*}$, we define

$W\cdot X:=W_{\mu}X^{\mu}\,,\qquad W^{2}:=W\cdot
W\,,\qquad|W|:=\sqrt{|W^{2}|}\,.$ (13)

We say $W$ is future-directed causal if it evaluates non-negatively on any
future-directed causal vector (or equivalently if $W^{2}\leq 0$ and the time
component $W_{0}$ is non-negative, or equivalently if the dual vector
$W^{\mu}$ is _past_ -directed causal); similarly for future-directed timelike.
We define $\mathfrak{j}^{+}\subset T^{*}$ as the set of future-directed causal
covectors at $x$, and $\mathfrak{i}^{+}\subset T^{*}$ for the future-directed
timelike covectors.

##### Covector pairings:

We will often make use of two real pairings between covectors. The first is
the norm of the 2-form $W\wedge X$:242424 We are employing the convention
where the norm of the $p$-form $\omega$ is defined in terms of its components
as
$|\omega|:=\sqrt{|\omega_{\mu_{1}\cdots\mu_{p}}\omega^{\mu_{1}\cdots\mu_{p}}|/p!}$.

$|W\wedge X|=\begin{cases}|W\cdot X|\,,\quad&W^{2}=0\text{ or }X^{2}=0\\\
|W||X_{\perp}|\,,\quad&W^{2}\neq 0\\\ |W_{\perp}||X|\,,\quad&X^{2}\neq
0\end{cases}$ (14)

where

$X_{\perp}:=X-\frac{X\cdot W}{W^{2}}W\,,\qquad W_{\perp}:=W-\frac{X\cdot
W}{X^{2}}X$ (15)

are the projections of $X$ perpendicular to $W$ and vice versa. The following
lemma gives some important basic properties of this function. (Although the
function $|W\wedge X|$ is symmetric, in the lemma we treat its arguments
asymmetrically, for reasons that will become clear below.)

######  Lemma 2.2.

(a) $|W\wedge X|$ is continuous and homogeneous in each argument. (b) For
fixed $W\in\mathfrak{j}^{+}$, $|W\wedge X|$ is a convex function of $X$. (c)
For fixed spacelike or null $X$, $|W\wedge X|$ is concave in $W$ on
$\mathfrak{j}^{+}$.

###### Proof.

(a) Clear. (b) For $W$ null, $|W\wedge X|=|W\cdot X|$, which is clearly convex
in $X$. For $W\in\mathfrak{i}^{+}$, $|X_{\perp}|$ is convex, being the norm of
$X_{\perp}$ within the spacelike hyperplane orthogonal to $W$. (c) For $X$
null, $|W\cdot X|$ is linear (and therefore concave) in $W$ on
$\mathfrak{j}^{+}$. For $X$ spacelike, $|W\wedge X|=|X||W_{\perp}|$, and
$|W_{\perp}|$ is concave, being the norm on the future solid light-cone within
the Lorentzian hyperplane orthogonal to $X$.252525 To see the convex-concave
character heuristically, plot the norm function on the vertical axis and the
space(time) directions on the horizontal axes. Then the norm function looks
like an “upright” cone (which is convex) when pertaining to spacelike
covector, but a “sideways” cone (which is concave) when pertaining to a
timelike covector. The constant-norm level sets are the conic sections which
respectively give spheres and hyperboloids in the corresponding space(time)
sections.  ∎

Second, we will need a function on $\mathfrak{j}^{+}\times T^{*}$ that is
concave in the first argument and convex in the second. $|W\wedge X|$ almost
works, except that it is not concave in $W$ when $X$ is timelike. This flaw is
remedied by taking the concave hull262626 The concave hull (or envelope) of a
function is the smallest concave function greater than or equal to the given
one. The proof that $|W\mathclap{\hskip 6.75pt\cdot}{\wedge}X|$ is the concave
hull of $|W\wedge X|$ is given in the proof of lemma 2.3. with respect to $W$,
yielding the following pairing, which we call “wedgedot” and denote $|\
\mathclap{\hskip 6.75pt\cdot}{\wedge}\ |$:272727 Note that we define only the
scalar quantity $|W\mathclap{\hskip 6.75pt\cdot}{\wedge}X|$; unlike $W\wedge
X$ and $W\cdot X$, $W\mathclap{\hskip 5.75pt\cdot}{\wedge}X$ by itself has no
meaning.

$|W\mathclap{\hskip 6.75pt\cdot}{\wedge}X|:=\max\\{|W\cdot X|,|W\wedge
X|\\}=\begin{cases}|W\cdot X|\,,\quad&X^{2}\leq 0\\\ |W\wedge
X|\,,\quad&X^{2}\geq 0\end{cases}\,.$ (16)

(See lemmas 4.5 and 4.6 in subsection 4.4.1 for two further ways of writing
$|W\mathclap{\hskip 6.75pt\cdot}{\wedge}X|$.) While this function may appear
from its definition to be symmetric in its two arguments, for our purposes it
is crucial that its domain $\mathfrak{j}^{+}\times T^{*}$ is _not_ symmetric.
The following lemma spells out several properties of the wedgedot pairing that
will play important roles in this paper.

######  Lemma 2.3.

On its domain $\mathfrak{j}^{+}\times T^{*}$, the function $|W\mathclap{\hskip
6.75pt\cdot}{\wedge}X|$ is continuous and homogeneous in each argument;
concave in $W$ for fixed $X$; and convex in $X$ for fixed $W$. It is also the
unique concave-convex function on $\mathfrak{j}^{+}\times T^{*}$ that equals
$|W\wedge X|$ for $X^{2}\geq 0$.

###### Proof.

For fixed $W$, homogeneity, continuity, and convexity in $X$ follow from the
fact that $|W\cdot X|$ and $|W\wedge X|$ both have those properties (see lemma
2.2), and those properties are inherited by the maximum.

For fixed spacelike $X$, the continuity, homogeneity, and concavity in $W$ are
given in lemma 2.2. For fixed future- or past-directed causal $X$, the
function $|W\cdot X|$ is linear in $W$ (on $\mathfrak{j}^{+}$) and therefore
continuous, homogeneous, and concave.

For the uniqueness of the concave-convex extension of $|W\wedge X|$, note
first that $|W\wedge X|=|W\cdot X|$ when either $W$ or $X$ is null. When
$X^{2}\leq 0$, $|W\cdot X|$ is a linear function of $W$ on its domain
$\mathfrak{j}^{+}$; any concave function of $W$ that equals $|W\wedge X|$ when
$W$ is null is therefore greater than or equal to $|W\cdot X|$. (This fact
makes $|W\cdot X|$ the concave hull of $|W\wedge X|$ with respect to $W$ for
fixed timelike $X$. For fixed spacelike $X$, $|W\wedge X|$ is already concave
with respect to $W$ and is therefore its own concave hull.) Similarly, any
convex function of $X$ that equals $|W\wedge X|$ when $X$ is null is, for
$X^{2}\leq 0$, less than or equal to $|W\cdot X|$. ∎

$W$$X$

$X$$W_{0}$$W$

$W_{0}=X$$W_{0}$$W$

Figure 5: Illustration of the constraint (17) in the 2-dimensional case.
[Left]: The set of allowed $X$ for fixed $W\in\mathfrak{i}^{+}$ (purple
arrow). [Middle]: The set of allowed $W$ for fixed spacelike $X$ (cyan arrow);
here $W_{0}\in\mathfrak{i}^{+}$ is normal to $X$ and of equal magnitude.
[Right]: The set of allowed $W$ for fixed $X\in\mathfrak{i}^{+}$; here
$W_{0}=X$.

##### Covector pairs:

An important role throughout the paper will be played by the set of pairs of
covectors $(W,X)$ obeying

$W\pm X\in\mathfrak{j}^{+}\,.$ (17)

Such pairs form a convex subset of $T^{*}\times T^{*}$. To get some intuition
for this set, first fix $W$ (which is necessarily in $\mathfrak{j}^{+}$). If
$W$ is timelike, then (17) is equivalent to
$|X_{\perp}|+|X_{\parallel}|\leq|W|$, where $X_{\perp},X_{\parallel}$ are the
projections of $X$ orthogonal and parallel respectively to $W$; or, to say it
another way, $X$ is required to be in the “causal diamond” within $T^{*}$ with
vertices $\pm W$. (See figure 5, left panel.) For $W$ null, the diamond
degenerates to a null line segment, and $X$ must be a convex combination of
$\pm W$ (and therefore must also be null). On the other hand, if we fix $X$,
then $W$ is restricted to the “causal future” (within $T^{*}$) of a point
$W_{0}\in\mathfrak{j}^{+}$ defined as follows: for spacelike $X$, $W_{0}$ is
the (unique) covector in $\mathfrak{j}^{+}$ obeying $X\cdot W_{0}=0$,
$|W_{0}|=|X|$; for $\pm X\in\mathfrak{j}^{+}$, $W_{0}=\pm X$. (See figure 5,
center and right panels.)

Further characterizations of the set of covector pairs obeying (17) are given
in subsection 4.4.1. It is also shown there that this set is closely related
to the wedgedot pairing defined above. Specifically, (17) is equivalent to
either of the following conditions:

$\forall\,Y\in\mathfrak{j}^{+}\,,\quad|Y\mathclap{\hskip
6.75pt\cdot}{\wedge}X|\leq-Y\cdot W$ (18) $W\in\mathfrak{j}^{+}\text{ and
}\forall\,Y\in T^{*}\,,\quad X\cdot Y\leq|W\mathclap{\hskip
6.75pt\cdot}{\wedge}Y|\,.$ (19)

Conversely, the wedgedot can be derived from the condition (17), in two
different ways. For any $W\in\mathfrak{j}^{+}$ and $X\in T^{*}$,

$|W\mathclap{\hskip 6.75pt\cdot}{\wedge}X|=\inf_{\begin{subarray}{c}Y:\\\ Y\pm
X\in\mathfrak{j}^{+}\end{subarray}}(-W\cdot Y)=\sup_{\begin{subarray}{c}Y:\\\
W\pm Y\in\mathfrak{j}^{+}\end{subarray}}Y\cdot X\,.$ (20)

This is another way to see that $|W\mathclap{\hskip 6.75pt\cdot}{\wedge}X|$ is
concave in $W$ and convex in $X$: $-W\cdot Y$ is concave in $W$ and the
pointwise infimum of a set of concave functions is concave; similarly, $Y\cdot
X$ is convex in $X$ and the pointwise supremum of a set of convex functions is
convex.

####  2.3.2 1-form fields

The vector field $W^{\mu}$ is divergenceless if and only if $*W$ is closed:

$\nabla_{\mu}W^{\mu}=0\quad\Longleftrightarrow\quad d{*W}=0\,.$ (21)

We will sometimes use the shorthand “$W$ is divergenceless” (strictly speaking
“co-closed” would be more correct). The flux of $W^{\mu}$ through a spacelike
or timelike hypersurface $H$ can also be expressed in terms of $*W$,

$\int_{H}\sqrt{h}\,N\cdot W=\int_{H}*W\,,$ (22)

where $h$ is the determinant of the induced metric on $H$ and $N$ is a unit
normal covector field, and similarly for the no-flux condition,

$N\cdot W|_{H}=0\quad\Longleftrightarrow\quad*W|_{H}=0\,,$ (23)

where $*W|_{H}$ is the pull-back of $*W$ onto $H$. The formulas in terms of
$*W$ are slightly more general than the ones in terms of $N\cdot W$, since
they apply even when $H$ is null. Even more generally, we will be using these
formulas on the boundaries $\mathcal{N}$, $\mathcal{I}^{\pm}$ of
$\bar{\mathcal{M}}$ where the metric, and therefore the Hodge star, are not
defined; implicitly, one is using the limiting value of $*W$ as the boundary
is approached. A more careful notation would instead use a $(D-1)$-form
$\tilde{W}$ on $\bar{\mathcal{M}}$ as the underlying variable, so that in
(21), (22), (23) we would have $d\tilde{W}$, $\int_{H}\tilde{W}$,
$\tilde{W}|_{H}$ respectively; and then define the 1-form $W:=*\tilde{W}$ on
$\mathcal{M}$ where the metric is defined.282828 If one is interested in
applying the V-flow and U-flow formulas discussed in section 4 in a setting
where the metric is not fixed a priori, such as metric reconstruction or
deriving the Einstein equation from the HRT formula, then it is probably
better to use the $(D-1)$-forms $\tilde{V}=*V$, $\tilde{U}=*U$, rather than
the 1-forms $V$, $U$ themselves, as the fundamental quantities defining a
flow.  One may think of $\tilde{W}$ as the collection of integral curves of
$W^{\mu}$, namely a 1-dimensional oriented structure which can be integrated
over a hypersurface to get a number, and in this sense $\tilde{W}$ naturally
captures the original notion of the bit threads. Nonetheless, at the cost of a
slight notational sloppiness we will stick to the more convenient 1-form
notation.

For the boundary of $\bar{\mathcal{M}}$, we will use an orientation in which
the normal covector $N$ is _inward_ -directed. This is slightly non-standard
but simplifies the homology relations (e.g. it makes the entanglement horizon
$\mathcal{H}(A)$ homologous to $D(A)$). In particular, $N$ is past-directed on
$\mathcal{I}^{+}$ and future-directed on $\mathcal{I}^{-}$. Hence the flux of
a future-directed 1-form is positive through $\mathcal{I}^{+}$ and negative
through $\mathcal{I}^{-}$. With this convention, Stokes’ theorem takes the
following form:

$\int_{\mathcal{M}}d{*W}+\int_{\mathcal{I}}*W+\int_{\mathcal{N}}*W=0\,.$ (24)

On a slice $\sigma$, the normal covector is chosen to be past-directed, so
that (as on $\mathcal{I}^{+}$) a future-directed covector has a positive flux.
In terms of homology, these orientations imply the following relations:

$\mathcal{I}^{+}\sim\sigma\sim-\mathcal{I}^{-}\text{ rel
}(\mathcal{I}^{0}\cup\mathcal{N})$ (25)

(the spacetime homology region between $\sigma$ and $\mathcal{I}^{\pm}$ being
$I^{\pm}(\sigma)$).

We will use a similar notation for 1-forms on a given hypersurface, but we
will denote them with lower-case letters $v,u$. The Hodge star and exterior
derivative applied to these forms are always the ones defined on the
hypersurface, not in the ambient space.

##  3 Games & relaxation

We start by recalling the maximin formula Wall:2012uf :

$S_{-}(A:B):=\sup_{\sigma\in\mathscr{S}}\
\operatorname*{inf\vphantom{p}}_{\gamma\in\Gamma_{\sigma}}\operatorname{area}(\gamma)\,,$
(26)

where $\Gamma_{\sigma}$ is the set of surfaces $\gamma$ in $\sigma$ homologous
to $A_{\sigma}$ relative to $\gamma^{0}_{\sigma}:=\mathcal{I}^{0}\cap\sigma$,
that do not intersect $\mathcal{N}$. The reason for the subscript on $S_{-}$
will become clear shortly. In this section we will derive a number of
variations on (26). In subsection 3.1, we will rewrite it in two ways: first,
by using the Riemannian max flow-min cut theorem, in terms of a flow localized
on a slice; and then in terms of the intersection of a slice and a time-sheet.
By switching the order of the maximization and minimization, this will then
allow us to obtain a “minimax” formula. The relation between such maximin-
minimax pairs of formulas is the subject of minimax theory, which is closely
related to game theory and which we briefly review in subsection 3.2. This
will lead us in subsection 3.3 to convex relax the two formulas, yielding a
third one, whose value sits between them. This convex-concave formula will be
the starting point for our derivation of flow formulas in section 4.

The maximin quantity is symmetric, $S_{-}(B:A)=S_{-}(A:B)$, and the same holds
for all of the quantities we derive in this section; this is hopefully clear
by inspection. Furthermore, since throughout this section we fix $A,B$, we
will from now on simply write $S_{-}$ for $S_{-}(A:B)$.

Throughout this section, as well as sections 4 and 5, we assume only the basic
structure for the bulk and boundary spacetimes described in subsection 2.2
(essentially global hyperbolicity), not any particular boundary conditions or
energy conditions.

###  3.1 Variations on maximin

Our first alternative formula for $S_{-}$ is obtained by applying the
Riemannian max flow-min cut theorem to replace, within each slice $\sigma$,
the minimization over surfaces by a maximization over flows.292929 Strictly
speaking, this situation does not quite fit the assumptions of the RMFMC
theorem proved in Headrick:2017ucz , which applies to compact Riemannian
manifolds. While $\sigma$ is compact, the metric on it does not extend to
$\mathcal{N}$ and may include null pieces. The first issue can be dealt with
by removing a neighborhood of $\mathcal{N}\cap\sigma$. The second issue can be
dealt with either by considering $\sigma$ as a limit of spacelike manifolds,
or by treating the null locus following the treatment of null manifolds in
Headrick:2017ucz  (but for max flow-min cut rather than min flow-max cut as in
that paper). Specifically, on the null locus there is a unique $(D-2)$-form
$\omega$ such that the area of any surface $m$ equals $|\int_{\gamma}\omega|$.
A flow is defined not by the 1-form $v$ but by the $(D-2)$-form $*v$, and the
constraint $|v|\leq 1$ is replaced by $*v=\alpha\omega$, where $|\alpha|\leq
1$. A 1-form $v$ on $\sigma$ is called a _$\sigma$ -flow_ if it has the
following properties:

$|v|\leq 1\,,\qquad d{*v}=0\,,\qquad*v|_{\gamma^{0}_{\sigma}}=0$ (27)

(where $d$ is the exterior derivative on $\sigma$, and $|\cdot|$ and $*$ are
defined with respect to the induced metric). We call the set of $\sigma$-flows
$\mathscr{F}_{\sigma}$. The RMFMC theorem (see Headrick:2017ucz  and
references therein) states that

$\inf_{\gamma\in\Gamma_{\sigma}}\operatorname{area}(\gamma)=\sup_{v\in\mathscr{F}_{\sigma}}\
\int_{A_{\sigma}}*v\,.$ (28)

$\mathscr{F}_{\sigma}$ is a convex set, and the objective functional
$\int_{A_{\sigma}}*v$ is linear; hence the right-hand side of (28) defines a
convex program.303030 Recall that a convex program is defined as the problem
of minimizing a convex function $f_{0}$ over a convex subset $X$ of an affine
space, subject to constraints $f_{i}(x)\leq 0$, $g_{i}(x)=0$, where the
$f_{i}$ are convex functions and the $g_{i}$ are affine functions on $X$. The
constraints defining $X$ are _implicit_ , while the constraints $f_{i}(x)\leq
0$, $g_{i}(x)=0$ are _explicit_. Thus, strictly speaking, the right-hand side
of (28) defines a convex program only after one has decided whether each of
the constraints (27) is implicit or explicit. Using (28), we can write $S_{-}$
in terms of a “maximax” formula:

$S_{-}=\sup_{\sigma\in\mathscr{S}}\ \sup_{v\in\mathscr{F}_{\sigma}}\
\int_{A_{\sigma}}*v\,.$ (29)

Alternatively, in order to put the space and time variations in (26) on an
equal footing, we can change the minimization domain so that it does not
depend on the maximization variable $\sigma$. This can be done by thinking of
the surface $\gamma$ as the intersection of the slice $\sigma$ with a time-
sheet $\tau\in\mathscr{T}$ (where $\mathscr{T}$ is the set of time-sheets
homologous to $D(A)$ relative to $\mathcal{I}$), as justified by the following
lemma:

######  Lemma 3.1.

Fix $\sigma\in\mathscr{S}$. For any $\tau\in\mathscr{T}$,
$\sigma\cap\tau\in\Gamma_{\sigma}$. Conversely, for any
$\gamma\in\Gamma_{\sigma}$, there exists a $\tau\in\mathscr{T}$ such that
$\sigma\cap\tau=\gamma$.

###### Proof.

Given $\tau\in\mathscr{T}$, $\sigma\cap\tau$ is homologous to $A_{\sigma}$ on
$\sigma$ (relative to $\gamma^{0}_{\sigma}$) via the intersection of $\sigma$
with the spacetime homology region between $D(A)$ and $\tau$; and it does not
intersect $\mathcal{N}$ (since $\tau$ does not intersect $\mathcal{N}$). Hence
it is in $\Gamma_{\sigma}$.

For the converse, let $t^{\mu}$ be a continuous future-directed timelike
vector field on $\bar{\mathcal{M}}$ that is tangent to $\mathcal{N}$ and
$\mathcal{I}^{0}$.313131 It is clear that such a vector field exists on a
sufficiently small patch of $\bar{\mathcal{M}}$. The following construction
shows that there is no global obstruction to its existence: start from an
atlas of charts for $\bar{\mathcal{M}}$, define such a vector field on each
chart, and average them on the overlaps using a partition of unity.
Alternatively, the standard argument for the existence of a globally defined
timelike vector field on a Lorentzian manifold, using an auxiliary Riemannian
metric (see e.g. p. 39 of Hawking:1973uf ), can be upgraded in the presence of
a boundary to ensure that the vector field is tangent to the timelike parts of
the boundary. The integral curves of $t^{\mu}$ pass through every point of
$\bar{\mathcal{M}}$, and each curve starts on $\mathcal{I}^{-}$, ends on
$\mathcal{I}^{+}$, and lies entirely in $\mathcal{M}$ (except its endpoints),
in $\mathcal{N}$, or in $\mathcal{I}^{0}$. Given $\gamma\in\Gamma_{\sigma}$,
let $r_{\sigma}$ be the homology region on $\sigma$ between $A_{\sigma}$ and
$\gamma$. The surface $\gamma$ can be extended into a time-sheet $\tau$ by
following the integral curves of $t^{\mu}$ in both directions. $\tau$ is
homologous to $D(A)$ (relative to $\mathcal{I}$) via the corresponding
extension of $r_{\sigma}$. ∎

With this lemma in hand, we can replace the minimization over $\gamma$ in (26)
with a minimization over $\tau$:

$S_{-}=\sup_{\sigma\in\mathscr{S}}\
\operatorname*{inf\vphantom{p}}_{\tau\in\mathscr{T}}\operatorname{area}(\sigma\cap\tau)\,.$
(30)

The formula (30) invites us to switch the order of the minimization and
maximization. We therefore also define the following “minimax” quantity:323232
We use the symbol $S$ here (and below, where we define a third quantity
$S_{\rm c}$) to emphasize the parallel to the maximin formula for holographic
EE. However, in this setting, which is much more general than standard
holographic spacetimes and where (as we will see) the three quantities
$S_{-}$, $S_{+}$, and $S_{\rm c}$ are not necessarily equal, we make no claim
for any of them being an entropy.,333333 Swapping the order of the
minimization and maximization in the maximin formula was also considered in
Agon:2019qgh .

$S_{+}:=\operatorname*{inf\vphantom{p}}_{\tau\in\mathscr{T}}\
\sup_{\sigma\in\mathscr{S}}\operatorname{area}(\sigma\cap\tau)\,.$ (31)

Recalling that $\Gamma_{\tau}$ is the set of surfaces $\gamma=\tau\cap\sigma$
for some $\sigma\in\mathscr{S}$, we can write this

$S_{+}=\operatorname*{inf\vphantom{p}}_{\tau\in\mathscr{T}}\
\sup_{\gamma\in\Gamma_{\tau}}\operatorname{area}(\gamma)\,,$ (32)

expressing the minimax quantity in a form analogous to (26).

By lemma 2.1, any achronal surface $\gamma^{\prime}$ (that is closed as a
subset of $\bar{\mathcal{M}}$ and does not intersect $\mathcal{I}^{\pm}$) is
contained in a slice $\sigma$. If $\gamma^{\prime}$ is contained in the time-
sheet $\tau$ then
$\gamma^{\prime}\subseteq\gamma:=\sigma\cap\tau\in\Gamma_{\tau}$ and
$\operatorname{area}(\gamma^{\prime})\leq\operatorname{area}(\gamma)$, so we
can replace the maximization in (32) with one over achronal surfaces:343434
The reader may wonder whether, by analogy with the maximax formula (29),
$S_{+}$ can be written in terms of a “minimin” formula. Indeed, one may be
tempted to apply the Lorentzian max cut-min flow theorem Headrick:2017ucz  to
the supremum in (32) in order to obtain a formula involving minimizing the
flux of a timelike flow on a time-sheet (a timelike flow being a covector
field $u$ obeying $u\in\mathfrak{i}^{+}$, $|u|\geq 1$, $d{*u}=0$,
$*u|_{\mathcal{I}^{0}\cap\tau}=0$). This would work if the supremum in (33)
were over surfaces $\gamma$ that are achronal _within_ the time-sheet $\tau$;
whereas it is only over surfaces that are achronal in the ambient spacetime
$\mathcal{M}$, a stronger condition. Nonetheless, in subsection 4.2.1, we will
define a “time-sheet-flow”, which is closely related to a Lorentzian flow
living on a time-sheet and gives a sort of minimin formula for $S_{+}$.

$S_{+}=\operatorname*{inf\vphantom{p}}_{\tau\in\mathscr{T}}\
\sup_{\begin{subarray}{c}\gamma\subset\tau\\\
\text{achronal}\end{subarray}}\operatorname{area}(\gamma)\,.$ (33)

###  3.2 Minimax theory

In order to better understand the relation between the two quantities $S_{-}$
and $S_{+}$ defined in the previous subsection, we now make a short digression
into minimax theory. This theory353535 Minimax theory, which was born with J.
von Neumann’s seminal work on game theory, continues to be an active area of
research in analysis, with applications to economics and many other fields.
For an overview, see du2013minimax . addresses the following question: Given
sets $X,Y$ and a function $f:X\times Y\to\mathbf{R}$, what can we say about
the relation between the maximin value $\sup_{x\in
X}\operatorname*{inf\vphantom{p}}_{y\in Y}f(x,y)$ and the minimax value
$\operatorname*{inf\vphantom{p}}_{y\in Y}\sup_{x\in X}f(x,y)$? We start with
three elementary general facts.

  * •

First, the maximin and minimax values are not necessarily equal. A simple
counterexample is given by setting $X=Y=\\{1,2\\}$ and $f(x,y)=(-1)^{x+y}$;
then

$\sup_{x\in X}\operatorname*{inf\vphantom{p}}_{y\in
Y}f(x,y)=-1\,,\qquad\operatorname*{inf\vphantom{p}}_{y\in Y}\sup_{x\in
X}f(x,y)=1\,.$ (34)

  * •

The maximin and minimax nonetheless do obey a relation. Clearly, for any
$x_{0}\in X$, $y_{0}\in Y$,

$\inf_{y\in Y}f(x_{0},y)\leq f(x_{0},y_{0})\leq\sup_{x\in X}f(x,y_{0})\,.$
(35)

Maximizing the left-hand side over $x_{0}$ and minimizing the right-hand side
over $y_{0}$ yields the _min-max inequality_ :

$\sup_{x\in X}\operatorname*{inf\vphantom{p}}_{y\in
Y}f(x,y)\leq\operatorname*{inf\vphantom{p}}_{y\in Y}\sup_{x\in X}f(x,y)\,.$
(36)

The min-max inequality can be understood in simple game-theory terms. Let $X$,
$Y$ represent the set of possible strategies for the two players respectively
in a single-round, zero-sum game, with payout $f(x,y)$ for player $X$ and
$-f(x,y)$ for player $Y$. Then the maximin is the best outcome for player $X$
if she plays first, while the minimax is her best outcome if she plays second
(assuming player $Y$ in each case is choosing his best strategy). The
inequality (36) expresses the fact that, in such a game, it is often better —
and never worse — to play second, allowing one to use knowledge of the other
player’s move to one’s advantage. Consider the example of (34), which
corresponds to the children’s game of evens-and-odds; normally in this game
the two players play simultaneously, for the simple reason that otherwise the
second player would always be able to win.

In the setting of Lagrange duality, the min-max inequality is responsible for
the weak duality property. With $f$ the Lagrangian function, the dual pair of
programs consists of maximizing $\inf_{y\in Y}f(x,y)$ and minimizing
$\sup_{x\in X}f(x,y)$. By the min-max inequality, the optimal value of the
maximization problem is bounded above by the optimal value of the minimization
problem.

  * •

Finally, if there exists a pair $(x_{0},y_{0})$ that saturates both
inequalities in (35), then both the minimax and maximin equal
$f(x_{0},y_{0})$, so the min-max inequality is saturated:

$\sup_{x\in X}\operatorname*{inf\vphantom{p}}_{y\in
Y}f(x,y)=f(x_{0},y_{0})=\operatorname*{inf\vphantom{p}}_{y\in Y}\sup_{x\in
X}f(x,y)\,.$ (37)

Such a point is called a _global saddle point_.

A minimax theorem is a theorem giving sufficient conditions on $X$, $Y$, and
$f$ for the maximin to equal the minimax. The first such theorem was von
Neumann’s, which applies to a zero-sum game with _mixed strategies_. A mixed
strategy for player $X$ is a vector of non-negative numbers $x_{i}$
($i=1,\ldots,n$) summing to 1, which can be thought of as a probability
distribution over a set of $n$ pure strategies; thus $X$ is the unit simplex
in $\mathbf{R}^{n}$. Similarly for player $Y$. Given a payout matrix $A_{ij}$
for the pure strategies, we assume that the payout function is the expectation
value of $A_{ij}$ over the joint distribution $x_{i}y_{j}$:

$f(x,y)=x_{i}A_{ij}y_{j}\,.$ (38)

According to von Neumann’s theorem, for such a game, the minimax equals the
maximin, hence the second-player advantage we saw before is erased. For
example, in the evens-and-odds game, player $X$’s best strategy if she plays
first is to choose equal weights for the two pure strategies,
$x_{1}=x_{2}=1/2$, leading to a payout $f=0$ regardless of player $Y$’s
strategy. The same holds for player $Y$, making it irrelevant who plays first.
(This is a simple example of a Nash equilibrium.)

In the case just discussed, the mixed-strategy maximin (and minimax) value 0
sits between the pure-strategy maximin $-1$ and minimax 1. This holds
generally:

$\max_{i}\min_{j}A_{ij}\leq\sup_{x\in X}\operatorname*{inf\vphantom{p}}_{y\in
Y}f(x,y)=\operatorname*{inf\vphantom{p}}_{y\in Y}\sup_{x\in
X}f(x,y)\leq\min_{j}\max_{i}A_{ij}\,.$ (39)

Let us prove the first inequality. Consider a particular pure strategy
$\hat{\imath}$ for player $X$. One possible mixed strategy $\hat{x}\in X$ is
to put all the weight on $\hat{\imath}$:
$\hat{x}_{i}:=\delta_{i\hat{\imath}}$. In that case

$\inf_{y\in Y}f(\hat{x},y)=\inf_{y\in
Y}A_{\hat{\imath}j}y_{j}=\min_{j}A_{\hat{\imath}j}\,,$ (40)

so $\sup_{x\in X}\inf_{y\in Y}f(x,y)\geq\min_{j}A_{\hat{\imath}j}$. This holds
for any $\hat{\imath}$, implying the first inequality.

Von Neumann’s theorem can be generalized to the case where $X$ and $Y$ are
convex subsets of affine spaces, at least one of them is compact, and $f$ is
concave-convex, i.e. concave in $x$ for fixed $y$ and convex in $y$ for fixed
$x$. There are many results extending this theorem in various ways, including
to the infinite-dimensional case, with additional technical assumptions; see
for example pjm/1103040253  and the review Simons1995 . As is our custom
throughout this paper, we will proceed rather naively, as far as the
functional analysis is concerned, when applying ideas from convex optimization
to function spaces.

###  3.3 Convex-concave formula

Figure 6:  Penrose diagram of a spacetime in which $S_{-}\neq S_{+}$. The
spacetime consists of a $(D-2)$-sphere fibered over a rectangular portion of
2-dimensional Minkowski space. The left and right sides of the rectangle are
$D(A)$ and $D(B)$ respectively, while the top and bottom are
$\mathcal{I}^{\pm}$ respectively. ($\mathcal{I}^{0}$ is empty in this
example.) The area of the sphere varies on the rectangle, being equal to
$a_{-}$ on the yellow portion and $a_{+}>a_{-}$ on the purple stripe. The
edges of the purple stripe are moving at a constant speed $\beta<1$, and
furthermore the point $x_{B}$ is timelike-related to $x_{A}$ ($x_{B}\in
I^{+}(x_{A})$) implying that no slice lies entirely within the purple stripe.
Since every slice $\sigma$ passes through the $a_{-}$ region,
$\inf_{\gamma\in\Gamma_{\sigma}}\operatorname{area}(\gamma)\leq a_{-}$;
furthermore, this inequality is saturated by spherically-symmetric slices, so
$S_{-}=a_{-}$. On the other hand, every time-sheet $\tau$ passes through the
$a_{+}$ region, so
$\sup_{\gamma\in\Gamma_{\tau}}\operatorname{area}(\gamma)\geq a_{+}$;
furthermore, this inequality is saturated by any time-sheet that is
spherically symmetric and does not have any seams, so $S_{+}=a_{+}$. As
expected, $S_{-}\leq S_{+}$. Note that this spacetime does not obey the null
energy condition, as can be seen from the fact that the area of a spherical
null congruence starting in the $a_{-}$ region initially has vanishing
expansion, yet, if it enters the $a_{+}$ region, its area increases. (Observe
however that, if we replaced the $a_{-}$ in the lower portion of the figure by
an area $a_{++}>a_{+}$, the maximin and minimax values would coincide and both
equal $a_{++}$.)

We now return to the setting at hand, and the maximin and minimax formulas
(30), (31) respectively. From the min-max inequality (36) we have

$S_{-}\leq S_{+}$ (41)

(thereby explaining the subscripts). Are there spacetimes where the two
quantities are unequal? As we will see, this depends on the assumptions one
makes about the spacetime. In the very general framework we use in this
section, as set out in subsection 2.2 (essentially just global hyperbolicity),
one can indeed construct spacetimes with a gap between the maximin and
minimax; an example is shown in figure 6. On the other hand, in section 6, we
will use Wall’s results Wall:2012uf  to show that, under standard assumptions
about holographic spacetimes (specifically, the null energy condition and AdS
boundary conditions), the HRT surface is a global saddle point, and therefore
its area equals both $S_{-}$ and $S_{+}$.

For the next few sections, we will instead close the gap between the maximin
and minimax by following von Neumann’s method of allowing “mixed strategies”,
in other words by convex-relaxing the domains $\mathscr{S}$ of slices and
$\mathscr{T}$ of time-sheets. The standard method for convex-relaxing such
sets of hypersurfaces is to introduce a scalar function obeying certain
boundary conditions, whose level sets represent a “smeared” hypersurface. (For
an overview and details, see Headrick:2017ucz .) Specifically, a convex
combination of slices is represented by a function $\phi$ on
$\bar{\mathcal{M}}$:363636 Note that the restriction of $\phi$ to lie in the
interval $[-1/2,1/2]$ is automatic, given the other constraints on $\phi$ and
the fact that every point in $\bar{\mathcal{M}}$ lies on a causal curve
starting in $\mathcal{I}^{+}$ and ending in $\mathcal{I}^{-}$.

$\mathscr{S}_{\rm
c}:=\bigg{\\{}\phi:\bar{\mathcal{M}}\to\left[-\frac{1}{2},\frac{1}{2}\right]\quad\left|\quad\phi|_{\mathcal{I}^{\pm}}=\pm\frac{1}{2}\,,\,d\phi\in\mathfrak{j}^{+}\right\\}.$
(42)

Given $\phi\in\mathscr{S}_{\rm c}$, for any $t\in(-1/2,1/2)$, the level set
$\sigma_{t}$, defined as the future boundary of the region where $\phi\leq t$,
is a slice. A single slice $\sigma$ is represented by the step function with
$\phi=\pm 1/2$ on $I^{\pm}(\sigma)$. (If $\sigma$ touches $\mathcal{I}^{\pm}$,
then imagine deforming it infinitesimally away from those boundaries.)
Similarly, a convex combination of time-sheets is represented by a function
$\psi$:

$\tilde{\mathscr{T}}_{\rm
c}:=\bigg{\\{}\psi:\bar{\mathcal{M}}\to\left[-\frac{1}{2},\frac{1}{2}\right]\quad\left|\quad\psi|_{D(A)}=-\frac{1}{2}\,,\,\psi|_{D(B)}=\frac{1}{2}\,,\,d\psi\text{
spacelike or $0$}\right\\}.$ (43)

Given $\psi\in\tilde{\mathscr{T}}_{\rm c}$, for any $s\in(-1/2,1/2)$, the
level set $\tau_{s}$ of $\psi$ is a time-sheet in $\mathscr{T}$. A single
time-sheet $\tau\in\mathscr{T}$ would be represented by the step function with
$\psi=-1/2$ on the spacetime homology region between $D(A)$ and $\tau$, and
$\psi=1/2$ on the complement. Given $\phi\in\mathscr{S}_{\rm c}$,
$\psi\in\tilde{\mathscr{T}}_{\rm c}$, the area of the intersection of the
respective level sets $\sigma_{t}$, $\tau_{s}$, averaged over $t$ and $s$,
can, by a generalization of the coarea formula, be expressed as an integral
over $\mathcal{M}$:

$f[\phi,\psi]:=\int_{-1/2}^{1/2}dt\int_{-1/2}^{1/2}ds\,\operatorname{area}(\sigma_{t}\cap\tau_{s})=\int_{\mathcal{M}}\sqrt{g}\,|d\phi\wedge
d\psi|\qquad(\phi\in\mathscr{S}_{\rm c},\psi\in\tilde{\mathscr{T}}_{\rm
c})\,.$ (44)

So far, so good. However, our job of convex relaxing is not finished: while
$\mathscr{S}_{\rm c}$ is a convex set, $\tilde{\mathscr{T}}_{\rm c}$ is not,
since the spacelike or vanishing covectors do not form a convex subset of the
cotangent space at a point on $\bar{\mathcal{M}}$. The convex hull of this
subset is the entire cotangent space, leading us to simply drop the condition
on $d\psi$, and define the relaxed time-sheet set as follows:373737 The
restriction of $\psi$ to lie in the interval $[-1/2,1/2]$ in the definition of
$\mathscr{T}_{\rm c}$ is actually optional. Dropping it simply leads to extra,
superfluous level sets, and doesn’t change the optimal value of any of the
functionals we consider. See subsection 4.4.2 for details.

$\mathscr{T}_{\rm
c}:=\bigg{\\{}\psi:\bar{\mathcal{M}}\to\left[-\frac{1}{2},\frac{1}{2}\right]\quad\left|\quad\psi|_{D(A)}=-\frac{1}{2}\,,\,\psi|_{D(B)}=\frac{1}{2}\right\\}.$
(45)

The level sets of $\psi\in\mathscr{T}_{\rm c}$ are still hypersurfaces
homologous to $D(A)$, but they are no longer necessarily everywhere timelike.
We must also correspondingly extend the definition of the objective functional
$f$ to $\mathscr{S}_{\rm c}\times\mathscr{T}_{\rm c}$. Recall that we want the
objective to be concave with respect to $\phi$ and convex with respect to
$\psi$. Luckily, it is possible to extend the definition (44) to this larger
domain while satisfying this condition. In fact, as shown in lemma 2.3, the
extension is unique, and given by the wedgedot pairing:

$f[\phi,\psi]:=\int_{\mathcal{M}}\sqrt{g}\,|d\phi\mathclap{\hskip
6.75pt\cdot}{\wedge}d\psi|\qquad(\phi\in\mathscr{S}_{\rm
c},\psi\in\mathscr{T}_{\rm c})\,.$ (46)

where $|W\mathclap{\hskip 6.75pt\cdot}{\wedge}X|:=\max\\{|W\wedge X|,|W\cdot
X|\\}$. (See lemma 2.3 for more properties of this function, which will play a
starring role throughout this paper.) This amounts to adjusting the area
functional in regions where the $\psi$ level set $\tau_{s}$ is not timelike:

$f[\phi,\psi]=\int_{-1/2}^{1/2}dt\int_{-1/2}^{1/2}ds\,\operatorname{area}^{\prime}(\sigma_{t},\tau_{s})\,,$
(47)

where

$\operatorname{area}^{\prime}(\sigma,\tau):=\int_{\sigma\cap\tau}\sqrt{h}\times\begin{cases}1\,&\text{($\tau$
timelike or null at $\sigma$)}\\\ \coth\chi\,&\text{($\tau$ spacelike at
$\sigma$)}\end{cases}\,,$ (48)

$h$ is the determinant of the metric on $\sigma\cap\tau$, and $\chi$ is the
dihedral boost angle between $\sigma$ and $\tau$ (in other words, the dot
product of their future-directed unit normals equals $-\cosh\chi$).

Since $f$ is concave-convex, we expect its maximin and minimax values to
agree,383838 Rigorously proving the equality of the minimax and maximin in
this setting is outside the scope of this paper. However, we can make a few
very crude remarks in this direction. Minimax theorems valid for infinite-
dimensional spaces, such as Sion’s theorem pjm/1103040253 , typically require,
in addition to the objective function being convex-concave (or some
generalization thereof), a continuity assumption on the objective as well as a
compactness assumption on the domain of at least one of the variables. Since
the integrand $|d\phi\mathclap{\hskip 6.75pt\cdot}{\wedge}d\psi|$ in the
definition of the functional $f$ is a continuous function of $d\phi$ and
$d\psi$, we would expect $f$ to be continuous with suitable mathematically
precise definitions of the function spaces $\mathscr{S}_{\rm c}$,
$\mathscr{T}_{\rm c}$ and topologies thereon. Furthermore, we would expect
those function spaces to be compact, since the functions’ domain
$\bar{\mathcal{M}}$ and range $[-1/2,1/2]$ are both compact. and we call this
value $S_{\rm c}$ (or $S_{\rm c}(A:B)$ if we need to make explicit the
dependence on the boundary regions):

$S_{\rm c}:=\sup_{\phi\in\mathscr{S}_{\rm c}}\
\operatorname*{inf\vphantom{p}}_{\psi\in\mathscr{T}_{\rm c}}\
f[\phi,\psi]=\operatorname*{inf\vphantom{p}}_{\psi\in\mathscr{T}_{\rm c}}\
\sup_{\phi\in\mathscr{S}_{\rm c}}\ f[\phi,\psi]\,.$ (49)

Furthermore, this value must sit between the non-convex maximin and minimax
values, the analogue of (39) in the game-theory setting:

$S_{-}\leq S_{\rm c}\leq S_{+}\,.$ (50)

We can prove (50) by the same method. For the first inequality, choose a slice
$\hat{\sigma}$, and let $\hat{\phi}$ be the corresponding step function,
$\hat{\phi}:=\pm 1/2$ on $I^{\pm}(\hat{\sigma})$. Then, for any
$\psi\in\mathscr{T}_{\rm c}$,

$f[\hat{\phi},\psi]=\int_{-1/2}^{1/2}ds\,\operatorname{area}^{\prime}(\hat{\sigma},\tau_{s})\geq\int_{-1/2}^{1/2}ds\,\operatorname{area}(\hat{\sigma}\cap\tau_{s})\geq\inf_{\gamma\in\Gamma_{\hat{\sigma}}}\operatorname{area}(\gamma)\,.$
(51)

In fact, these inequalities are tight, since, starting from the minimal
surface $\hat{\gamma}$ on $\hat{\sigma}$, by lemma 3.1, one can construct a
time-sheet $\hat{\tau}\in\mathscr{T}$ such that
$\hat{\sigma}\cap\hat{\tau}=\hat{\gamma}$, and from there set $\psi$ equal to
the corresponding step function. Hence

$\inf_{\psi\in\mathscr{T}_{\rm
c}}f[\hat{\phi},\psi]=\inf_{\gamma\in\Gamma_{\hat{\sigma}}}\operatorname{area}(\gamma)\,,$
(52)

which, by the definition of $S_{\rm c}$, implies

$S_{\rm
c}\geq\inf_{\gamma\in\Gamma_{\hat{\sigma}}}\operatorname{area}(\gamma)\,.$
(53)

Maximizing the right-hand side over $\hat{\sigma}\in\mathscr{S}$ gives the
first inequality in (50). The second inequality is proven the same way, using
the second formula for $S_{\rm c}$ in (49).

For a spacetime with a gap between $S_{-}$ and $S_{+}$, such as the one shown
in figure 6, $S_{\rm c}$ must obviously differ from at least one of them. So
we see that, in this case, convex relaxation is not an innocent operation:
smearing the slices and time-sheets can change the optimal value of the
objective. This is in contrast to the simpler Riemannian min cut and
Lorentzian max cut programs, where the convex relaxation does not change the
optimal value of the objective. In the next section, we will see in detail how
this works for the spacetime of figure 6. But first we will rewrite the
minimax and maximin formulas for $S_{\rm c}$ as pure minimization and
maximization programs, respectively.

##  4 Flows

In this section, we will derive, by Lagrange dualization starting from the
convex-concave formula (49), two new formulas for the quantity $S_{\rm c}$.
The first, in subsection 4.1, involves dualizing on $\psi$ for fixed $\phi$ to
obtain what we call the _V-flow_ program, while the second, in subsection 4.2,
involves dualizing on $\phi$ for fixed $\psi$ to obtain the _U-flow_ program.
In subsection 4.3, we consider the case where the boundary has been divided
into more than two regions, and show that $S_{\rm c}$ obeys the subadditivity
inequality. To avoid interrupting the narrative, most of the derivations and
proofs are relegated to subsection 4.4.

As in the previous section, we assume the spacetime setup laid out in
subsection 2.2. Except in subsection 4.3, we fix regions $A$, $B$ covering a
boundary slice, so we continue to suppress the dependence of them, e.g.
writing $S_{\rm c}$ rather than $S_{\rm c}(A:B)$.

###  4.1 V-flows

In the first formula for $S_{\rm c}$ in (49), the minimization over $\psi$,
for fixed $\phi$, defines a convex program. It is therefore natural to apply
Lagrange duality to this program to obtain a concave (maximization) program.
This dualization, which is carried out in subsection 4.4.2, is similar to the
dualization in the Riemannian setting of the relaxed min cut program to obtain
the max flow program (see Headrick:2017ucz ). However, this case is slightly
more complicated because the integrand $|d\phi\mathclap{\hskip
6.75pt\cdot}{\wedge}d\psi|$ in the objective functional (46) is more
complicated than the relaxed min cut objective, which is simply $|d\psi|$.
Nonetheless, the main features of the Riemannian max flow program remain: we
are maximizing the flux of a divergenceless 1-form (which we call $V$) subject
to a norm bound and a no-flux boundary condition on the relative-homology
boundary (in this case $\mathcal{I}$). However, the norm bound on $V$, which
in the Riemannian case is simply $|v|\leq 1$, is more complicated here, and
involves the 1-form $d\phi$. All in all, we have393939 It is possible to
dualize the program (54)–(56) on $\phi$, yielding a maximin formula in terms
of two 1-forms $U$ and $V$. However, this formula does not tell us anything
new. See subsection 4.4.2 for details.

$S_{\rm c}=\sup_{V\in\mathscr{F}}\int_{D(A)}*V\,,$ (54)

where $\mathscr{F}$ is the set of V-flows, and a V-flow is a 1-form $V$
obeying

$d{*V}=0\,,\qquad*V|_{\mathcal{I}}=0$ (55) $\exists\,\phi\in\mathscr{S}_{\rm
c}\text{ s.t. }d\phi\pm V\in\mathfrak{j}^{+}\,.$ (56)

We could have defined a V-flow as a pair $(V,\phi)$ obeying the constraints
(55), (56). However, we have chosen to treat $\phi$ instead as an auxiliary
variable whose job it is to enforce the norm bound. Indeed, as we will show in
subsection 4.1.3, it is possible to rewrite (56) without reference to $\phi$.

To understand the norm bound (56), it is important to understand the pointwise
condition ${d\phi\pm V\in\mathfrak{j}^{+}}$. Subsection 2.3.1 (see around
figure 5) discussed in detail properties of the set of covector pairs $(W,X)$
obeying the condition $W\pm X\in\mathfrak{j}^{+}$; we repeat the most
important points here in terms of $d\phi$ and $V$. First, this condition
implies that $d\phi\in\mathfrak{j}^{+}$ (as we already have from the
definition (42) of $S_{\rm c}$). For timelike $d\phi$, this condition is
equivalent to $|V_{\perp}|+|V_{\parallel}|\leq|d\phi|$, where
$V_{\perp},V_{\parallel}$ are the projections of $V$ orthogonal and parallel
respectively to $d\phi$; or, to say it another way, $V$ is required to be in
the “causal diamond” in the cotangent space with vertices $\pm d\phi$. Note
that this constraint allows $V$ to be timelike, spacelike, or null. For
$d\phi$ null, the diamond degenerates to a null line segment, and $V$ must be
a convex combination of $\pm d\phi$. In general, the larger $d\phi$ is at a
given point, the larger $V$ may be, and therefore the more flux can pass
through the point. However, given the constraints $d\phi\in\mathfrak{j}^{+}$,
$\phi|_{\mathcal{I}^{\pm}}=\pm 1/2$ in the definition of $\mathscr{S}_{\rm
c}$, $d\phi$ cannot be arbitrarily large everywhere in $\mathcal{M}$, which is
ultimately what limits the total flux of $V$ that can pass through the
spacetime from $D(A)$ to $D(B)$.

The symmetry of $S_{\rm c}$, $S_{\rm c}(B:A)=S_{\rm c}(A:B)$, can be
understood easily in the language of V-flows. First note that the constraints
(55), (56) defining the V-flows do not refer specifically to the region $A$;
the set $\mathscr{F}$ of V-flows is thus the same for the complementary region
$B$ (or, as we will discuss in subsection 4.3, any other boundary region for
the same regulated spacetime). Second, those constraints are invariant under
$V\to-V$, so if $V\in\mathscr{F}$ then $-V\in\mathscr{F}$. Finally, the
constraints (55) guarantee that the objectives match:
$\int_{D(A)}*V=\int_{D(B)}*(-V)$. Hence, for any max V-flow $V$ for $A$, $-V$
is a max V-flow for $B$ with the same value of the objective, and vice versa,
so the two programs must have equal maxima.

####  4.1.1 Slice-flows

Recall from (27) that, given an (everywhere spacelike) slice $\sigma$, a
$\sigma$-flow is a Riemannian flow with respect to the induced metric on
$\sigma$. A particular kind of V-flow is a _slice-flow_ , obtained from a
$\sigma$-flow $v$ on some slice $\sigma$ by setting

$\phi|_{I^{\pm}(\sigma)}=\pm\frac{1}{2}\,,\qquad
V=\delta(x^{0})v_{a}dx^{a}\,,$ (57)

where $\\{x^{0},x^{a}\\}$ is a set of Gaussian normal coordinates about
$\sigma$. Conversely, any V-flow such that $\phi=\pm 1/2$ on $I^{\pm}(\sigma)$
for some slice $\sigma$ necessarily takes the form
$V=\delta(x^{0})v_{a}dx^{a}$, where $v$ is a $\sigma$-flow. In fact, the full
set $\mathscr{F}$ of V-flows can be obtained by convex-relaxing the set of
slice-flows.404040 More precisly, $\mathscr{F}$ is the _pointwise_ convex hull
of the set of slice-flows. A slice-flow obeys, at each point in $\mathcal{M}$,
the conditions $|V|\leq|d\phi|$, $V\cdot d\phi=0$. The convex hull (in
$T^{*}\times T^{*}$) of the set of covector pairs $(W,V)$ obeying
$|V|\leq|W|$, $W\cdot V=0$ is the set obeying $W\pm V\in\mathfrak{j}^{+}$; see
lemma 4.1 in subsection 4.4.1. Since for any slice-flow $V$,
$\int_{D(A)}*V=\int_{A_{\sigma}}*v$, we have

$S_{-}=\sup_{V\text{ slice-flow}}\int_{D(A)}*V\,.$ (58)

As discussed at the end of section 3, since $S_{-}$ may be strictly less than
$S_{\rm c}$, this is a case where convex relaxation can lead to a higher
maximum.

####  4.1.2 Worked example

As an instructive example of the effect of convexity, consider the spacetime
of figure 6. The maximum flux of any slice-flow is $S_{-}=a_{-}$. However,
with a smeared slice we can do better. We will give a simple (although still
not optimal) construction.

To be concrete, let the metric on the spacetime be

$ds^{2}=-(dx^{0})^{2}+(dx^{1})^{2}+r(x^{0},x^{1})^{2}\,d\Omega_{D-2}^{2}\,,\qquad-\frac{T}{2}\leq
x^{0}\leq\frac{T}{2}\,,\qquad-\frac{L}{2}\leq x^{1}\leq\frac{L}{2}\,,$ (59)

with $T,L$ the dimensions of the rectangle, and let $T^{\prime}$ be the time
duration (at fixed $x^{1}$) of the stripe where the area of the sphere is
$a_{+}$ (the purple stripe in figure 6), and $\beta$ is the speed at which its
edges are moving. For the stripe to fit fully within the rectangle, we have

$T>\frac{L}{\beta}+T^{\prime}\,.$ (60)

Furthermore, the relation $x_{B}\in I^{+}(x_{A})$ implies

$L<\frac{L}{\beta}-T^{\prime}\,.$ (61)

Figure 7:  The V-flow $\hat{V}=V_{\equiv}+V_{{}_{\\!/\\!/}}$ on the spacetime
of figure 6. The horizontal lines represent the component $V_{\equiv}$ (62),
while the diagonal lines in the stripe represent the component
$V_{{}_{\\!/\\!/}}$ (63).

Convexity (or concavity) of an optimization problem guarantees that the
optimal configuration can be assumed to share any symmetries of the problem.
In what follows, we will use this fact to restrict our attention to
spherically symmetric configurations.

The function $\hat{\phi}=x^{0}/T$ represents a slice that is extended in the
$x^{1}$ and sphere directions and smeared uniformly in the $x^{0}$ direction.
The following 1-form obeys the constraints (55), (56) and has total flux
$a_{-}$:

$V_{\equiv}=\begin{cases}\frac{1}{T}\,dx^{1}\quad&(a_{-}\text{ region})\\\
\frac{a_{-}}{Ta_{+}}\,dx^{1}\quad&(a_{+}\text{ region})\end{cases}\,.$ (62)

Notice that this configuration saturates the constraint $d\phi\pm
V\in\mathfrak{j}^{+}$ in the $a_{-}$ region, but not in the $a_{+}$ region.
This gives us the opportunity to send additional flux through the $a_{+}$
stripe. Specifically, the following ansatz, with $\alpha$ an arbitrary
constant, represents a divergenceless 1-form that is uniform within the
stripe:

$V_{{}_{\\!/\\!/}}=\begin{cases}0\quad&(a_{-}\text{ region})\\\
\alpha\,(-dx^{0}+\beta dx^{1})\quad&(a_{+}\text{ region})\end{cases}\,.$ (63)

$V_{{}_{\\!/\\!/}}$ is divergenceless because the dual vector
$V_{{}_{\\!/\\!/}}^{\mu}$ is parallel to those edges. The flux of
$V_{{}_{\\!/\\!/}}$ is $\alpha\beta T^{\prime}a_{+}$. In order to maximize the
total flux of $\hat{V}=V_{\equiv}+V_{{}_{\\!/\\!/}}$, we should maximize
$\alpha$, subject to the constraints
$d\hat{\phi}\pm\hat{V}\in\mathfrak{j}^{+}$. In fact,
$d\hat{\phi}-\hat{V}\in\mathfrak{j}^{+}$ for any positive $\alpha$, so the
only constraint we have to worry about is
$d\hat{\phi}+\hat{V}\in\mathfrak{j}^{+}$, which requires

$\frac{a_{-}}{Ta_{+}}+\alpha\beta\leq\frac{1}{T}-\alpha\,.$ (64)

The maximum value of $\alpha$ subject to (64) is

$\alpha=\left(1-\frac{a_{-}}{a_{+}}\right)\frac{1}{T}\frac{1}{\beta+1}\,.$
(65)

All in all, the total flux for this configuration, illustrated in figure 7, is

$\int_{D(A)}*\hat{V}=a_{-}+\frac{T^{\prime}}{T}\frac{\beta}{\beta+1}(a_{+}-a_{-})\,.$
(66)

In particular, the flux lines of $\hat{V}$ are horizontal outside the stripe
but refract to have positive slope through the stripe, with the size of the
effect depending on the parameters: they remain spacelike when
$\frac{a_{-}}{a_{+}}>\frac{1-\beta}{2}$ but actually become timelike for
$\frac{a_{-}}{a_{+}}<\frac{1-\beta}{2}$.

Further study shows that this simple V-flow can be improved in various ways
(cf. appendix B) and in fact we don’t know what the maximal flux is for this
spacetime. The important point, however, is that convex relaxation has allowed
us to increase the flux beyond the non-convex value $a_{-}$. Note also that,
since $T^{\prime}<T$, the flux of $\hat{V}$ is less than $a_{+}$, as must be
the case since $S_{\rm c}\leq S_{+}=a_{+}$.

####  4.1.3 Bounds

To gain more intuition about V-flows, we will now derive several inequalities
involving $V$ from the constraints (55), (56). These inequalities have the
advantage that they do not involve $\phi$; however, they are non-local.

##### World-line bound:

First, in subsection 4.4.1 (see lemma 4.3), we show that, at any point in
$\mathcal{M}$ and for any covector $T\in\mathfrak{j}^{+}$, the condition
$d\phi\pm V\in\mathfrak{j}^{+}$ implies

$|T\mathclap{\hskip 6.75pt\cdot}{\wedge}V|\leq-T\cdot d\phi\,.$ (67)

Let $\mathpzc{q}$ be an inextendible causal curve parametrized by $t$, and
$\dot{x}$ the covector dual to the tangent vector $dx^{\mu}/dt$ of
$\mathpzc{q}$. (Recall that, with the tangent vector future-directed, the dual
covector is past-directed.) Then, setting $T=-\dot{x}$ in (67), integrating
over $t$, and using the fact that $\phi|_{\mathcal{I}^{\pm}}=\pm 1/2$ so
$\int{q}dt(\dot{x}\cdot d\phi)=1$, we have

$\int{q}dt\,|(-\dot{x})\mathclap{\hskip 6.75pt\cdot}{\wedge}V|\leq 1\,.$ (68)

(Note that, since the pairing $|\ \mathclap{\hskip 6.75pt\cdot}{\wedge}\ |$ is
homogeneous in the first argument, the left-hand side of (68) is independent
of the parametrization. Note also that, since the integrand is non-negative
and any extendible causal curve is part of an inextendible one, the bound (68)
applies also to extendible causal curves.)

If we specialize to the case where $\mathpzc{q}$ is everywhere timelike and
$t$ is a proper-time parameter (so $|\dot{x}|=1$), then, from (14) and (16),

$|V_{\perp}|\leq|(-\dot{x})\mathclap{\hskip 6.75pt\cdot}{\wedge}V|\,,$ (69)

(where $V_{\perp}$ is the projection of $V$ orthogonal to $\dot{x}$), so (68)
implies

$\int{q}dt\,|V_{\perp}|\leq 1\,.$ (70)

If furthermore $V$ is everywhere spacelike or null, then (69) is saturated, so
(70) is actually equivalent to (68). The situation is slightly more
complicated if $V$ is timelike in some region; then (68) is equivalent to (70)
applied to an almost-null timelike curve $\mathpzc{q}^{\prime}$ that zig-zags
close to $\mathpzc{q}$.414141 This is essentially a consequence of the fact
that $|(-\dot{x})\mathclap{\hskip 6.75pt\cdot}{\wedge}V|$ is linear in
$\dot{x}$ when both arguments are timelike, and approaches $|V_{\perp}|$ when