Time-symmetric initial data of large brane-localized black hole in RS-II model

Abstract

In the aim of shedding a new light on the classical black hole evaporation conjecture stating that a static brane-localized black hole (BH) larger than the bulk curvature scale does not exist in Randall-Sundrum II (RS-II) model, we investigate time-symmetric initial data with a brane-localized apparent horizon (AH) and analyzed its properties. We find that a three-parameter family of such initial data can be constructed by simply placing a brane on a constant time surface of Schwarzschild anti-de Sitter space. By this method, we unambiguously confirm that initial data with an arbitrarily large AH area do exist. We compare the ADM mass and the horizon area of our initial data with that of the black string (BS) solution. If there is a sequence of static brane-localized BH solutions, such solutions should be contained in the time-symmetric initial data. Moreover, if they are stable, it will have a smaller mass compared with the BS solution with the same horizon area. However, we find that any initial data constructed by this method do not have a smaller mass than the BS solution when the horizon area is larger than the size determined by the bulk curvature scale. We further investigate what kind of configuration realizes the minimum mass for the same AH area. The configuration that realizes the smallest mass turns out to be the one close to the BS truncated by a cap. One may think that this indicates the existence of a static brane-localized BH solution. However, since our three-parameter family of initial data does not include the configuration resembling the BS solution, this minimum of mass may just reflect the expected minimum of mass corresponding to the BS solution. We also demonstrate that the same method applies to construct initial data in (3+1)-dimensional RS-II brane world. In this case an exact solution of a brane-localized BH exists but BS solution does not. Nevertheless, the behavior of the initial data is quite similar in both cases. We find that the known exact solution always has a smaller mass than our initial data with the same horizon area. This result enforces the standard belief that the exact BH solution is the most stable black object in the four-dimensional RS-II model. These results are all consistent with the classical BH evaporation conjecture, but unfortunately it turns out that they do not provide a strong support of it.