527 research outputs found
A comment on the construction of the maximal globally hyperbolic Cauchy development
Under mild assumptions, we remove all traces of the axiom of choice from the
construction of the maximal globally hyperbolic Cauchy development in general
relativity. The construction relies on the notion of direct union manifolds,
which we review. The construction given is very general: any physical theory
with a suitable geometric representation (in particular all classical fields),
and such that a strong notion of "local existence and uniqueness" of solutions
for the corresponding initial value problem is available, is amenable to the
same treatment.Comment: Version 2: 9 (+epsilon; depending on compiler) pages; updated
references. Version 3: switched to revtex, 6 pages, version accepted for
publicatio
Stability and instability of expanding solutions to the Lorentzian constant-positive-mean-curvature flow
We study constant mean curvature Lorentzian hypersurfaces of
from the point of view of its Cauchy problem. We
completely classify the spherically symmetric solutions, which include among
them a manifold isometric to the de Sitter space of general relativity. We show
that the spherically symmetric solutions exhibit one of three (future)
asymptotic behaviours: (i) finite time collapse (ii) convergence to a time-like
cylinder isometric to some and (iii) infinite
expansion to the future converging asymptotically to a time translation of the
de Sitter solution. For class (iii) we examine the future stability properties
of the solutions under arbitrary (not necessarily spherically symmetric)
perturbations. We show that the usual notions of asymptotic stability and
modulational stability cannot apply, and connect this to the presence of
cosmological horizons in these class (iii) solutions. We can nevertheless show
the global existence and future stability for small perturbations of class
(iii) solutions under a notion of stability that naturally takes into account
the presence of cosmological horizons. The proof is based on the vector field
method, but requires additional geometric insight. In particular we introduce
two new tools: an inverse-Gauss-map gauge to deal with the problem of
cosmological horizon and a quasilinear generalisation of Brendle's Bel-Robinson
tensor to obtain natural energy quantities.Comment: Version 2: 60 pages, 1 figure. Changes mostly to fix typographical
errors, with the exception of Remark 1.2 and Section 9.1 which are new and
which explain the extrinsic geometry of the embedding in more detail in terms
of the stability result. Version 3: updated reference
A positive mass theorem for two spatial dimensions
We observe that an analogue of the Positive Mass Theorem in the
time-symmetric case for three-space-time-dimensional general relativity follows
trivially from the Gauss-Bonnet theorem. In this case we also have that the
spatial slice is diffeomorphic to \Real^2.Comment: 3 pages, more or less trivial. Posting it because I cannot find it in
the literature: comments and references to where this may have appeared
before are welcome! Version 2: fixed some typos, added some remarks based on
comments receive
A space-time characterization of the Kerr-Newman metric
In the present paper, the characterization of the Kerr metric found by Marc
Mars is extended to the Kerr-Newman family. A simultaneous alignment of the
Maxwell field, the Ernst two-form of the pseudo-stationary Killing vector
field, and the Weyl curvature of the metric is shown to imply that the
space-time is locally isometric to domains in the Kerr-Newman metric. The paper
also presents an extension of Ionescu and Klainerman's null tetrad formalism to
explicitly include Ricci curvature terms.Comment: (current version) 29 pages. Incorporated changes suggested by the
anonymous referee. Resubmitted to Annales Henri Poincar
Non-existence of multiple-black-hole solutions close to Kerr-Newman
We show that a stationary asymptotically flat electro-vacuum solution of
Einstein's equations that is everywhere locally "almost isometric" to a
Kerr-Newman solution cannot admit more than one event horizon. Axial symmetry
is not assumed. In particular this implies that the assumption of a single
event horizon in Alexakis-Ionescu-Klainerman's proof of perturbative uniqueness
of Kerr black holes is in fact unnecessary.Comment: Version 2: improved presentation; no changes to the result. Version
3: corrected an oversight in the historical review. Version 4: version
accepted for publicatio
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