On Geometro-thermodynamics of Dilaton Black Holes

In this talk we present the latest results from our ongoing project on geometrothermodynamics (also known as information geometry of thermodynamics or Ruppeiner geometry) of dilaton BHs in 4D in both Einstein and string frames and a dyonic dilaton BH and at the end we report very briefly results from this approach to the 2D dilaton BHs. The thermodynamic geometry, also known as Ruppeiner geometry (Ruppeiner [1979,1995]), of various BH families has been studied over the past few years (see, e.g. Åman et al. [2003][2007], Arcioni et al. [2005], Sarkar et al. [2006], Shen et al. [2006], Mirza et al. [2007], Ruppeiner [2007] and Quevedo [2007]). Our results so far have been physically suggestive, particularly in the MyersPerry Kerr BH case where the curvature singularities signal thermodynamic instability of the BH. The geometrical patterns are given by the curvature of the Ruppeiner metric defined as the Hessian of the entropy on the phase space of the thermodynamic system g ij = −∂i∂jS(M,N), (1) where M denotes mass (internal energy) and N are other parameters such as charge and spin. The minus sign arises because entropy is a concave function. Interpretations of the geometries associated with the metric are discussed in Ruppeiner [1995] and references therein. Even though most interesting Ruppeiner metrics that we encounter have curvature singularities which might be interpretable, there are some known flat Ruppeiner metrics that shed light on the understanding of thermodynamic geometries as a whole i.e. the structure of the entropy function (or mass) that gives a flat Ruppeiner geometry. In Åman et al. [2006] we proved a flatness theorem which states that Riemann curvature tensor constructed out of the negative of the Hessian of the entropy of the form S = Mf(Q/M) (2) [email protected] [email protected] [email protected] 4This metric is conformal to the so-called Weinhold (Weinhold [1975]) metric via g ij = Tg R ij where T is thermodynamic temperature of the system of interest.