20 research outputs found
On the Strength of Spacetime Singularities
New integral conditions are proposed that are sufficient for the existence of conjugate pointpairs along causal geodesics. As to maximal incomplete causal geodesics the upper bound on the rate of possible divergency of the tidal curvature is examined
On Killing vectors in initial value problems for asymptotically flat space-times
The existence of symmetries in asymptotically flat space-times are studied
from the point of view of initial value problems. General necessary and
sufficient (implicit) conditions are given for the existence of Killing vector
fields in the asymptotic characteristic and in the hyperboloidal initial value
problem (both of them are formulated on the conformally compactified space-time
manifold)
On the existence and convergence of polyhomogeneous expansions of zero-rest-mass fields
The convergence of polyhomogeneous expansions of zero-rest-mass fields in
asymptotically flat spacetimes is discussed. An existence proof for the
asymptotic characteristic initial value problem for a zero-rest-mass field with
polyhomogeneous initial data is given. It is shown how this non-regular problem
can be properly recast as a set of regular initial value problems for some
auxiliary fields. The standard techniques of symmetric hyperbolic systems can
be applied to these new auxiliary problems, thus yielding a positive answer to
the question of existence in the original problem.Comment: 10 pages, 1 eps figur
Polyhomogeneity and zero-rest-mass fields with applications to Newman-Penrose constants
A discussion of polyhomogeneity (asymptotic expansions in terms of and
) for zero-rest-mass fields and gravity and its relation with the
Newman-Penrose (NP) constants is given. It is shown that for spin-
zero-rest-mass fields propagating on Minkowski spacetime, the logarithmic terms
in the asymptotic expansion appear naturally if the field does not obey the
``Peeling theorem''. The terms that give rise to the slower fall-off admit a
natural interpretation in terms of advanced field. The connection between such
fields and the NP constants is also discussed. The case when the background
spacetime is curved and polyhomogeneous (in general) is considered. The free
fields have to be polyhomogeneous, but the logarithmic terms due to the
connection appear at higher powers of . In the case of gravity, it is
shown that it is possible to define a new auxiliary field, regular at null
infinity, and containing some relevant information on the asymptotic behaviour
of the spacetime. This auxiliary zero-rest-mass field ``evaluated at future
infinity ()'' yields the logarithmic NP constants.Comment: 19 page
Towards the classification of static vacuum spacetimes with negative cosmological constant
We present a systematic study of static solutions of the vacuum Einstein
equations with negative cosmological constant which asymptotically approach the
generalized Kottler (``Schwarzschild--anti-de Sitter'') solution, within
(mainly) a conformal framework. We show connectedness of conformal infinity for
appropriately regular such space-times. We give an explicit expression for the
Hamiltonian mass of the (not necessarily static) metrics within the class
considered; in the static case we show that they have a finite and well defined
Hawking mass. We prove inequalities relating the mass and the horizon area of
the (static) metrics considered to those of appropriate reference generalized
Kottler metrics. Those inequalities yield an inequality which is opposite to
the conjectured generalized Penrose inequality. They can thus be used to prove
a uniqueness theorem for the generalized Kottler black holes if the generalized
Penrose inequality can be established.Comment: the discussion of our results includes now some solutions of Horowitz
and Myers; typos corrected here and there; a shortened version of this
version will appear in Journal of Mathematical Physic
The Cauchy Problem for the Einstein Equations
Various aspects of the Cauchy problem for the Einstein equations are
surveyed, with the emphasis on local solutions of the evolution equations.
Particular attention is payed to giving a clear explanation of conceptual
issues which arise in this context. The question of producing reduced systems
of equations which are hyperbolic is examined in detail and some new results on
that subject are presented. Relevant background from the theory of partial
differential equations is also explained at some lengthComment: 98 page
Construction of anti-de Sitter-like spacetimes using the metric conformal Einstein field equations: the vacuum case
We make use of the metric version of the conformal Einstein field equations
to construct anti-de Sitter-like spacetimes by means of a suitably posed
initial-boundary value problem. The evolution system associated to this
initial-boundary value problem consists of a set of conformal wave equations
for a number of conformal fields and the conformal metric. This formulation
makes use of generalised wave coordinates and allows the free specification of
the Ricci scalar of the conformal metric via a conformal gauge source function.
We consider Dirichlet boundary conditions for the evolution equations at the
conformal boundary and show that these boundary conditions can, in turn, be
constructed from the 3-dimensional Lorentzian metric of the conformal boundary
and a linear combination of the incoming and outgoing radiation as measured by
certain components of the Weyl tensor. To show that a solution to the conformal
evolution equations implies a solution to the Einstein field equations we also
provide a discussion of the propagation of the constraints for this
initial-boundary value problem. The existence of local solutions to the
initial-boundary value problem in a neighbourhood of the corner where the
initial hypersurface and the conformal boundary intersect is subject to
compatibility conditions between the initial and boundary data. The
construction described is amenable to numerical implementation and should allow
the systematic exploration of boundary conditions.Comment: 29 pages, 1 figur
Symmetries of spacetime and their relation to initial value problems
We consider covariant metric theories of coupled gravity-matter systems
satisfying the following two conditions: First, it is assumed that, by a
hyperbolic reduction process, a system of first order symmetric hyperbolic
partial differential equations can be deduced from the matter field equations.
Second, gravity is supposed to be coupled to the matter fields by requiring
that the Ricci tensor is a smooth function of the basic matter field variables
and the metric. It is shown then that the ``time'' evolution of these type of
gravity-matter systems preserves the symmetries of initial data specifications.Comment: 12 pages, to appear in Class. Quant. Gra