Remark on the effective potential of the gravitational perturbation in the black hole background projected on the brane

摘要

The polar perturbation is examined when the spacetime is expressed by a 4d metric induced from higher-dimensional Schwarzschild geometry. Since the spacetime background is not a vacuum solution of 4d Einstein equation, the various general principles are used to understand the behaviour of the energy momentum tensor under the perturbation. It is found that although the general principles fix many components, they cannot fix two components of the energy momentum tensor. Choosing two components suitably, we derive the effective potential which has a correct 4d limit.