Quantum Gravitational Corrections to the Entropy of a Reissner-Nordström Black
Hole

Ruben Campos Delgado111email<EMAIL_ADDRESS>
Bethe Center for Theoretical Physics

and
Physikalisches Institut der Universität Bonn,

Nussallee 12, 53115 Bonn, Germany

Abstract

Starting from an effective action for quantum gravity, we calculate the
quantum gravitational corrections to the Wald entropy of a four dimensional
non-extremal Reissner-Nordström (RN) black hole in the limit of small electric
charge, generalising a previous calculation carried out by Calmet and Kuipers
[1] for a Schwarzschild black hole. We show that, at second order in the Ricci
curvature, the RN metric receives quantum corrections which shift the
classical position of the event horizon. We apply the Wald entropy formula by
integrating over the perimeter of the quantum corrected event horizon. We then
compute the quantum gravitational corrections to the temperature and the
pressure of the black hole.

##  1 Introduction

One of the greatest achievement in theoretical physics in the last fifty years
is the discovery that black holes have an entropy [2, 3]. The leading
classical contribution to the entropy of a black hole in Einstein’s general
relativity is equal to one quarter of the area of the event horizon. If the
theory of gravity is modified, then the entropy receives additional
contributions. Recently [1], effective field theory methods were used to
calculate the quantum gravitational corrections to the entropy of a
Schwarzschild black hole using the Wald entropy formula [4], which in four
dimensions reads

$S_{\text{Wald}}=-2\pi\int\limits_{r=r_{h}}d\Sigma\,\epsilon_{\mu\nu}\epsilon_{\rho\sigma}\frac{\partial\mathcal{L}}{\partial
R_{\mu\nu\rho\sigma}},$ (1.1)

where $\mathcal{L}$ is the Lagrangian density of the theory,
$d\Sigma=r^{2}\sin\theta d\theta d\phi$, $R_{\mu\nu\rho\sigma}$ is the Riemann
tensor and $\epsilon_{\mu\nu}$ is an antisymmetric tensor normalised as
$\epsilon_{\mu\nu}\epsilon^{\mu\nu}=-2$. The integral has to be evaluated at
the horizon radius $r_{h}$. An important observation is that the metric itself
may receive quantum gravitational corrections that could impact the position
of the horizon. The Schwarzschild metric does not receive corrections at
second order in curvature [5], but it does at third order [1].

In this work we apply the same effective field theory approach to quantum
gravity as in [1] to compute the quantum gravitational corrections at second
order in curvature to the entropy of a more realistic type of black holes,
namely the electrically charged ones, also known as Reissner-Nordström (RN)
black holes. The motivation behind adding electric charge is dictated not only
by theoretical interest. Indeed, real astrophysical black holes do have a tiny
electric charge, which can be approximately estimated as 100 Coulombs per
solar mass [6]. If one is interested only in the motion of photons and neutral
matter (neutrinos and gravitational waves) in the vicinity of the hole, then
the assumption of zero charge is justified. However, the dynamics of charged
particles like electrons and protons can be profoundly affected. The electric
field of the black hole can do work and accelerate the particles to large,
relativistic speeds. Incidentally, this could explain why cosmic rays are
detected on the Earth with very high energies (see M. Zajacek, A. Tursunov,
Electric charge of black holes: Is it really always negligible?,
arXiv:1904.04654 [astro-ph.GA]).

From the technical point of view, there is a difference between the result of
[1] and ours. Unlike the Schwarzschild black hole, which is a vacuum solution
of general relativity, the presence of electric charge for the RN black hole
implies a non-vanishing energy-momentum tensor. As a consequence, we found
that the classical RN metric receives quantum gravitational corrections
already at second order in curvature. We obtained these corrections by solving
the quantum corrected Einstein and Maxwell equations perturbatively in the
gravitational coupling $G_{N}$ up to order $\mathcal{O}(G^{2}_{N})$. We
present their derivation in Section 2. Quantum corrections to the RN metric
were already considered in the literature using alternative methods. In [7, 8]
it was assumed that it is quantum matter, and not quantum gravity, which is
responsible for the quantum effects. Accordingly, the energy-momentum tensor
was expanded in a power series in the fine structure constant $\alpha$ using
usual Feynman diagram techniques, then a Fourier transformation of the form
factors was performed to get the correction to the metric. In [9], quantum
corrections to the RN metric were obtained within the context of 2D
spherically symmetric dilaton gravity models.

To compute the entropy, we restricted to the case of a non-extremal RN black
hole and we additionally made the well-founded physical assumption that its
charge $Q$ is very small compared to its mass $M$. To be precise, in every
expressions we kept only terms up to order $\mathcal{O}\left(Q^{2}\right)$. In
this sense, our results can be thought of a small correction to those obtained
for the Schwarzschild black hole. As already mentioned, the Wald entropy
requires to integrate over the event horizon. We found that the corrections to
the metric imply a small shift of the horizon radius, which contributes to the
final expression of the entropy. These calculations are presented in Section
3. Previous works obtained the quantum corrections to the entropy of non-
extremal RN black holes in other ways. Most notably, by computing the stringy
$\alpha^{\prime}$ corrections [10], by applying the brick-wall method [11], by
counting the degrees of freedom near the horizon [12], or by immersing the
black hole in an isothermal bath within a cavity [13].

After computing the entropy, in Section 4 we have a look at the first law of
thermodynamics. We found that both the temperature and the pressure of RN
black holes receive quantum gravitational corrections at order
$\mathcal{O}\left(Q^{2}\right)$ in such a way that the first law remains
valid.

##  2 Quantum gravitational corrections to the Reissner-Nordström metric

###  2.1 Equations of motion from the effective action

Effective field theory is an organised procedure that separates out the
effects of high energy from those at low energy. Any ultra-violet completion
of quantum gravity that reduces to general relativity in its low energy regime
is described by an effective action, consisting of local and non-local terms
[14, 15, 16, 17, 18, 19, 20]. The local action is a curvature expansion
involving quantities invariant under general coordinate transformations. At
second order in curvature, it is

$\Gamma_{L}=\int d^{4}x\,\sqrt{-g}\,\bigg{(}\frac{R}{16\pi
G_{N}}+c_{1}(\mu)R^{2}+c_{2}(\mu)R_{\mu\nu}R^{\mu\nu}+c_{3}(\mu)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\bigg{)},$
(2.1)

where $\mu$ is an energy scale. The exact values of the coefficients
$c_{1},c_{2},c_{3}$ are unknown, as they depend on the nature of the ultra-
violet theory of quantum gravity. The non-local action is

$\Gamma_{NL}=-\int d^{4}x\sqrt{-g}\bigg{[}\alpha
R\ln\left(\frac{\Box}{\mu^{2}}\right)R+\beta
R_{\mu\nu}\ln\left(\frac{\Box}{\mu^{2}}\right)R^{\mu\nu}+\gamma
R_{\mu\nu\rho\sigma}\ln\left(\frac{\Box}{\mu^{2}}\right)R^{\mu\nu\rho\sigma}\bigg{]},$
(2.2)

where $\Box=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$ and
$\ln\left(\Box/\mu^{2}\right)$ is an operator with the following integral
representation [21]:

$\ln\left(\frac{\Box}{\mu^{2}}\right)=\int_{0}^{+\infty}ds\,\left(\frac{1}{\mu^{2}+s}-\frac{1}{\Box+s}\right).$
(2.3)

The numerical values of the coefficients $\alpha,\beta,\gamma$ are calculable,
see e.g. [22]. In four dimensions, the Gauss-Bonnet term

$\int
d^{4}x\,\sqrt{-g}\left(R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)$
(2.4)

is a topological invariant. Hence, as a simplification of the equations of
motion, it is possible to eliminate the Riemann tensor in the local action
(2.1) by redefining the coefficients as

$c_{1}\to\bar{c}_{1}=c_{1}-c_{3},\hskip
2.84526ptc_{2}\to\bar{c}_{2}=c_{2}+4c_{3},\hskip
2.84526ptc_{3}\to\bar{c}_{3}=0.$ (2.5)

At second order in curvature, there exists also a non-local version of the
Gauss-Bonnet theorem [5]:

$R_{\mu\nu\rho\sigma}\ln\left(\frac{\Box}{\mu^{2}}\right)R^{\mu\nu\rho\sigma}=4R_{\mu\nu}\ln\left(\frac{\Box}{\mu^{2}}\right)R^{\mu\nu}-R\ln\left(\frac{\Box}{\mu^{2}}\right)R.$
(2.6)

Therefore, it is possible to eliminate the Riemann tensor in the non-local
action by redefining the coefficients as

$\alpha\to\bar{\alpha}=\alpha-\gamma,\hskip
11.38109pt\beta\to\bar{\beta}=\beta+4\gamma,\hskip
11.38109pt\gamma\to\bar{\gamma}=0.$ (2.7)

Since we are considering an electrically charged black hole, to the purely
gravitational terms we have to add the Maxwell action

$\Gamma_{M}=-\frac{1}{4}\int d^{4}x\,\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu},$ (2.8)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic tensor and $A_{\mu}$ is the electromagnetic potential. The
full action is then

$\Gamma=\Gamma_{L}+\Gamma_{NL}+\Gamma_{M}=\int
d^{4}x\,\sqrt{-g}\,\mathcal{L}.$ (2.9)

The Maxwell equations, obtained by varying (2.9) with respect to $A_{\mu}$,
are

$g^{\mu\nu}\nabla_{\mu}F_{\nu\tau}=0.$ (2.10)

In addition, there is a Bianchi identity

$\nabla_{[\mu}F_{\nu\tau]}=0.$ (2.11)

The quantum corrected Einstein equations, obtained by varying (2.9) with
respect to the metric, are

$\frac{1}{8\pi
G_{N}}G_{\mu\nu}+2\left(H_{\mu\nu}+K_{\mu\nu}\right)=T_{\mu\nu}.$ (2.12)

The $T_{\mu\nu}$ on the right-hand side is the energy-momentum tensor for
electromagnetism,

$T_{\mu\nu}=\frac{1}{4\pi}\left(F_{\mu\rho}{F_{\nu}}^{\rho}-\frac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}\right).$
(2.13)

On the left-hand side, $G_{\mu\nu}$ is the usual Einstein tensor

$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu},$ (2.14)

while $H_{\mu\nu}$ and $K_{\mu\nu}$ are the quantum corrections to the
classical Einstein’s equations. The local contribution is

$\begin{gathered}H_{\mu\nu}=\bar{c}_{1}\bigg{(}2RR_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R^{2}-2\nabla_{\mu}\nabla_{\nu}R+2g_{\mu\nu}\Box
R\bigg{)}\\\
+\bar{c}_{2}\bigg{(}-\frac{1}{2}g_{\mu\nu}R_{\rho\sigma}R^{\rho\sigma}+2R^{\rho\sigma}R_{\mu\rho\nu\sigma}-\nabla_{\mu}\nabla_{\nu}R+\Box
R_{\mu\nu}+\frac{1}{2}g_{\mu\nu}\Box R\bigg{)}.\end{gathered}$ (2.15)

The non-local contribution is obtained by varying $\Gamma_{NL}$. In principle
one has to take the variation of the logarithm. However, $\delta/\delta
g^{\mu\nu}\ln\left(\Box/\mu^{2}\right)$ contains terms of higher order than
two in curvature [22, 23] so that they can be neglected here. The non-local
contribution is then

$\begin{gathered}K_{\mu\nu}=-2\bar{\alpha}\left(R_{\mu\nu}-\frac{1}{4}g_{\mu\nu}R+g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)\ln\left(\frac{\Box}{\mu^{2}}\right)R-\bar{\beta}\bigg{(}{\delta^{\rho}}_{\mu}R_{\nu\sigma}+{\delta^{\rho}}_{\nu}R_{\mu\sigma}\\\
-\frac{1}{2}g_{\mu\nu}{R^{\rho}}_{\sigma}+{\delta^{\rho}}_{\mu}g_{\nu\sigma}\Box+g_{\mu\nu}\nabla^{\rho}\nabla_{\sigma}-{\delta^{\rho}}_{\mu}\nabla_{\sigma}\nabla_{\nu}-{\delta^{\rho}}_{\nu}\nabla_{\sigma}\nabla_{\mu}\bigg{)}\ln\left(\frac{\Box}{\mu^{2}}\right){R^{\sigma}}_{\rho}.\end{gathered}$
(2.16)

###  2.2 Solution of the equations of motion

In this section we solve the quantum corrected Einstein and Maxwell equations
using perturbation theory in the gravitational constant $G_{N}$. The equations
are coupled together as the electromagnetic tensor $F_{\mu\nu}$ appears in the
Einstein equations through the energy-momentum tensor $T_{\mu\nu}$, while the
metric enters explicitly into Maxwell’s equations.

We consider a small perturbation $g^{q}_{\mu\nu}$ of order
$\mathcal{O}\left(G^{2}_{N}\right)$ around the classical RN solution:

$g_{\mu\nu}=g^{RN}_{\mu\nu}+g^{q}_{\mu\nu}.$ (2.17)

We recall that, in the canonical coordinates $x^{\mu}=\\{t,r,\theta,\phi\\}$,
the RN solution is

$\begin{gathered}ds^{2}_{RN}=g^{RN}_{\mu\nu}dx^{\mu}dx^{\nu}=-\left(1-\frac{2G_{N}M}{r}+\frac{G_{N}Q^{2}}{r^{2}}\right)dt^{2}\\\
+\left(1-\frac{2G_{N}M}{r}+\frac{G_{N}Q^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}.\end{gathered}$ (2.18)

Let us have a look at $g^{q}_{\mu\nu}$. We fix the gauge freedom by setting
$g^{q}_{\theta\theta}=g^{q}_{\phi\phi}=0$ and define two functions $\Delta(r)$
and $\Sigma(r)$ such that

$ds^{2}_{q}=g^{q}_{\mu\nu}dx^{\mu}dx^{\nu}=-G^{2}_{N}\Delta(r)dt^{2}-G^{2}_{N}\Sigma(r)dr^{2}.$
(2.19)

The full metric is then

$\begin{gathered}ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=-\left(1-\frac{2G_{N}M}{r}+\frac{G_{N}Q^{2}}{r^{2}}+G^{2}_{N}\Delta(r)\right)dt^{2}\\\
+\left(1-\frac{2G_{N}M}{r}+\frac{G_{N}Q^{2}}{r^{2}}+G^{2}_{N}\Sigma(r)\right)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}.\end{gathered}$ (2.20)

Notice that, at order $\mathcal{O}\left(G^{2}_{N}\right)$,

$\left(1-\frac{2G_{N}M}{r}+\frac{G_{N}Q^{2}}{r^{2}}\right)^{-1}-G^{2}_{N}\Sigma(r)=\left(1-\frac{2G_{N}M}{r}+\frac{G_{N}Q^{2}}{r^{2}}+G^{2}_{N}\Sigma(r)\right)^{-1}.$
(2.21)

The non-vanishing independent component of $F_{\mu\nu}$ for the classical RN
black hole is $F^{RN}_{tr}=-F^{RN}_{rt}=Q/r^{2}$. Since the Maxwell and
Einstein equations are coupled together, it is well possible that this
component receives a correction. We thus define a function $\Omega(r)$ such
that

$F_{tr}=-F_{rt}=\frac{Q}{r^{2}}+G^{2}_{N}\Omega(r).$ (2.22)

To proceed, we plug the metric (2.20) and the components of the
electromagnetic tensor (2.22) into the equations (2.10) - (2.12), keeping only
terms up to order $\mathcal{O}\left(G^{2}_{N}\right)$. Since $H_{\mu\nu}$ and
$K_{\mu\nu}$ are quadratic in curvature, this implies in particular that they
just need to be evaluated on the classical RN solution, i.e. (2.12) can be
expressed schematically as

$\frac{1}{8\pi
G_{N}}G_{\mu\nu}\left[g^{RN}+g^{q}\right]+2\left(H_{\mu\nu}\left[g^{RN}\right]+K_{\mu\nu}\left[g^{RN}\right]\right)=T_{\mu\nu}\left[g^{RN}+g^{q}\right].$
(2.23)

The Ricci scalar computed from the pure RN solution vanishes. Thus, in
$H_{\mu\nu}$ and $K_{\mu\nu}$ only the terms containing the Ricci and Riemann
tensors have to be considered. Special care is needed for the quantity
${L^{\rho}}_{\sigma}\equiv\ln\left(\frac{\Box}{\mu^{2}}\right){R^{\rho}}_{\sigma}$.
The components of the Ricci tensor for the classical RN solution are
${R^{t}}_{t}={R^{r}}_{r}=-{R^{\theta}}_{\theta}=-{R^{\phi}}_{\phi}=-G_{N}Q^{2}/r^{4}$.
Hence, it is necessary to compute $\ln\left(\Box/\mu^{2}\right)r^{-4}$. It was
shown [24] that the action of the logarithm on a smooth radial function $f(r)$
is

$\begin{gathered}\ln\left(\frac{\Box}{\mu^{2}}\right)f(r)=\frac{1}{r}\int_{0}^{+\infty}dr^{\prime}\,\frac{r^{\prime}}{r+r^{\prime}}f(r^{\prime})-\lim_{\epsilon\to
0^{+}}\bigg{\\{}\frac{1}{r}\int_{0}^{r-\epsilon}dr^{\prime}\,\frac{r^{\prime}}{r-r^{\prime}}f(r^{\prime})\\\
+\frac{1}{r}\int_{r+\epsilon}^{+\infty}dr^{\prime}\,\frac{r^{\prime}}{r^{\prime}-r}f(r^{\prime})+2f(r)\left[\gamma_{E}+\ln\left(\mu\epsilon\right)\right]\bigg{\\}},\end{gathered}$
(2.24)

where $\gamma_{E}$ is the Euler-Mascheroni constant. The function $r^{-4}$ is
smooth everywhere except at $r=0$, which causes some problems while evaluating
the integrals at $r=0$. However, as we show in Appendix A, the divergences can
be controlled and eventually disappear. The result is

$\ln\left(\frac{\Box}{\mu^{2}}\right)\frac{1}{r^{4}}=-\frac{2}{r^{4}}\left(\ln\left(\mu
r\right)+\gamma_{E}-\frac{3}{2}\right)\equiv-\frac{2}{r^{4}}\ln\left(\tilde{\mu}r\right),$
(2.25)

where we have defined for simplicity
$\tilde{\mu}=\mu\exp\left(\gamma_{E}-\frac{3}{2}\right)$.

We now write the explicit form of the equations. The $tt$ component of (2.23)
is

$-r^{4}\Sigma(r)-r^{5}\frac{d\Sigma(r)}{dr}+64\pi
Q^{2}\left[\left(3\bar{c}_{2}-5\bar{\beta}\right)+6\bar{\beta}\ln\left(\tilde{\mu}r\right)\right]=0,$
(2.26)

which is solved by

$\Sigma(r)=-\frac{64\pi
Q^{2}}{r^{4}}\left[\bar{c}_{2}-\bar{\beta}+2\bar{\beta}\ln\left(\tilde{\mu}r\right)\right].$
(2.27)

The $rr$ component of (2.23) is

$-r^{4}\Sigma(r)+64\pi
Q^{2}\left[\bar{c}_{2}+2\bar{\beta}\ln\left(\tilde{\mu}r\right)\right]-r^{5}\frac{d\Delta(r)}{dr}=0.$
(2.28)

Plugging (2.27) into (2.28) results in

$\Delta(r)=-\frac{32\pi
Q^{2}}{r^{4}}\left[\bar{c}_{2}+2\bar{\beta}\ln\left(\tilde{\mu}r\right)\right].$
(2.29)

The $\theta\theta$ and $\phi\phi$ components of (2.23) have the form

$256\pi Q^{2}\bar{c}_{2}+128\pi Q^{2}\bar{\beta}\left[-1+4\ln\left(\tilde{\mu
r}\right)\right]+r^{5}\left[\frac{d\Delta(r)}{dr}+\frac{d\Sigma(r)}{dr}+r\frac{d^{2}\Delta(r)}{dr^{2}}\right]=0.$
(2.30)

One can check that the expressions (2.27), (2.29) for $\Sigma(r)$ and
$\Delta(r)$ satisfy (2.30). Let us consider now the Maxwell equations. The
Bianchi identity (2.11) is automatically satisfied. The $t$ component of
(2.10) is

$4r\Omega(r)+Q\left[\frac{d\Delta(r)}{dr}-\frac{d\Sigma(r)}{dr}\right]+2r^{2}\frac{d\Omega(r)}{dr}=0.$
(2.31)

Plugging the solutions (2.27) and (2.29) into (2.31) gives

$\Omega(r)=\frac{16\pi
Q^{3}}{r^{6}}\left[\bar{c}_{2}-2\bar{\beta}+2\bar{\beta}\ln\left(\tilde{\mu}r\right)\right].$
(2.32)

We conclude that the set of equation obtained by varying the full action (2.9)
has the following solution, valid at order
$\mathcal{O}\left(G^{2}_{N}\right)$:

$\begin{gathered}ds^{2}=-f(r)dt^{2}+\frac{1}{g(r)}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2},\\\ F_{tr}=-F_{rt}=\frac{Q}{r^{2}}+\frac{16\pi
G^{2}_{N}Q^{3}}{r^{6}}\bigg{[}c_{2}+4c_{3}+2\left(\beta+4\gamma\right)\left(\ln\left(\mu
r\right)+\gamma_{E}-\frac{5}{2}\right)\bigg{]}\end{gathered}$ (2.33)

with

$f(r)=1-\frac{2G_{N}M}{r}+\frac{G_{N}Q^{2}}{r^{2}}-\frac{32\pi
G^{2}_{N}Q^{2}}{r^{4}}\bigg{[}c_{2}+4c_{3}+2\left(\beta+4\gamma\right)\left(\ln\left(\mu
r\right)+\gamma_{E}-\frac{3}{2}\right)\bigg{]},$ (2.34)
$g(r)=1-\frac{2G_{N}M}{r}+\frac{G_{N}Q^{2}}{r^{2}}-\frac{64\pi
G^{2}_{N}Q^{2}}{r^{4}}\Big{[}c_{2}+4c_{3}+2\left(\beta+4\gamma\right)\left(\ln\left(\mu
r\right)+\gamma_{E}-2\right)\Big{]}.$ (2.35)

In other words, both the the classical RN metric and the independent component
of the electromagnetic tensor receive quantum corrections at second order in
curvature. They are proportional to the charge, so in the limit $Q\to 0$ one
correctly recovers the classical Schwarzschild solution.

The components of the metric that we have just found seem to depend on the
arbitrary energy scale $\mu$. However, the renormalised constants $c_{1}$,
$c_{2}$, and $c_{3}$ carry an explicit scale dependence [25]:

$\begin{split}c_{1}(\mu)=c_{1}(\mu_{*})-\alpha\ln\left(\frac{\mu^{2}}{\mu^{2}_{*}}\right),\\\
c_{2}(\mu)=c_{2}(\mu_{*})-\beta\ln\left(\frac{\mu^{2}}{\mu^{2}_{*}}\right),\\\
c_{3}(\mu)=c_{3}(\mu_{*})-\gamma\ln\left(\frac{\mu^{2}}{\mu^{2}_{*}}\right),\\\
\end{split}$ (2.36)

where $\mu_{*}$ is some fixed scale where the effective theory is matched onto
the full theory. Inserting these expressions into (2.34) and (2.35), one sees
that the terms involving $\ln\mu$ cancel out. This in an indicator of the
correctness of our results.

##  3 Quantum gravitational corrections to the entropy of non-extremal
Reissner-Nordström black holes

The results obtained in the previous section are valid for a generic charge
$Q$. In this section we focus on a non-extremal black hole with the additional
physical condition the charge is much smaller than the mass, i.e. $Q^{2}\ll
GM^{2}$. Accordingly, in all the expressions we keep only terms up to order
$\mathcal{O}\left(Q^{2}\right)$.

The classical RN metric has two horizons located at the radii

$\begin{split}&r_{+}=G_{N}M+\sqrt{G^{2}_{N}M^{2}-G_{N}Q^{2}}=2G_{N}M-\frac{Q^{2}}{2M}+\mathcal{O}\left(Q^{4}\right),\\\
&r_{-}=G_{N}M-\sqrt{G^{2}_{N}M^{2}-G_{N}Q^{2}}=\frac{Q^{2}}{2M}+\mathcal{O}\left(Q^{4}\right).\end{split}$
(3.1)

The entropy is generally defined in terms of $r_{+}$ only, namely

$S_{RN}=\frac{\pi r^{2}_{+}}{G_{N}}=\frac{A}{4G_{N}}-2\pi
Q^{2}+\mathcal{O}\left(Q^{4}\right),$ (3.2)

where $A=16\pi G^{2}_{N}M^{2}$ is the classical area of the event horizon of a
Schwarzschild black hole. In this section we compute the quantum corrections
to this expression.

First of all, the corrections to the metric imply a shift of the horizon
radius:

$r_{h}=2G_{N}M-\frac{Q^{2}}{2M}+\frac{8\pi
Q^{2}}{G_{N}M^{3}}\Big{[}c_{2}+4c_{3}+2\left(\beta+4\gamma\right)\left(\ln\left(2G_{N}M\mu\right)+\gamma_{E}-2\right)\Big{]}+\mathcal{O}\left(Q^{4}\right).$
(3.3)

We now compute the entropy of a quantum non-extremal RN black hole using the
Wald formula (1.1) evaluated at the new radius (3.3). The $\epsilon_{\mu\nu}$
tensor is defined as

$\epsilon_{\mu\nu}=\begin{cases}\sqrt{f(r)/g(r)}\hskip
28.45274pt\text{if}\hskip 2.84526pt(\mu,\nu)=(t,r)\\\ -\sqrt{f(r)/g(r)}\hskip
21.33955pt\text{if}\hskip 2.84526pt(\mu,\nu)=(r,t)\\\ 0\hskip
85.35826pt\text{otherwise}.\end{cases}$ (3.4)

In the Wald formula, all the terms of the full effective action have to be
considered, without invoking the Gauss-Bonnet theorem. Up to orders
$\mathcal{O}(Q^{2})$ and $\mathcal{O}\left(G^{2}_{N}\right)$, the $rtrt$
component of the Riemann tensor and the $tt$ component of the Ricci tensor for
the metric (2.33) are respectively

$\begin{gathered}R^{trtr}=-\frac{2G_{N}M}{r^{3}}-\frac{3G_{N}Q^{2}}{r^{4}}-\frac{320\pi
G^{2}_{N}Q^{2}}{r^{6}}\bigg{[}(c_{2}+4c_{3})\\\ +2(\beta+4\gamma)\left(\ln(\mu
r)+\gamma_{E}-\frac{39}{20}\right)\bigg{]},\end{gathered}$ (3.5)
$\begin{gathered}R^{tt}=\frac{G_{N}Q^{2}}{r^{4}}+\frac{2G_{N}MQ^{2}}{r^{5}}-\frac{192\pi
G^{2}_{N}Q^{2}}{r^{6}}\bigg{[}(c_{2}+4c_{3})\\\ +2(\beta+4\gamma)\left(\ln(\mu
r)+\gamma_{E}-\frac{11}{12}\right)\bigg{]}\end{gathered}$ (3.6)

and the Ricci scalar is

$R=\frac{192\pi G^{2}_{N}Q^{2}(\beta+4\gamma)}{r^{6}}.$ (3.7)

Using the relations

$\frac{\partial R}{\partial
R_{\mu\nu\rho\sigma}}=\frac{1}{2}\left(g^{\mu\rho}g^{\nu\sigma}-g^{\mu\sigma}g^{\nu\rho}\right),$
(3.8) $\frac{\partial\left(R_{\alpha\beta}R^{\alpha\beta}\right)}{\partial
R_{\mu\nu\rho\sigma}}=g^{\mu\rho}R^{\nu\sigma}-g^{\nu\rho}R^{\mu\sigma},$
(3.9)
$\frac{\partial\left(R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}\right)}{\partial
R_{\mu\nu\rho\sigma}}=2R^{\mu\nu\rho\sigma}$ (3.10)

and the result of the action of $\ln\left(\Box/\mu^{2}\right)$ on
$1/r^{3},1/r^{5},1/r^{6},\ln(\mu r)/r^{6}$, collected in Appendix A, we find

$S_{\text{Wald}}=-8\pi\sqrt{\frac{f(r_{h})}{g(r_{h})}}\int\limits_{r=r_{h}}d\Sigma\,\frac{\partial\mathcal{L}}{\partial
R_{rtrt}}=\frac{A}{4G}-2\pi
Q^{2}+S_{0}+S_{2}Q^{2}+\mathcal{O}\left(Q^{4}\right),$ (3.11)

where

$S_{0}=64\pi^{2}c_{3}+64\pi^{2}\gamma\Big{[}\ln\left(4G^{2}_{N}M^{2}\mu^{2}\right)+2\gamma_{E}-2\Big{]},$
(3.12)
$S_{2}=\frac{4\pi^{2}}{G_{N}M^{2}}\Big{[}5c_{2}+20c_{3}+\beta\left(10\gamma_{E}-21\right)+8\gamma\left(5\gamma_{E}-11\right)+10\left(\beta+4\gamma\right)\ln\left(2G_{N}M\mu\right)\Big{]}\\\
+\frac{64\pi^{3}}{9G^{2}_{N}M^{4}}\bigg{\\{}54(\beta+4\gamma)\Big{[}c_{1}+2\alpha\ln\left(2G_{N}M\mu\right)\Big{]}+6(36\gamma_{E}-75)\beta\Big{[}c_{2}+2\beta\ln\left(2G_{N}M\mu\right)\Big{]}\\\
+48(48\gamma_{E}-97)\gamma\Big{[}c_{3}+2\gamma\ln\left(2G_{N}M\mu\right)\Big{]}+6\Big{[}c_{2}\gamma(120\gamma_{E}-251)+3c_{3}\beta(40\gamma_{E}-81)\\\
+4\beta\gamma(120\gamma_{E}-247)\ln\left(2G_{N}M\mu\right)\Big{]}+54\Big{[}c^{2}_{2}+4c_{2}\beta\ln\left(2G_{N}M\mu\right)+4\beta^{2}\ln^{2}\left(2G_{N}M\mu\right)\Big{]}\\\
+576\Big{[}c^{2}_{3}+4c_{3}\gamma\ln\left(2G_{N}M\mu\right)+4\gamma^{2}\ln^{2}\left(2G_{N}M\mu\right)\Big{]}+360\Big{[}c_{2}c_{3}+2c_{2}\gamma\ln\left(2G_{N}M\mu\right)\\\
+2c_{3}\beta\ln\left(2G_{N}M\mu\right)+4\beta\gamma\ln^{2}\left(2G_{N}M\mu\right)\Big{]}+(\beta+4\gamma)\Big{[}9\alpha(12\gamma_{E}-25)\\\
+18\beta\left(\gamma_{E}(6\gamma_{E}-25)+35+\pi^{2}\right)+4\gamma\big{(}6\gamma_{E}\left(24\gamma_{E}-97\right)+331+30\pi^{2}\big{)}\Big{]}\bigg{\\}}.$
(3.13)

The first expression $S_{0}$ reproduces the result obtained in [1] for a
Schwarzschild black hole. Plugging the explicit $\mu$ dependence of the
$c_{i}$ coefficients (2.36) into $S_{0}$ and $S_{2}$, one can check that the
terms involving $\mu$ cancel out. Hence, the corrections to the entropy are RG
invariant. Our results are consistent.

We conclude this section with a remark. The second part of $S_{2}$ is
proportional to $c_{i}c_{j}$ and $c_{i}\alpha_{j}$. In principle one could
reproduce this behaviour by including in the action also terms proportional to
$R^{4}$, like $c_{i}R^{2}c_{j}R^{2}$ and
$c_{i}R^{2}\alpha_{j}R\ln\left(\Box/\mu^{2}\right)R$. However, from
(3.5)-(3.7) the Riemann and Ricci tensor, and the Ricci scalar are
proportional to $G_{N}$ or $G^{2}_{N}$. This implies that the equations of
motion contain terms of at least order $\mathcal{O}\left(G^{3}_{N}\right)$.
For example, the equations of motion from $\Gamma_{4}=\int d^{4}x\,R^{4}$ are

$\begin{gathered}4R^{3}R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R^{4}-24R\left(\nabla_{\mu}R\right)\left(\nabla_{\nu}R\right)-12R^{2}\nabla_{\mu}\nabla_{\nu}R\\\
+12g_{\mu\nu}R^{2}\Box
R+24g_{\mu\nu}R\nabla_{\rho}R\nabla^{\rho}R=0.\end{gathered}$ (3.14)

Since we are working up to order $\mathcal{O}\left(G^{2}_{N}\right)$, we can
safely neglect those terms.

##  4 Quantum gravitational corrections to temperature and pressure

In this section we compute the quantum corrections to thermodynamic quantities
of physical interest, namely temperature and pressure. The first law of
thermodynamics for a Schwarzschild black hole is

$dE=dM=TdS,$ (4.1)

where $T$ and $S$ are the temperature and entropy of the black hole,
respectively. The total energy of a charged black hole is $E=M+Q\Phi$, where
$\Phi$ is the electric potential of the black hole. If the charge is fixed,
the first law gets modified as

$dE=dM+Qd\Phi=TdS,$ (4.2)

Classically, the first non-trivial contribution to the temperature that
contains the charge is a quartic term:

$T_{RN}=\frac{1}{4\pi}\Big{\lvert}\frac{dg^{RN}_{tt}}{dr}\Big{\rvert}_{r=r_{+}}=\frac{1}{8\pi
G_{N}M}-\frac{Q^{4}}{128\pi G^{3}_{N}M^{5}}+\mathcal{O}\left(Q^{6}\right).$
(4.3)

The corrections to the metric imply a correction to the temperature of order
$\mathcal{O}\left(Q^{2}\right)$:

$\begin{gathered}T=\frac{1}{4\pi}\sqrt{\frac{df(r)}{dr}\frac{dg(r)}{dr}}\bigg{\rvert}_{r=r_{h}}=\frac{1}{8\pi
G_{N}M}+\frac{Q^{2}}{4G^{3}_{N}M^{5}}\Big{[}2(c_{2}+4c_{3})\\\
+\beta\left(4\gamma_{E}-9\right)+4\gamma\left(4\gamma_{E}-9\right)+4\left(\beta+4\gamma\right)\ln\left(2G_{N}M\mu\right)\Big{]}+\mathcal{O}\left(Q^{4}\right).\end{gathered}$
(4.4)

The electric potential is

$\Phi=\int_{r_{h}}^{+\infty}dr^{\prime}\,F_{tr}=\int_{r_{h}}^{+\infty}dr^{\prime}\,\left(\frac{Q}{{r^{\prime}}^{2}}+G^{2}_{N}\Omega(r^{\prime})\right)=\frac{Q}{2G_{N}M}+\mathcal{O}\left(Q^{3}\right),$
(4.5)

meaning that it does not receive corrections at order
$\mathcal{O}\left(Q^{2}\right)$.

Even at the classical level, one finds that $TdS/dM-Qd\Phi/dM=1+\Delta(M,Q)$
i.e. the unity plus some additional terms. In order to preserve the first law,
one assumes that the charged black hole has a pressure $P$ such that

$T\frac{dS}{dM}-Q\frac{d\Phi}{dM}-P\frac{dV}{dM}=1+\Delta(M,Q),$ (4.6)

where $V=4/3\pi r^{3}_{h}$ is the volume of the black hole. The pressure is
then given by

$P=-\frac{\Delta(M,Q)}{\frac{dV}{dM}}=-\frac{T\frac{dS}{dM}-Q\frac{d\Phi}{dM}-1}{\frac{dV}{dM}}.$
(4.7)

The pressure on the outer horizon of a classical RN black hole is negative and
proportional to the charge squared [26]:

$P_{RN}\sim-\frac{Q^{2}}{r^{4}_{+}}.$ (4.8)

Evidently, a Schwarzschild black hole does not have a pressure at the
classical level. However, quantum effects do create a pressure [1]. Using
(4.7) we find that the pressure of a non-extremal RN black hole, including the
quantum corrections, is

$P=-\frac{Q^{2}}{64\pi
G^{4}_{N}M^{4}}-\frac{\gamma}{2G^{4}_{N}M^{4}}+P_{2}Q^{2}+\mathcal{O}\left(Q^{4}\right)$
(4.9)

with

$P_{2}=\frac{1}{32G^{5}_{N}M^{6}}\Big{[}c_{2}+4c_{3}+2\beta(\gamma_{E}-4)+8\gamma(\gamma_{E}-5)+2(\beta+4\gamma)\ln(2G_{N}M\mu)\Big{]}\\\
+\frac{\pi}{9G^{6}_{N}M^{8}}\bigg{\\{}54(\beta+4\gamma)\Big{[}c_{1}+2\alpha\ln\left(2G_{N}M\mu\right)\Big{]}+72(3\gamma_{E}-7)\beta\Big{[}c_{2}+2\beta\ln\left(2G_{N}M\mu\right)\Big{]}\\\
+768(3\gamma_{E}-7)\gamma\Big{[}c_{3}+2\gamma\ln\left(2G_{N}M\mu\right)\Big{]}+6\Big{[}c_{2}\gamma(120\gamma_{E}-287)+3c_{3}\beta(40\gamma_{E}-91)\\\
+160\beta\gamma(3\gamma_{E}-7)\ln\left(2G_{N}M\mu\right)\Big{]}+54\Big{[}c^{2}_{2}+4c_{2}\beta\ln\left(2G_{N}M\mu\right)+4\beta^{2}\ln^{2}\left(2G_{N}M\mu\right)\Big{]}\\\
+576\Big{[}c^{2}_{3}+4c_{3}\gamma\ln\left(2G_{N}M\mu\right)+4\gamma^{2}\ln^{2}\left(2G_{N}M\mu\right)\Big{]}+360\Big{[}c_{2}c_{3}+2c_{2}\gamma\ln\left(2G_{N}M\mu\right)\\\
+2c_{3}\beta\ln\left(2G_{N}M\mu\right)+4\beta\gamma\ln^{2}\left(2G_{N}M\mu\right)\Big{]}+(\beta+4\gamma)\Big{[}36\alpha(3\gamma_{E}-7)\\\
+9\beta\big{(}8\gamma_{E}(3\gamma_{E}-14)+95+4\pi^{2}\big{)}+\gamma\big{(}192\gamma_{E}(3\gamma_{E}-14)+2095+120\pi^{2}\big{)}\Big{]}\bigg{\\}}.$
(4.10)

The term $-\gamma/2G^{4}_{N}M^{4}$ is the only quantum correction for a
Schwarzschild black hole. A charged black hole has additional quantum
corrections proportional to the electric charge.

##  5 Conclusions and outlook

In this paper we have studied the quantum gravitational corrections to the
Wald entropy of a four dimensional, non-extremal Reissner-Nordström black
hole, starting from an effective action for quantum gravity supplemented by
the Maxwell action. The corrections to the entropy are RG invariant.
Furthermore, contrary to what happens for the Schwarzschild metric, we have
shown that the classical RN solution receives quantum corrections at second
order in curvature. These corrections shift the position of the event horizon.
Finally, we computed the quantum gravitational corrections to the temperature
and pressure. All these corrections are very small for astrophysical black
holes, for which $Q^{2}\ll GM^{2}$.

The research presented in this paper naturally lends itself to further
generalisations. For instance, one could try to compute the quantum
gravitational corrections to the entropy of a charged rotating (Kerr-Newman)
black hole, which is the most realistic model of an astrophysical black hole.
Alternatively, one could also have a look at AdS black holes, whose importance
lies in the context of the AdS/CFT correspondence.

##  6 Acknowledgements

The author thanks the Bethe Center for Theoretical Physics (BCTP) of the
University of Bonn for its hospitality, Saurabh Natu for valuable advice and
discussions about this research and the anonymous reviewers for their
comments, which improved the final version of the manuscript. This work was
supported by the Bonn-Cologne Graduate School of Physics and Astronomy (BCGS).

##  Appendix A Action of $\ln\left(\frac{\Box}{\mu^{2}}\right)$ on radial
functions

In this appendix we collect the result of the action of
$\ln\left(\Box/\mu^{2}\right)$ on various radial functions $f(r)$. If $r>0$,
$f(r)\neq 0$ and $\exists\epsilon>0$ such that $f(r^{\prime})$ is smooth for
$\lvert r-r^{\prime}\rvert\leq\epsilon$, then [24]

$\begin{gathered}\ln\left(\frac{\Box}{\mu^{2}}\right)f(r)=\frac{1}{r}\int_{0}^{+\infty}dr^{\prime}\,\frac{r^{\prime}}{r+r^{\prime}}f(r^{\prime})-\lim_{\epsilon\to
0^{+}}\bigg{\\{}\frac{1}{r}\int_{0}^{r-\epsilon}dr^{\prime}\,\frac{r^{\prime}}{r-r^{\prime}}f(r^{\prime})\\\
+\frac{1}{r}\int_{r+\epsilon}^{+\infty}dr^{\prime}\,\frac{r^{\prime}}{r^{\prime}-r}f(r^{\prime})+2f(r)\left[\gamma_{E}+\ln\left(\mu\epsilon\right)\right]\bigg{\\}}.\end{gathered}$
(A.1)

In order to derive the equations of motion from the effective action (2.9) in
Section 2.1, the non-local contribution $K_{\mu\nu}$ has to be evaluated on
the classical RN solution. Since the components of the Ricci tensor for the
classical RN solution are
${R^{t}}_{t}={R^{r}}_{r}=-{R^{\theta}}_{\theta}=-{R^{\phi}}_{\phi}=-G_{N}Q^{2}/r^{4}$,
the following quantity has to be computed:

$\ln\left(\frac{\Box}{\mu^{2}}\right)\frac{1}{r^{4}}.$ (A.2)

Owing to the singularity at $r=0$, the integrals in (A.1) need to be
regularised. A way to achieve this is by shifting the denominators of the
problematic integrands via $r^{\prime}\to r^{\prime}\pm i\epsilon$ and then
taking the real part of the result, as $\ln(\Box/\mu^{2})f(r)\in\mathbb{R}$.
Thus

$\begin{gathered}\ln\left(\frac{\Box}{\mu^{2}}\right)\frac{1}{r^{4}}=\text{Re}\lim_{\epsilon\to
0^{+}}\bigg{\\{}\frac{1}{r}\int_{0}^{+\infty}dr^{\prime}\,\frac{r^{\prime}}{r+r^{\prime}}\frac{1}{\left(r^{\prime}-i\epsilon\right)^{4}}-\frac{1}{r}\int_{0}^{r-\epsilon}dr^{\prime}\,\frac{r^{\prime}}{r-r^{\prime}}\frac{1}{\left(r^{\prime}+i\epsilon\right)^{4}}\\\
-\frac{1}{r}\int_{r+\epsilon}^{+\infty}dr^{\prime}\,\frac{r^{\prime}}{r^{\prime}-r}\frac{1}{{r^{\prime}}^{4}}-\frac{2}{r^{4}}\left[\gamma_{E}+\ln\left(\mu\epsilon\right)\right]\bigg{\\}}.\end{gathered}$
(A.3)

The divergences cancel out:

$\begin{gathered}\ln\left(\frac{\Box}{\mu^{2}}\right)\frac{1}{r^{4}}=\text{Re}\Big{[}\frac{1}{r}\left(\frac{3}{r^{3}}+i\frac{\pi}{r^{3}}-\frac{2\ln
r}{r^{3}}+\frac{2\ln\epsilon}{r^{3}}\right)-\frac{2\gamma_{E}}{r^{4}}-\frac{2\ln\epsilon}{r^{4}}-\frac{2\ln\mu}{r^{4}}\Big{]}=\\\
=-\frac{2}{r^{4}}\left(\ln\left(\mu
r\right)+\gamma_{E}-\frac{3}{2}\right).\end{gathered}$ (A.4)

In Section 3 we computed the corrections to the entropy using the Wald
formula. The derivative of the effective action with respect to the Riemann
tensor contains contributions from the Riemann and Ricci tensors of the metric
with quantum corrections (2.34), (2.35). These tensors contain the radial
functions $r^{-3},r^{-5},r^{-6},\ln(\mu r)r^{-6}$. Hence, it is necessary to
know the action of $\ln\left(\Box/\mu^{2}\right)$ on them. Let us consider for
example $\ln\left(\Box/\mu^{2}\right)r^{-3}$. We regularise the integrals in
(A.1) as

$\begin{gathered}\ln\left(\frac{\Box}{\mu^{2}}\right)\frac{1}{r^{3}}=\lim_{\epsilon\to
0^{+}}\bigg{\\{}\frac{1}{r}\int_{\sqrt{\epsilon
r}}^{+\infty}dr^{\prime}\,\frac{r^{\prime}}{r+r^{\prime}}\frac{1}{{r^{\prime}}^{3}}-\frac{1}{r}\int_{\sqrt{\epsilon
r}}^{r-\sqrt{\epsilon
r}}dr^{\prime}\,\frac{r^{\prime}}{r-r^{\prime}}\frac{1}{{r^{\prime}}^{3}}\\\
-\frac{1}{r}\int_{r+\sqrt{\epsilon
r}}^{+\infty}dr^{\prime}\,\frac{r^{\prime}}{r^{\prime}-r}\frac{1}{{r^{\prime}}^{3}}-\frac{2}{r^{3}}\left[\gamma_{E}+\ln\left(\mu\epsilon\right)\right]\bigg{\\}}.\end{gathered}$
(A.5)

Again, all divergences cancel out and the result is

$\ln\left(\frac{\Box}{\mu^{2}}\right)\frac{1}{r^{3}}=-\frac{2}{r^{3}}\left(\ln\left(\mu
r\right)+\gamma_{E}-1\right).$ (A.6)

With analogous calculations we find

$\ln\left(\frac{\Box}{\mu^{2}}\right)\frac{1}{r^{5}}=-\frac{2}{r^{5}}\left(\ln(\mu
r)+\gamma_{E}-\frac{47}{12}\right),$ (A.7)
$\ln\left(\frac{\Box}{\mu^{2}}\right)\frac{1}{r^{6}}=-\frac{2}{r^{6}}\left(\ln(\mu
r)+\gamma_{E}-\frac{25}{12}\right),$ (A.8)
$\ln\left(\frac{\Box}{\mu^{2}}\right)\frac{\ln\left(\tilde{\mu}r\right)}{r^{6}}=\\\
=-\frac{2\ln\left(\tilde{\mu}r\right)}{r^{6}}\left(\ln(\mu
r)+\gamma_{E}-\frac{25}{12}\right)+\\\
+\frac{1}{r^{6}}\left(\frac{205}{72}-\frac{\pi^{2}}{3}\right).$ (A.9)

## References

  * [1] X. Calmet and F. Kuipers, “Quantum gravitational corrections to the entropy of a Schwarzschild black hole,” Phys. Rev. D 104 no. 6, (2021) 066012, arXiv:2108.06824 [hep-th].
  * [2] S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. 43 (1975) 199–220. [Erratum: Commun.Math.Phys. 46, 206 (1976)].
  * [3] J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D 7 (1973) 2333–2346.
  * [4] R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D 48 no. 8, (1993) R3427–R3431, arXiv:gr-qc/9307038.
  * [5] X. Calmet, “Vanishing of Quantum Gravitational Corrections to Vacuum Solutions of General Relativity at Second Order in Curvature,” Phys. Lett. B 787 (2018) 36–38, arXiv:1810.09719 [hep-th].
  * [6] J. Bally and E. R. Harrison, “The electrically polarized universe,” Astrophysical Journal 220 (1978) 743.
  * [7] J. F. Donoghue, B. R. Holstein, B. Garbrecht, and T. Konstandin, “Quantum corrections to the Reissner-Nordström and Kerr-Newman metrics,” Phys. Lett. B 529 (2002) 132–142, arXiv:hep-th/0112237. [Erratum: Phys.Lett.B 612, 311–312 (2005)].
  * [8] B. R. Holstein, “Metric modifications for a massive spin 1 particle,” Phys. Rev. D 74 (2006) 084030, arXiv:gr-qc/0607051.
  * [9] M. Buric and V. Radovanovic, “Quantum corrections for the Reisner-Nordstrom black hole,” Class. Quant. Grav. 16 (1999) 3937–3951, arXiv:gr-qc/9907036.
  * [10] P. A. Cano, S. Chimento, R. Linares, T. Ortín, and P. F. Ramírez, “$\alpha^{\prime}$ corrections of Reissner-Nordström black holes,” JHEP 02 (2020) 031, arXiv:1910.14324 [hep-th].
  * [11] Z.-h. Li, “Quantum corrections to the entropy of a Reissner-Nordstrom black hole due to spin fields,” Phys. Rev. D 62 (2000) 024001.
  * [12] M. Yoon, J. Ha, and W. Kim, “Entropy of Reissner-Nordstrom Black Holes with Minimal Length Revisited,” Phys. Rev. D 76 (2007) 047501, arXiv:0706.0364 [gr-qc].
  * [13] M. M. Akbar and S. Das, “Entropy corrections for Schwarzschild and Reissner-Nordström black holes,” Class. Quant. Grav. 21 (2004) 1383–1392, arXiv:hep-th/0304076.
  * [14] S. Weinberg, “Ultraviolet divergences in quantum theories of gravitation,” in General Relativity: An Einstein Centenary Survey, pp. 790–831.  Univ. Pr., Cambridge, UK, 1980.
  * [15] A. A. Starobinskii, “Evolution of small perturbations of isotropic cosmological models with one-loop quantum gravitational corrections,” ZhETF Pisma Redaktsiiu 34 (1981) 460–463.
  * [16] A. O. Barvinsky and G. A. Vilkovisky, “The generalized Schwinger-DeWitt technique and the unique effective action in quantum gravity,” Phys. Lett. B 131 (1983) 313–318.
  * [17] A. O. Barvinsky and G. A. Vilkovisky, “The Generalized Schwinger-DeWitt Technique in Gauge Theories and Quantum Gravity,” Phys. Rept. 119 (1985) 1–74.
  * [18] A. O. Barvinsky and G. A. Vilkovisky, “Beyond the Schwinger-Dewitt Technique: Converting Loops Into Trees and In-In Currents,” Nucl. Phys. B 282 (1987) 163–188.
  * [19] A. O. Barvinsky and G. A. Vilkovisky, “Covariant perturbation theory. 2: Second order in the curvature. General algorithms,” Nucl. Phys. B 333 (1990) 471–511.
  * [20] J. F. Donoghue, “General relativity as an effective field theory: The leading quantum corrections,” Phys. Rev. D 50 (1994) 3874–3888, arXiv:gr-qc/9405057.
  * [21] J. F. Donoghue and B. K. El-Menoufi, “Covariant non-local action for massless QED and the curvature expansion,” JHEP 10 (2015) 044, arXiv:1507.06321 [hep-th].
  * [22] J. F. Donoghue and B. K. El-Menoufi, “Nonlocal quantum effects in cosmology: Quantum memory, nonlocal FLRW equations, and singularity avoidance,” Phys. Rev. D 89 no. 10, (2014) 104062, arXiv:1402.3252 [gr-qc].
  * [23] A. Codello and R. K. Jain, “On the covariant formalism of the effective field theory of gravity and its cosmological implications,” Class. Quant. Grav. 34 no. 3, (2017) 035015, arXiv:1507.07829 [astro-ph.CO].
  * [24] X. Calmet, R. Casadio, and F. Kuipers, “Quantum Gravitational Corrections to a Star Metric and the Black Hole Limit,” Phys. Rev. D 100 no. 8, (2019) 086010, arXiv:1909.13277 [hep-th].
  * [25] B. K. El-Menoufi, “Quantum gravity of Kerr-Schild spacetimes and the logarithmic correction to Schwarzschild black hole entropy,” JHEP 05 (2016) 035, arXiv:1511.08816 [hep-th].
  * [26] Y.-H. Wei, “Understanding first law of thermodynamics of black holes,” Phys. Lett. B 672 (2009) 98–100.