High and low dimensions in the black hole negative mode

Abstract

The negative mode of the Schwarzschild black hole is central to Euclidean quantum gravity around hot flat space and for the Gregory-Laflamme black string instability. We analyze the eigenvalue as a function of space-time dimension λ = λ(d) by constructing two perturbative expansions: one for large d and the other for small d − 3, and determining as many coefficients as we are able to compute analytically. Joining the two expansions we obtain an interpolating rational function accurate to better than 2% through the whole range of dimensions including d = 4.

Quasinormal modes of (anti-)de Sitter black holes in the 1/D expansion

“…In figure 2 we compare this result to the values found in [4] from the numerical solution of the problem. Forn = 2, e.g., a black string in D = 6, the numerical value is k GL = 1.269, while (4.28) gives k GL = 1.238, which is off by 2.4%.…”

“…(4.27) gives a better overall fit to numerical calculations of the curve Ω + (k) only whenn 8. 4 Note however that (5.5) implies that the accuracy at small k can be considerably better than indicated by (5.7). We elaborate this point in the next section.…”

“…In more detail, when D is very large the gravitational field of a black hole gets strongly localized close to the horizon [4,5]. The existence of a well-defined 'near-horizon region' [6] splits the quasinormal spectrum into two distinct sets that capture very different black hole dynamics.…”

Analytic evidence for the Gubser–Mitra conjecture

“…The resultant expression of perturbation equation is so simple that it allows for analytic investigations or accurate numerical ones. For example, an approximation formula of the dimensional dependence of GL marginal mode, which is sufficiently accurate in all dimensions, was obtained in [17], while solving the perturbation equation for general dimensions is difficult even in such a gauge.…”

LargeDlimit of dimensionally continued gravity

“…It would be interesting to study the instability of p-brane solutions of this theory in a similar way to what has been done in Refs. [16,17] at large D. The analysis of mechanical stability, on the other hand, can hardly be accomplished for these theories. This is mainly because of two reasons: First, the higher-curvature terms in the action introduce higher powers of the derivatives that make the complexity of the equations to grow dramatically even for large D. Secondly, the special theories that are being selected by demanding (6) have the property of having a unique maximally symmetric vacuum, and this produces that the equations of motion factorize in a way that the first orders in perturbation theory identically vanish, making necessary to go beyond the linear approximation.…”