Late-Time Tails of Wave Propagation in Higher Dimensional Spacetimes

TL;DR

This paper analyzes how massless fields decay over time near higher-dimensional Schwarzschild black holes, revealing that decay rates depend on whether the spacetime dimension is odd or even, with implications for theories involving extra dimensions.

Contribution

It provides a detailed characterization of late-time tail decay rates of massless fields in higher-dimensional Schwarzschild spacetimes, highlighting the dependence on spacetime dimensionality and the presence of extra dimensions.

Findings

Odd D>3 fields decay as t^[-(2l+D-2)]

Even D>4 fields decay as t^[-(2l+3D-8)]

D=4 case exhibits t^[-(2l+3)] decay behavior

Abstract

We study the late-time tails appearing in the propagation of massless fields (scalar, electromagnetic and gravitational) in the vicinities of a D-dimensional Schwarzschild black hole. We find that at late times the fields always exhibit a power-law falloff, but the power-law is highly sensitive to the dimensionality of the spacetime. Accordingly, for odd D>3 we find that the field behaves as t^[-(2l+D-2)] at late times, where l is the angular index determining the angular dependence of the field. This behavior is entirely due to D being odd, it does not depend on the presence of a black hole in the spacetime. Indeed this tails is already present in the flat space Green's function. On the other hand, for even D>4 the field decays as t^[-(2l+3D-8)], and this time there is no contribution from the flat background. This power-law is entirely due to the presence of the black hole. The D=4 case is special and exhibits, as is well known, the t^[-(2l+3)] behavior. In the extra dimensional scenario for our Universe, our results are strictly correct if the extra dimensions are infinite, but also give a good description of the late time behaviour of any field if the large extra dimensions are large enough.