Static black holes with a negative cosmological constant: Deformed horizon and anti–de Sitter boundaries

Using perturbative techniques, we investigate the existence and properties of a static solution for the Einstein equation with a negative cosmological constant, which we call the deformed black hole. We derive a solution for a static and axisymmetric perturbation of the Schwarzschild–anti-de Sitter black hole that is regular in the range from the horizon to spacelike infinity. The key result is that this perturbation simultaneously deforms the two boundary surfaces—i.e., both the horizon and spacelike two-surface at infinity. Then we discuss the Abbott-Deser mass and the Ashtekar-Magnon one for the deformed black hole, and according to the Ashtekar-Magnon definition, we construct the thermodynamic first law of the deformed black hole. The first law has a correction term which can be interpreted as the work term that is necessary for the deformation of the boundary surfaces. Because the work term is negative, the horizon area of the deformed black hole becomes larger than that of the Schwarzschild–anti-de Sitter black hole, if compared under the same mass, indicating that the quasistatic deformation of the Schwarzschild–anti-de Sitter black hole may be compatible with the thermodynamic second law (i.e., the area theorem).