214 research outputs found
Higher Dimensional Bondi Energy with a Globally Specified Background Structure
A higher (even spacetime) dimensional generalization of the Bondi energy has
recently been proposed by gr-qc/0304054 within the framework of conformal
infinity and Hamiltonian formalizm. The gauge condition employed in
gr-qc/0304054 to derive the Bondi energy expression is, however, peculiar in
the sense that cross-sections of null infinity specified by that gauge are
anisotropic and in fact non-compact. For this reason, that gauge is difficult
to use for explicit computations of the Bondi energy in general, asymptotically
flat radiative spacetimes. Also it is not clear, under that gauge condition,
whether apparent difference between the expressions of higher dimensional Bondi
energy and the 4-dimensional one is due to the choice of gauges or
qualitatively different nature of higher dimensional gravity from 4-dimensional
gravity. In this paper, we consider instead, Gaussian null conformal gauge as
one of more natural gauge conditions that admit a global specification of
background structure with compact, spherical cross-sections of null infinity.
Accordingly, we modify the previous definition of higher dimensional news
tensor so that it becomes well-defined in the Gaussian null conformal gauge and
derive, for vacuum solutions, the expression for the Bondi energy-momentum in
the new gauge choice, which takes a universal form in arbitrary (even
spacetime) dimensions greater than or equal to four.Comment: 23 pages, no figures, references are added, minor corrections,
numerous minor revisions, to appear in Class. Quantum Gra
Master equations for perturbations of generalised static black holes with charge in higher dimensions
We extend the formulation for perturbations of maximally symmetric black
holes in higher dimensions developed by the present authors in a previous paper
(hep-th/0305147) to a charged black hole background whose horizon is described
by an Einstein manifold. For charged black holes, perturbations of
electromagnetic fields are coupled to the vector and scalar modes of metric
perturbations non-trivially. We show that by taking appropriate combinations of
gauge-invariant variables for these perturbations, the perturbation equations
for the Einstein-Maxwell system are reduced to two decoupled second-order wave
equations describing the behaviour of the electromagnetic mode and the
gravitational mode, for any value of the cosmological constant. These wave
equations are transformed into Schr\"odinger-type ODEs through a Fourier
transformation with respect to time. Using these equations, we investigate the
stability of generalised black holes with charge. We also give explicit
expressions for the source terms of these master equations with application to
the emission problem of gravitational waves in mind.Comment: 46 pages in the PTP-TEX style including 7 figures. The published
versio
Perturbations and Stability of Static Black Holes in Higher Dimensions
In this chapter we consider perturbations and stability of higher dimensional
black holes focusing on the static background case. We first review a
gauge-invariant formalism for linear perturbations in a fairly generic class of
(m+n)-dimensional spacetimes with a warped product metric, including black hole
geometry. We classify perturbations of such a background into three types, the
tensor, vector and scalar-type, according to their tensorial behaviour on the
n-dimensional part of the background spacetime, and for each type of
perturbations, we introduce a set of manifestly gauge invariant variables. We
then introduce harmonic tensors and write down the equations of motion for the
expansion coefficients of the gauge invariant perturbation variables in terms
of the harmonics. In particular, for the tensor-type perturbations a single
master equation is obtained in the (m+n)-dimensional background, which is
applicable for perturbation analysis of not only static black holes but also
some class of rotating black holes as well as black-branes. For the vector and
scalar type, we derive a set of decoupled master equations when the background
is a (2+n)-dimensional static black hole in the Einstein-Maxwell theory with a
cosmological constant. As an application of the master equations, we review the
stability analysis of higher dimensional charged static black holes with a
cosmological constant. We also briefly review the recent results of a
generalisation of the perturbation formulae presented here and stability
analysis to static black holes in generic Lovelock theory.Comment: Invited review for Prog. Theor. Phys. Suppl, 45 pages, 2 figures, 1
table, v2: references added, the notations slightly modified to match PTPS
published versio
Stability of Higher-Dimensional Schwarzschild Black Holes
We investigate the classical stability of the higher-dimensional
Schwarzschild black holes against linear perturbations, in the framework of a
gauge-invariant formalism for gravitational perturbations of maximally
symmetric black holes, recently developed by the authors. The perturbations are
classified into the tensor, vector, and scalar-type modes according to their
tensorial behaviour on the spherical section of the background metric, where
the last two modes correspond respectively to the axial- and the polar-mode in
the four-dimensional situation. We show that, for each mode of the
perturbations, the spatial derivative part of the master equation is a
positive, self-adjoint operator in the -Hilbert space, hence that the
master equation for each tensorial type of perturbations does not admit
normalisable negative-modes which would describe unstable solutions.
On the same Schwarzschild background, we also analyse the static perturbation
of the scalar mode, and show that there exists no static perturbation which is
regular everywhere outside the event horizon and well-behaved at spatial
infinity. This checks the uniqueness of the higher-dimensional spherically
symmetric, static, vacuum black hole, within the perturbation framework.
Our strategy for the stability problem is also applicable to the other
higher-dimensional maximally symmetric black holes with non-vanishing
cosmological constant. We show that all possible types of maximally symmetric
black holes (thus, including the higher-dimensional Schwarzschild-de Sitter and
Schwarzschild-anti-de Sitter black holes) are stable against the tensor and the
vector perturbations.Comment: 19 pages, 9 figures, references and comments on the generalised black
hole case are added, minor changes in text, version to appear in PT
A master equation for gravitational perturbations of maximally symmetric black holes in higher dimensions
We show that in four or more spacetime dimensions, the Einstein equations for
gravitational perturbations of maximally symmetric vacuum black holes can be
reduced to a single 2nd-order wave equation in a two-dimensional static
spacetime for a gauge-invariant master variable, irrespective of the mode of
perturbations. Our formulation applies to the case of vanishing as well as
non-vanishing cosmological constant Lambda. The sign of the sectional curvature
K of each spatial section of equipotential surfaces is also kept general. In
the four-dimensional Schwarzschild background, this master equation for a
scalar perturbation is identical to the Zerilli equation for the polar mode and
the master equation for a vector perturbation is identical to the Regge-Wheeler
equation for the axial mode. Furthermore, in the four-dimensional
Schwarzschild-anti-de Sitter background with K=0,1, our equation coincides with
those derived by Cardoso and Lemos recently. As a simple application, we prove
the perturbative stability and uniqueness of four-dimensional non-extremal
spherically symmetric black holes for any Lambda. We also point out that there
exists no simple relation between scalar-type and vector-type perturbations in
higher dimensions, unlike in four dimensions. Although we only treat maximally
symmetric black holes in the present paper, the final master equations are
valid even when the hirozon geometry is described by a generic Einstein
manifold.Comment: 22 pages in the PTP TeX style, no figure. The published versio
Perturbations and Stability of Static Black Holes in Higher Dimensions
In this chapter we consider perturbations and stability of higher dimensional
black holes focusing on the static background case. We first review a
gauge-invariant formalism for linear perturbations in a fairly generic class of
(m+n)-dimensional spacetimes with a warped product metric, including black hole
geometry. We classify perturbations of such a background into three types, the
tensor, vector and scalar-type, according to their tensorial behaviour on the
n-dimensional part of the background spacetime, and for each type of
perturbations, we introduce a set of manifestly gauge invariant variables. We
then introduce harmonic tensors and write down the equations of motion for the
expansion coefficients of the gauge invariant perturbation variables in terms
of the harmonics. In particular, for the tensor-type perturbations a single
master equation is obtained in the (m+n)-dimensional background, which is
applicable for perturbation analysis of not only static black holes but also
some class of rotating black holes as well as black-branes. For the vector and
scalar type, we derive a set of decoupled master equations when the background
is a (2+n)-dimensional static black hole in the Einstein-Maxwell theory with a
cosmological constant. As an application of the master equations, we review the
stability analysis of higher dimensional charged static black holes with a
cosmological constant. We also briefly review the recent results of a
generalisation of the perturbation formulae presented here and stability
analysis to static black holes in generic Lovelock theory.Comment: Invited review for Prog. Theor. Phys. Suppl, 45 pages, 2 figures, 1
table, v2: references added, the notations slightly modified to match PTPS
published versio
- …