Novel Remarks on Point Mass Sources, Firewalls, Null Singularities and Gravitational Entropy

Abstract

A continuous family of static spherically symmetric solutions of Einstein’s vacuum field equations with a spatial singularity at the origin $r = 0$ is found. These solutions are parametrized by a real valued parameter $\lambda$ (ranging from 0 to 1) and such that the radial horizon’s location is displaced continuously towards the singularity ($r = 0$) as $\lambda$ increases. In the extreme limit $\lambda = 1$, the location of the singularity and horizon merges leading to a null singularity. In this extreme case, any infalling observer hits the null singularity at the very moment he/she crosses the horizon. This fact may have important consequences for the resolution of the fire wall problem and the complementarity controversy in black holes. An heuristic argument is provided how one might avoid the Hawking particle emission process in this extreme case when the 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 corresponding to those solutions is evaluated explicitly. It is found that the Euclidean action is precisely equal to the black hole entropy (in Planck area units). This result holds in any dimensions $D \ge 3$.

Appendix: Schwarzschild-Like Solutions in \(D > 3 \)

In this Appendix we follow closely the calculations of the static spherically symmetric vacuum solutions to Einstein’s equations in any dimension $D > 3$. Let us start with the line element with signature $( -, +, +, +, \ldots , +)$

$$\begin{aligned} ds^{2}=-e^{\mu (r)}(dt )^{2}+e^{\nu (r)}(dr)^{2}+R^{2}(r)\tilde{g} _{ij}d\xi ^{i}d\xi ^j , \end{aligned}$$

(3.1)

where the areal radial function $\rho ( r )$ is now denoted by R ( r ) and which must not be confused with the scalar curvature $\mathcal{R }$. Here, the metric $\tilde{g}_{ij}$ corresponds to a homogeneous space and $i,j=3,4,\ldots ,D-2$ and the temporal and radial indices are denoted by 1, 2 respectively. In our text we denoted the temporal index by 0 . The only non-vanishing Christoffel symbols are given in terms of the following partial derivatives with respect to the r variable and denoted with a prime

$$\begin{aligned} \begin{array}{llllll} \Gamma _{21}^{1}&{} =\frac{1}{2}\mu ^{\prime }, &{} \quad \Gamma _{22}^{2}&{} =\frac{1}{2} \nu ^{\prime }, &{} \quad \Gamma _{11}^{2}&{} =\frac{1}{2}\mu ^{ \prime }e^{\mu -\nu },\\ \Gamma _{ij}^{2}&{} =-e^{-\nu }RR^{\prime }\tilde{g}_{ij}, &{} \quad \Gamma _{2j}^{i}&{} = \frac{R^{\prime }}{R}\delta _{j}^{i}, &{} \quad \Gamma _{jk}^{i}&{} =\tilde{\Gamma } _{jk}^{i}, \end{array} \end{aligned}$$

(3.2)

and the only nonvanishing Riemann tensor are

$$\begin{aligned} \begin{array}{llll} \mathcal {R}_{212}^{1}&{}=-\frac{1}{2}\mu ^{\prime \prime }-\frac{1}{4}\mu ^{\prime 2}+\frac{1}{4}\nu ^{\prime }\mu ^{\prime }, &{}\quad \mathcal {R} _{i1j}^{1}&{}=-\frac{1}{2}\mu ^{\prime }e^{-\nu }RR^{\prime }\tilde{g}_{ij}, \\ \mathcal {R}_{121}^{2}&{}=e^{\mu -\nu }\left( \frac{1}{2}\mu ^{\prime \prime }+\frac{1 }{4}\mu ^{\prime 2}-\frac{1}{4}\nu ^{\prime }\mu ^{\prime }\right) , &{}\quad \mathcal {R }_{i2j}^{2}&{}=e^{-\nu }\left( \frac{1}{2}\nu ^{\prime }RR^{\prime }-RR^{\prime \prime }\right) \tilde{g}_{ij}, \\ \mathcal {R}_{jkl}^{i}&{}=\tilde{R}_{jkl}^{i}-R^{\prime 2}e^{-\nu }(\delta _{k}^{i}\tilde{g}_{jl}-\delta _{l}^{i}\tilde{g}_{jk}). &{} \end{array} \end{aligned}$$

(3.3)

The vacuum field equations are

$$\begin{aligned} \mathcal {R}_{11}=e^{\mu -\nu }\left( \frac{1}{2}\mu ^{\prime \prime }+\frac{1}{4} \mu ^{\prime 2}-\frac{1}{4}\mu ^{\prime }\nu ^{\prime }+\frac{(D-2)}{2}\mu ^{\prime }\frac{R^{\prime }}{R}\right) =0, \end{aligned}$$

(3.4)

$$\begin{aligned} \mathcal {R}_{22}=-\frac{1}{2}\mu ^{\prime \prime }-\frac{1}{4}\mu ^{\prime 2}+\frac{1}{4}\mu ^{\prime }\nu ^{\prime }+(D-2)\left( \frac{1}{2}\nu ^{\prime } \frac{R^{\prime }}{R}-\frac{R^{\prime \prime }}{R}\right) =0, \end{aligned}$$

(3.5)

and

$$\begin{aligned} \mathcal {R}_{ij}={ e^{-\nu }\over R^2 } \left( \frac{1}{2}(\nu ^{\prime }-\mu ^{\prime })RR^{\prime }-RR^{\prime \prime }-(D-3)R^{\prime 2}\right) \tilde{g}_{ij}+{k\over R^2} (D-3) \tilde{g}_{ij}=0,\nonumber \\ \end{aligned}$$

(3.6)

where $k=\pm 1$, depending if $\tilde{g}_{ij}$ refers to positive or negative curvature. From the combination $e^{-\mu +\nu }\mathcal{R}_{11}+\mathcal{R}_{22}=0$ we get

$$\begin{aligned} \mu ^{\prime }+\nu ^{\prime }=\frac{2R^{\prime \prime }}{R^{\prime }}. \end{aligned}$$

(3.7)

The solution of this equation is

$$\begin{aligned} \mu +\nu ~=~\ln R^{\prime 2}~+~C, \end{aligned}$$

(3.8)

where C is an integration constant that one sets to zero if one wishes to recover the flat Minkowski spacetime metric in spherical coordinates in the asymptotic region $r \rightarrow \infty$.

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

$$\begin{aligned} e^{-\nu }(\nu ^{\prime }RR^{\prime }-2RR^{\prime \prime }-(D-3)R^{\prime 2}) = -k(D-3) \end{aligned}$$

(3.9)

or

$$\begin{aligned} \gamma ^{\prime }RR^{\prime }+2\gamma RR^{\prime \prime }+(D-3)\gamma R^{\prime 2}=k(D-3), \end{aligned}$$

(3.10)

where

$$\begin{aligned} \gamma =e^{-\nu }. \end{aligned}$$

(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

$$\begin{aligned} \gamma&= \left( 1 -\frac{ 16 \pi G_D M}{ ( D- 2) \Omega _{D - 2 } R^{D-3}}\right) \left( { \frac{ dR }{dr }}\right) ^{ -2}\\ \nonumber&\Rightarrow g_{rr} = e^\nu = \left( 1-\frac{ 16\pi G_D M}{ ( D- 2)\Omega _{D-2} R^{D-3}}\right) ^{ -1} \left( {\frac{dR}{dr}}\right) ^{2}, \end{aligned}$$

(3.12)

where $\Omega _{D-2}$ is the appropriate solid angle in $D -2$-dim and $G_D$ is the D-dim gravitational constant whose units are $(length)^{D-2}$. Thus $G_D M$ has units of $(length)^{D - 3}$ as it should. When $D = 4$ as a result that the 2-dim solid angle is $\Omega _2 = 4 \pi$ one recovers from Eq. (3.12) the 4-dim Schwarzchild solution. The solution in Eq. (3.12) is consistent with Gauss law and Poisson’s equation in $D-1$ spatial dimensions obtained in the Newtonian limit.

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

$$\begin{aligned} -\nu =\ln \left( k-\frac{\beta _D G_D M }{ R^{D-3}}\right) -2\ln R^{\prime }. \end{aligned}$$

(3.13)

$\beta _D$ is a constant equal to $16\pi / ( D- 2) \Omega _{D-2}$, where $\Omega _{D -2}$ is the solid angle in the $D-2$ transverse dimensions to r, t and is given by $( D - 1) \pi ^{ ( D - 1) / 2 } / \Gamma [ ( D + 1) / 2 ]$.

Thus, according to (3.8) we get

$$\begin{aligned} \mu =\ln \left( k-\frac{\beta _D G_D M }{ R^{D-3}}\right) + constant. \end{aligned}$$

(3.14)

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

$$\begin{aligned} ds^{2}= & {} -\left( k-\frac{\beta _D G_D M }{ R^{D-3}}\right) (dt)^{2} + \frac{ (dR/dr)^2 }{\left( k-\frac{\beta _D G_D M }{ R^{D-3}}\right) }(dr)^{2} + R^{2}(r)\tilde{g}_{ij}d\xi ^{i}d\xi ^j \nonumber \\= & {} -\left( k-\frac{\beta _D G_D M }{ R^{D-3}}\right) (dt)^{2} + \frac{ 1}{\left( k-\frac{\beta _D G_D M }{ R^{D-3}}\right) }(dR)^{2} +~ R^{2}(r)\tilde{g}_{ij}d\xi ^{i}d\xi ^j\nonumber \\ \end{aligned}$$

(3.15)

One can verify, that Eqs. (3.4)–(3.6), leading to Eqs. (3.9)–(3.10), do not determine the form R(r). It is also interesting to observe that the only effect of the homogeneous metric ${\tilde{g}}_{ij}$ is reflected in the $k = \pm 1$ parameter, associated with a positive (negative) constant scalar curvature of the homogeneous $D-2$-dim space. $k = 0$ corresponds to a spatially flat $D-2$-dim section. The metric solution in Eq. (1.1) is associated to a different signature than the one chosen in this Appendix, and corresponds to $D = 4$ and $k = 1$.