Robert Beig

Killing vectors in asymptotically flat space–times. I. Asymptotically translational Killing vectors and the rigid positive energy theorem

We study Killing vector fields in asymptotically flat space–times. We prove the following result, implicitly assumed in the uniqueness theory of stationary black holes. If the conditions of the rigidity part of the positive energy theorem are met, then in such space–times there are no asymptotically null Killing vector fields, except if the initial data set can be embedded in Minkowski space–time. We also give a proof of the nonexistence of nonsingular (in an appropriate sense) asymptotically flat space–times that satisfy an energy condition and that have a null ADM four‐momentum, under conditions weaker than previously considered.

Einstein's equations near spatial infinity

A new class of space-times is introduced which, in a neighbourhood of spatial infinity, allows an expansion in negative powers of a radial coordinate. Einsteins vacuum equations give rise to a hierarchy of linear equations for the coefficients in this expansion. It is demonstrated that this hierarchy can be completely solved provided the initial data satisfy certain constraints.

KIDs are Non-Generic

Abstract.We prove that the space-time developments of generic solutions of the vacuum constraint Einstein equations do not possess any global or local Killing vectors, when Cauchy data are prescribed on an asymptotically flat Cauchy surface, or on a compact Cauchy surface with mean curvature close to a constant, or for CMC asymptotically hyperbolic initial data sets. More generally, we show that nonexistence of global symmetries implies, generically, non-existence of local ones. As part of the argument, we prove that generic metrics do not possess any local or global conformal Killing vectors.

Proof of a Multipole Conjecture due to Geroch

A result, first conjectured by Geroch, is proved to the extent, that the multipole moments of a static space-time characterize this space-time uniquely. As an offshoot of the proof one obtains an essentially coordinate-free algorithm for explicitly writing down a geometry in terms of its moments in a purely algebraic manner. This algorithm seems suited for symbolic manipulation on a computer.

The multipole structure of stationary space‐times

A definition of multipole moments for stationary asymptotically flat solutions of Einstein’s equations is proposed. It is shown that these moments characterize a given space‐time uniquely. Conversely, they can be arbitrarily prescribed, i.e., they generate power series for the field variables which satisfy the field equations to all orders. Despite their apparently rather different origin, they are shown to be identical with the Geroch–Hansen ones.

Static self-gravitating elastic bodies in Einstein gravity

We prove that given a stress-free elastic body there exists, for sufficiently small values of the gravitational constant, a unique static solution of the Einstein equations coupled to the equations of relativistic elasticity. The solution constructed is a small deformation of the relaxed configuration. This result yields the first proof of existence of static solutions of the Einstein equations without symmetries. c � 2008 Wiley Periodicals, Inc.

Time-Independent Gravitational Fields

In this article we want to describe what is known about time independent spacetimes from a global point of view. The physical situations we want to treat are isolated bodies at rest or in uniform rotation in an otherwise empty universe. In such cases one expects the gravitational field to have no “independent degrees of freedom”. Very loosely speaking, the spacetime geometry should be uniquely determined by the matter content of the model under consideration. In a similar way, for a given matter model (such as that of a perfect fluid), there should be a one-to-one correspondence between Newtonian solutions and general relativistic ones.

On the uniqueness of static perfect-fluid solutions in general relativity

Following earlier work of Masood-ul-Alam, we consider a uniqueness problem for non-rotating stellar models. Given a static, asymptotically flat perfectfluid spacetime with barotropic equation of state θ(p), and given another such spacetime which is spherically symmetric and has the same θ(p) and the same surface potential: we prove that both are identical provided θ(p) satisfies a certain differential inequality. This inequality is more natural and less restrictive than the conditions required by Masood-ul-Alam.

The stationary gravitational field near spatial infinity

Any stationary, asymptotically flat solution to Einsteins equation is shown to asymptotically approach the Kerr solution in a precise sense. As an application of this result we prove a technical lemma on the existence of harmonic coordinates near infinity.

Late time behavior of the maximal slicing of the Schwarzschild black hole

A time-symmetric Cauchy slice of the extended Schwarzschild spacetime can evolve into a foliation of the