335 research outputs found
Existence of CMC Cauchy surfaces and spacetime splitting
In this paper, we review results on the existence (and nonexistence) of
constant mean curvature spacelike hypersurfaces in the cosmological setting,
and discuss the connection to the spacetime splittng problem. It is a pleasure
to dedicate this paper to Robert Bartnik, who has made fundamental
contributions to this area.Comment: 16 page
Maximum Principles for Null Hypersurfaces and Null Splitting Theorems
A maximum principle for C^0 null hypersurfaces is obtained and used to derive
a splitting theorem for spacetimes which contain null lines. As a consequence
of this null splitting theorem, it is proved that an asymptotically simple
vacuum (Ricci flat) spacetime which contains a null line is isometric to
Minkowski space.Comment: 26 pages, latex2
Rigidity of outermost MOTS - the initial data version
In [5], a rigidity result was obtained for outermost marginally outer trapped
surfaces (MOTSs) that do not admit metrics of positive scalar curvature. This
allowed one to treat the "borderline case" in the author's work with R. Schoen
concerning the topology of higher dimensional black holes [8]. The proof of
this rigidity result involved bending the initial data manifold in the vicinity
of the MOTS within the ambient spacetime. In this note we show how to
circumvent this step, and thereby obtain a pure initial data version of this
rigidity result and its consequence concerning the topology of black holes.Comment: 8 pages; v2: minor changes; version to appear in GR
dS/CFT and spacetime topology
Motivated by recent proposals for a de Sitter version of the AdS/CFT
correspondence, we give some topological restrictions on spacetimes of de
Sitter type, i.e., spacetimes with , which admit a regular past
and/or future conformal boundary. For example we show that if , , is a globally hyperbolic spacetime obeying suitable energy conditions,
which is of de Sitter type, with a conformal boundary to both the past and
future, then if one of these boundaries is compact, it must have finite
fundamental group and its conformal class must contain a metric of positive
scalar curvature. Our results are closely related to theorems of Witten and Yau
hep-th/9910245 pertaining to the Euclidean formulation of the AdS/CFT
correspondence.Comment: 16 pages, Latex2e, v2: reference corrected, v3: reference added,
material added to the introductio
Some uniqueness results for dynamical horizons
We first show that the intrinsic, geometrical structure of a dynamical
horizon is unique. A number of physically interesting constraints are then
established on the location of trapped and marginally trapped surfaces in the
vicinity of any dynamical horizon. These restrictions are used to prove several
uniqueness theorems for dynamical horizons. Ramifications of some of these
results to numerical simulations of black hole spacetimes are discussed.
Finally several expectations on the interplay between isometries and dynamical
horizons are shown to be borne out.Comment: 26 pages, 4 figures, v4: references updated, minor corrections, to
appear in Advances in Theoretical and Mathematical Physic
Topology and singularities in cosmological spacetimes obeying the null energy condition
We consider globally hyperbolic spacetimes with compact Cauchy surfaces in a
setting compatible with the presence of a positive cosmological constant. More
specifically, for 3+1 dimensional spacetimes which satisfy the null energy
condition and contain a future expanding compact Cauchy surface, we establish a
precise connection between the topology of the Cauchy surfaces and the
occurrence of past singularities. In addition to (a refinement of) the Penrose
singularity theorem, the proof makes use of some recent advances in the
topology of 3-manifolds and of certain fundamental existence results for
minimal surfaces.Comment: 8 pages; v2: minor changes, version to appear in CM
Cosmological singularities in Bakry-\'Emery spacetimes
We consider spacetimes consisting of a manifold with Lorentzian metric and a
weight function or scalar field. These spacetimes admit a Bakry-\'Emery-Ricci
tensor which is a natural generalization of the Ricci tensor. We impose an
energy condition on the Bakry-\'Emery-Ricci tensor and obtain singularity
theorems of a cosmological type, both for zero and for positive cosmological
constant. That is, we find conditions under which every timelike geodesic is
incomplete. These conditions are given by "open" inequalities, so we examine
the borderline (equality) cases and show that certain singularities are avoided
in these cases only if the geometry is rigid; i.e., if it splits as a
Lorentzian product or, for a positive cosmological constant, a warped product,
and the weight function is constant along the time direction. Then the product
case is future timelike geodesically complete while, in the warped product
case, worldlines of certain conformally static observers are complete. Our
results answer a question posed by J Case. We then apply our results to the
cosmology of scalar-tensor gravitation theories. We focus on the Brans-Dicke
family of theories in 4 spacetime dimensions, where we obtain "Jordan frame"
singularity theorems for big bang singularities.Comment: 15 pages; The wording of Theorem 1.5 is slightly clarified over the
wording in the published version, with no change in the resul
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