Signatures of quantum geometry from exponential corrections to the black hole entropy

Abstract

It has been recently shown in Chatterjee and Ghosh (Phys Rev Lett 125:041302, 2020, https://doi.org/10.1103/PhysRevLett.125.041302) that microstate counting carried out for quantum states residing on the horizon of a black hole leads to a correction of the form $\exp (-A/4l_p^2)$ in the Bekenstein-Hawking form of the black hole entropy. In this paper, we develop a novel approach to obtain the possible form of the spacetime geometry from the entropy of the black hole for a given horizon radius. The uniqueness of this solution for a given energy-momentum tensor has also been discussed. Remarkably, the black hole geometry reconstructed has striking similarities to that of noncommutative-inspired Schwarzschild black holes (Nicolini et al. in Phys Lett B 632:547, 2006). We also obtain the matter density functions using Einstein field equations for the geometries we reconstruct from the thermodynamics of black holes. These also have similarities to that of the matter density function of a noncommutative-inspired Schwarzschild black hole. The conformal structure of the metric is briefly discussed and the Penrose–Carter diagram is drawn. We then compute the Komar energy and the Smarr formula for the effective black hole geometry and compare it with that of the noncommutative-inspired Schwarzschild black hole. We also discuss some astrophysical implications of the solutions. Finally, we propose a set of quantum Einstein vacuum field equations, as a solution of which we obtain one of the spacetime solutions obtained in this work. We then show a direct connection between the quantum Einstein vacuum field equations and the first law of black hole thermodynamics.

Appendices

Apppendix 1: Quantum corrected Einstein equations for $h(r)=r^2$

In the earlier part of our analysis after obtaining Eq. (37), we have considered the classical Einstein equation with matter density function $\rho (r)$ given in Eq. (48) to prove the uniqueness of our solution. In this section, we look for quantum corrections to the Einstein equations. We now propose a set of modified Einstein vacuum field equations. The quantum Einstein vacuum field equations involving the $\{t,t\}$ and $\{r,r\}$ components of the quantum modified Einstein tensor $\bar{G}_{tt}$ and $\bar{G}_{rr}$ are proposed as follows

$$\begin{aligned} \begin{aligned} r^2f(r)\bar{G}_{tt}=-r^2f(r)\bar{G}_{rr}=0 \end{aligned} \end{aligned}$$

(89)

where

$$\begin{aligned} r^2f(r)\bar{G}_{tt}=\left[ 1-f(r)-rf'(r)\right] -\left( 1+\frac{l_p^2}{2\pi r^2}\right) e^{-\frac{\pi r^2}{l_p^2}}. \end{aligned}$$

(90)

Similarly, the other two quantum modified vacuum field equations are proposed as

$$\begin{aligned} \begin{aligned} \bar{G}_{\theta \theta }=-\frac{1}{\sin ^2\theta }\bar{G}_{\phi \phi }=0 \end{aligned} \end{aligned}$$

(91)

where

$$\begin{aligned} \bar{G}_{\theta \theta }=\frac{r}{2}(2f'(r)+rf''(r))-\left( \frac{\pi r^2}{l_p^2}+\frac{1}{2}+\frac{l_p^2}{2\pi r^2}\right) e^{-\frac{\pi r^2}{l_p^2}}. \end{aligned}$$

(92)

In Eqs. (89, 91), $\bar{G}_{tt}$, $\bar{G}_{rr}$, $\bar{G}_{\theta \theta }$, and $\bar{G}_{\phi \phi }$ denotes the quantum modified Einstein tensor. Solving Eq. (89) or Eq. (91) perturbatively, we obtain the form of the metric f(r) given in Eq. (37). It is important to observe that both Eqs. (89, 91) are vacuum field equations.

It is important to note that there are no solid justifications for the quantum Einstein equations in Eqs. (89–92) at this moment. In order to obtain the quantum corrections, we have absorbed the energy-momentum tensor into the left hand side of the Einstein field equations and proposed the resultant object as the quantum vacuum Einstein field equation giving rise to the solution f(r) in Eq. (37).

Appendix 2: Quantum corrected Einstein equations and its connection to the first law of black hole thermodynamics for $h(r)=r^2$

We shall now exploit the analytical form of $r^2f(r)\bar{G}_{rr}$ given in Eq. (89) around the horizon radius $\tilde{r}_+$ given in Eq. (23). In the limit $r\rightarrow \tilde{r}_+$, Eq. (89) can be expressed (upto $\mathcal {O}\left( \exp \left( -\pi r_+^2/l_p^2\right) \right)$) as follows

$$\begin{aligned} \tilde{r}_+f'(\tilde{r}_+)-1+\left( 1+\frac{l_p^2}{2\pi r_+^2}\right) e^{-\frac{\pi r_+^2}{l_p^2}}=0. \end{aligned}$$

(93)

We now multiply both sides of the above equation with $\frac{c^4}{2G}d\tilde{r}_+$ and obtain the following relation

$$\begin{aligned} \frac{c^4}{2G}f'(\tilde{r}_+)\tilde{r}_+d\tilde{r}_+&\simeq \frac{c^4}{2G}\left[ d\tilde{r}_+-\left( 1+\frac{l_p^2}{2\pi r_+^2}\right) e^{-\frac{\pi r_+^2}{l_p^2}}dr_+\right] \nonumber \\&=\frac{c^4}{2G}dr_+\nonumber \\&=c^2dM. \end{aligned}$$

(94)

The left hand side of the above equation can be rearranged as

$$\begin{aligned} \frac{c^4}{2G}f'(\tilde{r}_+)\tilde{r}_+d\tilde{r}_+=\left( \frac{\hbar cf'(\tilde{r}_+)}{4\pi k_B}\right) \frac{k_Bc^3}{4\hbar G}d(4\pi \tilde{r}_+^2). \end{aligned}$$

(95)

In this relation the term in the parenthesis is the temperature of the black hole whose metric is given by Eq. (37). Invoking the form of the horizon from Eq. (23), we finally obtain the following relation

$$\begin{aligned} Td\left( \frac{k_BA}{4l_p^2}-k_Be^{-\frac{A}{4l_p^2}}\right) =c^2dM \end{aligned}$$

(96)

where we have identified $A=4\pi r_+^2$ with $r_+$ being the usual Schwarzschild radius. Now from Eq. (18), we can see that for $n=0$ case, the modified entropy formula is $S=\frac{k_BA}{4l_p^2}-k_Be^{-\frac{A}{4l_p^2}}$. With this identification, we can recast Eq. (96) given as

$$\begin{aligned} TdS=c^2dM=dE \end{aligned}$$

(97)

which is the usual first law of black hole thermodynamics. The above analysis establishes a direct connection between geometry and thermodynamics [47]. It is also important to identify that the pressure term at the horizon is zero leading to the conclusion that we are dealing with a vacuum field equation. This indeed is a nice consistency check of the proposed quantum Einstein equations in Eqs. (89, 92).