Nonassociative black holes in R-flux deformed phase spaces and relativistic models of Perelman thermodynamics

Abstract

This paper explores new classes of black hole (BH) solutions in nonassociative and noncommutative gravity, focusing on features that generalize to higher dimensions. The theories we study are modelled on (co) tangent Lorentz bundles with a star product structure determined by R-flux deformations in string theory. For the nonassociative vacuum Einstein equations we consider both real and complex effective sources. In order to analyze the nonassociative vacuum Einstein equations we develop the anholonomic frame and connection deformation methods, which allows one to decoupled and solve these equations. The metric coefficients can depend on both space-time coordinates and energy-momentum. By imposing conditions on the integration functions and effective sources we find physically important, exact solutions: (1) 6-d Tangherlini BHs, which are star product and R-flux distorted to 8-d black ellipsoids (BEs) and BHs; (2) nonassocitative space-time and co-fiber space double BH and/or BE configurations generalizing Schwarzschild-de Sitter metrics. We also investigate the concept of Bekenstein-Hawking entropy and find it applicable only for very special classes of nonassociative BHs with conventional horizons and (anti) de Sitter configurations. Finally, we show how analogs of the relativistic Perelman W-entropy and related geometric thermodynamic variables can be defined and computed for general classes of off-diagonal solutions with nonassociative R-flux deformations.