Topological interpretation of the horizon temperature

Abstract

A class of metrics g_ab(xⁱ) describing spacetimes with horizons (and associated thermodynamics) can be thought of as a limiting case of a family of metrics g_ab(xⁱ;λ) without horizons when λ→0. We construct specific examples in which the curvature corresponding gab(xⁱ;λ) becomes a Dirac delta function and gets concentrated on the horizon when the limit λ→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 behavior. In particular, the Rindler spacetime can be thought of as the limiting case of (horizon-free) metrics of the form [g₀₀=ε²+a²x²; g_μν=-δ_μν] or [g₀₀=-g^xx=(ε²+4a²x²) ^½, g_yy=g_zz=−1] when ε→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 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.