Topological interpretation of the horizon temperature

Abstract

A class of metrics $g_{ab}(x^i)$ describing spacetimes with horizons (and associated thermodynamics) can be thought of as a limiting case of a family of metrics $g_{ab}(x^i;\lambda)$ {\it without horizons} when $\lambda\to 0$. I construct specific examples in which the curvature corresponding $g_{ab}(x^i;\lambda)$ becomes a Dirac delta function and gets concentrated on the horizon when the limit $\lambda\to 0$ is taken, but the action remains finite. When the horizon is interpreted in this manner, one needs to remove the corresponding surface from the Euclidean sector, leading to winding numbers and thermal behaviour. In particular, the Rindler spacetime can be thought of as the limiting case of (horizon-free) metrics of the form [$g_{00}=\epsilon^2+a^2x^2; g_{\mu\nu}=-\delta_{\mu\nu}$] or [$g_{00} = - g^{xx} = (\epsilon^2 +4 a^2 x^2)^{1/2}, g_{yy}=g_{zz}=-1]$ when $\epsilon\to 0$. In the Euclidean sector, the curvature gets concentrated on the origin of $t_E-x$ plane in a manner analogous to Aharanov-Bohm effect (in which the the vector potential is a pure gauge everywhere except at the origin) and the curvature at the origin leads to nontrivial topological features and winding number.