561 research outputs found
On the Bartnik extension problem for the static vacuum Einstein equations
We develop a framework for understanding the existence of asymptotically flat
solutions to the static vacuum Einstein equations with prescribed boundary data
consisting of the induced metric and mean curvature on a 2-sphere. A partial
existence result is obtained, giving a partial resolution of a conjecture of
Bartnik on such static vacuum extensions. The existence and uniqueness of such
extensions is closely related to Bartnik's definition of quasi-local mass.Comment: 33 pages, 1 figure. Minor revision of v2. Final version, to appear in
Class. Quantum Gravit
Einstein equations in the null quasi-spherical gauge
The structure of the full Einstein equations in a coordinate gauge based on
expanding null hypersurfaces foliated by metric 2-spheres is explored. The
simple form of the resulting equations has many applications -- in the present
paper we describe the structure of timelike boundary conditions; the matching
problem across null hypersurfaces; and the propagation of gravitational shocks.Comment: 12 pages, LaTeX (revtex, amssymb), revision 18 pages, contains
expanded discussion and explanations, updated references, to appear in CQ
Embedding spherical spacelike slices in a Schwarzschild solution
Given a spherical spacelike three-geometry, there exists a very simple
algebraic condition which tells us whether, and in which, Schwarzschild
solution this geometry can be smoothly embedded. One can use this result to
show that any given Schwarzschild solution covers a significant subset of
spherical superspace and these subsets form a sequence of nested domains as the
Schwarzschild mass increases. This also demonstrates that spherical data offer
an immediate counter example to the thick sandwich `theorem'
Multidimensional Gravity on the Principal Bundles
The multidimensional gravity on the total space of principal bundle is
considered. In this theory the gauge fields arise as nondiagonal components of
multidimensional metric. The spherically symmetric and cosmology solutions for
gravity on SU(2) principal bundle are obtained. The static spherically
symmetric solution is wormhole-like solution located between two null surfaces,
in contrast to 4D Einstein-Yang-Mills theory where corresponding solution
(black hole) located outside of event horizon. Cosmology solution (at least
locally) has the bouncing off effect for spatial dimensions. In spirit of
Einstein these solutions are vacuum solutions without matter.Comment: REVTEX, 13pages, 2 EPS figure
Trapped Surfaces in Vacuum Spacetimes
An earlier construction by the authors of sequences of globally regular,
asymptotically flat initial data for the Einstein vacuum equations containing
trapped surfaces for large values of the parameter is extended, from the time
symmetric case considered previously, to the case of maximal slices. The
resulting theorem shows rigorously that there exists a large class of initial
configurations for non-time symmetric pure gravitational waves satisfying the
assumptions of the Penrose singularity theorem and so must have a singularity
to the future.Comment: 14 page
Just how long can you live in a black hole and what can be done about it?
We study the problem of how long a journey within a black hole can last.
Based on our observations, we make two conjectures. First, for observers that
have entered a black hole from an asymptotic region, we conjecture that the
length of their journey within is bounded by a multiple of the future
asymptotic ``size'' of the black hole, provided the spacetime is globally
hyperbolic and satisfies the dominant-energy and non-negative-pressures
conditions. Second, for spacetimes with Cauchy surfaces (or an
appropriate generalization thereof) and satisfying the dominant energy and
non-negative-pressures conditions, we conjecture that the length of a journey
anywhere within a black hole is again bounded, although here the bound requires
a knowledge of the initial data for the gravitational field on a Cauchy
surface. We prove these conjectures in the spherically symmetric case. We also
prove that there is an upper bound on the lifetimes of observers lying ``deep
within'' a black hole, provided the spacetime satisfies the
timelike-convergence condition and possesses a maximal Cauchy surface. Further,
we investigate whether one can increase the lifetime of an observer that has
entered a black hole, e.g., by throwing additional matter into the hole.
Lastly, in an appendix, we prove that the surface area of the event horizon
of a black hole in a spherically symmetric spacetime with ADM mass
is always bounded by , provided
that future null infinity is complete and the spacetime is globally hyperbolic
and satisfies the dominant-energy condition.Comment: 20 pages, REVTeX 3.0, 6 figures included, self-unpackin
Late time behaviour of the maximal slicing of the Schwarzschild black hole
A time-symmetric Cauchy slice of the extended Schwarzschild spacetime can be
evolved into a foliation of the -region of the spacetime by maximal
surfaces with the requirement that time runs equally fast at both spatial ends
of the manifold. This paper studies the behaviour of these slices in the limit
as proper time-at-infinity becomes arbitrarily large and gives an analytic
expression for the collapse of the lapse.Comment: 18 pages, Latex, no figure
Constant mean curvature solutions of the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds
We follow the approach employed by Y. Choquet-Bruhat, J. Isenberg and D.
Pollack in the case of closed manifolds and establish existence and
non-existence results for the Einstein-scalar field constraint equations on
asymptotically hyperbolic manifolds.Comment: 15 page
A new geometric invariant on initial data for Einstein equations
For a given asymptotically flat initial data set for Einstein equations a new
geometric invariant is constructed. This invariant measure the departure of the
data set from the stationary regime, it vanishes if and only if the data is
stationary. In vacuum, it can be interpreted as a measure of the total amount
of radiation contained in the data.Comment: 5 pages. Important corrections regarding the generalization to the
non-time symmetric cas
The Yamabe invariant for axially symmetric two Kerr black holes initial data
An explicit 3-dimensional Riemannian metric is constructed which can be
interpreted as the (conformal) sum of two Kerr black holes with aligned angular
momentum. When the separation distance between them is large we prove that this
metric has positive Ricci scalar and hence positive Yamabe invariant. This
metric can be used to construct axially symmetric initial data for two Kerr
black holes with large angular momentum.Comment: 14 pages, 2 figure
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