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 ) 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 ^ { 2 } x ^ { 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 .