A new look at black holes via thermal dimensions and the complex coordinates/ temperature vectors correspondence

Abstract

The action of active diffeomorphisms (diffs) [Formula: see text] on the Schwarzschild metric leads to metrics which are also static spherically symmetric solutions of the Einstein vacuum field equations. It is shown how in a [Formula: see text] case it allows to introduce a deformation of the manifold such that [Formula: see text], and [Formula: see text] corresponding, respectively, to the spacelike singularity and horizon of the Schwarzschild metric. In doing so, one ends up with a spherical [Formula: see text] surrounding the singularity at [Formula: see text]. In order to explore the “interior” region of this void, we introduce complex radial coordinates whose imaginary components have a direct link to the inverse Hawking temperature, and which furnish a path that provides access to interior region. In addition, we show that the black hole entropy [Formula: see text] (in Planck units) is equal to the [Formula: see text] of a rectangular strip in the [Formula: see text] radial-coordinate plane associated to this path. The gist of the physical interpretation behind this construction is that there is an emergence of thermal dimensions which unfolds as one plunges into the interior void region via the use of complex coordinates, and whose imaginary components capture the span of the thermal dimensions. Namely, the filling of the void leads to an [Formula: see text] internal/ thermal dimension via the imaginary part [Formula: see text] of the complex radial variable [Formula: see text].