THE MASTER EQUATIONS IN THE EUCLIDEAN SCHWARZSCHILD–TANGHERLINI METRIC OF A SMALL STATIC PERTURBATION
Abstract
The master equations in the Euclidean Schwarzschild–Tangherlini space–time of a small static perturbation are studied. For each harmonic mode on the sphere there are two solutions that behave differently at infinity. One solution goes like the power 2-l-n of the radial variable, the other solution goes like the power l. These solutions occur in power series.
The second main statement of the paper is that any eigentensor of the Lichnerowicz operator in a Euclidean Schwarzschild space–time with an eigenvalue different from zero is essentially singular at infinity. Possible applications of the stability of instantons are discussed.
We present the analysis of a small static perturbation of the Euclidean Schwarzschild–Tangherlini metric tensor. The higher order perturbations will appear later. We determine independently the static perturbations of the Schwarzschild quantum black hole in dimension 1+n≥4, where the system of equations is reduced to master equations — ordinary differential equations. The solutions are hypergeometric functions which in some cases can be reduced to polynomials.
In the same Schwarzschild background, we analyze static perturbations of the scalar mode and show that there does not exist any static perturbation that is regular everywhere outside the event horizon and is well-behaved at the spatial infinity. This confirms the uniqueness of the spherically symmetric static empty quantum black hole, within the perturbation framework.
Our strategy for treating the stability problem is also applicable to other symmetric quantum black holes with a nonzero cosmological constant.
| You currently do not have access to the full text article. |
|---|
