The Euclidean Gravitational Action as Black Hole Entropy, Singularities and Spacetime Voids

We argue why the static spherically symmetric (SSS) vacuum solutions of Einstein’s equations described by the textbook Hilbert metric gμν(r) is not diffeomorphic to the metric gμν(|r|) corresponding to the gravitational field of a point mass delta function source at r = 0. By choosing a judicious radial function R(r) = r + 2G|M |Θ(r) involving the Heaviside step function, one has the correct boundary condition R(r = 0) = 0 , while displacing the horizon from r = 2G|M | to a location arbitrarily close to r = 0 as one desires, rh → 0, where stringy geometry and quantum gravitational effects begin to take place. We solve the field equations due to a delta function point mass source at r = 0, and show that the Euclidean gravitational action (in h̄ units) is precisely equal to the black hole entropy (in Planck area units). This result holds in any dimensions D ≥ 3 . In the Reissner-Nordsrom (massive-charged) and Kerr-Newman black hole case (massive-rotating-charged) we show that the Euclidean action in a bulk domain bounded by the inner and outer horizons is the same as the black hole entropy. When one smears out the point-mass and point-charge delta function distributions by a Gaussian distribution, the areaentropy relation is modified. We postulate why these modifications should furnish the logarithmic corrections (and higher inverse powers of the area) to the entropy of these smeared Black Holes. To finalize, we analyse the Bars-Witten stringy black hole in 1 + 1 dim and its relation to the maximal acceleration principle in phase spaces and Finsler geometries.