# Report on GRG18, Session A3
Mathematical Studies of the Field Equations

Lars Andersson∗<EMAIL_ADDRESS>Albert Einstein Institute, Am Mühlenberg 1,
D-14476 Potsdam, Germany and Department of Mathematics, University of Miami,
Coral Gables, FL 33124, USA

###### Abstract.

In this report I will give a summary of some of the main topics covered in
Session A3, mathematical studies of the field equations, at GRG18, Sydney.
Unfortunately, due to length constraints, some of the topics covered at the
session will be very briefly mentioned or left out altogether. The summary is
mainly based on extended abstracts submitted by the speakers and some of those
who presented posters at the session. I would like to thank all participants
for their contributions and help with this summary.

∗ Supported in part by the NSF, under contract no. DMS 0407732.

##  1\. The Buchdahl inequality

The Buchdahl inequality, which is included in most textbooks on general
relativity, states that for a static, self-gravitating body,

(1) $2M/R\leq 8/9,$

where $M$ is the ADM mass and $R$ the area radius of the boundary of the
static body. The proof by Buchdahl, cf. [16] of (1) assumed that the energy
density is non-increasing outwards and that the pressure is isotropic. A bound
on $2M/R$ has an immediate observational consequence: if $2M/R<8/9$ then the
gravitational red shift is less than 2 but if $2M/R$ approaches 1 the red
shift is unbounded. The assumptions used to derive the inequality are very
restrictive, and as e.g. pointed out by Guven and Ó Murchhada [27] neither of
them hold in a simple soap bubble and they do not approximate any known
topologically stable field configuration.

In addition to the restrictions implied by the hypotheses made by Buchdahl
there are two other disadvantages associated with this inequality: it refers
to the boundary of the body (i.e. the interior is excluded) and the solution
which gives equality in the inequality is the Schwarzschild interior solution
which has constant energy density and for which the pressure blows up at the
centre. In particular it violates the dominant energy condition.

Håkan Andréasson explained recent work [2], where all four restrictions
described above are eliminated. He considered matter models for which the
energy density $\rho$ and the radial pressure $p$ is non-negative and which
satisfies $p+2p_{T}\leq\Omega\rho,$ where $\Omega$ is a non-negative constant
and $p_{T}$ is the tangential pressure, and showed that

(2) $\sup_{r>0}\frac{2m(r)}{r}\leq\frac{(1+2\Omega)^{2}-1}{(1+2\Omega)^{2}}.$

Here $m$ is the quasi-local mass so that $M=m(R).$ The case $\Omega=1$ gives
the bound $2m/r\leq 8/9$. Among the matter models which satisfy the conditions
stated are Vlasov matter and matter which satisfies the dominant energy
condition and has non-negative pressure. Inequality (2) is sharp in the sense
of measures and there are examples which come arbitrarily close to saturating
the inequality. Andréasson has recently generalized the inequality to the case
of charged spheres [3].

##  2\. Initial-boundary value problems

A long standing problem in the analytical and numerical study of the Einstein
equations is the choice of boundary conditions for boundaries near “infinity”.
As is well known, the Einstein equations in harmonic coordinates reduces to a
quasilinear system of wave equations. It was proved in previous work by Kreiss
and Winicour cf. [36] that the initial-boundary value problem is well posed
for such systems with the Sommerfeld outgoing boundary condition. The proof
relied on pseudo-differential operator techniques.

Oscar Reula reported on a recent proof [35] of this fact which makes use only
of partial integrations. The proof is much more transparent than the previous
proof, and the argument based on partial integration is well adapted to the
analysis of discretizations of the wave equation based on difference operators
which have good “partial summation” properties.

Let $(M,g_{ab})$ be a spacetime with timelike boundary $\mathcal{T}$. Let $M$
be foliated by spacelike surfaces $\Sigma_{t}$, with timelike unit normal
$n^{a}$. Consider the initial boundary value problem for the wave equation
$g^{ab}\nabla_{a}\nabla_{b}\phi=F$ with boundary condition of the form
$(T^{b}+aN^{b})\nabla_{b}\phi\big{|}_{\mathcal{T}}=q$, $a>0$, where $T^{a}$ is
a timelike unit vector field which is tangent to $\mathcal{T}$ and $N^{a}$ is
the outward pointing normal to the boundary
$\partial\Sigma_{t}=\Sigma_{t}\cap\mathcal{T}$ in $\Sigma_{t}$. The Sommerfeld
condition is given by the choice $a=1$, which corresponds to a condition on
the derivative of $\phi$ in an outgoing null direction. The proof proceeds by
considering the fluxes of energy defined with respect to a vector field
$u^{a}=T^{a}+\delta N^{a}$, where $\delta>0$ is a parameter to be chosen. Let
$\Theta^{a}{}_{b}=\nabla^{a}\phi\nabla_{b}\phi-\frac{1}{2}\delta^{a}{}_{b}\nabla_{c}\phi\nabla^{c}\phi$
be the energy-momentum tensor for the scalar field. The energy $E$ and outward
flux $\mathcal{F}$ are defined by integration over $\Sigma_{t}$ and
$\partial\Sigma_{t}$, with respect to the densities
$u^{b}\Theta^{a}{}_{b}n_{a}$ and $u^{b}\Theta^{a}{}_{b}N_{a}$, respectively.
Using integration by parts one derives by a straightforward argument an energy
estimate bounding $\partial_{t}E$ in terms of $E$, $F$ and $\mathcal{F}$.
Expanding out the flux density $u^{b}\Theta^{a}{}_{b}N_{a}$ and using the
boundary condition to eliminate terms, one finds that for an appropriate
choice of $\delta=\delta(a)>0$, it contains a negative multiple of the square
gradient of $\phi$ on $\partial\Sigma$ plus terms which can be estimated in
terms of $E,F,\mathcal{F}$. Inserting this in the above mentioned energy
estimate yields a form of the energy estimate which gives the boundary
regularity required for well-posedness. Based on these ideas, the proof of
well-posedness for the initial boundary can now be completed along standard
lines.

##  3\. Moving punctures and black holes

During the time since GRG17, Dublin 2004, there have been tremendous advances
in numerical relativity, in particular in the binary black hole problem. The
first breakthrough was the work by Frans Pretorius, using a constraint
stabilized harmonic coordinate formulation, with excision. Using this code he
was during the spring of 2005 able to evolve stably several orbits of a binary
black hole system. This was followed shortly thereafter by announcements from
several groups using different methods, that they were able to achieve similar
results. One of the methods which has emerged as a successful alternative to
the harmonic formulation is based on the Baumgarte-Shapiro-Shibata-Nakamura
(BSSN) formulation of the Einstein evolution equations, cf. [49, 7], see also
[48]. Due to various issues arising from a combination of several factors
(gauge choice, excision etc.), codes based on BSSN initially suffered from
instabilities which prevented more than partial orbits to be calculated.
However when the proper techniques are applied one may successfully use the
BSSN formulation to stably evolve black hole spacetimes, even without
excision. The black holes (BHs) are thus included in the domain of
computation, and one effectively treats the BHs as essentially weak
singularities in the Cauchy slices. This approach is now one of the most
popular in the numerical GR community. In spite of the success in using this
formulation, the reason why it actually works has remained mysterious.

Sascha Husa discussed recent work, cf. [28, 15], which sheds light on this
issue. The moving puncture method starts with BHs represented by “filling”
their interiors with wormholes, reduced to singular (“puncture”) points by
spatial compactification. The moving-puncture method [17, 6] is only a minor
modification of previous approaches, in that no attempt is made to factor out
the singular puncture geometry during the simulation, but surprisingly in
numerical evolutions of a Schwarzschild BH using the BAM code [15], Husa et
al. found that this leads to a drastic change in the geometry of the time
slices [28]. They have constructed an analytic solution for the stationary
state of a nonspinning BH that matches the moving-puncture gauge, and found
that the numerical data asymptote to this solution.

The key result about the geometry of moving punctures is that for a
Schwarzschild black hole the numerical evolutions approach a stationary slice
that neither reaches an internal asymptotically flat end nor hits the physical
singularity, as might be expected for a stationary slice with non-negative
lapse. Rather, the slice ends at a throat at finite Schwarzschild radius
($R_{0}\approx 1.31M$), but infinite proper distance from the apparent
horizon. This changes the singularity structure of the “puncture”. It is still
a puncture in that there is a coordinate singularity at a single point in the
numerical coordinates, but it does not correspond to an asymptotically flat
end. In the course of Schwarzschild evolutions Husa et al. have found that the
throat does collapse to the origin. Where one would have expected an inner and
an outer horizon, they find only one zero in the norm of
$(\frac{\partial}{\partial t})^{a}$, corresponding to the outer horizon. An
under-resolved region does develop in the spacetime (it is the region between
the throat and the interior spacelike infinity), but we are pushed out of
causal contact with it. The throat itself has receded to infinite proper
distance from the outer horizon. Matter fields or gravitational radiation will
be trapped between the inner horizon and the throat, because unlike the gauge
their propagation is limited by the speed of light; Husa et al. will consider
this issue in future work.

##  4\. Isolated systems

The analysis of the asymptotic structure of asymptotically flat spacetimes is
of central importance in mathematical general relativity. The conformal
compactification setting introduced by Penrose allows one to give a stringent
analysis of the asymptotic properties of spacetimes, including conserved
quantities. Further, the weak cosmic censorship conjecture has a clear
formulation in this setting. The work of Klainerman et al. [18, 33] shows that
asymptotic fall-off conditions on Cauchy data can be introduced so that the
Cauchy development has a conformal compactification with any finite
regularity. It is known that there are large classes of spacetimes which have
a conformal compactification which is regular at spatial infinity. It is an
interesting open question whether the assumption of smooth null infinity
implies some type of rigidity for the spacetime [21, 19]. The regular
conformal field equations of Friedrich give a clear cut formulation of this
problem, and there is good hope of eventually finding sharp conditions for
regularity.

###  4.1. Stability of Minkowski space

The talk of Lydia Bieri addressed the global, nonlinear stability of solutions
of the Einstein equations in General Relativity. In particular, she discussed
results on the initial value problem for the Einstein vacuum equations, which
generalizes the results of Christodoulou and Klainerman [18]. Every strongly
asymptotically flat, maximal, initial data which is globally close to the
trivial data gives rise to a solution which is a complete spacetime tending to
the Minkowski spacetime at infinity along any geodesic. In [11], Bieri has
considered the Cauchy problem with more general, asymptotically flat initial
data. In particular, under the assumptions in [11], the spacetime curvature
fails to be continuous. In order to show decay of the spacetime curvature and
the corresponding geometrical quantities, the Einstein equations are
decomposed with respect to adequate foliations of the spacetime. In the proof
of Bieri, one encounters borderline estimates for certain quantities with
respect to decay, indicating that the conditions concerning decay at infinity
imposed on the initial data are sharp.

###  4.2. Rigidity for asymptotically simple spacetimes

Juan Antonio Valiente Kroon reported on progress [53] towards a proof of the
following conjecture concerning asymptotically simple spacetimes: If an
asymptotically Euclidean, conformally flat, time symmetric initial data set
for the Einstein vacuum equations gives rise to a development admitting a
smooth conformal compactification at null infinity, then the initial data must
be Schwarzschildean in a neighbourhood of infinity.  It should be noted that
the Schwarzschild spacetime is the only stationary spacetime admitting
conformally flat slices. There are further indications of generalisations of
this conjecture for more general classes of data, cf. [54, 55]. A possible
proof of the above conjecture requires a detailed understanding of the
structure of certain asymptotic expansions of the development of the initial
data near null and spatial infinity. The tools to obtain these expansions are
the “extended conformal Einstein equations” and the representation of spatial
infinity known as “the cylinder at spatial infinity” which have been
introduced by Friedrich in [25]. The asymptotic expansions are obtained by
solving, for a given initial data set, a hierarchy of interior equations at
the cylinder at spatial infinity. These interior equations allows to transport
initial data from the initial hypersurface up to the “critical sets” where
null infinity “touches” spatial infinity. The structure of the interior
equations suggests that their solutions should contain very specific types of
logarithmic divergences at the critical sets unless the initial data is
Schwarzschildean. The explicit existence of these obstructions has been shown
in [53].

##  5\. Singularities

###  5.1. Stochastic aspects of generic singularities

The proposal of Belinskiǐ, Khalatnikov and Lifshitz [9, 10, 32] (BKL) on the
structure of generic cosmological singularities states roughly that the
essential properties of the asymptotic dynamics of the gravitational field
along typical timelines can be understood by considering certain spatially
homogeneous models. Writing the Einstein equations in terms of Hubble
normalized scale invariant frame variables due to Uggla et al. [52, 1, 47] or
the approach of Damour et al. [22], based on Iwasawa decomposition, each of
which relies on a long history of previous formulations, gives a description
of the asymptotics of the gravitational field in terms of billiard dynamical
systems. In the case of the Hubble normalized formulation, one gets a billiard
in the Kasner plane, cf. [56], while for the Iwasawa formulation one gets for
the case of 3+1 gravity, a billiard in a domain of hyperbolic space, which is
analogous to the Misner-Chitre billiard.

Claes Uggla presented work [30], building on [52, 1, 47], which generalizes
and makes rigorous some aspects of the previous work of BKL and others on the
stochastic aspects of the system which arise due to the chaotic nature of the
asymptotic billiard systems. Using a combination of dynamical and statistical
analyses and in part heuristic arguments, this work describes the generic
properties of a “billiard attractor”. It turns out that several degrees of
freedom, which a priori could have been of relevance for the asymptotic
behavior are, generically, statistically suppressed. The dynamical and
statistical arguments of Uggla et al. may be contrasted with results
concerning asymptotic behavior in Bianchi type IX obtained using purely
dynamical arguments [46]. The results presented by Uggla suggest that the role
of Bianchi type IX as a model for the asymptotic dynamics of generic
singularities should be re-evaluated, in view of the fact that generic
singularities exhibit features that are not shared by Bianchi type IX.

###  5.2. Kinematic and Weyl singularities

For expanding Bianchi cosmologies with a tilted perfect fluid with linear
equation of state $p=(\gamma-1)\rho$, all timelike and null geodesics are
complete into the future. However, the fluid congruence may be incomplete into
the future (called a congruence singularity), accompanied by the blow-up of
kinematic quantities associated with the fluid congruence. Woei-Chet Lim
discussed recent work [20, 37] on the nature of such singularities. Much
emphasis has been placed on the blow-up of the components of the Weyl tensor
associated with the fluid congruence. However, as shown by examples due to Lim
et al., this phenomenon is independent of the incompleteness of the fluid
congruence. Hence it is necessary to differentiate congruence singularities
into kinematic singularities and Weyl singularities. In particular, the fluid
may become extremely titled or the kinematic or Weyl components may blow up,
depending on whether $\gamma$ exceeds certain critical values.

###  5.3. Perturbations of naked singularity spacetimes

The scalar field may be considered as a toy model for perturbations of a
background spacetime. Brien Nolan discussed some rigorous mathematical results
that probe the linear stability of naked singularities in self-similar
collapse. He showed that the multipoles of a minimally coupled massless scalar
field propagating on a spherically symmetric self-similar background space-
time admitting a naked singularity maintain finite $H^{1,2}$ norm as they
impinge on the Cauchy horizon. Further, each multipole obeys a point-wise
bound at the horizon, as does its locally observed energy density [40]. The
null energy condition is assumed to hold in the background spacetime. In order
to view the scalar field as a toy model for perturbations of a background
spacetime, the matter model must be specified. To study such perturbations,
the matter content must be specified. Nolan considers what in this context is
the simplest case: that of null dust - i.e. the stability of the Cauchy
horizon in self-similar Vaidya space-time is studied. The results for the
scalar field carry over to odd-parity linear perturbations at the level of
both the metric and the curvature [41]. For even parity perturbations, the
linearised Einstein equations can be reduced to a 5-dimensional first order
symmetric hyperbolic system. The components of the state vector are gauge
invariant metric and matter perturbation quantities. Nolan shows that the
$L^{2}$ norm of the state vector blows up at the Cauchy horizon and that
solutions for which the $L^{\infty}$ norm of the state vector does not blow up
at the Cauchy horizon are unstable in the space of test function initial data.
This provides strong evidence that the Cauchy horizon of the self-similar
Vaidya space-time is unstable [39].

###  5.4. Spacetime boundaries and properties of singularities

In his talk, Benjamin Whale reported on joint work with Sue Scott, where the
a-boundary and the a-boundary singularity theorem are applied to analyze the
physical properties of singularities.

##  6\. Quasi-local aspects

###  6.1. Towards the quasi-localization of canonical GR

Perhaps the most natural way of introducing the conserved quantities in
asymptotically flat spacetimes is the canonical/Hamiltonian one. A key
observation in the Hamiltonian formulation of GR, due to Arnowitt, Deser and
Misner [5], is that the evolution parts of Einstein’s equations can be
recovered formally as canonical equations of motion in the phase space, in
which the constraint function (i.e. whose vanishing is just the constraint
parts of Einstein’s equations) play the role of the Hamiltonian. However, as
Regge and Teitelboim stressed, if we want to recover the correct evolution
parts of Einstein’s equations as partial differential equations for smooth
fields rather than some distributional generalization of them, then the
Hamiltonian in the canonical equations of motion must be functionally
differentiable with respect to the canonical variables [45]. Regge and
Teitelboim showed that such a Hamiltonian can be obtained by adding an
appropriate 2-surface integral to the constraint functions. The remarkable
property of this Hamiltonian is that the ADM energy-momentum and angular
momentum can be recovered as the value of the Hamiltonian on the constraint
surface with appropriately chosen lapse and shift. On the other hand, the
investigations of Beig and Ó Murchadha showed that for given boundary
conditions on the canonical variables the asymptotic form of the lapse and the
shift is already determined if we require that the evolution equations
preserve the boundary conditions [8]. Moreover, they also showed that both the
constraint functions and the Hamiltonians close to Poisson algebras, the
former being an ideal in the latter, and their quotient, the so-called algebra
of observables, is isomorphic to the Poincare algebra. The ADM conserved
quantities then can be considered as appropriate coordinates on this algebra
of observables.

László Szabados considered in his talk the boundary conditions for canonical
vacuum GR at the quasi-local level, i.e. when the spacelike hypersurface on
which the canonical variables are defined is compact with smooth 2-boundary
${\mathcal{S}}$ [50] . The key ideas used by Szabados are found by analogy
with the Hamiltonian analysis in the asymptotically flat case above, namely
the requirement of the functional differentiability of all the functions whose
Poisson bracket must be calculated, and the requirement of the compatibility
of the evolution equations and the boundary conditions both on the canonical
variables and the lapse and the shift. It has been shown by Szabados [51] that
fixing the area element on the 2-surface ${\mathcal{S}}$ (rather than the
whole induced 2-metric) is enough to have a well defined constraint algebra, a
well defined Poisson algebra of basic Hamiltonians parameterized by lapses
that are vanishing on ${\mathcal{S}}$ and shifts that are tangent to and
divergence free on ${\mathcal{S}}$. Their quotient is a Lie algebra of a class
of 2-surface observables.

The evolution equations preserve these boundary conditions, and the
observables (realized as the value of the basic Hamiltonians on the constraint
surface) are 2+2–covariant, gauge-invariant and provide a representation of
the Lie algebra of the divergence-free vector fields on ${\mathcal{S}}$.
Szabados pointed out, that in special, standard situations the observables
appear to behave as angular momentum.

###  6.2. The Einstein scalar field at finite infinity

The ‘finite infinity’ paradigm of Ellis was proposed to study isolated
gravitational systems, in our universe, where we experience the presence of
other local matter fields and a cosmological background [24]. In this context
it is not possible to make the assumption of asymptotic flatness at infinity,
under which boundary conditions on matter have been studied in detail.
Instead, a smooth timelike surface $\mathcal{F}$, is introduced. The timelike
surface $\mathcal{F}$ should be located at a large finite distance from the
centre of the local system of interest, with the aim to study the fields
generated from this on $\mathcal{F}$.

William Clavering in his talk examined the behaviour of the spherically
symmetric Einstein scalar field at $\mathcal{F}$. He considered the evolution
of a scalar field over a domain bounded by an initial hypersurface,
characteristic curves, and $\mathcal{F}$. Using Hawking’s mass formula [29],
he has studied the consequences of imposing conditions on the mass-energy flux
at $\mathcal{F}$. Two examples were discussed; each a perturbation of a static
exact solution. For a perturbation of Schwarzschild there is no mass-energy
flux of the scalar field. Clavering conjectures that this is the case in all
vacuum spacetimes. The analysis has been repeated for the pure scalar field
case of the Wyman spacetime [57]. In this case, non-trivial expressions for
the mass-energy flux in terms of the scalar field at $\mathcal{F}$ were
obtained.

###  6.3. Quasi-local energy inside the event horizon

Pointlike objects cause many of the divergences that afflict physical
theories. For instance, the gravitational binding energy of a point particle
in Newtonian mechanics is infinite. In general relativity, the analog of a
point particle is a black hole and the notion of binding energy must be
replaced by quasilocal energy. The quasilocal energy (QLE) derived by York,
and elaborated by Brown and York [13], is finite outside the horizon but it
was not considered how to evaluate it inside the horizon.

Andrew Lundgren presented a prescription for finding the QLE inside a horizon,
and showed that it is finite at the singularity for a variety of types of
black hole. It turns out that the energy is typically concentrated just inside
the horizon, not at the central singularity. It is impossible to localize
gravitational energy due to the equivalence principle, so it is meaningless to
define gravitational energy at a single point. This problem is avoided by
considering the gravitational energy in a region of space. The boundary of the
region is a two-dimensional surface, and in fact the quasilocal energy is
defined only in terms of quantities defined on the enclosing surface
(specifically the induced metric and extrinsic curvature). The QLE is defined
only in terms of the gravitational degrees of freedom and makes no mention of
any other fields that are present. However, since gravity couples to
everything, the QLE counts all energy that is present in the region.

Lundgren et al. [38] have considered only spherically-symmetric static black
holes. The definition of the QLE in [13] was extended to be valid inside the
event horizon, which is only a coordinate singularity. In the Schwarzschild
case, the QLE at the central singularity is zero. The analogous quantity for a
classical field of a point particle diverges at the center. The gravitational
field itself carries energy which gravitates, causing the nonlinearity of
general relativity. Lundgren remarked that the nonlinearity somehow provides a
mechanism that cures the divergence expected of a point particle field. The
definition of the QLE presented by Lundgren can be applied also to the
deSitter case and the case with non-vanishing charge.

##  7\. Black holes

###  7.1. Black hole rigidity and spacetimes with compact Cauchy horizon

Jim Isenberg reported on recent work with Vince Moncrief which studies
analytic solutions of Einstein’s equations containing nondegenerate compact
Cauchy horizons with non closed generators. If certain hypotheses are added,
the spacetimes necessarily admit an isometry which acts along the generators
of the horizon. Isenberg showed how to use these results to prove that
stationary (non static) analytic black holes in arbitrary dimensions
necessarily admit symmetries which are independent of the presumed
stationarity symmetry. This work is closely related to work of Hollands et al.
[31].

###  7.2. Electromagnetic field on the background of high dimensional black
holes

In the last years various generalizations of black hole solutions to the high
dimensional gravity have been found. One of them describes a generally
rotating black hole with NUT charges. This solution possesses several nice
properties as, e.g., the existence of the Killing-Yano tensor, the complete
integrability of the geodesic motion or the separability of the Hamilton-
Jacobi and Klein-Gordon equations. Its generalization including the
acceleration of the black holes or the electromagnetic field is not, however,
straightforward.

In four dimensions it is possible to generalize the black hole solution to
describe the accelerated black holes (the Plebański-Demiński solution). This
transition is achieved by the conformal rescaling of the metric accompanied by
a modification of some metric functions. Pavel Krtous showed in his talk that
such a procedure is not sufficiently general in higher dimensions—only the
maximally symmetric spacetimes in ‘accelerated’ coordinates are obtained.
Further, he presented general algebraically special solutions of the Maxwell
equations on the background of the mentioned high dimensional generally
rotating black hole. They are adjusted to the algebraic structure of the
background which is reflected by the existence of the principal Killing-Yano
tensor. Such electromagnetic fields generalize the field of charged black
holes in four dimensions. However, one may show that in higher dimensions such
fields cannot be easily modified in such a way that they would satisfy full
Maxwell-Einstein equations.

##  8\. Post-Newtonian expansions

Post-Newtonian expansions (PNEs) are expansions of the Einstein equations in
the parameter $1/c$, around $c=\infty$, where $c$ is lightspeed. The limit
$c=\infty$ is the Newtonian limit of general relativity. Such expansions have
been studied since the discovery of general relativity and there is a large
literature available. Recently, post-Newtonian expansions have been used to
calculate wave forms for the gravitational wave emissions from binary black
hole inspirals. These have been compared with numerically calculated wave
forms and turn out to be highly accurate until the last stage of the inspiral.

The majority of results on PNEs are based on formal expansions which are used
to calculate approximate values of physical quantities, see eg. [23, 12] and
references therein. However, this formal approach does not deal with the
question of convergence. In the absence of a precise notion of convergence, it
becomes unclear to what extent the formal expansions actually approximate
relativistic solutions. In view of the importance of the applications of PNEs,
it is interesting to give a rigorous foundation to the procedure.

Todd Oliynyk discussed his recent work on post-Newtonian expansions for the
Einstein-Euler equations. Oliynyk considers expansions of solutions to these
equations in the parameter $\epsilon=v_{T}/c$ about $\epsilon=0$, where
$v_{T}$ is a characteristic velocity scale associated with the fluid matter.
By rescaling the gravitational and matter variables, the Einstein-Euler
equations can be written as

(3) $G^{ij}=2\kappa\epsilon^{4}T^{ij}\quad\text{and}\quad\nabla_{i}T^{ij}=0$

where $\kappa$ $=$ $4\pi G\rho_{T}/v_{T}^{2}$, $v_{i}v^{i}$ $=$
$-\epsilon^{-2}$, $\rho_{T}$ is a characteristic value for the fluid density,
and $t$ $=$ $x^{4}/v_{T}$ is a “Newtonian” time coordinate. In his talk,
Oliynyk presented results on PNEs for a class of polytropes, which go beyond
formal considerations [43, 44]. By introducing suitable renormalized variables
for which the limit $\epsilon\to 0$ makes sense, and introducing a suitable
gauge, Oliynyk derives a family of symmetric hyperbolic systems, depending on
the parameter $\epsilon$. This system is studied in a class of weighted
Sobolev spaces $H^{k}_{\delta,\epsilon}$, which interpolate between the
weighted spaces $H^{k}_{\delta}$ and the standard spaces $H^{k}$. Using these
spaces, it is possible to prove $\epsilon$-dependent energy estimates for
solutions to the Einstein-Euler equations. These estimates are then used to
prove the existence of convergent expansions in $\epsilon$ for suitably chosen
initial data. Oliynyk discussed the relation between his convergent expansions
and the formal PNEs. In order to recover the PNEs to a certain order requires
the initial data to be chosen correctly. The method used to choose initial
data is based on ideas in [14]. Oliynyk discussed how to recoved the post-
Newtonian expansion to $2^{\text{nd}}$ order in his framework.

##  9\. Miscellaneous

### Wave tails in curved spacetimes

Risto Tammelo reported on joint work with Laas and Mankinen dealing with the
wave tails of scalar and electromagnetic fields. Their work shows that the
intensity of the tail of the electromagnetic wave pulse emitted by a wave
source within a compact binary in the vicinity of the geometric shadow of the
source of gravitation can be of the same magnitude as the intensity of the
direct pulse. The energy carried away by the tail can amount to approximately
10% of the energy of the low-frequency modes of the direct pulse. As an
example of an exactly solvable model system, a scalar wave field on a
particular Robertson-Walker spacetime has been considered. In the case of
minimal coupling, if the metric describes the Friedman dust-dominated
universe, the higher-order multipole solutions do not contain a wave tail
term. A massless nonconformally-coupled scalar field is also considered in a
class of Robertson-Walker spacetimes. Tammelo et al. show that an initially
massless scalar field in the Robertson-Walker universe can obtain a mass in
the corresponding asymptotic Minkowski space region.

### Ricci flow and the Einstein equations

Eric Woolgar described work on the Ricci flow of asymptotically flat manifolds
in the rotationally symmetric case [42]. This paper shows that if no minimal
hypersphere is present initially, then one never forms. The flow then exists
for all future time and converges to flat spacetime in the limit of infinite
time, the limit being taken in the Cheeger-Gromov sense. The mass does not
change during the flow, but the quasilocal mass within any fixed hypersphere
about the origin (defined either by fixing the proper radius, the surface
area, or the enclosed volume) evaporates away smoothly. Mohammad Akbar
discussed relations between Ricci solitons and solutions to the Einstein-
scalar field equations.

### Posters

Several interesting abstracts had to be relegated to the poster session. Among
these were abstracts by Håkan Andréasson on the formation of black holes in
spherically symmetric gravitational collapse [4], Jim Isenberg on the AVTD
behavior of $T^{2}$ symmetric solutions of the Einstein vacuum equations,
Makoto Narita on cylindrically symmetric expanding spacetimes, Oscar Reula,
[34], showing that the Robinson Trautman Maxwell equations do not constitute a
well posed initial value problem, and Juan Antonio Valiente Kroon on a
characterization of Schwarzschild initial data sets [26].

## References

  * [1] L. Andersson, H. van Elst, W. C. Lim, and C. Uggla, _Asymptotic Silence of Generic Cosmological Singularities_ , Physical Review Letters 94 (2005), no. 5, 051101.
  * [2] H. Andréasson, _Sharp bounds on $2m/r$ of general spherically symmetric static objects_, 2007, arXiv.org:gr-qc/0702137.
  * [3] by same author, _Sharp bounds on the critical stability radius for relativistic charged spheres: I_ , 2007, arXiv.org:0708.0219.
  * [4] H. Andreasson, M. Kunze, and G. Rein, _The formation of black holes in spherically symmetric gravitational collapse_ , 2007, arXiv.org:0706.3787.
  * [5] R. Arnowitt, S. Deser, and C.W. Misner, _The dynamics of general relativity_ , Gravitation, an Introduction to Current Research (L. Witten, ed.), Wiley, New York, U.S.A., 1962, arXiv.org:gr-qc/0405109, pp. 227–265.
  * [6] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, _Gravitational wave extraction from an inspiraling configuration of merging black holes_ , Phys. Rev. Lett. 96 (2006), 111102.
  * [7] Thomas W. Baumgarte and Stuart L. Shapiro, _On the numerical integration of Einstein’s field equations_ , Phys. Rev. D59 (1999), 024007.
  * [8] R. Beig and N. Ó Murchadha, _The Poincaré group as the symmetry group of canonical general relativity_ , Ann. Phys. (N.Y.) 174 (1987), 463–498.
  * [9] V. A. Belinskiǐ, I. M. Khalatnikov, and E. M. Lifshitz, _Oscillatory approach to a singular point in the relativistic cosmology._ , Advances in Physics 19 (1970), 525–573.
  * [10] by same author, _A general solution of the Einstein equations with a time singularity._ , Advances in Physics 31 (1982), 639–667.
  * [11] L. Bieri, _An extension of the stability theorem of the Minkowski space in general relativity_ , Ph.D.thesis. ETH Zurich, Switzerland, 2007.
  * [12] L. Blanchet, G. Faye, and S. Nissanke, _Structure of the post-Newtonian expansion in general relativity_ , Phys. Rev. D (3) 72 (2005), no. 4, 044024, 10.
  * [13] J. D. Brown and J. W. York, Jr., _Quasilocal energy and conserved charges derived from the gravitational action_ , Phys. Rev. D47 (1993), 1407–1419.
  * [14] G. Browning and H.-O. Kreiss, _Problems with different time scales for nonlinear partial differential equations_ , SIAM J. Appl. Math. 42 (1982), no. 4, 704–718.
  * [15] B. Bruegmann, J. A. Gonzalez, M. Hannam, S. Husa, U. Sperhake, and W. Tichy, _Calibration of moving puncture simulations_ , 2006, arXiv.org:gr-qc/0610128.
  * [16] H. A. Buchdahl, _General Relativistic Fluid Spheres_ , Physical Review 116 (1959), 1027–1034.
  * [17] M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, _Accurate evolutions of orbiting black-hole binaries without excision_ , Phys. Rev. Lett. 96 (2006), 111101.
  * [18] D. Christodoulou and S. Klainerman, _The global nonlinear stability of the Minkowski space_ , Princeton Mathematical Series, vol. 41, Princeton University Press, Princeton, NJ, 1993.
  * [19] Piotr T. Chrusciel and Erwann Delay, _Existence of non-trivial, vacuum, asymptotically simple space-times_ , Class. Quant. Grav. 19 (2002), L71.
  * [20] A. A. Coley, S. Hervik, and W. C. Lim, _Fluid observers and tilting cosmology_ , Classical Quantum Gravity 23 (2006), no. 10, 3573–3591.
  * [21] Justin Corvino and Richard M. Schoen, _On the asymptotics for the vacuum Einstein constraint equations_ , J. Differential Geom. 73 (2006), no. 2, 185–217.
  * [22] T. Damour, M. Henneaux, and H. Nicolai, _Cosmological billiards_ , Classical Quantum Gravity 20 (2003), no. 9, R145–R200.
  * [23] J. Ehlers, _On limit relations between, and approximative explanations of, physical theories_ , Logic, methodology and philosophy of science, VII (Salzburg, 1983), Stud. Logic Found. Math., vol. 114, North-Holland, Amsterdam, 1986, pp. 387–403.
  * [24] G. F. R. Ellis, _Cosmology and local physics_ , New Astron. Rev. 46 (2002), 645–657.
  * [25] H. Friedrich, _Gravitational fields near space-like and null infinity_ , J. Geom. Phys. 24 (1998), no. 2, 83–163.
  * [26] A. Garcia-Parrado Gomez-Lobo and J. A. Valiente Kroon, _Initial data sets for the schwarzschild spacetime_ , Physical Review D 75 (2007), 024027\.
  * [27] J. Guven and N. Ó Murchadha, _Bounds on $2m/R$ for static spherical objects_, Phys. Rev. D (3) 60 (1999), no. 8, 084020, 8.
  * [28] M. Hannam, S. Husa, D. Pollney, B. Bruegmann, and N. O’Murchadha, _Geometry and regularity of moving punctures_ , 2006, arXiv.org:gr-qc/0606099.
  * [29] S. Hawking, _Gravitational radiation in an expanding universe_ , J. Math. Phys. 9 (1968), 598–604.
  * [30] J. M. Heinzle, C. Uggla, and N. Rohr, _The cosmological billiard attractor_ , 2007, arXiv.org:gr-qc/0702141.
  * [31] S. Hollands, A. Ishibashi, and R. M. Wald, _A higher dimensional stationary rotating black hole must be axisymmetric_ , Commun. Math. Phys. 271 (2007), 699–722.
  * [32] I. M. Khalatnikov, E. M. Lifshitz, K. M. Khanin, L. N. Shchur, and Ya. G. Sinaĭ, _On the stochastic properties of relativistic cosmological models near the singularity_ , General relativity and gravitation (Padova, 1983), Fund. Theories Phys., Reidel, Dordrecht, 1984, pp. 343–349.
  * [33] S. Klainerman and F. Nicolò, _Peeling properties of asymptotically flat solutions to the Einstein vacuum equations_ , Classical Quantum Gravity 20 (2003), no. 14, 3215–3257.
  * [34] C. Kozameh, O. Reula, and H.-O. Kreiss, _On the well posedness of robinson trautman maxwell solutions_ , 2007, arXiv.org:0708.1933.
  * [35] H. O. Kreiss, O. Reula, O. Sarbach, and J. Winicour, _Well-posed initial-boundary value problem for the harmonic einstein equations using energy estimates_ , 2007, arXiv.org:0707.4188.
  * [36] H.-O. Kreiss and J. Winicour, _Problems which are well posed in a generalized sense with applications to the Einstein equations_ , Classical Quantum Gravity 23 (2006), no. 16, S405–S420.
  * [37] W. C. Lim, A. A. Coley, and S. Hervik, _Kinematic and Weyl singularities_ , Classical Quantum Gravity 24 (2007), no. 3, 595–604.
  * [38] A. P. Lundgren, B. S. Schmekel, and Jr. J. W. York, _Self-renormalization of the classical quasilocal energy_ , Physical Review D (Particles, Fields, Gravitation, and Cosmology) 75 (2007), no. 8, 084026.
  * [39] B. C. Nolan, _Instability of the Cauchy horizon in self-similar Vaidya space-time_ , In preparation.
  * [40] by same author, _Bounds for scalar waves on self-similar naked-singularity backgrounds_ , Classical Quantum Gravity 23 (2006), no. 13, 4523–4537.
  * [41] by same author, _Odd-parity perturbations of self-similar Vaidya spacetime_ , Classical Quantum Gravity 24 (2007), no. 1, 177–200.
  * [42] T. Oliynyk and E. Woolgar, _Asymptotically flat ricci flows_ , to appear in Comm. Anal. Geom., 2006, arxiv:math/0607438.
  * [43] T.A. Oliynyk, _The newtonian limit for perfect fluids_ , to appear in  Comm. Math. Phys.
  * [44] by same author, _Post-newtonian expansions for perfect fluids_ , in preparation.
  * [45] T. Regge and C. Teitelboim, _Role of surface integrals in the Hamiltonian formulation of general relativity_ , Ann. Phys. (N.Y.) 88 (1974), 286–318.
  * [46] H. Ringström, _The Bianchi IX attractor_ , Ann. Henri Poincaré 2 (2001), no. 3, 405–500.
  * [47] N. Röhr and C. Uggla, _Conformal regularization of Einstein’s field equations_ , Classical Quantum Gravity 22 (2005), no. 17, 3775–3787.
  * [48] Olivier Sarbach, Gioel Calabrese, Jorge Pullin, and Manuel Tiglio, _Hyperbolicity of the BSSN system of Einstein evolution equations_ , Phys. Rev. D66 (2002), 064002.
  * [49] M. Shibata and T. Nakamura, _Evolution of three-dimensional gravitational waves: Harmonic slicing case_ , Phys. Rev. D52 (1995), 5428–5444.
  * [50] L. B. Szabados, _Quasi-local energy-momentum and angular momentum in GR: A review article_ , Living Rev. Relativity 7 (2004), no. 4, 1–140, http://www.livingreviews.org/lrr-2004-4.
  * [51] by same author, _On a class of 2-surface observables in general relativity_ , Class. Quantum Grav. 23 (2006), 2291–2302, arXiv.org:gr-qc/0511059.
  * [52] C. Uggla, H. van Elst, J. Wainwright, and G. F. Ellis, _Past attractor in inhomogeneous cosmology_ , Phys. Rev. D 68 (2003), no. 10, 103502–+.
  * [53] J. A. Valiente Kroon, _A new class of obstructions to the smoothness of null infinity_ , Comm. Math. Phys. 244 (2004), no. 1, 133–156.
  * [54] by same author, _Time asymmetric spacetimes near null and spatial infinity. I. Expansions of developments of conformally flat data_ , Classical Quantum Gravity 21 (2004), no. 23, 5457–5492.
  * [55] by same author, _Time asymmetric spacetimes near null and spatial infinity. II. Expansions of developments of initial data sets with non-smooth conformal metrics_ , Classical Quantum Gravity 22 (2005), no. 9, 1683–1707.
  * [56] J. Wainwright and G. F. R. Ellis, _Dynamical Systems in Cosmology_ , Dynamical Systems in Cosmology, Edited by J. Wainwright and G. F. R. Ellis, pp. 357. ISBN 0521673526. Cambridge, UK: Cambridge University Press, June 2005., June 2005.
  * [57] M. Wyman, _Static spherically symmetric scalar fields in general relativity_ , Phys. Rev. D24 (1981), 839–841.