The Einstein–Hilbert action is generalized to higher dimensions in the form
This is a straightforward generalization, and the only aspect that deserves some attention is the implicit definition of Newton’s constantMass, angular momenta, and other conserved charges of isolated systems are defined through comparison to the field created near asymptotic infinity by a weakly gravitating system ([154] gives a careful Hamiltonian analysis of conserved charges in higher-dimensional asymptotically-flat spacetimes). The Einstein equations for a small perturbation around flat Minkowski space
in linearized approximation take the conventional form where Since the sources are localized and we work at linearized perturbation order, the fields in the asymptotic
region are the same as those created by point-like sources of mass and angular momentum
with antisymmetric matrix
, at the origin
of flat space in Cartesian coordinates,
Given the abundance of black-hole solutions in higher dimensions, one is interested in comparing
properties, such as the horizon area , of different solutions characterized by the same set of parameters
. A meaningful comparison between dimensionful magnitudes requires the introduction
of a common scale, so the comparison is made between dimensionless magnitudes obtained
by factoring out this scale. Since classical general relativity in vacuum is scale invariant, the
common scale must be one of the physical parameters of the solutions, and a natural choice is
the mass. Thus we introduce dimensionless quantities for the spins
and the area
,
Note that, with our definition of the gravitational constant , both the Newtonian gravitational
potential energy,
To warm up before dealing with black holes, we follow John Michell and Simon de Laplace and compute,
using Newtonian mechanics, the radius at which the escape velocity of a test particle in this field reaches
the speed of light. The kinetic energy of a particle of unit mass with velocity is
, so the equation
that determines the Michell–Laplace ‘horizon’ radius is
Consider the linearized solution above for a static source (14) in spherical coordinates, and pass to a gauge
where
is the area radius,
Having this elementary class of black-hole solutions, it is easy to construct other vacuum solutions with
event horizons in . The direct product of two Ricci-flat manifolds is itself a Ricci-flat manifold. So,
given any vacuum black-hole solution
of the Einstein equations in
dimensions, the metric
These black brane spacetimes are not (globally) asymptotically flat, so we only introduce them insofar as they are relevant for understanding the physics of asymptotically-flat black holes.
The stability of the Schwarzschild solution against linearized gravitational perturbations can be
analyzed by decomposing such perturbations into scalar, vector and tensor types according to how they
transform under the rotational-symmetry group
[105
, 163
, 151
]. Assuming a time
dependence
and expanding in spherical harmonics on
, the equations governing each type of
perturbation reduce to a single ODE governing the radial dependence. This equation can be
written in the form of a time-independent Schrödinger equation with “energy” eigenvalue
.
In investigating stability, we consider perturbations that are regular on the future horizon and
outgoing at infinity. An instability would correspond to a mode with . For such
modes, the boundary conditions at the horizon and infinity imply that the left-hand side (LHS)
of the Schrödinger equation is self-adjoint, and hence
is real. Therefore, an unstable
mode must have negative imaginary
. For tensor modes, the potential in the Schrödinger
equation is manifestly positive, hence
and there is no instability [105]. For vectors and
scalars, the potential is not everywhere positive. Nevertheless, it can be shown that the operator
appearing on the LHS of the Schrödinger equation is positive, hence
and there is no
instability [151
]. In conclusion, the
Schwarzschild solution is stable against linearized gravitational
perturbations.
The instabilities of black strings and black branes [118, 119] have been reviewed in [167, 128], so we shall
be brief in this section and only mention the features that are most relevant to our subject. We shall only
discuss neutral black holes and black branes; when charges are present, the problem becomes more
complex.
This instability is the prototype for situations in which the size of the horizon is much larger in some
directions than in others. Consider, as a simple, extreme case of this, the black string obtained by adding a
flat direction to the Schwarzschild solution. One can decompose linearized gravitational perturbations
into scalar, vector and tensor types according to how they transform with respect to transformations of the
Schwarzschild coordinates. Scalar and vector perturbations of this solution are stable [117
]. Tensor
perturbations that are homogeneous along the
-direction are also stable, since they are the same as
tensor perturbations of the Schwarzschild black hole. However, there appears to be an instability for
long-wavelength tensor perturbations with nontrivial dependence on
; the frequency
of
perturbations
acquires a positive imaginary part when
, where
is the
Schwarzschild horizon radius. Thus, if the string is compactified on a circle of length
, it
becomes unstable. Of the unstable modes, the fastest one (with the largest imaginary frequency) occurs for
roughly one half of
. The instability creates inhomogeneities along the direction of the string.
Their evolution beyond the linear approximation has been followed numerically in [42]. It is unclear yet
what the endpoint is; the inhomogeneities may well grow until a sphere pinches down to a
singularity.3
In this case, the Planck scale will be reached along the evolution, and fragmentation of the black string into
black holes, with a consistent increase in the total horizon entropy, may occur.
Another important feature of this phenomenon is the appearance of a zero-mode (i.e., static)
perturbation with . Perturbing the black string with this mode yields a new static solution
with inhomogeneities along the string direction [117
, 120
]. Following numerically these static
perturbations beyond the linear approximation has given a new class of inhomogeneous black
strings [248
].
These results easily generalize to black -branes; for a wavevector
along the
directions tangent
to the brane, the perturbations
with
are unstable. The value of
depends on the codimension of the black brane, but not on
.
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