Questions and progress in mathematical general relativity

Jim Isenberg, University of Oregon
jim-at-newton.uoregon.edu

Maxwell's equations are vitally important for describing the physical
universe but raise very few serious questions mathematically. There is
considerable mathematical interest in the global behavior of wave maps but
there is little evidence that wave maps play a direct role in modeling
important physical phenomena. One of the very few field equation systems
which is both a vital tool for modeling physics and a very rich source of
serious research problems in math is that of Einstein.
From a mathematical point of view, the study of Einstein's equations lies in the realm of geometric analysis, which analyzes PDEs (partial differential equations) that involve structures from differential geometry. For some of the PDEs of geometric analysis, such as the minimal surface equation and the harmonic map equation, the PDE system is elliptic (potential-like); for others, like Ricci flow and mean curvature flow, the PDEs are parabolic (heat-like) ; while for still others, such as wave maps and Yang-Mills, the PDEs are hyperbolic (wave-like). Interestingly in the case of Einstein's equations, there are aspects of all three types: the Einstein constraints are essentially elliptic, the full system is essentially hyperbolic, and when certain ansatze are applied (such as that of Robinson-Trautman) parabolic analysis is called for. As a consequence of this feature, the mathematical study of the Einstein equation system intersects with a much wider variety of areas in geometric analysis than does that of most other geometric analysis PDEs.

Rather than attempting any sort of an overview, I want to focus now on a
number of mathematically interesting questions which have arisen from the
study of Einstein's equations. In almost all of them there has been
substantial progress in recent years.
**Positive Mass and the Penrose Inequality**: It is relatively
straightforward to define an integral quantity which measures the mass of
an isolated gravitational system. One of the most intensely pursued
questions of mathematical GR during the 1960's and 1970's was whether one
could prove that such a quantity is either positive or zero, and zero if
and only the spacetime is flat. A theorem to that effect was proven
finally in 1979 by Schoen-Yau [1] and in 1981 by Witten
[2] independently. Since then, interest has shifted to showing that
if a spacetime contains black holes, then the mass must be greater than or
equal to a quantity involving the square root of the areas of the horizons
(the ``Penrose Inequality"). Two very innovative proofs of this
conjecture, *for the time symmetric case* (initial data with
vanishing extrinsic curvature), have been produced in recent years by Bray
[3] and by Huisken-Ilmanen [4]. One would very much like
to extend this work to the non time symmetric case; this appears to be
very difficult. We note that in the time symmetric case, the question can
be analyzed purely in terms of Riemannian geometry. This is not true of
the more general case.

**Shielding Gravitational Effects**: In Maxwell's theory, a conductor
can be used to mask from the view of outside observers many of the details
of a charge configuration. Since there are no known negative masses in
gravitational physics, it has long been thought that there are no such
conductor-like objects for shielding from external view the details of a
mass or gravitational field configuration. The elliptic character of the
Einstein constraints has reinforced this conjecture. Recent work on the
gluing of solutions of the Einstein constraints now belies this idea. In
particular the work of Corvino-Schoen [5,6] shows that
for essentially any asymptotically Euclidean initial data for Einstein's
equations, one can cut out the region outside of some ball and replace it
by a smooth extension which, some finite distance out, is exactly data for
the Schwarzschild or Kerr solution. The interior details are lost to
observers who are far enough away. The gluing results of
Chrusciel-Isenberg-Mazzeo-Pollack [7,8] allow
other quite surprising joins of disparate sets of initial data. One finds
as a consequence of their work that a given region may contain an
arbitrary number of wormholes without affecting at all the gravitational
fields some distance away from the region. All of these gluing results
exploit the underdetermined nature of the Einstein constraint equations
(more fields than equations). There is likely much more that can be done
to exploit this feature. In particular, one would like to know if it
allows one to always extend a given solution of the constraints on a ball
to an asymptotically Euclidean solution on . This question plays a
role in the Bartnik approach to defining the ``quasilocal mass"of a given
region.

**Cosmic Censorship and the Nature of Singularities**: The
Hawking-Penrose singularity theorems indicate that singularities (in the
sense of geodesic incompleteness) generically occur in solutions of
Einstein's equations , but they say little about the nature of these
singularities. Roughly speaking, one expects the singularities to be
characterized either by curvature blowup or by the breakdown of causality
(marked by the formation of a Cauchy horizon). The Strong Cosmic
Censorship Conjecture (SCC) suggests that curvature blowup occurs
generically, and Cauchy horizons develop only in very special cases.
Proving SCC is a big challenge, requiring detailed knowledge of how
solutions evolve generically. There has, however, been progress in proving
the conjecture in limited families of solutions, defined by the presence
of symmetries. The most recent work in this direction, done by Ringstrom
[9] proves that SCC holds for the class of Gowdy solutions on the
torus. One of the key steps used by Ringstrom (as well as his
predecessors) is the verification that the Gowdy solutions are velocity
dominated near the singularity. This approach may be useful for certain
less specialized families of spacetimes which also appear to be velocity
dominated, like the polarized solutions with symmetry, While more
general families of solutions are likely to not have this property,
numerical studies indicate that they may be oscillatory in the BKL sense,
and the challenge now is to verify this, and use it to prove SCC.

The very recent work of Dafermos [10] on the stability of the Cauchy horizons found in the Reissner-Nordstrom solutions (spherically symmetric charged black holes) is also relevant to the question of SCC. Interestingly, he finds stability for certain differentiability classes of perturbations, but not for others. Further development of this approach should be very useful

The Weak Cosmic Censorship Conjecture (WCC), which is *not* a
consequence of SCC, concerns a very different question: If a singularity
develops as a result of collapse in an asymptotically flat solution, does
an event horizon generally develop and cover it from view by observers at
infinity? As with SCC, WCC is a conjecture concerning the behavior of
*generic* solutions, not every solution. While interest in this
question is strong, there have not been any important recent results
relevant to WCC.

**Long Time Behavior of Solutions and Stability**: For any nonlinear
hyperbolic PDE system with a well-posed Cauchy problem, one of the
questions of primary interest is whether one can characterize those
initial data sets for which solutions exist for all (proper) time. Since
singularities do generically develop, one cannot expect long time
existence in both time directions; but there is no reason to expect
singularities in both directions. For a system as nonlinear as Einstein's
equations, long time existence is a very difficult problem. As a first
step towards its study, one approach is to investigate the stability of
long time existence about solutions like Minkowski space for which it is
known to hold. Some years ago, Christodoulou and Klainerman [11]
showed that indeed Minkowski space is stable in this sense. The recent
works of Klainerman-Niccolo [12] and of Lindblad-Rodnianski
[13] prove roughly the same result, but with stronger control of
the asymptotic properties of the spacetimes, with different choices of
gauge, and with the use of techniques which are much simpler. One would
like to extend these results to Schwarzschild and to Kerr, but there is to
date no progress in this direction.

Andersson and Moncrief [13] have studied the stability of long time existence for a different sort of spacetime: They have proven stability for the expanding, spatially compact spacetimes one obtains by compactifying the constant negative curvature hyperboloids in the future light cone of the origin in Minkowski spacetime. It is not known what the spacetime developments of the perturbed data do in the contracting direction toward the singularity, but in the future direction, they find that in fact the perturbed spacetimes asymptotically approach the flat spacetimes from which they have been perturbed. Similar stability results have been obtained by Choquet-Bruhat and Moncrief [14] for expanding symmetric spacetimes, and it is intriguing to speculate that for rapid enough expansion, stability of long time existence is a general feature.

Another line of research related to the long time behavior of solutions focuses on determining which sets of asymptotically Euclidean initial data develop into spacetimes which can be conformally compactified so as to include the familiar ``scri" structure at null infinity. Until recently, the only spacetimes known to have this complete structure were stationary. Combining the gluing work of Corvino-Schoen discussed above with Friedrich's [15] work on the generation of scri from sufficiently small initial data on an asymptotically hyperbolic spacelike surface, one now knows that there are non stationary, radiating solutions with a complete scri. The recent work of Kroon [16] shows that the nature of scri can, for general spacetimes, become quite complicated (including ``polyhomogeneous" behavior.)

Finally, we note that one approach towards obtaining long time existence
for a hyperbolic system is to try to reduce the level of regularity
(number of derivatives) needed to prove well-posedness, and then find a
conserved norm compatible with that level of regularity.
Klainerman-Rodnianski [17] and Smith-Tataru [18] have
focused on this approach, and have managed to prove well-posedness for
data with the metric in the Sobolev space
, and the
extrinsic curvature in
. The first group is working very
hard to remove the .

There are a number of other very interesting areas of mathematical study of the Einstein equations, including work to establish an initial value-boundary value formulation of the system, efforts to understand and parameterize non constant mean curvature solutions of the Einstein constraint equations, attempts to obtain an analytical understanding of the critical solutions found in Choptuik's numerical studies of gravitational collapse, and continued studies of the ``static stars are spherical" conjecture. There is every expectation that the recent record of progress in this area will continue for some time to come.

**References:**

[1] R. Schoen and S.T. Yau, *On the proof of the positive
mass conjecture in general relativity*, Comm. Math. Phys. **65** (1979),
45-76.

[2] E. Witten, *A simple proof of the positive energy
theorem*, Comm. Math. Phys. **80** (1981), 381.

[3] H. Bray, *Proof of the Riemannian Penrose inequality
using the positive mass theorem*, J. Diff. Geom. **59** (2001), 177-267.

[4] G. Huisken and T. Ilmanen, *The Inverse mean curvature
flow and the Riemannian Penrose inequality*, J. Diff. Geom. **59**
(2001), 353-437.

[5] J. Corvino,*Scalar curvature deformation and a gluing
construction for the Einstein constraint equations*, Comm. Math. Phys.
**214** (2000), 137-189.

[6] J. Corvino and R. Schoen, *On the asymptotics for the
vacuum Einstein constraint equations*,
gr-qc/0301071.

[7] J. Isenberg, R. Mazzeo, and D. Pollack, *Gluing and
wormholes for the Einstein constraint equations*, Comm. Math. Phys. **
231** (2002), 529-568.

[8] P. Chrusciel, J. Isenberg, and D. Pollack, *Initial
data engineering*, gr-qc/0403066.

[9] H. Ringstrom, *Asymptotic expansions close to the
singularity in Gowdy spacetimes*, Class. Quan. Grav. **21** (2004),
S305-S322.

[10] M. Dafermos, *The interior of charged black holes and
the problem of uniqueness in general relativity*,
gr-qc/0307013.

[11] D. Christodoulou and S. Klainerman, *The Global
nonlinear stability of the Minkowski space*, Princeton Math Series
**41**, Princeton U. Press, Princeton, N.J., (1993).

[12] S. Klainerman and F. Nicolo, *Peeling properties of
asymptotically flat solutions to the Einstein vacuum equations*, Class.
Quan. Grav. **20** (2003), 3215-3257.

[13] H. Lindblad and I. Rodnianski, *Global existence for
the Einstein vacuum equations in wave coordinates*,
math.AP/0312479.

[14] L. Andersson and V. Moncrief, *Future complete vacuum
spacetimes*, gr-qc/0303045.

[15] Y. Choquet-Bruhat and V. Moncrief, *Future complete
Einsteinian spacetimes with isometry group*, Ann. H. Poincare **
2** (2001), 1007-1064.

[16] H. Friedrich *On the existence of n-geodesically
complete or future complete solutions of the Einstein field equations with
smooth asymptotic structure*, Comm. Math. Phys. **107** (1986), 587-609.

[17] J. Kroon, *Polyhomogeneous expansions close to null and
spatial infinity*, gr-qc/0202001.

[18] S. Klainerman and I. Rodnianski, *Rough solutions of
the Einstein vacuum equations*, C. R. Math. Acad. Sci. Paris **334**
(2002), 125-130.

[19] H. Smith and D. Tataru, *Sharp local well-posedness
results for the nonlinear wave equation*, preprint (2001).

Jorge Pullin 2004-09-10