Related papers: Mean Curvature, Singularities and Time Functions i…
The global properties of spatially homogeneous cosmological models with collisionless matter are studied. It is shown that as long as the mean curvature of the hypersurfaces of homogeneity remains finite no singularity can occur in finite…
We prove Hawking's singularity theorem for spacetime metrics of local Lipschitz regularity. The proof rests on (1) new estimates for the Ricci curvature of regularising smooth metrics that are based upon a quite general Friedrichs-type…
The singularity theorems of Penrose, Hawking, and Geroch predict the existence of incomplete inextendible causal geodesics in a wide range of physically adequate spacetimes modeling the gravitational collapse of stars and the expanding…
We study the geometry of stable maximal hypersurfaces in a variety of spacetimes satisfying various physically relevant curvature assumptions, for instance the Timelike Convergence Condition (TCC). We characterize stability when the target…
The stability of cosmological event and Cauchy horizons of spacetimes associated with plane symmetric domain walls are studied. It is found that both horizons are not stable against perturbations of null fluids and massless scalar fields;…
Many results in mathematical relativity, including results for both the initial data problem and for the evolution problem, rely on the existence of a constant mean curvature (CMC) Cauchy surface in the underlying spacetime. However, it is…
Since the solution of the so-called folk problems of smoothability, there has been a special interest in the properties of classical time and volume functions of spacetimes. Here we supply some information that complements the one provided…
The classical singularity theorems of R. Penrose and S. Hawking from the 1960s show that, given a pointwise energy condition (and some causality as well as initial assumptions), spacetimes cannot be geodesically complete. Despite their…
In this paper we provide under certain geometric and physical assumptions new uniqueness and non-existence results for complete spacelike hypersurfaces of constant mean curvature in spatially open Generalized Robertson-Walker spacetimes.…
The problem of existence of spacelike hypersurfaces with constant mean curvature in asymptotically flat spacetimes is considered for a class of asymptotically Schwarzschild spacetimes satisfying an interior condition. Using a barrier…
In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to the Einstein-vacuum equations in high dimensions, stable, approximately…
We consider spacetimes with compact Cauchy hypersurfaces and with Ricci tensor bounded from below on the set of timelike unit vectors, and prove that the results known for spacetimes satisfying the timelike convergence condition, namely,…
This article presents a comprehensive and rigorous overview of spacetime singularities within the framework of classical General Relativity. Singularities are defined through the failure of geodesic completeness, reflecting the limits of…
A theorem about local in time existence of spacelike foliations with prescribed mean curvature in cosmological spacetimes will be proved. The time function of the foliation is geometrically defined and fixes the diffeomorphism invariance…
We consider complete spacelike hypersurfaces with constant mean curvature in the open region of de Sitter space known as the steady state space. We prove that if the hypersurface is bounded away from the infinity of the ambient space, then…
A rigidity theorem that applies to 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 was given…
We present a way of understanding the curvature of space-time, the basic philosophy being that the (linear) geometry of any space is determined by the (linear) functionals on the algebra(s) of any fields defined on the space. It is known…
A key test for any quasi-local energy in general relativity is that it be nonnegative and satisfy a rigidity property; if it vanishes, the region enclosed is flat. We show that the Hawking energy, when evaluated on its natural…
We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on…
We show that there exist maximal globally hyperbolic solutions of the Einstein-dust equations which admit a constant mean curvature Cauchy surface, but are not covered by a constant mean curvature foliation.