Higher-dimensional black hole Geometric Thermodynamics

The pseudo-Riemannian Geometry is utilized in Mathematical Optimization, Thermodynamics or in Statistics as an important tool for recent research. Given a pseudo-Riemannian manifold (M, g) and a smooth function f : M -> R, whose Hessian with respect to g is non-degenerate, one can define on M the associated pseudo-Riemannian Hessian metric h = Hess(g)f. In the following, we apply this methodology for describing the geometrical properties of some interesting mathematical objects like the higher dimensional Reissner-Nordstrom black holes. The paper is organized as follows. Section 1 reviews the history of pseudo-Riemannian Geometry and points out which Hessian is more convenient for physical problems. Section 2 gives the Christoffel symbols and the system of geodesics of pseudo-Riemannian manifold (M, h = Hess(g)f), establishes the relation between the components of the curvature. tensors field of (M,h) and (M,g), and determines the PDEs representing the coincidence between the Christoffel symbols of (AY, h) and the Christoffel symbols of (ill,, g). The last section presents the comparison of null-length curves trajectories obtained with the two metrics for a 5-dimensional RN black hole.