225 research outputs found
Spacetime Ehlers group: Transformation law for the Weyl tensor
The spacetime Ehlers group, which is a symmetry of the Einstein vacuum field
equations for strictly stationary spacetimes, is defined and analyzed in a
purely spacetime context (without invoking the projection formalism). In this
setting, the Ehlers group finds its natural description within an infinite
dimensional group of transformations that maps Lorentz metrics into Lorentz
metrics and which may be of independent interest. The Ehlers group is shown to
be well defined independently of the causal character of the Killing vector
(which may become null on arbitrary regions). We analyze which global
conditions are required on the spacetime for the existence of the Ehlers group.
The transformation law for the Weyl tensor under Ehlers transformations is
explicitly obtained. This allows us to study where, and under which
circumstances, curvature singularities in the transformed spacetime will arise.
The results of the paper are applied to obtain a local characterization of the
Kerr-NUT metric.Comment: LaTeX, 25 pages, no figures, uses Amstex. Accepted for publication in
Classical and Quantum Gravit
Global and uniqueness properties of stationary and static spacetimes with outer trapped surfaces
Global properties of maximal future Cauchy developments of stationary,
m-dimensional asymptotically flat initial data with an outer trapped boundary
are analyzed. We prove that, whenever the matter model is well posed and
satisfies the null energy condition, the future Cauchy development of the data
is a black hole spacetime. More specifically, we show that the future Killing
development of the exterior of a sufficiently large sphere in the initial data
set can be isometrically embedded in the maximal Cauchy development of the
data. In the static setting we prove, by working directly on the initial data
set, that all Killing prehorizons are embedded whenever the initial data set
has an outer trapped boundary and satisfies the null energy condition. By
combining both results we prove a uniqueness theorem for static initial data
sets with outer trapped boundary.Comment: 38 pages, 2 figures, Late
On the Penrose inequality for dust null shells in the Minkowski spacetime of arbitrary dimension
A particular, yet relevant, particular case of the Penrose inequality
involves null shells propagating in the Minkowski spacetime. Despite previous
claims in the literature, the validity of this inequality remains open. In this
paper we rewrite this inequality in terms of the geometry of the surface
obtained by intersecting the past null cone of the original surface S with a
constant time hyperplane and the "time height" function of S over this
hyperplane. We also specialize to the case when S lies in the past null cone of
a point and show the validity of the corresponding inequality in any dimension
(in four dimensions this inequality was proved by Tod). Exploiting properties
of convex hypersurfaces in Euclidean space we write down the Penrose inequality
in the Minkowski spacetime of arbitrary dimension n+2 as an inequality for two
smooth functions on the sphere. We finally obtain a sufficient condition for
the validity of the Penrose inequality in the four dimensional Minkowski
spacetime and show that this condition is satisfied by a large class of
surfaces.Comment: 25 pages, 2 figures, Late
Geometry of normal graphs in Euclidean space and applications to the Penrose inequality in Minkowski
The Penrose inequality in Minkowski is a geometric inequality relating the
total outer null expansion and the area of closed, connected and spacelike
codimension-two surfaces S in the Minkowski spacetime, subject to an additional
convexity assumption. In a recent paper, Brendle and Wang find a sufficient
condition for the validity of this Penrose inequality in terms of the geometry
of the orthogonal projection of S onto a constant time hyperplane. In this
work, we study the geometry of hypersurfaces in n-dimensional euclidean space
which are normal graphs over other surfaces and relate the intrinsic and
extrinsic geometry of the graph with that of the base hypersurface. These
results are used to rewrite Brendle and Wang's condition explicitly in terms of
the time height function of S over a hyperplane and the geometry of the
projection of S along its past null cone onto this hyperplane. We also include,
in an Appendix, a self-contained summary of known and new results on the
geometry of projections along the Killing direction of codimension
two-spacelike surfaces in a strictly static spacetime.Comment: 15 pages, 1 figure, Late
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