Late-time tails of wave propagation in higher dimensional spacetimes

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 $D>\mathrm{}3\mathrm{}$ 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 tail is already present in the flat space Green’s function. On the other hand, for even $D>\mathrm{}4\mathrm{}$ 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 $D=4\mathrm{}$ case is special and exhibits, as is well known, $t^{-\mathrm{}(2l+3)\mathrm{}}$ 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 behavior of any field if the large extra dimensions are large enough.