# Corrections to Extremal Black Holes from Iyer-Wald Formalism

Lars Aalsma<EMAIL_ADDRESS>Department of Physics, University of Wisconsin-
Madison, 1150 University Avenue, Madison, Wisconsin 53706, USA

###### Abstract

We present a general method of computing corrections to the extremality bound
and entropy of Kerr-Newman black holes due to an arbitrary perturbation using
the Iyer-Wald formalism. In this method, corrections to the extremality bound
are given by an integral over an effective stress tensor which, in particular
cases of interest, reduces to the usual stress tensor. This clarifies the
relation between extremality corrections and energy conditions. In particular,
we show that a necessary condition to decrease the mass of an extremal black
hole in a canonical ensemble, as required by the weak gravity conjecture, is
that the perturbation violates the dominant energy condition. As an
application of our method, we compute higher-derivative corrections to charged
black holes in anti-de Sitter space and Kerr black holes.

##  I Introduction

In the absence of experiments that directly probe Planckian energies, a regime
where quantum gravity effects become important, it is useful to understand if
quantum gravity puts any constraints on low-energy effective field theories
(EFTs). While one might naïvely expect that any such effect is suppressed by
the Planck scale and is therefore irrelevant at low energies, it turns out
that the structure of quantum gravity is more interesting than that. Even in
systems with low energies and curvatures, quantum gravity still seems to put
some constrains on EFTs. Two famous examples of this effect are the black hole
information paradox, which suggests corrections to the standard semiclassical
picture of black holes to be consistent with unitarity, and the absence of
global symmetries in quantum gravity.

The goal of the swampland program is to make precise what constraints quantum
gravity puts on EFTs (see Brennan  _et al._ (2017); Palti (2019) for reviews).
One such constraint is the weak gravity conjecture (WGC), which states that
any EFT with a U(1) gauge field coupled to gravity should contain a state that
has a charge-to-mass ratio that exceeds the one for extremal black holes in
that theory Arkani-Hamed  _et al._ (2007). Kinematically, this allows extremal
black holes to shed their charge by emitting superextremal states. The so-
called mild form of the WGC suggests that black holes themselves can be the
states that satisfy the WGC, which is possible because higher-derivative
corrections modify the black hole extremality bound Kats  _et al._ (2007).
Depending on the sign of the Wilson coefficients multiplying the higher-
derivative corrections, this allows extremal black holes to have a charge-to-
mass ratio that exceeds the one in the uncorrected theory, thereby satisfying
the WGC. This mild form of the WGC has been studied for many different black
hole solutions Cheung  _et al._ (2018); Hamada  _et al._ (2019); Cheung  _et
al._ (2019); Loges  _et al._ (2020a); Bellazzini  _et al._ (2019); Cano  _et
al._ (2020a, b); Charles (2019); Jones and McPeak (2020); Cremonini  _et al._
(2020, 2021a); Loges  _et al._ (2020b); Melo and Santos (2021); Aalsma  _et
al._ (2021); Arkani-Hamed  _et al._ (2021); Cremonini  _et al._ (2021b) and it
has been demonstrated that, at least under some assumptions about the UV
theory from which the higher-derivative corrections originate, unitarity and
causality constrain the signs to be compatible with the WGC Hamada  _et al._
(2019); Bellazzini  _et al._ (2019); Arkani-Hamed  _et al._ (2021).
Additionally, in particular cases the mild form and particle form of the WGC
can be shown to be intimately connected Aalsma  _et al._ (2019).

In Cheung  _et al._ (2018); Hamada  _et al._ (2019), an interesting
perspective has been put on the mild form of the WGC by demonstrating that
corrections to the extremality bound in a canonical ensemble are related to
corrections to the entropy in a microcanonical ensemble. This
entropy/extremality relationship follows quite generally from thermodynamics
Loges  _et al._ (2020a); Goon and Penco (2020) and reformulates the WGC as a
statement that (higher-derivative) corrections should increase the
microcanonical entropy of an extremal black hole.

In this paper, we will further investigate this relation for four-dimensional
Kerr-Newman black holes and give a new derivation of it based on the Iyer-Wald
formalism. Similar to a thermodynamic approach, this has the advantage that it
is not necessary to explicitly solve for the corrected metric which can be a
laborious task. Using this approach we obtain a relation that relates
corrections to the extremality bound to an integral over an effective stress
tensor whose positivity properties determines the sign of the corrections. We
show that this integral is directly related to corrections to the area of the
black hole that, in many cases of interest, gives the dominant contribution to
the microcanonical black hole entropy. The effective stress tensor captures
any perturbation to the action and not just higher-derivative corrections. In
special cases, the effective stress tensor is equivalent to the standard
stress tensor. This occurs for Reissner-Nordström and Kerr black holes and we
show that in those cases a necessary condition to decrease the mass of an
extremal black hole in the canonical ensemble is that the perturbation to the
stress tensor violates the dominant energy condition (DEC). However, in
general this condition is not sufficient.

As an application and consistency check of our method, we compute higher-
derivative corrections to four-dimensional charged black holes in anti-de
Sitter space and Kerr black holes. In those cases, we investigate to what
extent energy conditions can be used to constrain the sign of the higher-
derivative corrections. We find that a violation of the DEC only fixes the
sign of the $(F_{ab}F^{ab})^{2}$ operator. This is the same operator
constrained by unitarity and combining this with the null energy condition
(NEC) the WGC follows for charged black holes that are small with respect to
the AdS radius. However, due to the angular dependence of rotating black
holes, energy conditions do not constrain the sign of higher-derivative
corrections to Kerr black holes.

This paper is organized as follows. In Sec. II we use the Iyer-Wald formalism
to derive a general relation between corrections to Kerr-Newman black holes
and the effective stress tensor. Continuing, in Sec. III we discuss how this
relation relates to energy conditions and apply it to four-dimensional higher-
derivative corrected charged black holes in AdS and Kerr black holes. Finally,
we comment on the formulation of the WGC in AdS space in Sec. IV and conclude
in Sec. V.

##  II Black Hole Corrections from Iyer-Wald

###  II.1 Iyer-Wald Formalism

In this section, we use the Iyer-Wald formalism Wald (1993); Iyer and Wald
(1994) to derive how an arbitrary perturbation corrects four-dimensional Kerr-
Newman black holes. We start by considering a general Lagrangian $d$-form
$L(\phi)$ that depends on an arbitrary set of matter fields $\phi$. Varying
with respect to the matter fields yields the following general form.

$\delta L(\phi)=E(\phi)+\mathrm{d}\theta(\phi,\delta\phi)~{}.$ (1)

$E(\phi)$ collectively denotes the equations of motion and
$\mathrm{d}\theta(\phi,\delta\phi)$ are boundary terms. When the equations of
motion are satisfied $E(\phi)=0$. We define a symplectic current $(d-1)$-form
as follows:

$\omega(\phi;\delta_{1}\phi,\delta_{2}\phi)=\delta_{1}\theta(\phi,\delta_{2}\phi)-\delta_{2}\theta(\phi,\delta_{1}\phi)~{}.$
(2)

For some vector field $\xi$ we can define a Noether current $J_{\xi}(\phi)$:

$J_{\xi}(\phi)=\theta(\phi;{\cal L}_{\xi}\phi)-\iota_{\xi}L(\phi)~{}.$ (3)

Here $\iota_{\xi}$ denotes the interior product. At the same time, it can be
shown Iyer and Wald (1995) that the Noether current can also be written as

$J_{\xi}(\phi)=C_{\xi}+\mathrm{d}Q_{\xi}~{}.$ (4)

Here $C_{\xi}\equiv\xi^{a}C_{a}$ captures the constraints of the theory such
that $C_{a}=0$ when $E(\phi)=0$. $Q_{\xi}$ is the Noether charge associated
with $\xi$. By varying the Noether current and equating (3) and (4) we obtain

$\displaystyle\mathrm{d}\left(\delta
Q_{\xi}-\iota_{\xi}\theta(\phi,\delta\phi)\right)$
$\displaystyle=\omega(\phi;\delta\phi,{\cal L}_{\xi}\phi)-\delta C_{\xi}$ (5)
$\displaystyle-\iota_{\xi}(E(\phi)\delta\phi)~{}.$

By assuming that $\xi$ is a symmetry of the action, such that ${\cal
L}_{\xi}\phi=0$, we find that when integrated over a Cauchy surface $\Sigma$
the relation (5) can be written on shell as

$\int_{\partial\Sigma}\left(\delta
Q_{\xi}-\iota_{\xi}\theta(\phi,\delta\phi)\right)=-\int_{\Sigma}\delta
C_{\xi}~{}.$ (6)

We will now evaluate this relation for the case of primary interest, which is
Einstein-Maxwell theory.

###  II.2 Einstein-Maxwell Theory

The Lagrangian of Einstein-Maxwell theory is

$L=\frac{1}{2}\left(R-2\Lambda\right)\epsilon-\frac{1}{4}F_{ab}F^{ab}\epsilon~{}.$
(7)

Note that we work in units where $8\pi G_{d}=1$ and $\epsilon$ is the volume
$d$-form. By explicitly varying the action we find the equations of motion

$\displaystyle 0$ $\displaystyle=G_{ab}+\Lambda g_{ab}+T^{A}_{ab}~{},$ (8)
$\displaystyle 0$ $\displaystyle=\nabla_{a}F^{ab}~{},$

and the symplectic potential of the metric and gauge field

$\displaystyle\theta^{g}_{abc}$
$\displaystyle=\frac{\epsilon_{abcd}}{2}g^{de}\left(\nabla^{f}\delta
g_{ef}-g^{hi}\nabla_{e}\delta g_{hi}\right)~{},$ (9)
$\displaystyle\theta^{A}_{abc}$ $\displaystyle=-\epsilon_{abcd}F^{de}\delta
A_{e}~{}.$

The electromagnetic stress tensor is

$T^{A}_{ab}=F_{ac}F_{b}^{\,\,c}-\frac{1}{4}g_{ab}F_{cd}F^{cd}~{}.$ (10)

Using (3) we find that the Noether current can be written precisely in the
form $\eqref{eq:Noether2}$. The constraints are given by

$\displaystyle C^{g}_{abc}$
$\displaystyle=\epsilon_{abce}\xi^{d}(T^{g})_{d}^{\,\,e}~{},$ (11)
$\displaystyle C^{A}_{abc}$
$\displaystyle=\epsilon_{abce}(\xi^{d}A_{d})j^{e}~{},$

where the tensor $T^{g}_{ab}$ is defined by subtracting the electromagnetic
part from the stress tensor and we introduced the electromagnetic current
$j^{a}$:

$\displaystyle T^{g}_{ab}$ $\displaystyle=G_{ab}+\Lambda
g_{ab}-T^{A}_{ab}~{},$ (12) $\displaystyle j^{a}$
$\displaystyle=\nabla_{b}F^{ab}~{}.$

The Noether charges are given by

$\displaystyle Q_{ab}^{g}$
$\displaystyle=\frac{\epsilon_{abcd}}{2}\nabla^{c}\xi^{d}~{},$ (13)
$\displaystyle Q_{ab}^{A}$
$\displaystyle=-\frac{\epsilon_{abcd}}{2}F^{cd}A_{e}\xi^{e}~{}.$

Let us consider a stationary solution to the source-free equations of motion
(that is $T^{g}_{ab}=j_{a}=0$). We assume the existence of a Killing vector
$K^{a}=t^{a}+\Omega\phi^{a}$ with $t^{a}$ a timelike Killing vector,
$\phi^{a}$ the axial Killing vector and $\Omega$ the angular potential.
Notably this is the case for Kerr-Newman black holes, whose metric is given by

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-\frac{\Delta(r)}{\rho(r)^{2}}\left(\mathrm{d}t-a\sin^{2}\theta\mathrm{d}\phi\right)^{2}+\rho(r)^{2}\left(\frac{\mathrm{d}r^{2}}{\Delta(r)}+\mathrm{d}\theta^{2}\right)$
(14)
$\displaystyle+\frac{\sin^{2}\theta}{\rho(r)^{2}}\left((r^{2}+a^{2})\mathrm{d}\phi-a\mathrm{d}t\right)^{2}~{}.$

The gauge field is given by

$A=-\frac{\sqrt{2}rr_{Q}}{\rho(r)^{2}}\left(\mathrm{d}t-a\sin^{2}\theta\mathrm{d}\phi\right)~{},$
(15)

and

$\displaystyle\rho(r)^{2}$ $\displaystyle=r^{2}+a^{2}\cos^{2}\theta~{},$ (16)
$\displaystyle\Delta(r)$ $\displaystyle=r^{2}-r_{s}r+a^{2}+r_{Q}^{2}~{},$

with $a=J/M$, $r_{s}=M/(4\pi)$ and $r_{Q}=Q/(4\sqrt{2}\pi)$. The two horizons
of the black hole, obtained by solving $\Delta(r)=0$, are given by

$r_{\pm}=\frac{1}{2}\left(r_{s}\pm\sqrt{r_{s}^{2}-4(a^{2}+r_{Q}^{2})}\right)~{}.$
(17)

The angular and electric potential are

$\displaystyle\Omega$
$\displaystyle=\frac{a}{\left(\frac{M}{8\pi}\right)^{2}+a^{2}}~{},$ (18)
$\displaystyle\Phi$ $\displaystyle=\frac{2MQ}{(8\pi a)^{2}+M^{2}}~{}.$

We can now evaluate (6). To do so, we note that $\partial\Sigma$ consists of
two parts: a surface at infinity and the bifurcation surface of the black
hole, which is the surface where the past and future horizon intersect. We
will assume that the matter fields fall off sufficiently rapidly at infinity
such that they do not contribute to the surface integral at infinity. This
implies a particular gauge choice for the gauge potential $A_{a}$. Although
this might lead to singular behaviour of $A_{a}$ on the horizon, its variation
$\delta A_{a}$ is smooth Sorce and Wald (2017), and this does not pose any
further problems. Using the explicit form of the metric, see (14), we find

$\int_{\infty}\left(\delta
Q_{K}-\iota_{K}\theta(\phi,\delta\phi)\right)=\delta M-\Omega\delta J~{},$
(19)

where $M$ and $J$ are the ADM mass and angular momentum, respectively.
Although (14) corresponds to an asymptotically flat black hole this relation
also holds for AdS black holes. The matter fields will contribute to the
integral over the bifurcation surface $B$. For the gravitational part this
leads to the well-known result Wald (1993)

$\int_{B}\left(\delta Q^{g}_{K}-\iota_{K}\theta^{g}\right)=\kappa\delta
A_{B}~{}.$ (20)

Here $\kappa$ is the surface gravity at the horizon and $\delta A_{B}$ is
variation of the bifurcation surface.111Strictly speaking, only nonextremal
black holes have bifurcation surfaces so to obtain a similar expression for
extremal black holes we should take a limit from a nonextremal black hole.
This is implicit in what follows. To arrive at this relation we used
$\left.K^{a}\right|_{B}=0$. Because the temperature of a black hole is given
by $T=\kappa/(2\pi)$, we can write the right-hand side as $T\delta S_{\rm
BH}$. It is important to note that this is the variation of the Bekenstein-
Hawking entropy and not the Wald entropy, because possible higher-derivative
corrections are considered to be perturbations that appear in $\delta C_{K}$.
Similarly, for the electromagnetic part we find

$\displaystyle\int_{B}\left(\delta Q^{A}_{K}-\iota_{K}\theta^{A}\right)$
$\displaystyle=\frac{\Phi}{2}\int_{B}\delta\left(\epsilon_{abcd}F^{cd}\right)~{},$
(21) $\displaystyle=\frac{\Phi}{2}\delta Q~{}.$

Here $\Phi\equiv-\left.A_{a}K^{a}\right|_{B}$ is the electric potential at the
horizon. To arrive at the right-hand side, which we recognize as being
proportional to the electric flux $\delta Q$ through the horizon, we again
used $\left.K^{a}\right|_{B}=0$. However, note that $\Phi$ is constant.
Computing the variation of $C_{K}$ and performing an integration by parts (6)
becomes

$\displaystyle\delta M-\Omega\delta J-T\delta S_{\rm BH}-\Phi\delta Q=$ (22)
$\displaystyle\qquad-\int_{\Sigma}\epsilon_{ebcd}\left[\delta(T^{g})_{a}^{\,\,e}+F_{af}\delta
F^{ef}\right]K^{a}~{}.$

From this relation we can read off corrections to the different quantities of
the black hole. Before we do so, let us make a remark. Although we did not
necessarily assume that the Kerr-Newman black holes we are considering are
only electrically charged, possible magnetic charges do not appear in (22).
The reason for this, as was recently nicely explained in Mitsios  _et al._
(2021), is that the Iyer-Wald formalism only picks up the variation of charges
that are associated with a gauge symmetry. This means that if we would like to
consider magnetically charged or dilatonic black holes, we would need to take
into account additional variations that lead to the conserved charges
associated with the magnetic and scalar field. We leave this generalization to
future work. Alternatively, one could use a thermodynamic approach based on
the Euclidean action Loges  _et al._ (2020a) to compute such corrections.

###  II.3 Corrections to Extremal Black Holes

We will now see how (22) can be employed to deduce corrections to the black
hole extremality bound. First of all, we find it convenient to write the
right-hand side as an “effective” stress tensor

$\delta T_{ab}^{\rm eff}\equiv\delta T^{g}_{ab}+F_{ac}\delta
F_{b}^{\,\,c}~{}.$ (23)

This quantity can be interpreted as the stress tensor of some perturbation in
special cases that we will discuss later. We see this by rewriting the
effective stress tensor as

$\delta T_{ab}^{\rm eff}=\delta T_{ab}+\left(\frac{1}{2}g_{ab}F_{cd}\delta
F^{cd}-F_{ac}\delta F_{b}^{\,\,c}\right)~{},$ (24)

where the total stress tensor is the sum of the gravitational and
electromagnetic part: $\delta T_{ab}=\delta T^{g}_{ab}+\delta T^{A}_{ab}$.
Only when the terms in brackets vanish, this effective stress tensor coincides
with the standard stress tensor of the perturbation we are considering. To
explicitly evaluate the integral over the stress tensor, it is useful to write
it has

$\int_{\Sigma}\epsilon_{ebcd}\delta\left(T_{a}^{\,\,e}\right)^{\rm
eff}K^{a}=-\int_{\Sigma}\tilde{\epsilon}_{bcd}\delta T_{ae}^{\rm
eff}K^{a}n^{e}~{},$ (25)

where $\tilde{\epsilon}$ is the volume form on $\Sigma$ and $n^{a}$ the
timelike unit normal to $\Sigma$. The relation of interest now takes the form

$\delta M-\Omega\delta J-\Phi\delta Q-T\delta S_{\rm
BH}=\int_{\Sigma}\tilde{\epsilon}_{bcd}\delta T_{ae}^{\rm eff}K^{a}n^{e}~{}.$
(26)

To evaluate (26) we use the explicit form of the Kerr metric given in (14). As
mentioned before, we use the asymptotically flat form, but the same relation
holds in AdS space. If we now define

$\Xi:=M-\Omega J-\Phi Q~{},$ (27)

we note that extremal Kerr-Newman black holes without perturbation ($\delta
T_{ab}^{g}=j_{a}=0$) obey

$\delta\Xi=\delta M-\Omega\delta J-\Phi\delta Q=0~{}.$ (28)

Thus, we see that when we send $T\to 0$ the correction to the integral over
the effective stress tensor determines the correction to the extremality
bound. Also, if we focus on black holes that obey $\delta\Xi=0$, the
correction to the entropy is given by minus the integral over the effective
stress tensor. In summary, we have

$\displaystyle\lim_{T\to 0}\left(\delta M-\Omega\delta J-\Phi\delta Q\right)$
$\displaystyle=\lim_{T\to 0}\int_{\Sigma}\tilde{\epsilon}_{bcd}\delta
T_{ae}^{\rm eff}K^{a}n^{e}~{},$ (29) $\displaystyle\left(T\delta S_{\rm
BH}\right)$ $\displaystyle=-\int_{\Sigma}\tilde{\epsilon}_{bcd}\delta
T_{ae}^{\rm eff}K^{a}n^{e}~{}.$

The second relation involving the entropy seems similar to the relation
between the microcanonical entropy and the correction to the Gibbs free energy
(see for example Eq. (17) of Melo and Santos (2021)), but we stress again that
(29) involves the Bekenstein-Hawking entropy and not the Wald entropy. To
relate our correction to the Wald entropy, one can use the first law $\delta
M=T\delta S_{\rm Wald}+\Omega\delta J+\Phi\delta Q$, where $\delta S_{\rm
Wald}$ is the variation of the Wald entropy, which leads to

$T\left(\delta S_{\rm Wald}-\delta S_{\rm
BH}\right)=\int_{\Sigma}\tilde{\epsilon}_{bcd}\delta T_{ae}^{\rm
eff}K^{a}n^{e}~{}.$ (30)

From the first law, it is clear that, when $\delta\Xi=0$, we obtain the
derived correction to the Bekenstein-Hawking entropy.

While these equations already give a relationship between entropy and
extremality corrections, we note that when $\delta\Xi=0$ the entropy
correction must diverge in the limit $T\to 0$ to yield a finite result. For
this reason it is more insightful to evaluate the entropy corrections in a
microcanonical ensemble (fixed mass, angular momentum and electric charge).
From (29) we then find that the microcanonical entropy is given by

$\left(T\delta S_{\rm
BH}\right)_{M,J,Q}=-\int_{\Sigma}\tilde{\epsilon}_{bcd}\delta T_{ae}^{\rm
eff}K^{a}n^{e}~{}.$ (31)

At the same time, the mass correction at $T=0$ in a canonical ensemble (fixed
temperature, angular momentum and electric charge) is given by

$\lim_{T\to 0}\left(\delta M\right)_{T,J,Q}=\lim_{T\to
0}\int_{\Sigma}\tilde{\epsilon}_{bcd}\delta T_{ae}^{\rm eff}K^{a}n^{e}~{}.$
(32)

Thus, we can relate the mass correction in the canonical ensemble in the
extremal limit ($T\to 0$) to the microcanonical entropy correction in the
extremal limit ($\Xi\to 0$) :

$\lim_{T\to 0}\left(\delta M\right)_{T,J,Q}=-\lim_{\Xi\to 0}\left(T\delta
S_{\rm BH}\right)_{M,J,Q}~{}.$ (33)

Similar relationships between entropy and extremality corrections have
appeared before in the literature in various guises and have been derived by
studying explicit corrections to black hole solutions as well as by using a
thermodynamic approach Cheung  _et al._ (2018); Hamada  _et al._ (2019);
Cheung  _et al._ (2019); Loges  _et al._ (2020a); Goon and Penco (2020);
Cremonini  _et al._ (2020); Aalsma  _et al._ (2021); Arkani-Hamed  _et al._
(2021)

Let us now comment upon the validity and use of (33). First, it should be
stressed that the order of limits in the entropy correction is important. We
should first consider a variation of the entropy, for example with respect to
a small expansion parameter proportional to the correction we are considering,
and subsequently take the limit $\Xi\to 0$. As has been carefully explained in
Cremonini  _et al._ (2020) (see also Reall and Santos (2019)) this is
important because these two limits do not necessarily commute: the
microcanonical entropy need not to be analytic in the expansion parameter when
$\Xi=0$. Despite this subtlety, even when we first consider the limit $\Xi\to
0$ and then perform the variation with a suitable expansion parameter, there
still is a relationship between entropy corrections in a microcanonical
ensemble and mass corrections in the canonical ensemble Cremonini  _et al._
(2020) that is only modified by an order one constant.

Second, one might be puzzled about topological terms, like the Gauss-Bonnet
term in four dimensions. Because these terms appear as total derivatives when
varying the action they do not contribute to the equations of motion, the
effective stress tensor, or the mass and entropy correction. However, because
they appear as boundary terms they have support on the bifurcation surface of
the black hole and therefore do contribute to the Wald entropy. This is
consistent with (33) which only involves the Bekenstein-Hawking entropy, but
seemingly in conflict with entropy/extremality relationships involving the
Wald entropy Goon and Penco (2020). The resolution lies in the fact that in a
microcanonical ensemble, higher-derivative corrections only shift the
temperature of a black hole when its area is corrected. Topological terms do
not contribute to the area, so even when $\delta S$ is finite $\lim_{\Xi\to
0}(T\delta S)_{M,J,Q}=0$ because the temperature goes to zero in this limit.

This does imply, as remarked in Bobev  _et al._ (2021), that there is no
direct relationship between corrections to the extremality bound and the sign
of the Wilson coefficients multiplying topological terms in the action. In
particular, the WGC does not follow from requiring entropy corrections to be
positive in the presence of topological terms. Nonetheless, we still believe
(33) to be interesting because in many cases of interest the correction to the
temperature of an extremal black hole in a microcanonical ensemble scales as
$T\sim{\cal O}(\sqrt{\alpha_{i}})$, where $\alpha_{i}$ denotes a Wilson
coefficient. In that case, the shift in the area also scales with a square
root leading to $\delta S_{\rm BH}\sim{\cal O}(\sqrt{\alpha_{i}})$ such that
$\delta S=\delta S_{\rm BH}+{\cal O}(\alpha_{i})$. Possible corrections from
topological terms appear at order ${\cal O}(\alpha_{i})$ and are subdominant.
From this perspective, it might be more appropriate to refer to (33) as an
area/extremality relation.

##  III Energy Conditions and Extremality Corrections

The fact that the mass correction is given by the integral over the effective
stress suggests a possible relation between the WGC and energy conditions on
the perturbations we are considering, at least in cases where the notion of
the effective stress tensor coincides with the usual stress tensor. As
mentioned before, this happens when

$\frac{1}{2}g_{ab}F_{cd}\delta F^{cd}-F_{ac}\delta F_{b}^{\,\,c}=0~{}.$ (34)

Notably, this is the case for Kerr black holes, but after performing a
contraction with $K^{a}$ we find that this term also vanishes for electrically
charged Reissner-Nordström black holes. Thus, in these cases we find that the
correction to the mass in a canonical ensemble is given by

$\lim_{T\to 0}\left(\delta M\right)_{T,J,Q_{e}}=\lim_{T\to
0}\int_{\Sigma}\tilde{\epsilon}_{bcd}\delta T_{ae}K^{a}n^{e}~{},$ (35)

A decrease in the mass (as required for the WGC to be satisfied by charged
higher-derivative corrected black holes) therefore implies that at least at
some point along the Cauchy surface $\delta T_{ab}K^{a}n^{b}<0$. When $K^{a}$
and $n^{a}$ are two cooriented timelike vectors, this corresponds to a
violation of the DEC for the stress tensor of the perturbation, which is
$\delta T_{ab}$. Because $n^{a}$ is the unit normal to the Cauchy surface it
is a future pointing timelike vector by definition. Similarly, away from the
horizon the Killing vector $K^{a}$ is everywhere future-pointing and timelike
for the Reissner-Nordström black hole, but not for Kerr black holes. In that
case, the nature of $K^{a}$ depends on the azimuthal angle $\theta$. However,
for an energy condition to be violated it is sufficient to find a violation at
just a single point so for simplicity we can focus on the surface $\theta=0$.
In that case, $K^{a}$ is everywhere timelike away from the horizon and we come
to the same conclusion that the matter generating $\delta T_{ab}$ must violate
the DEC to satisfy the WGC. Here, we should make it clear that a violation of
the DEC is only required for the additional matter perturbing the
(uncorrected) black hole. Typically, the total stress tensor which includes
the (unperturbed) electromagnetic stress tensor does satisfy the DEC, simply
because of the positive energy density of the electric field which is only
perturbed slightly by adding matter.

However, while a violation of the DEC is a necessary condition for the WGC to
be satisfied it is not sufficient. We still need to perform the integral over
the Cauchy surface and one could imagine a situation where $\delta
T_{ab}K^{a}n^{b}<0$ at some point(s), but the entire integral gets overwhelmed
by positive contributions in such a way to yield a positive result.
Prohibiting this situation either requires the contraction of the stress
tensor to have a definite sign or a more general energy condition that bounds
the integral over the stress tensor. However, integrated energy conditions
typically hold for timelike or null trajectories Kontou and Sanders (2020), so
it seems difficult to bound the integral over the spacelike Cauchy surface
this way. We will now examine the stress tensor and extremality corrections
corresponding of higher-derivative corrected Reissner-Nordström and Kerr black
holes.

###  III.1 AdS-Reissner-Nordström Black Hole

We now consider an electrically charged Reissner-Nordström black hole in AdS
space perturbed by the following parity-even four-derivative higher-derivative
operators.

$\displaystyle L$
$\displaystyle=\Big{(}\frac{1}{2}R+\frac{3}{\ell^{2}}-\frac{1}{4}F_{ab}F^{ab}+\frac{a_{1}}{4}(F_{ab}F^{ab})^{2}$
(36)
$\displaystyle\quad+\frac{a_{2}}{2}F_{ab}F_{cd}W^{abcd}\Big{)}\epsilon~{}.$

We note that this is not the most general set of leading higher-derivative
corrections, which would require us to include a Weyl-squared operator
$W_{abcd}W^{abcd}$. However, in AdS space this term falls off as ${\cal
O}(1/r^{4})$ at large $r$, which is not fast enough to not contribute to a
boundary term that was an assumption in the derivation of (22). Therefore
additional care must taken to take into account the Weyl-squared term and for
simplicity we will drop it and leave this possibility for future work. In
Cremonini  _et al._ (2020) a thermodynamic approach based on holographic
renormalization was used to compute the effect of such corrections.
Nonetheless, if we consider black holes that are small with respect to the AdS
radius, it is justified to ignore the Weyl-squared term because in that case
it can be rewritten in terms of the other operators using the equations of
motion.

The metric and gauge field that solve the equations of motion at the two-
derivative level are given by

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-f(r)\mathrm{d}t^{2}+f(r)^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega_{k,2}^{2}$
(37) $\displaystyle f(r)$
$\displaystyle=k-\frac{M}{4\pi}+\frac{Q^{2}}{32\pi^{2}r^{2}}+\frac{r^{2}}{\ell^{2}}$
$\displaystyle A$ $\displaystyle=-\frac{Q}{4\pi r}\mathrm{d}t~{}.$

The parameter $k=(-1,0,1)$ determines the horizon geometry that is,
respectively, hyperbolic, planar or spherical. Thus, for $k=0$ this metric
describes a black brane. $\mathrm{d}\Omega_{k,2}^{2}$ denotes the respective
area element. As discussed, we take a gauge for the gauge field in which it
vanishes at infinity. It will be useful to write $f(r)$ in terms of its roots
as

$\displaystyle f(r)$
$\displaystyle=\frac{(r+r_{0})(r-r_{-})(r-r_{+})(r-r_{0}+r_{-}+r_{+})}{\ell^{2}r^{2}}~{},$
(38) $\displaystyle 2r_{0}$
$\displaystyle=\sqrt{-4k\ell^{2}-3r_{-}^{2}-2r_{-}r_{+}-3r_{+}^{2}}+r_{-}+r_{+}~{}.$

The relationship with the ADM mass and electric charge is

$\displaystyle Q^{2}$
$\displaystyle=\frac{32\pi^{2}r_{0}r_{-}r_{+}(r_{-}+r_{+}-r_{0})}{\ell^{2}}~{},$
(39) $\displaystyle M$
$\displaystyle=\frac{4\pi(r_{+}-r_{0})(r_{0}-r_{-})(r_{-}+r_{+})}{\ell^{2}}.$

Varying with respect to the metric and gauge field, we obtain the expression
for the perturbed stress tensor

$\displaystyle\delta
T^{g}_{ab}=\frac{a_{1}}{4}\left(g_{ab}(F_{cd}F^{cd})^{2}-8F_{cd}F^{cd}F_{a}^{\,\,\,e}F_{be}\right)$
(40)
$\displaystyle+\frac{a_{2}}{2}\bigg{[}g_{ab}R_{cdef}F^{cd}F^{ef}-6F_{cb}F^{de}R^{c}_{\,\,\,ade}$
$\displaystyle-4\nabla_{d}\nabla_{c}\big{(}F^{c}_{\,\,\,a}F^{d}_{\,\,\,b}\big{)}-2g_{ab}R^{cd}F_{ce}F_{d}^{\,\,\,e}+8R_{bc}F_{ad}F^{cd}$
$\displaystyle+4R^{cd}F_{ca}F_{db}+2g_{ab}\nabla_{c}\nabla_{d}\big{(}F^{c}_{\,\,\,e}F^{de}\big{)}-4\nabla_{c}\nabla_{b}\big{(}F_{ad}F^{cd}\big{)}$
$\displaystyle+2\square\big{(}F_{ac}F_{b}^{\,\,\,c}\big{)}+\frac{1}{3}g_{ab}RF_{cd}F^{cd}-\frac{4}{3}RF_{a}^{\,\,\,c}F_{bc}$
$\displaystyle-\frac{2}{3}F_{cd}F^{cd}R_{ab}+\frac{2}{3}\nabla_{a}\nabla_{b}\big{(}F_{cd}F^{cd}\big{)}-\frac{2}{3}g_{ab}\square\big{(}F_{cd}F^{cd}\big{)}\bigg{]}~{},$

and the electric source

$j^{a}=2\nabla_{b}\left(a_{1}F_{cd}F^{cd}F^{ab}+a_{2}W^{abcd}F_{cd}\right)~{}.$
(41)

To check if a violation of the DEC is a sufficient condition for the WGC to be
satisfied, we can evaluate $\delta T_{ab}n^{a}K^{b}$. In general, this leads
to a rather lengthy expression, so we summarize the situation. First of all,
we note that for $k=-1$ the stress tensor always has an indefinite sign for
fixed Wilson coefficients due to the negative curvature of the horizon. Also,
for $k=(0,1)$ the term proportional to $a_{2}$ similarly does not have a
definite sign, but the term proportional to $a_{1}$ is given by

$\displaystyle\left.\frac{5\ell^{4}}{12r_{+}^{3}\Omega_{0,2}}\delta
T_{ab}n^{a}K^{b}\right|_{a_{2}=0}=-\frac{15r_{+}^{5}\sqrt{r^{4}-4rr_{+}^{3}+3r_{+}^{4}}}{\ell
r^{9}}a_{1}~{},$ (42)
$\displaystyle\left.\frac{5\ell^{4}}{12r_{+}^{3}\Omega_{1,2}}\delta
T_{ab}n^{a}K^{b}\right|_{a_{2}=0}=-\frac{5r_{+}\left(\ell^{2}+3r_{+}^{2}\right)^{2}(r-r_{+})}{3\ell
r^{9}}a_{1}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\times\sqrt{\ell^{2}+r^{2}+2rr_{+}+3r_{+}^{2}}~{}.$

$\Omega_{k,2}$ denotes the unit area element, which diverges for $k=(-1,0)$.
In these cases, one can still consider corrections to the energy and entropy
density. Thus, a violation of the DEC only fixes the sign of the contribution
of the $(F_{ab}F^{ab})^{2}$ operator. We note that the condition $a_{1}\geq 0$
also follows from unitarity, while $a_{2}$ cannot be argued to take a
particular sign this way Hamada  _et al._ (2019); Arkani-Hamed  _et al._
(2021). However, in addition to the DEC we can also consider the NEC. Using
the null vector $N^{a}=(\sqrt{-1/g_{tt}},\sqrt{1/g_{rr}},0,0)$, we find

$k=(0,1):\qquad\delta T_{ab}N^{a}N^{b}\geq 0\quad\to\quad a_{2}\leq 0~{},$
(43)

To see how these conditions can be used to constrain the corrections to the
extremality bound we now perform the integral over the stress tensor.

$\displaystyle\frac{5\ell^{4}}{12r_{+}^{3}\Omega_{k,2}}\int_{\Sigma}\tilde{\epsilon}_{bcd}\delta
T_{ae}K^{a}n^{e}=$ (44)
$\displaystyle-\frac{\left(k\ell^{2}+3r_{+}^{2}\right)\left(2a_{1}\left(k\ell^{2}+3r_{+}^{2}\right)-a_{2}\left(k\ell^{2}-2r_{+}^{2}\right)\right)}{6r_{+}^{4}}~{}.$

For $k\neq 0$ the combination of Wilson coefficients that determines the
correction differs for small and large black holes.222Here, small or large
refers to the size of the black hole with respect to the AdS radius. The small
black holes we are studying should not be confused with black holes whose
horizon size is so small that the higher-derivative expansion breaks down.
Small black holes ($r_{+}/\ell\ll 1$) with either a spherical or hyperbolic
horizon ($k=\pm 1$) don’t notice the AdS curvature, so the combination of
Wilson coefficients controlling corrections to the extremality bound is the
same as in flat space Kats  _et al._ (2007)

$\frac{5\ell^{4}}{12r_{+}^{3}\Omega_{\pm
1,2}}\int_{\Sigma}\tilde{\epsilon}_{bcd}\delta
T_{ae}K^{a}n^{e}=-\frac{\left(2a_{1}-a_{2}\right)\ell^{4}}{6r_{+}^{4}}~{}.$
(45)

On the other hand, for large black holes $r_{+}/\ell\gg 1$ and we find that

$\frac{5\ell^{4}}{12r_{+}^{3}\Omega_{\pm
1,2}}\int_{\Sigma}\tilde{\epsilon}_{bcd}\delta
T_{ae}K^{a}n^{e}=-\left(3a_{1}+a_{2}\right)~{},$ (46)

which is also the correction for black branes:

$\frac{5\ell^{4}}{12r_{+}^{3}\Omega_{0,2}}\int_{\Sigma}\tilde{\epsilon}_{bcd}\delta
T_{ae}K^{a}n^{e}=-\left(3a_{1}+a_{2}\right)~{}.$ (47)

Note that for hyperbolic black holes ($k=-1$) there is a particular size for
which all corrections vanish, that is $r_{+}^{2}=\ell^{3}/3$, and for
spherical black holes ($k=+1$) the correction proportional to $a_{2}$ vanishes
when $r_{+}^{2}=\ell^{2}/2$. The sign of the corrections to the mass in a
canonical ensemble is therefore controlled by a different combination of
Wilson coefficient depending on the horizon geometry and the size of the black
hole.

For small black holes in AdS space for which ignoring the Weyl squared term is
justified we find that unitarity $(a_{1}\geq 0)$ in combination with the NEC
$(a_{2}\leq 0)$ requires the extremal mass to decrease in a canonical
ensemble, consistent with the WGC.

###  III.2 Kerr Black Hole

We also consider the leading higher-derivative corrections to the Kerr black
hole. Although there is no WGC-like motivation that constrains the sign of
higher-derivative corrections to Kerr black holes, due to the Penrose process
extremal Kerr black holes are unstable, it is still interesting to see how
their properties are corrected. The leading parity-even higher-derivative term
has six derivatives and the Lagrangian is given by333Additional terms
contribute if one includes light degrees of freedom in addition to the metric
Cano and Ruipérez (2019).

$L=\frac{1}{2}\left(R+\alpha
R_{abcd}R^{cdef}R_{ef}^{\,\,\,\,\,ab}\right)\epsilon~{}.$ (48)

At the two-derivative level, the Kerr solution is given by (14) with
$r_{Q}=0$. Performing the variation with respect to the metric, we find that
the perturbed stress tensor is

$\displaystyle\frac{\delta T_{ab}}{\alpha}$
$\displaystyle=\frac{1}{2}g_{ab}W_{cd}^{\,\,gh}W^{cdef}W_{efgh}-6(\nabla_{c}W_{bfde})(\nabla^{f}W_{a}^{\,\,cde})$
(49)
$\displaystyle+R_{a}^{\,\,cde}\left[3R_{bd}^{\,\,fg}R_{cefg}+12{R_{b}^{\,\,f}}_{d}^{\,\,g}R_{cfeg}-3R_{bc}^{\,\,fg}R_{defg}\right.$
$\displaystyle\left.+3{R_{b}^{\,\,f}}_{c}^{\,\,g}R_{defg}\right]~{}.$

The corrections are now easily found by evaluating the stress tensor. As we
already remarked before, due to the dependence on the azimuthal angle $\theta$
the contraction of the stress tensor $\delta T_{ab}n^{a}K^{b}$ does not have a
definite sign for a fixed $\alpha$. This means that violation of the DEC is a
necessary but not sufficient condition to fix the sign of the mass correction
in a canonical ensemble. For the same reason, the NEC is always violated for
some value of $\theta$.

Performing the integral we find

$\displaystyle\int_{\Sigma}\tilde{\epsilon}_{bcd}\delta T_{ae}K^{a}n^{e}=$
(50)
$\displaystyle\frac{4\pi\alpha}{35r_{+}^{3}(r_{-}+r_{+})^{3}}\left(-37r_{-}^{3}+357r_{-}^{2}r_{+}-455r_{-}r_{+}^{2}+175r_{+}^{3}\right)~{}.$

Interestingly, for fixed $\alpha$ the right-hand side does have a definite
sign which implies that the correction to the Bekenstein-Hawking entropy also
has a definite sign even away from extremality. In the extremal limit the
correction reduces to

$\int_{\Sigma}\tilde{\epsilon}_{bcd}\delta
T_{ae}K^{a}n^{e}=\frac{4\pi\alpha}{7r_{+}^{3}}~{}.$ (51)

Reproducing the result in Reall and Santos (2019). We see that at extremality
the mass in a canonical ensemble decreases when $\alpha<0$. However, in
contrast to charged black holes no energy condition fixes the sign of $\alpha$
to be negative. Perhaps, this is not so surprising because, as discussed,
there is no WGC-like motivation that constrains the sign of higher-derivative
corrections to Kerr. Only when the rotation of a black hole can be related to
charge, such a connection might arise Aalsma  _et al._ (2019, 2021).

##  IV Weak Gravity Conjecture in Anti-de Sitter Space

We will now consider the derived corrections to the extremality bound of AdS
black holes in the light of a formulation of the WGC in AdS space. For
(electrically) charged asymptotically flat black holes it is clear that the
mild form of the WGC requires higher-derivative corrections to decrease the
mass in a canonical ensemble. This has been used to derive bounds on Wilson
coefficients. Vice versa, the WGC has been derived by using bounds on Wilson
coefficients coming from unitarity and causality.

However, for black holes that are asymptotically AdS the correct formulation
of the WGC is less clear although proposals exist, such as Harlow (2016);
Montero (2019). Recently, a version of the WGC in AdS space has been
formulated Aharony and Palti (2021) that is inspired by the repulsive force
conjecture (RFC) Palti (2017); Heidenreich  _et al._ (2019). The RFC states
that, for two identical charged particles, their total long-range force should
be repulsive rather than attractive (or vanish). For particles that only
experience gravitational and electromagnetic forces, this conjecture is
identical to the WGC in flat space. However, when massless scalar fields are
included the RFC and WGC become distinct statements. It is now interesting to
understand if there is a mild version of the RFC where the self-repulsive
states are black holes themselves. In flat space, this question has recently
been discussed in Cremonini  _et al._ (2021b), where it was shown in several
examples that the RFC cannot be satisfied by black holes along all charge
directions. This calls into question the validity of a mild form of the RFC.

Also, in AdS space formulating the RFC is difficult because there is always a
contribution from the cosmological constant to the force acting on a state.
Therefore, Aharony and Palti (2021) (see also Antipin  _et al._ (2021))
suggested a positive binding conjecture, which requires the existence of a
charged state that has non-negative self-binding energy to avoid the formation
of a large tower of bound states. In the dual conformal field theory (CFT)
this was phrased as a convexity condition for the scaling dimensions of
charged CFT operators. Given this proposal, one might wonder if it possible
that a mild form of this convex charge conjecture (CCC) is satisfied by
higher-derivative corrected black holes in AdS. Although the CCC does not
necessarily require black hole states to satisfy the convexity condition, it
is an interesting questions to ask if they could. In fact, it was already
observed in Aharony and Palti (2021) that large black holes in AdS, dual to
heavy operators in the CFT, are convex. So, does convexity put any constraints
on the sign of the Wilson coefficients of higher-derivative operators? For
large black holes in AdS4 ($r_{+}/\ell\gg 1$) the extremality bound scales as
$M\sim A(a_{i})Q^{3/2}$, where $A(a_{i})$ is a charge-independent positive
constant that depends on the Wilson coefficients, see Fig. 1.

Figure 1: Cartoon of the extremality bound of spherical black holes in AdS.
Higher-derivative corrections lead to a modification of the extremality bound.
The mass of large extremal black holes in AdS (red) is a convex function of
charge irrespective of the sign of the Wilson coefficients $a_{i}$. The mass
of small black holes in AdS (green) is a concave function of charge if they
satisfy the WGC ($B(a_{i})>0$).

On the other hand, small extremal black holes in AdS space are marginally
convex at the two-derivative level since their mass scales linearly with
charge. Now, higher-derivative corrections that are consistent with the WGC
decrease the mass ($B(a_{i})>0$) at fixed charge resulting in a concave
function. As observed in Aharony and Palti (2021) this implies that, when the
WGC is satisfied for small black holes, they violate the convexity condition.
However, since these black holes are unstable they do not lead to a large
tower of stable states. On the other hand, large black holes are always convex
irrespective of the sign of the Wilson coefficients. Therefore, although the
CCC is automatically satisfied for large black holes in AdS, it does not put
any constraints on the sign of Wilson coefficients of higher-derivative
corrections.

##  V Conclusions

In this paper, we considered corrections to extremal black holes by arbitrary
perturbations using the Iyer-Wald formalism. The main result is a relation
between the correction to the extremality bound and an effective stress
tensor. In particular cases of interest, which we clarified, the effective
stress tensor reduces to the usual stress tensor. Furthermore, we gave a new
derivation of the entropy/extremality relation that has appeared before in the
literature. Because we relate corrections to the extremality bound to the
stress tensor, it clarifies the connection between the WGC and energy
conditions. In particular, we showed that a necessary condition to decrease
the mass of an extremal black hole in a canonical ensemble is that the
perturbation violates the DEC. Combining a unitarity bound with the NEC we
showed that the mild form of the WGC follows for charged black holes.

In future work, it would be interesting to generalize the relation we derived
to include black holes with magnetic charges and couplings to additional
(scalar) fields. This requires some special attention because those
contributions do not appear in the standard way the Iyer-Wald formalism is
applied Mitsios  _et al._ (2021). Additionally, to derive our main result we
assumed that the perturbations we considered decay sufficiently fast at
infinity. Notably, this is not the case for all four-derivative operators in
AdS space and taking into account these perturbations, which will contribute
additional boundary terms, also seems like a worthwhile endeavor. Moreover, it
would be interesting to understand better the relation between energy
conditions and the WGC. In particular, while it is reasonable to expect that
higher-derivative corrections generated by integrating out healthy matter at
tree level satisfies (some) energy conditions, quantum effects are known to
violate them. Nonetheless, in that case it is still expected that some
averaged condition holds Kontou and Sanders (2020).

To conclude, we hope that the general method we presented here to compute
extremal black hole corrections will provide useful in further studies of
black holes and the WGC.

## Acknowledgments

I am grateful to Alex Cole, Eleni Kontou, Greg Loges and Gary Shiu for useful
discussions. I especially thank Greg Loges for carefully explaining some of
the subtleties of black hole thermodynamics and Eleni Kontou for feedback on
an earlier version of this paper. This work is supported by the DOE under
Grant No. DE-SC0017647.

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