24 research outputs found
A Note on the Instability of Lorentzian Taub-NUT-Space
I show that there are no SU(2)-invariant (time-dependent) tensorial
perturbations of Lorentzian Taub-NUT space. It follows that the spacetime is
unstable at the linear level against generic perturbations. I speculate that
this fact is responsible for so far unsuccessful attempts to define a sensible
thermodynamics for NUT-charged spacetimes.Comment: 13 pages, no figure
Self-gravitating Klein-Gordon fields in asymptotically Anti-de-Sitter spacetimes
We initiate the study of the spherically symmetric Einstein-Klein-Gordon
system in the presence of a negative cosmological constant, a model appearing
frequently in the context of high-energy physics. Due to the lack of global
hyperbolicity of the solutions, the natural formulation of dynamics is that of
an initial boundary value problem, with boundary conditions imposed at null
infinity. We prove a local well-posedness statement for this system, with the
time of existence of the solutions depending only on an invariant H^2-type norm
measuring the size of the Klein-Gordon field on the initial data. The proof
requires the introduction of a renormalized system of equations and relies
crucially on r-weighted estimates for the wave equation on asymptotically AdS
spacetimes. The results provide the basis for our companion paper establishing
the global asymptotic stability of Schwarzschild-Anti-de-Sitter within this
system.Comment: 50 pages, v2: minor changes, to appear in Annales Henri Poincar\'
Semi-classical stability of AdS NUT instantons
The semi-classical stability of several AdS NUT instantons is studied.
Throughout, the notion of stability is that of stability at the one-loop level
of Euclidean Quantum Gravity. Instabilities manifest themselves as negative
eigenmodes of a modified Lichnerowicz Laplacian acting on the transverse
traceless perturbations. An instability is found for one branch of the
AdS-Taub-Bolt family of metrics and it is argued that the other branch is
stable. It is also argued that the AdS-Taub-NUT family of metrics are stable. A
component of the continuous spectrum of the modified Lichnerowicz operator on
all three families of metrics is found.Comment: 18 pages, 3 figures; references adde
The Simplicial Ricci Tensor
The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of
gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the
moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the
Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton
to define a non-linear, diffusive Ricci flow (RF) that was fundamental to
Perelman's proof of the Poincare conjecture. Analytic applications of RF can be
found in many fields including general relativity and mathematics. Numerically
it has been applied broadly to communication networks, medical physics,
computer design and more. In this paper, we use Regge calculus (RC) to provide
the first geometric discretization of the Ric. This result is fundamental for
higher-dimensional generalizations of discrete RF. We construct this tensor on
both the simplicial lattice and its dual and prove their equivalence. We show
that the Ric is an edge-based weighted average of deficit divided by an
edge-based weighted average of dual area -- an expression similar to the
vertex-based weighted average of the scalar curvature reported recently. We use
this Ric in a third and independent geometric derivation of the RC Einstein
tensor in arbitrary dimension.Comment: 19 pages, 2 figure
Investigating Off-shell Stability of Anti-de Sitter Space in String Theory
We propose an investigation of stability of vacua in string theory by
studying their stability with respect to a (suitable) world-sheet
renormalization group (RG) flow. We prove geometric stability of (Euclidean)
anti-de Sitter (AdS) space (i.e., ) with respect to the simplest
RG flow in closed string theory, the Ricci flow. AdS space is not a fixed point
of Ricci flow. We therefore choose an appropriate flow for which it is a fixed
point, prove a linear stability result for AdS space with respect to this flow,
and then show this implies its geometric stability with respect to Ricci flow.
The techniques used can be generalized to RG flows involving other fields. We
also discuss tools from the mathematics of geometric flows that can be used to
study stability of string vacua.Comment: 29 pages, references added in this version to appear in Classical and
Quantum Gravit
A proof of the Geroch-Horowitz-Penrose formulation of the strong cosmic censor conjecture motivated by computability theory
In this paper we present a proof of a mathematical version of the strong
cosmic censor conjecture attributed to Geroch-Horowitz and Penrose but
formulated explicitly by Wald. The proof is based on the existence of
future-inextendible causal curves in causal pasts of events on the future
Cauchy horizon in a non-globally hyperbolic space-time. By examining explicit
non-globally hyperbolic space-times we find that in case of several physically
relevant solutions these future-inextendible curves have in fact infinite
length. This way we recognize a close relationship between asymptotically flat
or anti-de Sitter, physically relevant extendible space-times and the so-called
Malament-Hogarth space-times which play a central role in recent investigations
in the theory of "gravitational computers". This motivates us to exhibit a more
sharp, more geometric formulation of the strong cosmic censor conjecture,
namely "all physically relevant, asymptotically flat or anti-de Sitter but
non-globally hyperbolic space-times are Malament-Hogarth ones".
Our observations may indicate a natural but hidden connection between the
strong cosmic censorship scenario and the Church-Turing thesis revealing an
unexpected conceptual depth beneath both conjectures.Comment: 16pp, LaTeX, no figures. Final published versio
A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds
We consider Kerr spacetimes with parameters a and M such that |a|<< M,
Kerr-Newman spacetimes with parameters |Q|<< M, |a|<< M, and more generally,
stationary axisymmetric black hole exterior spacetimes which are sufficiently
close to a Schwarzschild metric with parameter M>0, with appropriate geometric
assumptions on the plane spanned by the Killing fields. We show uniform
boundedness on the exterior for sufficiently regular solutions to the scalar
homogeneous wave equation. In particular, the bound holds up to and including
the event horizon. No unphysical restrictions are imposed on the behaviour of
the solution near the bifurcation surface of the event horizon. The pointwise
estimate derives in fact from the uniform boundedness of a positive definite
energy flux. Note that in view of the very general assumptions, the
separability properties of the wave equation on the Kerr background are not
used.Comment: 71 pages, 3 figure
Ricci solitons, Ricci flow, and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua
The elliptic Einstein-DeTurck equation may be used to numerically find
Einstein metrics on Riemannian manifolds. Static Lorentzian Einstein metrics
are considered by analytically continuing to Euclidean time. Ricci-DeTurck flow
is a constructive algorithm to solve this equation, and is simple to implement
when the solution is a stable fixed point, the only complication being that
Ricci solitons may exist which are not Einstein. Here we extend previous work
to consider the Einstein-DeTurck equation for Riemannian manifolds with
boundaries, and those that continue to static Lorentzian spacetimes which are
asymptotically flat, Kaluza-Klein, locally AdS or have extremal horizons. Using
a maximum principle we prove that Ricci solitons do not exist in these cases
and so any solution is Einstein. We also argue that Ricci-DeTurck flow
preserves these classes of manifolds. As an example we simulate Ricci-DeTurck
flow for a manifold with asymptotics relevant for AdS_5/CFT_4. Our maximum
principle dictates there are no soliton solutions, and we give strong numerical
evidence that there exists a stable fixed point of the flow which continues to
a smooth static Lorentzian Einstein metric. Our asymptotics are such that this
describes the classical gravity dual relevant for the CFT on a Schwarzschild
background in either the Unruh or Boulware vacua. It determines the leading
O(N^2) part of the CFT stress tensor, which interestingly is regular on both
the future and past Schwarzschild horizons.Comment: 48 pages, 7 figures; Version 2 - section 2.2.1 on manifolds with
boundaries substantially modified, corrected and extended. Discussion in
section 3.1 amended. References added and minor change