Abstract
We consider linear gravitational perturbations of the Kerr brane, an exact solution of vacuum Einstein's equations in dimensions higher than four and a low-energy solution of string theory. Decomposing the perturbations in tensor harmonics of the transverse Ricci-flat space, we show that tensor- and vector-type metric perturbations of the Kerr brane satisfy respectively a massive Klein-Gordon equation and a Proca equation on the four-dimensional Kerr space, where the mass term is proportional to the eigenvalue of the harmonics. Massive bosonic fields trigger a well-known superradiant instability on a Kerr black hole. We thus establish that Kerr branes in dimensions $D\geq6$ are gravitationally unstable due to superradiance. These solutions are also unstable against the Gregory-Laflamme instability and we discuss the conditions for either instability to occur and their rather different nature. When the transverse dimensions are compactified and much smaller than the Kerr horizon, only the superradiant instability is present, with a time scale much longer than the dynamical time scale. Our formalism can be also used to discuss other types of higher-dimensional black objects, taking advantage of recent progress in studying linear perturbations of four-dimensional black holes.
Highlights
JHEP09(2015)209 in contrast to the GL instability, the superradiant instability is expected to occur even when the length scale of the transverse dimensions is smaller than the horizon radius of the rotating hole
Decomposing the perturbations in tensor harmonics of the transverse Ricci-flat space, we show that tensor- and vector-type metric perturbations of the Kerr brane satisfy respectively a massive Klein-Gordon equation and a Proca equation on the four-dimensional Kerr space, where the mass term is proportional to the eigenvalue of the harmonics
We first note that metric perturbations of the Kerr brane in D = 4 + n ≥ 7 dimensions can in general be classified into three types according to their tensorial behavior on the n-dimensional transverse Ricci flat dimensions: the tensor, vector, and scalar-type in terminology of ref. [37]. (For n = 2, only two classes exist: the vector- and scalar-type, whereas for n = 1, i.e., Kerr-string, only scalar-type perturbations can be defined.2) In the present paper, we will focus on tensor- (n ≥ 3) and vector- (n ≥ 2) type perturbations
Summary
The Kerr brane is the direct product of the four-dimensional spacetime described by the Kerr metric and an n-dimensional Ricci-flat space K, ds2 = gMN dxM dxN = gaKberrdyadyb + R2γij dzidzj ,. Where (for instance, in Boyer-Lindquist coordinates) ya = (t, r, θ, φ), γij is the metric of K, and R is a constant with dimensions of a length. The Ricci-flatness of γij and the fact that gaKberr is a solution of the four-dimensional vacuum Einstein’s equations guarantee that the full metric gMN is a solution of the D = (4+n)-dimensional Einstein’s equations in vacuum
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