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Foundations of PhysicsAn International Journal Devoted tothe Conceptual Bases and FundamentalTheories of Modern Physics ISSN 0015-9018 Found PhysDOI 10.1007/s10701-015-9954-x

Novel Remarks on Point Mass Sources,Firewalls, Null Singularities andGravitational Entropy

Carlos Castro Perelman

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Found PhysDOI 10.1007/s10701-015-9954-x

Novel Remarks on Point Mass Sources, Firewalls, NullSingularities and Gravitational Entropy

Carlos Castro Perelman1,2

Received: 14 March 2015 / Accepted: 11 September 2015© Springer Science+Business Media New York 2015

Abstract A continuous family of static spherically symmetric solutions of Einstein’svacuum field equations with a spatial singularity at the origin r = 0 is found. Thesesolutions are parametrized by a real valued parameter λ (ranging from 0 to 1) and suchthat the radial horizon’s location is displaced continuously towards the singularity(r = 0) as λ increases. In the extreme limit λ = 1, the location of the singularityand horizon merges leading to a null singularity. In this extreme case, any infallingobserver hits the null singularity at the very moment he/she crosses the horizon. Thisfact may have important consequences for the resolution of the fire wall problem andthe complementarity controversy in black holes. An heuristic argument is providedhow onemight avoid the Hawking particle emission process in this extreme case whenthe singularity and horizon merges. The field equations due to a delta-function point-mass source at r = 0 are solved and the Euclidean gravitational action correspondingto those solutions is evaluated explicitly. It is found that the Euclidean action is pre-cisely equal to the black hole entropy (in Planck area units). This result holds in anydimensions D ≥ 3.

Keywords General relativity · Black holes · Strings

B Carlos Castro Perelmanperelmanc@hotmail.com

1 Universidad Tecnica Particular de Loja, San Cayetano Alto, 1101608 Loja, Ecuador

2 Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, GA 30314,USA

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1 Family of Static Spherically Symmetric Solutions

It is not widely known that there are static spherically symmetric (SSS) vacuumsolutions of Einstein’s equations [7], beyond the Hilbert [14,30] and Schwarzschild[23] solutions,

(ds)2 =(1 − 2GM

r

)(dt)2 −

(1 − 2GM

r

)−1

(dr)2 − r2(d�)2. (1.1a)

and which are given by a family of metrics parametrized by a family of areal radialfunctions ρλ(r) (in c = 1units), in terms of a real parameter 0 ≤ λ < 1, as follows

(ds)2(λ) =(1 − 2GM

ρλ(r)

)(dt)2 −

(1 − 2GM

ρλ(r)

)−1

(dρλ)2 − ρ2

λ(r)(d�)2, (1.1b)

where (dρλ)2 = (dρλ(r)/dr)2(dr)2 and the solid angle infinitesimal element is

(d�)2 = (dφ)2 + sin2(φ)(dθ)2. The surface area enclosed by each point at r isnow given by 4π(ρλ(r))2 so that ρλ(r) plays the role of an effective radius and hencethe name of “areal-radial” function for ρλ(r).

In the Appendix we show explicitly that the metric (1.1b) is a solution to Einstein’svacuum field equations. This expression for the family of metrics is given in terms ofthe family of areal radial functions ρλ(r) which does not violate Birkhoff’s theoremsince themetric (1.1b) expressed in terms of the areal radial functionsρλ(r) has exactlythe same functional form as that required by Birkoff’s theorem. It is well known thatthe extended Schwarzschild metric solution for r < 0 with M > 0, correspondsto a solution in the region r > 0 with M < 0. Negative masses are associated withrepulsive gravity. For this reason, the domain of values of r will be chosen to span thewhole real axis −∞ ≤ r ≤ ∞.

One should emphasize that despite that the metric (1.1b) has the same appearanceas the Schwarzschild solution (1.1a), by replacing r → ρλ(r) due to a symmetry ofEinstein’s equations, it is very di f f erent . We have not relabeled the radial variabler by giving it another name and calling it “ρλ”, because ρλ(r) is itself a function of r .The metric (1.1b) leads to modifications of the Newtonian potential at short distances.One recovers the Newtonian potential in the asymptotic regime when ρλ(r) ∼ r .

We must stress also that the radial metric components grr of the Hilbert–Schwarzschild solution (1.1a) is not given by a radial reparametrization of the gρρ

component of the metric (1.1b). Let us suppose that one had a radial reparametrizationof the form r → ρ(r). The transformation law for the metric components is given by

grr (r) = −(1 − 2GM/r)−1 = −(1 − 2GM/ρ(r))−1(dρ(r)

dr

)2

= gρρ

(dρ(r)

dr

)2

(1.2a)

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An integration of Eq. (1.2a), after relabeling the variables as r ′, ρ′, respectively, gives

I =∫ ρ

ρ′=0

dρ′√1 − 2GM/ρ′ =

∫ r

r ′=0

dr ′√1 − 2GM/r ′ ⇒ I (ρ) − I (0) = I (r) − I (0)

(1.2b)

where the integral I (ρ) is

I (ρ) = ρ

√1 − 2GM

ρ+ GM log

∣∣∣∣∣(2ρ[

√1 − 2GM

ρ+ 1] − 2GM

)∣∣∣∣∣ (1.2c)

and an identical integral for I (r). I (0) = GM log| − 2GM |. Since the functionalforms of I in the last terms of Eq. (1.2b) are identical, and because one learnsfrom Eq. (1.2b) that I (ρ(r)) = I (r), then this leads to the trivial reparametrizationρ(r) = r . Therefore, the metric (1.1b) is not a radial reparametrization of the Hilbert–Schwarzschild solution (1.1a) because one has in general that (1−2GM/r)−1( drdρ )2 =(1 − 2GM/ρ)−1 when ρ(r) = r . Furthermore, a radial reparametrization of theHilbert–Schwarzschild solution does not affect the angular components of the met-ric, so that onewould have had r2(d�)2 → r2(d�)2 = ρ2

λ(r)(d�)2 unless ρλ(r) = r .The particular choice for the function ρ(r) = r yields the standard Hilbert–

Schwarzschild solution, whereas the infinite family of metric solutions (1.1b) areassociated to an infinite number of modi f ications to the gravitational Newtonianpotential, in the weak field limit. Experiments will tell what type of corrections to theNewtonian potential at shorter distances occur and which will fix the functional formof the areal radial function ρ(r) (which in the Appendix we denote by R(r) and mustnot be confused with the scalar curvature R(r)).

The boundary conditions obeyed by the areal radial function ρλ(r) must be ρλ(r =0) = 0, ρλ(r = ∞) = ∞. The Hilbert textbook (black hole) solution [14,30] whenρ(r) = r obeys the boundary conditions but the Abrams–Brillouin [2] choice ρ(r) =r + 2GM does not . The original solution of 1916 found by Schwarzschild for ρ(r)did not obey the boundary condition ρ(r = 0) = 0 as well. The condition ρ(r = 0) =2GM has a serious f law and is : how is it possible for a point-mass at r = 0 to havea non-zero area 4π(2GM)2 and a zero volume simultaneously ?; so it seems thatone is forced to choose the Hilbert areal radial function ρ(r) = r . It is known thatfractals have unusual properties related to their lengths, areas, volumes, dimensionsbut we are not focusing on fractal spacetimes at the moment. For instance, one couldhave a fractal surface of infinite area but zero volume (space-filling fractal surface).The finite area of 4π(2GM)2 could then be seen as a regularized value of the infinitearea of a “fractal horizon”.

The Hilbert choice for the areal radial function ρ(r) = r is ultimately linked to theactual form of the Newtonian potential VN = −(Gm1m2/r). In the last few decadescorrections to Newton’s law of gravitation and constraints on them have become thesubject of considerable study, see the monograph [8]. Yukawa-type corrections toNewton’s gravitational law from two recent measurements of the Casimir interac-

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tion between metallic surfaces was studied by [15]. A Yukawa-like correction to theNewtonian potential could be chosen to be

V (r) = −Gm1m2

r(1 − λe−r/2GM ), λ > 0 (1.3a)

where λ and ro = 2GM are the strength and interaction range of the Yukawa-typecorrection. One may notice that the potential (1.3a) can be rewritten in terms of anareal-radial function ρ(r) as

V (r) = −Gm1m2

ρ(r), ρ(r) = r

1 − λe−r/2GM, λ = 1 (1.3b)

One has the correct boundary conditions for the areal radial function when λ = 1

ρ(r = 0) = 0; ρ(r = rh) = 2GM; 0 ≤ rh ≤ 2GM (1.3c)

so that the location of the horizon radius rh has been shi f ted towards the singularity.In the asymptotic regime one has as expected ρ(r → ∞) → r , so that the areal-radialfunction tends to r (as in the Hilbert choice) and the expression for the potential isasymptotic to the Newtonian one.

One may notice in Eq. (1.3b) that when λ = 1 then ρ(λ=1)(r = 0) = 2GM afterusing L’Hopital’s rule. Since ρ(r = 0) is not zero one must rule out the value λ = 1.Therefore, the range of values for the parameter λ must be 0 ≤ λ < 1. When λ = 0one recovers the ordinary Hilbert solution ρ(r) = r . Also, the derivatives (dρλ(r)/dr)are finite (1/1 − λ) at r = 0 but blow up at r = 0 when λ = 1.

Instead of the Yukawa-type areal radial function (1.3b), one could have had a dif-ferent one-parameter family of areal-radial functions ρλ(r).1 For example, the metricsolutions (1.1b) are invariant under the transformations r → −r; M → −M forour particular choice of the areal radial functions ρλ(r) given in Eq. (1.3b) and due tothe condition ρλ(−r,−M) = −ρλ(r, M). This allows us to extended the solutions tothe r < 0 region.

For a recent analysis of the properties of the maximal extensions (in regions r < 0)of the Kerr and Kerr–Newman spacetimes with negative mass, see [18]. Negativemass (or regions of negative mass density) imply violations of one or another variantof the positive energy condition of Einstein’s general theory of relativity; however,the positive energy condition is not a required condition for the mathematical consis-tency of the theory as shown by [29]. The authors [27] pointed out that the quantummechanics of the Casimir effect can be used to produce a locally mass-negative regionof space-time. In this article, and subsequent work by others, they showed that nega-tive matter could be used to stabilize a wormhole. Therefore, the solutions describedabove make sense once one takes into consideration the negative and positive domainof values of r = ±√

x2 + y2 + z2, consistent with the ± signs under the square root.The Penrose diagrams associated with the solutions described in (1.1b) are the

same as the diagrams corresponding to the extended Schwarzchild solutions with the

1 We thank Matej Pavsic for a discussion on other choices for the radial functions.

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only difference that we must replace the radial variable r for ρ. The horizons at theradial locations r (λ)

h all correspond to the unique value of the areal radial function

ρ(r (λ)h ) = 2GM and t = ±∞. The spatial singularity is located at ρλ(r = 0) = 0.

The Fronsdal–Kruskal–Szekeres [10,17,27] change of coordinates that permit an ana-lytical extension into the interior region of the black hole has the same functional formas before after replacing r for ρ. In the exterior region ρ(r) > 2GM one has

U =(

ρ(r)

2GM− 1

) 12

eρ(r)/4GMcosh

(t

4GM

),

V =(

ρ(r)

2GM− 1

) 12

eρ(r)/4GMsinh

(t

4GM

); (1.4a)

and the change of coordinates in the interior region ρ(r) < 2GM is

U =(1 − ρ(r)

2GM

) 12

eρ(r)/4GMsinh

(t

4GM

),

V =(1 − ρ(r)

2GM

) 12

eρ(r)/4GMcosh

(t

4GM

)(1.4b)

In the overlap ρ(r) = 2GM region one hasU = ±V when t = ±∞, andU = V = 0for f ini te t .

The coordinate transformations lead to a well behaved metric (except at ρ(r =0) = 0)

ds2 = 4(2GM)3

ρ(U, V )e−ρ(U,V )/2GM (dV 2 − dU 2) − ρ(U, V )2(d�)2. (1.4c)

When ρ(r = rhori zon) = 2GM and d� = 0, the above interval displacement ds2 = 0is null along the lines U = ±V ⇒ dU = ±dV . The interval is singular ds2 = ∞at ρ(r = 0) = 0; i.e. the singularity r = 0 corresponds to the spacelike linesV 2 −U 2 = 1 ⇒ dV 2 − dU 2 = −dU 2/(U 2 + 1) < 0.

In the extreme limiting case λ → 1 any infalling observer reaches a horizon r (λ)h

whose location approaches arbitrarily close to the singularity r = 0. To model thescenario when the horizon merges precisely with the singularity one needs an arearadial function defined as follows

ρ(r = 0) = 0, ρ(r) = r

1 − e−r/2GM, r > 0 (1.5)

Under r → −r; M → −M one has that ρ(r) → −ρ(r) so one can ensure theinvariance of the metric (1.1b) under these transformations and extend the solutionsto the r < 0 region.

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Hence, we have in Eq. (1.5) that ρ(r = 0+) = 2GM but ρ(r = 0) = 0 since apoint mass must have zero area and zero volume. The horizon is located at rh = 0+and the singularity at r = 0. There is a discontinuity of ρ(r) at r = 0. Because a pointmass is an infinitely compact source there is nothing wrong with the possibility ofhaving a discontinui ty of the metric at the location of the singularity r = 0. In thisextreme case, when the the location of the horizon merges with the singularity, thereis a null-line singularity at r = 0 and a null-surface at r = 0+. This may sound quiteparadoxically but it is a consequence of themetric discontinui ty at r = 0, the locationof the point mass (singularity). This key fact may have important consequences forthe resolution of the fire wall problem and the complementarity controversy in blackholes [1,17].

In 1975,Hawking andBekenstein showed how the black holes should slowly radiateaway energy, which poses a problem. From the no hair theorem, one would expect theHawking radiation to be completely independent of the material entering the blackhole. Nevertheless, if the material entering the black hole were a pure quantum state,the transformation of that state into themixed state ofHawking radiationwould destroyinformation about the original quantum state. This violates Liouville’s theorem andpresents a physical paradox. Hawking remained convinced that the equations of blackhole thermodynamics together with the no-hair theorem led to the conclusion thatquantum information may be destroyed. This is the so-called Black Hole InformationParadox.

An heuristic explanation by Hawking and described by Penrose in his book [21]is the following. Virtual particle–antiparticle pairs are constantly being created out ofthe vacuum but then annihilated in a very short time. But very near the horizon of ablack hole, it’s possible for one particle to fall in before the annihilation can happen, inwhich case the other one escapes as Hawking radiation. For this, the virtual particlesboth become real, and energy conservation demands that the ingoing particles havenegative energy. This they can do because the Killing vector κa becomes spacelikeinside the horizon. If κa is spacelike the conserved energy paκa (being assesed frominfinity) can be negative, where pa is the particle’s four momentum.

However, the solutions described in this work allow for the horizon to be displacedarbitrarily close to the singularity, and in the limiting case when the horizon and (null)singularity merge, the interior region disappears such that there is no longer room forthe ingoing particle to go and acquire a negative energy. The virtual particles in thiscase would be annihilated at (or outside) the (null) singularity. Thus it is plausible, inprinciple, to avoid the Hawking emission process in this scenario.

Susskind [25] proposed a radical resolution to the black hole information paradoxbyclaiming that the information is both reflected at the event horizon and passes throughthe event horizon and cannot escape, with the catch being no observer can confirm bothstories simultaneously. However, according to an infalling observer, nothing specialhappens at the event horizon itself, and both the observer and the information will hitthe singularity. This is not to say there are two copies of the information lying about oneat or just outside the horizon, and the other inside the black hole as that would violatethe no cloning theorem. Instead, an observer can only detect the information at thehorizon itself, or inside, but never both simultaneously. Black hole complementarity

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is a feature of the quantum mechanics of noncommuting observables, and Susskindproposed that both stories are complementary in the quantum sense.

A black hole firewall is a hypothetical phenomenon where an observer that fallsinto an old black hole encounters high-energy quanta at (or near) the event horizon.The “firewall” phenomenon was proposed in 2012 by Almheiri et al. [1] as a possiblesolution to an apparent inconsistency in black hole complementarity. The use of afirewall to resolve this inconsistency remains controversialwith high-energy physicistsdivided as to the solution to the paradox [31].

According to Quantum Field Theory in curved spacetime, a single emission ofHawking radiation involves two mutually entangled particles. The outgoing particleescapes and is emitted as a quantum of Hawking radiation; the infalling particle isswallowed by the black hole. An outgoing particle emitted at time t must be entangledwith all the Hawking radiation the black hole has previously emitted. This createsa paradox: a principle called “monogamy of entanglement” requires that, like anyquantum system, the outgoing particle cannot be fully entangled with two independentsystems at the same time; yet here the outgoing particle appears to be entangled withboth the infalling particle and, independently, with past Hawking radiation [31].

In order to resolve the paradox, onemay eventually be forced to give up one of threetime-tested theories: Einstein’s equivalence principle, unitarity, or existing quantumfield theory [1]. It is suggested that the entanglement must somehow get immedi-ately broken between the infalling particle and the outgoing particle. Breaking thisentanglement would release inconceivable amounts of energy, thus creating a sear-ing “black hole firewall” at the black hole event horizon. This resolution requires aviolation of Einstein’s equivalence principle, which states that free-falling is indistin-guishable from floating in empty space. The firewall would exist at the black hole’sevent horizon, and would be invisible to observers outside the event horizon. Matterpassing through the event horizon into the black hole would immediately be “burnedto a crisp” by an arbitrarily hot “seething maelstrom of particles” at the firewall [31].

To finalize this section, we point out that in Eqs. (1.3b, 1.3c), for a judicious choiceof λP < 1, one may have that the horizon radius r (λP )

h coincides with the Planck

length LP , but the value of the areal radial function is still given by ρ(r (λP )h ) = 2GM

and which is not small for macroscopic masses. This interplay between a microscopiclength LP (where QuantumGravity effects are relevant) and amuch larger scale 2GM(where classical gravity is valid) corresponding to the horizon radius of the standardblack hole solutions (when ρ(r) = r ) might be relevant to the Hawking radiationphenomenon and in viewing a black hole as a macroscopic quantum system due to aBose Einstein condensate of a very large number of soft gravitons [6].

2 Point Mass Sources and Euclidean Gravitational Action as Entropy

To finalize we shall study explicitly the case when a delta function type of singularityfor the energy-momentum tensor (mass density and pressure) associated with the pointmass is the source of a curvature singularity at r = 0. A rigorous correct treatment ofpointmass distributions inGeneralRelativity has been provided based onColombeau’s[4,5,11,13,24] theory of nonlinear distributions, generalized functions and nonlinear

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calculus. This permits the proper multiplication of distributions since the old Schwarztheory of linear distributions is invalid in nonlinear theories like General Relativity.Colombeau’s nonlinear distributional geometry supersedes the no-go results ofGerochand Traschen [10] stating that there is no proper framework to study distributions ofmatter of co-dimensions higher than two (neither points nor strings in D = 4) inGeneral Relativity. Colombeau’s theory of Nonlinear Distributions (and NonstandardAnalysis) is the proper way to deal with point-mass sources in nonlinear theories likeGravity and where one may rigorously solve the problem without having to introducea boundary of spacetime at r = 0.

Let us begin now with the temporal and radial components of a continuous metricat r = 0 and whose signature is chosen to be (−,+,+,+)

gtt =−(1 − 2GM

|r |)

=−(1 − 2GM

r

r

|r |)

=−(1− 2GM

rf (r)

); f (r) ≡ r

|r | .(2.1)

the function f (r) = r|r | is by definition the same as the sign function sgn(r) = f (r).

Because it is an odd function of r it vanishes at r = 0, and is ±1 when r > 0, r < 0,respectively. The mass is chosen to be positive M > 0. The other metric componentsare

grr = − 1

gtt, gφφ = r2, gθθ = r2sin2(φ). (2.2)

In Sect. 1 we stressed the fact that the values of r may span the region −∞ ≤ r ≤ ∞and despite that the sign and Heaviside functions are not differentiable in the ordinarysense at r = 0, they are differentiable under the generalized notion of differentiationin distribution theory [32–34]. In particular, the derivative of the sign function is [33]

f (r) ≡ sgn(r) ⇒ f ′(r) = d f (r)

dr= 2δ(r); f ′′(r) = d2 f (r)

dr2= 2δ′(r), (2.3)

and one learns that the curvature scalar R corresponding to the metric (2.1, 2.2) isnow nonvanishing at r = 0

R = −2GM

[f ′′(r)r

+ 2f ′(r)r2

](2.4)

= −4GM

[δ′(r)r

+ 2δ(r)

r2

]= −8πGT,

where T in Eq. (2.4) is the trace of the stress energy tensor gμνTμν in the Einstein’sfield equations due to the presence of matter. The Ricci tensor associated to the metric(2.1, 2.2) can be obtained by using Eqs. (3.4–3.6) in the Appendix

Rt t = 2GM

r2

(1 − 2GM

|r |)

δ(r), Rrr = −2GM

r2

(1 − 2GM

|r |)−1

δ(r)

(2.5a)

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Rθθ = 0, Rφφ = 0 (2.5b)

The Einstein tensor corresponding to Eqs. (2.1, 2.2) is

Rt t− 1

2gttR= 2GM

r2

(1− 2GM

|r |)

δ(r)− 1

2

(−

(1− 2GM

|r |)) (

−4GMδ(r)

r2

)=0

(2.6a)

Rrr − 1

2grr R = −2GM

r2

(1 − 2GM

|r |)−1

δ(r)

−1

2

(1 − 2GM

|r |)−1 (

−4GMδ(r)

r2

)= 0 (2.6b)

Rθθ − 1

2gθθ R = gθθ

2GMδ(r)

r2= 8πG gθθ pθ ⇒ pθ = 2Mδ(r)

8πr2(2.6c)

Rφφ − 1

2gφφ R = gφφ

2GMδ(r)

r2= 8πG gφφ pφ ⇒ pφ = 2Mδ(r)

8πr2(2.6d)

Therefore, the Einstein tensor associated with the metric (2.1, 2.2) corresponds to amatter distribution given by

ρ = 0, pr = 0, pθ = pφ = 2Mδ(r)

8πr2(2.7)

Equation (2.7) is consistent with Eq. (2.4) as a result of the important distributionalrelation rδ′(r) = −δ(r). In general, when n derivatives are involved one has the rela-tion rnδ(n)(r) = (−1)nn!δ(r) [33] and which enables to write everything in Eq. (2.4)directly in terms of the delta function δ(r) as shown in Eq. (2.7).

The weak energy conditions [30] ρ ≥ 0, ρ + pi ≥ 0, for i = 1, 2, 3, are satisfiedby (2.7). Also the strong energy conditions ρ + ∑

i pi ≥ 0 are satisfied. While on theother hand, the dominant energy conditions ρ ≥ |pi | are not satisfied. The non-trivialEinstein–Hilbert action is

S = − 1

16πG

∫ ∫R 4πr2 dr dt = 1

16πG

∫ ∫4GM δ(r)

r24πr2 dr dt

= 1

16πG

∫ ∫16πGM δ(r) dr dt =

∫ ∫M δ(r) dr dt (2.8)

Because the radial integral (2.8) is symmetric in r due to δ(−r) = δ(r), the radialintegral from r = 0 to r = ∞ can be rewritten as one half the integral from r = −∞to r = ∞

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S =∫ r=∞

r=0

∫M δ(r) dr dt = 1

2

∫ r=∞

r=−∞δ(r) dr

∫M dt = 1

2

∫M dt

(2.9)

The metric in Eqs. (2.1, 2.2) has a well defined notion of surface gravity at r = 2GM ,which is the location of the standard horizon, and the radial derivatives of GM/|r | arewell defined and finite at r = 2GM . Therefore, the concepts of entropy and Hawkingtemperature [12] are meaningful in this case.

The Euclideanized Einstein–Hilbert action (2.8, 2.9) associated with the non-trivialscalar curvature is obtained after a compactification of the temporal direction alonga circle S1 whose net Euclidean time integration interval is 2π tE . The latter intervalcan be defined in terms of the Hawking temperature TH and the Boltzman constant kBas 2π tE = (1/kBTH ) = 8πGM . The temperature TH also agrees with the Unruh–Rindler temperature h̄

2π |a| (in units h̄ = c = 1), where |a| = 14GM is the magnitude

of the surface gravity (acceleration) at the standard horizon location r = 2GM . Inte-grating with respect to the Euclidean temporal coordinate, the Euclidean gravitationalaction becomes then

SE =(M

2

)(2π tE ) = 4π G M2 = 1

4

4π(2GM)2

G= Area

4 L2P

. (2.10)

which is precisely the black bole Entropy in Planck area units G = L2P ( h̄ = c = 1 ).

This result that the Euclideanized gravitational action (associated with a non-trivialscalar curvature involving delta functions due to point-mass sources) is the same as theblack hole entropy can be generalized to higher dimensions. In theReissner–Nordsrom(massive-charged) and Kerr–Newman black hole case (massive-rotating-charged) [3]has shown that the Euclidean action in a bulk domain bounded by the inner and outerhorizons is the same as the black hole entropy. These findings should be compared toVerlinde’s enthropic gravity proposal [28] based on the holographic principle.

We finalize by adding some remarks [3] about how a f uzzy point mass may admit ashort distance cut-off of the Brillouin form ρ(r = 0) = 2GM (instead of zero) if onehas a Noncommutative spacetime coordinates algebra [xμ, xν] = i�μν, [pμ, pν] =0, [xμ, pν] = i h̄ημν where �μν are c-numbers of (Planck length)2 magnitude.A change of coordinates in phase space x ′μ = xμ + 1

2�μν pν leads to commuting

coordinates x ′μ and allows to define r ′(r) =√

(xi + 12 �iρ pρ) (xi + 1

2 �iτ pτ ).

One can select �μν such that r ′(xi = 0) = r ′(r = 0) = 2GM , after using the on-shell condition pμ pμ = M2. Therefore one recovers the cut-off corresponding to theBrillouin area radial function ρ(r) = r + 2GM ⇒ ρ(r = 0) = 2GM . Thus a fuzzypoint mass has non-zero area and volume.

To conclude, one has to wait for a theory of Quantum Gravity to fully address theseissues of the avoidance of singularities due to the noncommutativity of spacetimecoordinates. Another relevant topic is to explore the Nonperturbative RenormalizationGroup flow [19,20,22] of the metric gμv[k] in terms of the momentum scale k and itsrelationship (if any) to the family of metrics (1.1) parametrized by λ.

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Acknowledgments This work was sponsored by the Prometeo Project of the Secretary of Higher Edu-cation, Science, Technology and Innovation of the Republic of Ecuador. We are indebted to M. Bowers forassistance.

Appendix: Schwarzschild-Like Solutions in D > 3

In this Appendix we follow closely the calculations of the static spherically symmetricvacuum solutions to Einstein’s equations in any dimension D > 3. Let us start withthe line element with signature (−,+,+,+, . . . ,+)

ds2 = −eμ(r)(dt)2 + eν(r)(dr)2 + R2(r)g̃i j dξ i dξ j , (3.1)

where the areal radial function ρ(r) is now denoted by R(r) and which must not becon f used with the scalar curvature R. Here, the metric g̃i j corresponds to a homo-geneous space and i, j = 3, 4, . . . , D − 2 and the temporal and radial indices aredenoted by 1, 2 respectively. In our text we denoted the temporal index by 0. Theonly non-vanishing Christoffel symbols are given in terms of the following partialderivatives with respect to the r variable and denoted with a prime

�121 = 1

2μ′, �2

22 = 12ν

′, �211 = 1

2μ′eμ−ν,

�2i j = −e−νRR′g̃i j , �i

2 j = R′R δij , �i

jk = �̃ijk,

(3.2)

and the only nonvanishing Riemann tensor are

R1212 = − 1

2μ′′ − 1

4μ′2 + 1

4ν′μ′, R1

i1 j = − 12μ

′e−νRR′g̃i j ,

R2121 =eμ−ν

( 12μ

′′+ 14μ

′2 − 14ν

′μ′) , R2i2 j =e−ν

( 12ν

′RR′−RR′′) g̃i j ,Ri

jkl = R̃ijkl − R′2e−ν(δik g̃ jl − δil g̃ jk).

(3.3)

The vacuum field equations are

R11 = eμ−ν

(1

2μ′′ + 1

4μ′2 − 1

4μ′ν′ + (D − 2)

2μ′ R′

R

)= 0, (3.4)

R22 = −1

2μ′′ − 1

4μ′2 + 1

4μ′ν′ + (D − 2)

(1

2ν′ R′

R− R′′

R

)= 0, (3.5)

and

Ri j = e−ν

R2

(1

2(ν′ − μ′)RR′ − RR′′ − (D − 3)R′2

)g̃i j + k

R2 (D − 3)g̃i j = 0,

(3.6)

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where k = ±1, depending if g̃i j refers to positive or negative curvature. From thecombination e−μ+νR11 + R22 = 0 we get

μ′ + ν′ = 2R′′

R′ . (3.7)

The solution of this equation is

μ + ν = ln R′2 + C, (3.8)

where C is an integration constant that one sets to zero if one wishes to recover theflat Minkowski spacetime metric in spherical coordinates in the asymptotic regionr → ∞.

Substituting (3.7) into Eq. (3.6) we find

e−ν(ν′RR′ − 2RR′′ − (D − 3)R′2) = −k(D − 3) (3.9)

or

γ ′RR′ + 2γ RR′′ + (D − 3)γ R′2 = k(D − 3), (3.10)

where

γ = e−ν . (3.11)

The solution of (3.10) for an ordinary D-dim spacetime (one temporal dimension)corresponding to a D − 2-dim sphere for the homogeneous space can be written as

γ =(1 − 16πGDM

(D − 2)�D−2RD−3

) (dR

dr

)−2

(3.12)

⇒ grr = eν =(1 − 16πGDM

(D − 2)�D−2RD−3

)−1 (dR

dr

)2

,

where�D−2 is the appropriate solid angle in D−2-dim andGD is the D-dim gravita-tional constant whose units are (length)D−2. Thus GDM has units of (length)D−3 asit should. When D = 4 as a result that the 2-dim solid angle is �2 = 4π one recoversfrom Eq. (3.12) the 4-dim Schwarzchild solution. The solution in Eq. (3.12) is consis-tent with Gauss law and Poisson’s equation in D − 1 spatial dimensions obtained inthe Newtonian limit.

For the most general case of the D − 2-dim homogeneous space we should write

− ν = ln

(k − βDGDM

RD−3

)− 2 ln R′. (3.13)

βD is a constant equal to 16π/(D − 2)�D−2, where �D−2 is the solid angle in theD − 2 transverse dimensions to r, t and is given by (D − 1)π(D−1)/2/�[(D + 1)/2].

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Thus, according to (3.8) we get

μ = ln

(k − βDGDM

RD−3

)+ constant. (3.14)

we can set the constant to zero, and this means the line element (3.1) can be writtenas

ds2 = −(k − βDGDM

RD−3

)(dt)2 + (dR/dr)2(

k − βDGDMRD−3

) (dr)2 + R2(r)g̃i j dξ i dξ j

= −(k − βDGDM

RD−3

)(dt)2 + 1(

k − βDGDMRD−3

) (dR)2 + R2(r)g̃i j dξ i dξ j

(3.15)

One can verify, that Eqs. (3.4)–(3.6), leading to Eqs. (3.9)–(3.10), do not determinethe form R(r). It is also interesting to observe that the only effect of the homogeneousmetric g̃i j is reflected in the k = ±1 parameter, associated with a positive (negative)constant scalar curvature of the homogeneous D − 2-dim space. k = 0 correspondsto a spatially flat D − 2-dim section. The metric solution in Eq. (1.1) is associated toa different signature than the one chosen in this Appendix, and corresponds to D = 4and k = 1.

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