Holographic stress tensors for Kerr-AdS black holes

Abstract

We use the counterterm subtraction method to calculate the action and the stress-energy-momentum tensor for (Kerr) rotating black holes in ${\mathrm{AdS}}_{n+1},$ for $n=2,$ 3, and 4. We demonstrate that the expressions for the total energy for the Kerr-${\mathrm{AdS}}_{3}$ and Kerr-${\mathrm{AdS}}_{5}$ spacetimes, in the limit of vanishing black hole mass, are equal to the Casimir energies of the holographically dual n-dimensional conformal field theories. In particular, for Kerr-${\mathrm{AdS}}_{5}$ spacetimes, dual to the case of the four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory on the rotating Einstein universe, we explicitly verify the equality of the zero mass stress tensor from the two sides of the correspondence, and present the result for a general mass as a prediction from gravity. Amusingly, it is observed in four dimensions that while the trace of the stress tensor defined using the standard counterterms does not vanish, its integral does, thereby keeping the action free of ultraviolet divergences. Using a different regularization scheme ``another'' stress tensor can be defined, which is traceless.