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  and in 1981 by Witten  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  and by Huisken-Ilmanen . 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  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  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  showed that indeed Minkowski space is stable in this sense. The recent works of Klainerman-Niccolo  and of Lindblad-Rodnianski  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  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  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  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  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  and Smith-Tataru  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.
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