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

**Authors:** Carlos Castro

We argue why the static spherically symmetric (SSS) vacuum solutions of Einstein's
equations described by the textbook Hilbert metric g_{μν}(r) is not diffeomorphic
to the metric g_{μν}(|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, r_{h} → 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 ℏ 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.

**Comments:** 41 pages, This article appeared in the Journal of Mathematical Physics, vol 49 (2008) 042501.

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### Submission history

[v1] 26 Aug 2009

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