Mass of Rotating Black Holes in Gauged Supergravities

TL;DR

This work evaluates the masses of several rotating black holes in gauged supergravities using the AMD conformal mass definition, demonstrating exact agreement with masses obtained from the first law of thermodynamics across D=5 (minimal and U(1)^3) and higher-dimensional cases (D=4, D=7). It also analyzes the Abbott-Deser method, showing consistency in some cases but critical ambiguities when scalar fields are present, and introduces scalar-field–related corrections that restore agreement with AMD for the problematic solutions. The Euclidean action and quantum statistical relation are checked for the 5D minimal case, reinforcing the thermodynamic–statistical consistency. The results underscore the robustness of the AMD approach (which avoids background/deviation decompositions) while clarifying the limitations and necessary refinements of AD in the presence of scalars, with implications for energy definitions in AdS/CFT contexts. Overall, the paper clarifies how conserved-quantity integrals encode black hole energetics in gauged supergravities and provides practical prescriptions for computing masses in a broad class of AdS rotating black holes.

Abstract

The masses of several recently-constructed rotating black holes in gauged supergravities, including the general such solution in minimal gauged supergravity in five dimensions, have until now been calculated only by integrating the first law of thermodynamics. In some respects it is more satisfactory to have a calculation of the mass that is based directly upon the integration of a conserved quantity derived from a symmetry principal. In this paper, we evaluate the masses for the newly-discovered rotating black holes using the conformal definition of Ashtekar, Magnon and Das (AMD), and show that the results agree with the earlier thermodynamic calculations. We also consider the Abbott-Deser (AD) approach, and show that this yields an identical answer for the mass of the general rotating black hole in five-dimensional minimal gauged supergravity. In other cases we encounter discrepancies when applying the AD procedure. We attribute these to ambiguities or pathologies of the chosen decomposition into background AdS metric plus deviations when scalar fields are present. The AMD approach, involving no decomposition into background plus deviation, is not subject to such complications. Finally, we also calculate the Euclidean action for the five-dimensional solution in minimal gauged supergravity, showing that it is consistent with the quantum statistical relation.