On the Rigidity Theorem for Spacetimes with a Stationary Event Horizon or a Compact Cauchy Horizon

We consider smooth electrovac spacetimes which represent either (A) an asymptotically flat, stationary black hole or (B) a cosmological spacetime with a compact Cauchy horizon ruled by closed null geodesics. The black hole event horizon or, respectively, the compact Cauchy horizon of these spacetimes is assumed to be a smooth null hypersurface which is non-degenerate in the sense that its null geodesic generators are geodesically incomplete in one direction. In both cases, it is shown that there exists a Killing vector field in a one-sided neighborhood of the horizon which is normal to the horizon. We thereby generalize theorems of Hawking (for case (A)) and Isenberg and Moncrief (for case (B)) to the non-analytic case.Comment: 16 pages, no figure

Lsdiff M and the Einstein Equations

We give a formulation of the vacuum Einstein equations in terms of a set of volume-preserving vector fields on a four-manifold ${\cal M}$. These vectors satisfy a set of equations which are a generalisation of the Yang-Mills equations for a constant connection on flat spacetime.Comment: 5 pages, no figures, Latex, uses amsfonts, amssym.def and amssym.tex. Note added on more direct connection with Yang-Mills equation

Intrinsic time gravity and the Lichnerowicz-York equation

We investigate the effect on the Hamiltonian structure of general relativity of choosing an intrinsic time to fix the time slicing. 3-covariance with momentum constraint is maintained, but the Hamiltonian constraint is replaced by a dynamical equation for the trace of the momentum. This reveals a very simple structure with a local reduced Hamiltonian. The theory is easily generalised; in particular, the square of the Cotton-York tensor density can be added as an extra part of the potential while at the same time maintaining the classic 2 + 2 degrees of freedom. Initial data construction is simple in the extended theory; we get a generalised Lichnerowicz-York equation with nice existence and uniqueness properties. Adding standard matter fields is quite straightforward.Comment: 4 page

Conformal ``thin sandwich'' data for the initial-value problem of general relativity

The initial-value problem is posed by giving a conformal three-metric on each of two nearby spacelike hypersurfaces, their proper-time separation up to a multiplier to be determined, and the mean (extrinsic) curvature of one slice. The resulting equations have the {\it same} elliptic form as does the one-hypersurface formulation. The metrical roots of this form are revealed by a conformal ``thin sandwich'' viewpoint coupled with the transformation properties of the lapse function.Comment: 7 pages, RevTe

The Cauchy problem in General Relativity: An algebraic characterization

In this paper we shall analyse the structure of the Cauchy Problem (CP briefly) for General Relativity (GR briefly) by applying the theory of first order symmetric hyperbolic systems