# Slowly rotating and the accelerating $\alpha^{\prime}$-corrected black holes
in four and higher dimensions

Felipe Agurto-Sepúlveda1,2, Mariano Chernicoff3,4, Gaston Giribet5, Julio
Oliva2 and Marcelo Oyarzo2 1Departamento de Astronomía, Universidad de
Concepción, Casilla 160-C, Chile;  2Departamento de Física, Universidad de
Concepción, Casilla, 160-C, Concepción, Chile.  3Departamento de Física,
Facultad de Ciencias, Universidad Nacional Autónoma de México
A.P. 70-542, CDMX 04510, México;  4Departament de Física Quàntica i
Astrofísica and Institut de Ciències del Cosmos (ICC),
Universitat de Barcelona, Martí Franquès 1, ES-08028, Barcelona, Spain.
5Departamento de Física, Universidad de Buenos Aires
Ciudad Universitaria, pabellón 1, 1428, Buenos Aires, Argentina;

###### Abstract

We consider the low-energy effective action of string theory at order
$\alpha^{\prime}$, including $R^{2}$-corrections to the Einstein-Hilbert
gravitational action and non-trivial dilaton coupling. By means of a
convenient field redefinition, we manage to express the theory in a frame that
enables us to solve its field equations analytically and perturbatively in
$\alpha^{\prime}$ for a static spherically symmetric ansatz in an arbitrary
number of dimensions. The set of solutions we obtain is compatible with
asymptotically flat geometries exhibiting a regular event horizon at which the
dilaton is well-behaved. For the 4-dimensional case, we also derive the
stationary black hole configuration at first order in $\alpha^{\prime}$ and in
the slowly rotating approximation. This yields string theory modifications to
the Kerr geometry, including terms of the form $a$, $a^{2}$, $\alpha^{\prime}$
and $a\alpha^{\prime}$. In addition, we obtain the first $\alpha^{\prime}$
correction to the C-metrics, which accommodates accelerating black holes. We
work in the string frame and discuss the connection to the Einstein frame, for
which rotating black holes have already been obtained in the literature.

##  I Introduction

Higher-curvature corrections to Einstein’s general relativity (GR) are
ubiquitous in any sensible approach to quantum gravity, and they are a solid
prediction of string theory GreenSchwarzWitten . Even before the formulation
of the latter, effective actions containing higher order contractions of the
Riemann tensor were known to emerge in quantum field theory on curved
spacetime BD  and in the semiclassical approach to quantum gravity. Such
actions are the natural generalization of Einstein-Hilbert action, thus
correcting GR in the ultraviolet (UV) regime. Also, from the mathematical
point of view it was early understood that higher-curvature terms were natural
in higher dimensions Lanczos ; Lovelock ; Lovelock2 ; and, on general grounds,
it is widely accepted that any attempt to formulate a sensible UV-complete
theory will involve higher-curvature corrections in a way or another. In 1976,
Stelle argued that gravitational actions which include terms quadratic in the
curvature tensor are renormalizable Stelle . This is due to the fact that non-
linear renormalization of the graviton and the ghost fields suffices to absorb
the non-gauge-invariant divergences that might arise. Stelle explained how
these and other divergences may be eliminated in a way that simplifies the
renormalization procedure, even when matter fields are coupled. Nevertheless,
renormalizability is not the only issue: The inclusion of quadratic-curvature
terms in the gravitational action typically introduces massive local degrees
of freedom, apart from the massless graviton of GR Stelle2 . These extra modes
organize themselves as a massive spin-2 and a massive spin-0 excitations,
yielding a total of 8 local degrees of freedom. The massive spin-2 part of the
field has negative energy, and this is the reason why it is usually asserted
that, with exception of a few remarkable cases Lovelock ; Lovelock2 ; Critical
; Critical2 , augmenting the Einstein-Hilbert action with a finite set of
higher-curvature terms yields ghosts when the theory is expanded about
maximally symmetric vacua. The observation of Stelle2  motivated, in the early
80s, the search for ghost-free higher-curvature theories and consistent UV
completions. Since then, actions containing higher-curvature terms were
considered in the context of cosmology Starobinsky , black hole physics
BoulwareDeser , and string theory Zwiebach . In 1985, Zwiebach studied the
compatibility between the presence of curvature squared terms and the absence
of ghost modes in the low energy limit of string theory Zwiebach . He argued
that the so-called Einstein-Gauss-Bonnet (EGB) action was a good candidate for
string effective action as it yields a ghost-free non-trivial gravitational
self-interacting theory in any dimension greater than four, $d>4$. The EGB
action is made out of a dimensionally extended version of the quadratic Chern-
Gauss-Bonnet topological invariant, which, while being dynamically trivial in
$d\leq 4$, does yield a UV correction to GR in $d>4$ with a single massless
spin-2 excitation and with field equations of second order. The latter
property makes EGB theory free of Ostrogradski instabilities. Still, in
BoulwareDeser  Boulware and Deser showed that the EGB model, proposed in
Zwiebach  as a stringy action, contains, in addition to flat spacetime, a
second non-perturbative anti-de Sitter (AdS) vacuum which turns out to be
unstable due to the presence of ghosts. This is nothing but the fact that
actions that are polynomials of degree $k$ in contractions of the Riemann
tensor generically yield $k$ different vacua, many of them being artifacts of
the truncation of the effective theory. In BoulwareDeser2  the authors noticed
that the inclusion of the dilaton field in the EGB effective action suffices
to remove the spurious (A)dS vacuum permitted in its absence. They also showed
that the spherically symmetric static solutions to the dilatonic EGB theory
might have a well-defined asymptotic behaviour, being non-trivial, and being
compatible with the existence of a regular event horizon at which the dilaton
is well-behaved111Higher-curvature black holes were also studied in the
context of thermodynamics Jacobson1  and many other subjects, like the
holography Buchel2 , the weak-gravity conjecture Motl , among others. For
related early works on this paper, see Moura1 ; Moura2 ; Callan ; Myersfoot ;
Cheung1 ; Cheung2 ; Cheung3 . This was later confirmed by explicit examples,
and here we will also provide a concrete realization of it.

Soon after Zwiebach , in a foundational paper of string theory GrossWitten ,
Gross and Witten finally proved that the gravitational field equations of
string theory actually contain higher-curvature corrections to GR. More
precisely, they derived the modifications of the classical gravitational
equations for the type II string theory by studying tree-level gravitational
scattering amplitudes, and they determined the effective gravitational action
up to quartic order in the curvature tensor, which corresponds to order
$\mathcal{O}(\alpha’^{3})$ string corrections. Unlike bosonic string theory,
type II superstring theory in $d=10$ dimensions does not contain quadratic
corrections to GR, and the cubic ones can be set to zero by fields
redefinition – although quadratic corrections can actually appear in Calabi-
Yau compactification of the quartic actions, with the moduli playing the role
of the couplings cf. Ferrara1 ; Ferrara2 ; Guica –. In contrast, quadratic
corrections do appear in critical bosonic and heterotic string theories. They
were studied in Metsaev ; Metsaev2  by Metsaev and Tseytlin, who checked the
equivalence of the string equations of motion and the $\sigma$-model Weyl
invariance conditions at order $\mathcal{O}(\alpha’)$. They obtained the
functional dependence on the dilaton, the graviton, and the antisymmetric
tensor. To do so, they first determined the $\mathcal{O}(\alpha’)$ terms in
the string effective action starting from the expressions for the 3- and
4-point string scattering amplitudes; then, they computed the 2-loop
$\beta$-function in the worldsheet $\sigma$-model. This results in an
effective gravity action with quadratic-curvature ($R^{2}$) corrections
coupled to the other massless fields of the theory; see also the important
works Bergshoeff1 ; Bergshoeff2 , and for modern developments on
$\alpha^{\prime}$ corrections in relation to $T$-duality and Double Field
Theory see Hohm1 ; Hohm2 ; Hohm3 ; Hohm4 ; Hohm5 ; Hohm6 ; Hohm7 ; Marques1 ;
Marques2 ; Marques3 ; Marques4  and references therein and thereof.

In recent years, with the advent of AdS/CFT correspondence and its
ramifications, higher-curvature terms were reconsidered in the context of
holography and the interest on them was revived. Probably the best-known
example of this is the discussion of the higher-curvature terms in relation to
the Kovtun-Starinets-Son (KSS) viscosity bound Brigante ; Brigante0 , which
showed that, for a class of conformal field theories (CFT) with a gravity dual
with the EGB action, the shear viscosity to entropy density ratio could
violate the conjectured KSS lower bound. This proved that the presence of
higher-curvature terms could result in qualitatively new phenomena; see also
Edelstein1 ; Edelstein2 . Microcausality violation in the CFT was also studied
in the same type of scenario Brigante , which was rapidly interpreted as
evidence supporting the idea of a universal lower bound on the shear viscosity
to entropy density ratio for all consistent theories. This triggered a long
series of works devoted to check the consistency conditions of effective
theories with higher-curvature modifications. For example, in Buchel  the
authors discussed causality conditions in $R^{2}$ theories; they study
causality violation in holographic hydrodynamics focusing on the EGB theory as
a working example. In the latter theory, the value of the only $R^{2}$
coupling constant is related to the difference between the two central charges
of the dual 4-dimensional CFT, and the authors of Buchel  showed that, when
such difference is sufficiently large, causality is violated. This problem was
also studied in Hofman , where the author discussed the relation between
causality constraints in the bulk theory and the condition of energy
positivity in the dual CFT. He specifically argued that special care is needed
when solving the classical equations of motion in the higher-curvature gravity
theory, for which the study of causality problems may be subtle. Holography in
presence of EGB gravity actions have been further studied in Buchel2  and in
references thereof. The authors of Buchel2  studied the problem in arbitrary
number of dimensions $d$ and established a holographic dictionary that relates
the couplings of the gravitational theory to the universal numbers in the
correlators of the stress tensor of the dual CFT, cf. BuchelMyers . This
allowed the authors to examine constraints on the gravitational couplings by
demanding consistency of the CFT, and this yielded a much more general set of
causality constraints.

Both in the context of AdS/CFT and in other scenarios, the consistency
conditions for higher-curvature theories were intensively studied in the last
fifteen years. This line of research has continued and a much more general
picture of the set of consistency conditions has been accomplished. Causality,
locality, stability, hyperbolicity and other aspects were revisited. In
Gruzinov , it was shown how causality constrains the sign of the stringy
$R^{4}$ corrections to the Einstein-Hilbert action, giving a general
restriction on candidate theories of quantum gravity. In Gomberoff , a special
type of pathology that the truncated EGB theory exhibits was studied. This is
a phase transition driven by non-perturvative effects that might take place in
gravitational theories whenever higher-curvature corrections with no extra
fields are considered. In Camanho , Maldacena et al. studied causality
constraints on corrections to the graviton 3-point coupling. They considered
higher-curvature corrections to the graviton vertex in a weakly coupled
gravity theory and they derived stringent causality constraints. By
considering high energy scattering processes, they noticed a potential
causality violation that might occur whenever additional Lorentz invariant
structures are included in the graviton 3-point vertex. They argued that such
a violation could be cured by the addition of an infinite tower of extra
massive higher-spin fields such as those predicted by string theory. This
problem was later reconsidered by many authors, cf. Reall .

Motivated by this renewed interest in higher-curvature gravity, in the last
years there have been important developments in the subject, and many new
higher-curvature models were proposed and studied. The list includes the
quasitopological theories quasitopological ; quasitopological2 ;
quasitopological3 , the critical gravity theories in AdS Critical ; Critical2
, the so-called Einsteinian cubic gravity Cubic ; Cubic2 , and their
generalizations quasitopologicalinfty ; quasitopologicalinfty2 . Black holes
have recently been studied in all these setups quasitopological2 ; Chernicoff
; CubicBH ; Grandi , as well as in string theory inspired scenarios Cano1 ;
Cano2 ; Cano3 ; see also Maeda2009 ; Giribeto ; Ohta  and references therein
and thereof. Here, we will present and study analytic, static, spherically
symmetric solutions to the $\alpha^{\prime}$-corrected gravity action in
arbitrary dimension $d$ and including a non-vanishing dilaton coupling. We
will consider the graviton-dilaton sector of the low-energy effective action
of string theory with $R^{2}$ terms in a specific frame that will enable us to
solve the problem explicitly to order $\mathcal{O}(\alpha^{\prime})$ in the
entire spacetime. Our solutions manifestly show that the theory is compatible
with static, spherically symmetric solutions which are asymptotically flat and
exhibit a regular event horizon at which the dilaton is well-behaved. The
paper is organized as follows: In section II, we present the gravity theory in
a convenient frame. We briefly discuss the field redefinition ambiguity to the
relevant order, and we use it to solve the adequate ansatz. The field
equations are written down and solved, and the black hole solution for $d=4$
is presented. In section III, we study the black hole thermodynamics. This
amounts to work out the Wald entropy formula, which, as usual in this type of
setup, yields corrections to the Bekenstein-Hawking area law. The mass of the
solution may then be inferred from the first law of black hole mechanics. In
section IV, we perform a consistency check of the previous formulae by
explicitly computing the black hole mass by means of the Iyer-Wald method for
conserved charges, which shows perfect agreement. We also show the agreement
with the Euclidean action approach. In section V, we generalize our result by
introducing angular momentum in the slowly rotating approximation. We derive a
stationary metric that represents stringy modifications to the Kerr geometry.
In section VI, we obtained the $\alpha^{\prime}$ correction to the C-metric,
which accommodates accelerating black holes. While we work in the string
frame, in section VII we discuss the frame transformation that maps out theory
to the Einstein frame, including the higher curvature corrections. In the
latter frame, rotating solutions have already been studied in the literature,
and we discuss the precise relation between the two frames. In section VIII,
we generalize the static solution by presenting the explicit form of the
dilatonic black hole solution in arbitrary dimension $d$.

##  II Dilatonic black hole

We consider the low-energy effective action of string theory including
$\alpha^{\prime}$ corrections to the graviton-dilaton sector; namely Metsaev ;
Metsaev2

$I\left[g_{\mu\nu},\phi\right]=\int_{M}d^{d}x\sqrt{-g}e^{-2\phi}\left[R+4\left(\nabla\phi\right)^{2}+\alpha\
R_{\mu\nu\lambda\rho}R^{\mu\nu\lambda\rho}+\mathcal{O}\left(\alpha^{2}\right)\right]\
,$ (1)

where we denoted $\alpha=\frac{1}{8}\alpha^{\prime}$. We are not considering
the dependence on the $B$-field here. Performing field redefinition
$g_{\mu\nu}\rightarrow g_{\mu\nu}+\delta g_{\mu\nu}$,
$\phi\rightarrow\phi+\delta\phi$ with

$\displaystyle\delta\phi=-\frac{\alpha}{2}\left(R+4(2d-5)\partial_{\mu}\phi\partial^{\mu}\phi\right)\,,$
(2) $\displaystyle\delta
g_{\mu\nu}=-4\alpha\left(R_{\mu\nu}-4\partial_{\mu}\phi\partial_{\nu}\phi+4g_{\mu\nu}\partial_{\alpha}\phi\partial^{\alpha}\phi\right)\,,$
(3)

one obtains the action in a frame that is convenient for the computation we
want to undertake; namely

$\displaystyle I\left[g_{\mu\nu},\phi\right]$ $\displaystyle=\int
d^{d}x\sqrt{-g}e^{-2\phi}\left[R+4\left(\nabla\phi\right)^{2}\right.$
$\displaystyle\left.+\alpha\left(R_{\mu\nu\lambda\rho}R^{\mu\nu\lambda\rho}-4R_{\mu\nu}R^{\mu\nu}+R^{2}-16\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{2}\right)+\mathcal{O}\left(\alpha^{2}\right)\right]\
,$ (4)

up to six-derivative operators of order $\mathcal{O}\left(\alpha^{\prime
2}\right)$, cf. Maeda2011 . As the $R^{2}$-terms take the form of the $4d$
Euler characteristic, the field equations of the theory in this frame are of
second order in an explicit manner. Let us consider first the case in $d=4$.
The field equations derived from (4) are given by

$\displaystyle
G_{\mu\nu}+4\partial_{\mu}\phi\partial_{\nu}\phi-2g_{\mu\nu}\partial_{\rho}\phi\partial^{\rho}\phi+2S_{\mu\nu}-2g_{\mu\nu}S_{\
\rho}^{\rho}+\alpha H_{\mu\nu}\left.=\right.0\ ,$ (5) $\displaystyle
R+4\partial_{\rho}\phi\partial^{\rho}\phi+4S_{\ \mu}^{\mu}+\alpha
L_{GB}-32\alpha\left(\nabla^{\mu}\left(\partial_{\rho}\phi\partial^{\rho}\phi\right)\partial_{\mu}\phi+\left(\partial_{\rho}\phi\partial^{\rho}\phi\right)S_{\
\mu}^{\mu}+\frac{1}{2}\left(\partial_{\rho}\phi\partial^{\rho}\phi\right)^{2}\right)\left.=\right.0\
,$

where $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$ is the Einstein tensor,
and where

$\displaystyle S_{\rho\sigma}\equiv
e^{2\phi}\nabla_{\rho}\left(e^{-2\phi}\nabla_{\sigma}\phi\right)\ ,\ \
L_{GB}\equiv
R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^{2}\ ,$ (7)

and

$\displaystyle H_{\mu\nu}$ $\displaystyle=$ $\displaystyle S_{\mu\nu}R-4S_{\
(\mu}^{\sigma}R_{\nu)\sigma}+2S_{\
\sigma}^{\sigma}R_{\mu\nu}+2S^{\sigma\lambda}R_{\mu\sigma\lambda\nu}-8\left(\partial_{\rho}\phi\partial^{\rho}\phi\right)\partial_{\mu}\phi\partial_{\nu}\phi$
$\displaystyle+g_{\mu\nu}\left(2\left(\partial_{\rho}\phi\partial^{\rho}\phi\right)^{2}-S_{\
\sigma}^{\sigma}R+2S_{\ \lambda}^{\sigma}R_{\ \ \sigma}^{\lambda}\right)\ .$

Lagrangian $L_{GB}$ is the integrand of the $4$-dimensional Chern-Gauss-Bonnet
topological invariant which, in the absence of the dilaton and in $d=4$,
yields the Euler characteristic; this is the EGB quadratic gravity Lagrangian.

We are interested in solving the equations above for a static spherically
symmetric spacetime with non-trivial dilaton profile (later, in Section V, we
will generalize the solution to the stationary, non-static case). In order to
do so, we work perturbatively at order $\mathcal{O}(\alpha)$, and propose the
ansatz

$\displaystyle\phi\left(r\right)$
$\displaystyle=\phi_{0}+\alpha\phi_{1}\left(r\right)\ ,$ (9) $\displaystyle
ds^{2}$ $\displaystyle=-\left(1+\alpha
N_{1}\left(r\right)\right)^{2}\left(1-\frac{\mu}{r}+\alpha
f_{1}\left(r\right)\right)dt^{2}+\frac{dr^{2}}{1-\frac{\mu}{r}+\alpha
f_{1}\left(r\right)}+r^{2}d\Omega^{2}\ ,$ (10)

where $\phi_{1}(r)$, $N_{1}(r)$, and $f_{1}(r)$ are functions of the radial
coordinate $r$ to be determined; $\mu$ is an arbitrary constant; $d\Omega^{2}$
is the constant-curvature metric on the unit sphere. The solution we will find
in this way will be valid up to order $\mathcal{O}(\alpha)$. Plugging this
ansatz in the field equations and expanding up to first order in $\alpha$, we
obtain a remarkably simple system of equations which lead to the following
general solution

$\phi\left(r\right)=\phi_{0}+\alpha\left(A+B\log\left(\frac{r-\mu}{r}\right)-\frac{2}{\mu
r}-\frac{1}{r^{2}}-\frac{2\mu}{3r^{3}}\right)$ (11)

with $A$ and $B$ being two arbitrary constants; the former constant appears
merely as a shift of $\phi_{0}$ which does not enter in the metric, and so it
can be absorbed by redefining $\bar{\phi}_{0}=\phi_{0}+\alpha A$, which gives
the value that the dilaton takes at infinity; notice that, at infinity, (11)
goes like $\phi\simeq\bar{\phi}_{0}+\mathcal{O}(1/r)$. Up to
$\mathcal{O}(\alpha)$ terms, for the metric we find

$\displaystyle g^{rr}$ $\displaystyle=1-\frac{\mu}{r}+\alpha\left(-\frac{\mu
B}{r}\log\left(\frac{r-\mu}{r}\right)+\frac{C}{r}+\frac{2}{r^{2}}+\frac{\mu}{r^{3}}-\frac{10}{3}\frac{\mu^{2}}{r^{4}}\right)\
,$ $\displaystyle g_{tt}$
$\displaystyle=\frac{\mu}{r}-1-\alpha\left(\frac{B\left(2r-3\mu\right)}{r}\log\left(\frac{r-\mu}{r}\right)+D+\frac{4}{r^{2}}+\frac{5\mu}{3r^{3}}+\frac{2\mu^{2}}{r^{4}}-\frac{\mu^{2}D-\mu
C+2\mu^{2}B+8}{\mu\,r}\right)$

where $D$ and $C$ are other two integration constants. The former can be
eliminated by rescaling the time coordinate as $t\to t/({1+\alpha D})$.

If we define $r_{+}=\mu+\alpha\mu_{1}$, we can easily find the
$\alpha$-corrected location of the event horizon by solving for $\mu_{1}$ as a
function of the integration constants. This amounts to demand
$g^{rr}\left(r_{+}\right)=g_{tt}\left(r_{+}\right)=0$, which is actually
required for the horizon to be regular. Expanding up to first order in
$\alpha$, this yields

$\displaystyle\left(B\log\left(\frac{\mu}{\alpha\mu_{1}}\right)+\frac{\mu_{1}+C}{\mu}-\frac{1}{3\mu^{2}}\right)\alpha+O\left(\alpha^{2}\right)$
$\displaystyle=0\ ,$ (12)
$\displaystyle\left(B\log\left(\frac{\mu}{\alpha\mu_{1}}\right)-2B+\frac{\mu_{1}+C}{\mu}-\frac{1}{3\mu^{2}}\right)\alpha+O\left(\alpha^{2}\right)$
$\displaystyle=0\ ,$ (13)

from which we consequently obtain that $B=0$. Therefore, the
$\alpha^{\prime}$-corrected black hole configuration reads

$\displaystyle g^{rr}(r)$
$\displaystyle=1-\frac{\mu}{r}+\alpha\left(\frac{C}{r}+\frac{2}{r^{2}}+\frac{\mu}{r^{3}}-\frac{10}{3}\frac{\mu^{2}}{r^{4}}\right)+\mathcal{O}(\alpha^{2})\
,$ (14) $\displaystyle g_{tt}(r)$
$\displaystyle=\frac{\mu}{r}-1-\alpha\left(\frac{4}{r^{2}}+\frac{5\mu}{3r^{3}}+\frac{2\mu^{2}}{r^{4}}-\frac{8-\mu
C}{\mu\,r}\right)+\mathcal{O}(\alpha^{2})\ ,$ (15) $\displaystyle{\phi}(r)$
$\displaystyle=\bar{\phi}_{0}-\alpha\left(\frac{2}{\mu
r}+\frac{1}{r^{2}}+\frac{2\mu}{3r^{3}}\right)+\mathcal{O}(\alpha^{2})\ ,$ (16)

and, up to $\mathcal{O}(\alpha)$ corrections, the location of the horizon is

$r_{+}=\mu+\alpha\left(\frac{1}{3\mu}-C\right)\,.$ (17)

The solution we have just derived is asymptotically flat, and it exhibits a
smooth event horizon at $r_{+}$, where the dilaton remains finite:

$\phi(r_{+})=\bar{\phi}_{0}-\frac{11}{3}\frac{\alpha}{r_{+}^{2}}+\mathcal{O}(\alpha^{2})\,;\
\ \ \phi(\infty)=\bar{\phi}_{0}+\mathcal{O}(\alpha^{2}).$ (18)

In the following sections, we will analyze the physical properties of this
solution, we will compute its conserved charges, and, finally, we will
generalize it to $d\geq 4$ dimensions.

##  III Thermodynamics

The thermodynamics of higher-curvature black holes has been studied for a long
time Jacobson1 ; Jacobson2 ; Jacobson3 , and in a vast number of contexts.
Here, we will focus on the properties of the black hole solution we just
presented. The Wald formula gives the entropy as a Noether charge computed at
the horizon. This is given by the following integral on the horizon
$\mathcal{H}$

$S=\frac{\beta}{4}\int_{\mathcal{H}}\,\sqrt{-g}\,\epsilon_{\mu\nu\rho\sigma}\,q^{\mu\nu}dx^{\rho}\wedge
dx^{\sigma},$ (19)

in $d=4$ spacetime dimensions, with $\beta$ being the periodicity of the
Euclidean time. The Noether pre-potential associated to this charge is given
by

$q^{\mu\nu}\equiv-2\left(E^{\mu\nu\rho\sigma}\nabla_{\rho}\xi_{\sigma}+2\xi_{\rho}\nabla_{\sigma}E^{\mu\nu\rho\sigma}\right)$
(20)

and

$E_{\ \ \rho\sigma}^{\mu\nu}\equiv\frac{\partial\mathcal{L}}{\partial R_{\ \
\mu\nu}^{\rho\sigma}}\ .$ (21)

For the action (4), the tensor (21) and the Noether pre-potential take the
following form

$\displaystyle E_{\ \ \rho\sigma}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-2\phi}\left(\delta_{\rho\sigma}^{\mu\nu}+\alpha\delta_{\rho\,\sigma\,\nu_{3}\nu_{4}}^{\mu\,\nu\,\mu_{3}\mu_{4}}R_{\
\ \ \
\mu_{3}\mu_{4}}^{\nu_{3}\nu_{4}}\right)+\mathcal{O}\left(\alpha^{2}\right)\ ,$
(22) $\displaystyle q^{\mu\nu}$ $\displaystyle=$ $\displaystyle
2e^{-2\phi}T^{\mu\nu}+\alpha e^{-2\phi}\left(4T^{\mu\nu}R+16T^{\sigma[\mu}R_{\
\sigma}^{\nu]}+4T^{\rho\sigma}R_{\ \
\rho\sigma}^{\mu\nu}\right)+\mathcal{O}\left(\alpha^{2}\right)\ ,$ (23)

respectively, where we have defined $T^{\rho\sigma}\equiv
4\xi^{[\rho}\nabla^{\sigma]}\phi-\nabla^{[\rho}\xi^{\sigma]}$. Evaluating the
Wald entropy (19) for our solution, we obtain

$S=16\pi^{2}e^{-2\bar{\phi}_{0}}\mu^{2}-32\pi^{2}\alpha\,e^{-2\bar{\phi}_{0}}\left(C\mu-8\right)+\mathcal{O}\left(\alpha^{2}\right)\
.$ (24)

If we naturally identify

$e^{-2\bar{\phi}_{0}}=\frac{1}{16\pi G}\ ,$ (25)

$G$ being the $4d$ Newton constant, the leading term in (24) reproduces the
Bekenstein-Hawking entropy, while the order $\mathcal{O}(\alpha)$ terms yield
corrections to it. More precisely, we find

$\displaystyle
S=\frac{\pi\mu^{2}}{G}+\frac{16\pi\alpha}{G}\left(1-\frac{1}{8}C\mu\right)+\mathcal{O}\left(\alpha^{2}\right)\,.$
(26)

Notice that (26) depends both on $\mu$ and $C$. The dependence of $C$ can be
traced back to the fact that $\alpha(8-C\mu)/\mu$ is the $\mathcal{O}(\alpha)$
correction to the parameter in front of the Newtonian piece $\sim 1/r$ in the
component $g_{tt}$ of the metric, cf. (15). Then, using (17), the entropy can
also be written as

$\displaystyle S=\frac{\pi
r_{+}^{2}}{G}+\frac{46\pi\alpha}{3G}+\mathcal{O}\left(\alpha^{2}\right)\,.$
(27)

Notice that the potential term linear in $r_{+}$ (i.e. the one that could come
from the term linear in $\mu$ in (26)) has cancelled out. In fact, at order
$\mathcal{O}(\alpha)$, in virtue of the field equations, the computation
reduces to that of the full action evaluated on the undeformed GR solution
$f_{1}=N_{1}=\phi_{1}=0$. This means that, at that order, the only correction
to the area law $S=\frac{A}{4G}$ is given by a positive constant. On the same
grounds, corrections of the form $\mathcal{O}(\alpha r_{+}^{d-4}/G)$ are
expected in higher dimensions.

Next, let us compute the Hawking temperature. We can do this by resorting to
the Euclidean formalism. However, it is convenient to first simplify the
expressions a bit. We can write $\mu$ as a function of $r_{+}$ by simply
inverting (17), which yields

$\displaystyle g_{tt}(r)$ $\displaystyle=$
$\displaystyle-1+\frac{1}{r}\left(r_{+}+\frac{23\alpha}{3r_{+}}\right)-\alpha\left(\frac{4}{r^{2}}+\frac{5r_{+}}{3r^{3}}+\frac{2r_{+}^{2}}{r^{4}}\right)+\mathcal{O}\left(\alpha^{2}\right)\
,$ (28) $\displaystyle g^{rr}(r)$ $\displaystyle=$ $\displaystyle
1-\frac{1}{r}\left(r_{+}-\frac{\alpha}{3r_{+}}\right)+\alpha\left(\frac{2}{r^{2}}+\frac{r_{+}}{r^{3}}-\frac{10r_{+}^{2}}{3r^{4}}\right)+\mathcal{O}\left(\alpha^{2}\right)\
,$ (29) $\displaystyle\phi(r)$ $\displaystyle=$
$\displaystyle\bar{\phi}_{0}-\alpha\left(\frac{2}{r_{+}r}+\frac{1}{r^{2}}+\frac{2r_{+}}{3r^{3}}\right)+\mathcal{O}\left(\alpha^{2}\right)\
.$ (30)

This gives the periodicity condition for the real section of the Euclidean
geometry to be regular at $r=r_{+}$; namely

$\beta=4\pi
r_{+}+\frac{44\pi\alpha}{3r_{+}}+\mathcal{O}\left(\alpha^{2}\right)\ ,$ (31)

which results in the black hole temperature

$T=\frac{1}{4\pi
r_{+}}\left(1-\frac{11}{3}\frac{\alpha}{r_{+}^{2}}\right)+\mathcal{O}\left(\alpha^{2}\right)\,.$
(32)

This corrects the Hawking formula for GR at scales $r_{+}\simeq\alpha^{1/2}$.
This result, together with the expression (27) for the entropy, yields the
first law type relation

$\delta E\equiv T\delta
S=\delta\left(\frac{r_{+}}{2G}+\frac{11\alpha}{6Gr_{+}}\right)+\mathcal{O}\left(\alpha^{2}\right)\,,$
(33)

from which, up to subleading orders in $\alpha$, we can obtain the
gravitational energy

$E-E^{(0)}=\frac{r_{+}}{2G}\left(1+\frac{11}{3}\frac{\alpha}{r_{+}^{2}}\right)$
(34)

$E_{0}$ being an integration constant that corresponds to the energy of the
reference background. Below, we will confirm this result by rederiving the
gravitational energy using the Iyer-Wald method for computing Noether charges.
It is also worth noticing that, if we insist in extrapolating the formulae
above for small values of $r_{+}$, which is not well justified as higher-order
terms are expected to be relevant in that regime, then the formula obtained
for the specific heat changes its sign and becomes positive within the range
$\frac{11}{3}\alpha<r_{+}^{2}<{11}\alpha$; the black hole temperature (32)
vanishes at the lower bound $r_{+}^{2}=\frac{11}{3}\alpha$.

##  IV Conserved charges

In order to compute the gravitational energy of the solution, we have to
supplement the bulk action with the appropriated boundary terms. In the case
of the higher curvature action (4), the boundary term to be added reads

$\displaystyle I_{BT}\equiv\int_{\partial
M}d^{3}x\sqrt{-h}e^{-2\phi}\left[2K+4\alpha\delta^{\mu_{1}\mu_{2}\mu_{3}}_{\nu_{1}\nu_{2}\nu_{3}}K^{\nu_{1}}_{\mu_{1}}\left(\frac{1}{2}\mathcal{R}^{\nu_{2}\nu_{3}}_{\
\
\mu_{2}\mu_{3}}-\frac{1}{3}K^{\nu_{2}}_{\mu_{2}}K^{\nu_{3}}_{\mu_{3}}\right)\right]\equiv\int_{\partial
M}d^{3}x\sqrt{-h}\mathcal{B}$ (35)

where $K$ is the trace of the extrinsic curvature $K^{\mu}_{\nu}$, and
$\mathcal{R}^{\mu\nu}_{\ \rho\sigma}$ and $h_{\mu\nu}$ are the intrinsic
curvature and the induced metric on $\partial M$, respectively; cf. Nathalie .
The contribution (35) renders the variational principle well-posed. Then, the
energy of the spacetime, which corresponds to the black hole mass, is given by
the following integral on the sphere at infinity, $S^{2}_{\infty}$; namely

$\displaystyle
M=\int_{S^{2}_{\infty}}\left(\boldsymbol{Q}[\boldsymbol{t}]-\boldsymbol{t}\cdot\boldsymbol{B}\right)\
,$ (36)

where $\boldsymbol{Q}[\boldsymbol{t}]$ is the Hodge dual of the Noether pre-
potential for the killing vector $\boldsymbol{t}=\partial_{t}$ and

$\boldsymbol{B}=\frac{1}{3!}\,\mathcal{B}\,n^{\sigma}\,\sqrt{-g}\,\epsilon_{\mu\nu\rho\sigma}\,dx^{\mu}\wedge
dx^{\nu}\wedge dx^{\rho}\ .$ (37)

In flat spacetime, the trace of the extrinsic curvature is
${K}^{(0)}=\frac{2}{r}$ which gives a divergent piece in the action principle
as the volume element contributes with $r^{2}$. To obtain a finite action
principle and a finite energy definition, we have to subtract the extrinsic
curvature of flat spacetime to each piece of the extrinsic curvature appearing
in the formulae above. In other words, we have to define

$\bar{K}_{\mu\nu}\equiv K_{\mu\nu}-{K}^{(0)}_{\mu\nu}\,,$

using flat space as a background reference; this corresponds to set
$E^{(0)}=0$ for Minkowski spacetime. According to this, the energy content of
the spacetime, as defined in (36), precisely gives

$M=\frac{r_{+}}{2G}\left(1+\frac{11}{3}\frac{\alpha}{r_{+}^{2}}\right)\,,$
(38)

which agrees with (34).

Another crosscheck for this result can be done by means of the Euclidean
action formalism. In the saddle point approximation, the on-shell Euclidean
action gives the partition function; namely

$\log{Z}\simeq I^{E}+I^{E}_{BT}\,,$ (39)

where the superscript $E$ stands for Euclidean. It is worth emphasizing that,
at order $\mathcal{O}(\alpha)$, the computation of the Euclidean action
reduces to the evaluation of the full action $I^{E}+I^{E}_{BT}$ on the
undeformed GR solution. Therefore, the energy of the configuration can be
simply derived from (39) by computing

$\bar{E}=-\frac{\partial\log{Z}}{\partial\beta}\ .$ (40)

The on-shell action computed with $\bar{K}_{\mu\nu}$ for the configuration
(28)-(30) with the Euclidean time periodicity (31) turns out to be finite, and
it reads

$\displaystyle I^{E}+I^{E}_{BT}=-\frac{\pi
r_{+}^{2}}{G}-\frac{10\alpha\pi}{3G}\,.$ (41)

From this expression, we easily find

$\bar{E}=\frac{r_{+}}{2G}\left(1+\frac{11}{3}\frac{\alpha}{r_{+}^{2}}\right)\,,$
(42)

which, again, exactly reproduces (34) at the right order. This results in an
$\mathcal{O}(\alpha)$, $r_{+}$-dependent correction to the GR Smarr formula;
namely

$T\,S\,-\,\frac{1}{2}\,\bar{E}\,=\,\frac{2\alpha}{Gr_{+}}\,.$ (43)

At order $\mathcal{O}(\alpha)$ this is equivalent to an additive constant in
the entropy.

##  V Adding angular momentum

Black hole solution (10) can be generalized to the stationary non-static case,
and the analytic expression in the slowly rotating approximation can also be
found following the similar perturbative method as before. At first order in
$\alpha$ and including the rotation parameter in linear and quadratic terms as
well as in terms of the form $a\alpha$, the solutions reads

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{\mu}{r}+\frac{\mu
a^{2}\cos^{2}\theta}{r^{3}}+\alpha
f_{1}\left(r\right)\right)dt^{2}+2a\left(-\frac{\mu\sin^{2}\theta}{r}+\alpha\,h_{t\varphi}\left(r,\theta\right)\right)dtd\varphi\,+$
$\displaystyle\ \ \ \ \ \left(\frac{1}{1-\frac{\mu}{r}+\alpha
g_{1}\left(r\right)}-\frac{\left(\left(\mu-r\right)\cos^{2}\theta+2r\right)a^{2}}{\left(r-\mu\right)^{2}r}\right)dr^{2}\,+$
$\displaystyle\ \ \
\left(r^{2}+a^{2}\cos^{2}\theta\right)d\theta^{2}+\left(\left(r^{2}+a^{2}\right)\sin^{2}\theta+\frac{a^{2}\mu\sin^{4}\theta}{r}\right)d\varphi^{2}\,,$

with

$\displaystyle f_{1}\left(r\right)$
$\displaystyle=\frac{2\mu^{2}}{r^{4}}+\frac{5\mu}{3r^{3}}+\frac{4}{r^{2}}-\frac{8}{\mu
r}\ ,$ (45) $\displaystyle g_{1}\left(r\right)$
$\displaystyle=-\frac{40}{3r^{4}}+\frac{\mu}{r^{3}}+\frac{2}{r^{2}}\ ,$ (46)
$\displaystyle h_{t\varphi}\left(r\right)$
$\displaystyle=\sin^{2}\theta\left(\frac{\hat{C}}{r}+\frac{2\mu^{2}+3\mu
r+6r^{2}}{r^{4}}\right)\,,$ (47)

and with $\hat{C}$ being a new integration constant that, at this order, comes
to renormalize the angular momentum; see (51) below. The scalar configuration
is

$\phi\left(r\right)=\phi_{0}-\alpha\left(\frac{2\mu}{3r^{3}}+\frac{1}{r^{2}}+\frac{2}{\mu
r}\right)\ .$ (48)

One can verify that, expanding both in the Gauss-Bonnet coupling, $\alpha$,
and in the rotation parameter, $a$, all the field equations are solved at the
right order; namely

$E_{\mu\nu}=\mathcal{O}\left(\alpha a^{2},\alpha^{2}\right)\ .$ (49)

The angular momentum can be computed by using the Wald formalism, which yields
a form

$J=-\int_{S^{2}_{\infty}}\boldsymbol{Q}[\partial_{\varphi}]$ (50)

with $\boldsymbol{Q}[\partial_{\varphi}]$ representing the Hodge dual of the
Noether pre-potential for the Killing vector $\partial_{\varphi}$. The angular
momentum of the spacetime is given by

$J=\frac{a\mu}{2G}\left(1-\frac{\alpha\hat{C}}{\mu}\right)\,\,.$ (51)

Solution (V)-(47) gives string theory modification to Kerr geometry. In
particular, we see order $\mathcal{O}(a\alpha)$ modifications to the off-
diagonal term in the Boyer-Lindquist coordinates. This will result in
deviations from the GR prediction of the Lense-Thirring precession. It will
also induce modifications to the spheroidal shape of the shadow of a rotating
black hole; see Eiroa  and references thereof.

##  VI Accelerating black holes

Let us consider the following ansatz for the metric and for the dilaton

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{\Omega\left(x,y\right)}{A^{2}\left(x+y\right)^{2}}\left(-F\left(y\right)dt^{2}+\frac{dy^{2}}{F\left(y\right)}+\frac{dx^{2}}{G\left(x\right)}+G\left(x\right)d\varphi^{2}\right)\
,$ (52) $\displaystyle\phi$ $\displaystyle=$
$\displaystyle\phi\left(x,y\right)\ .$ (53)

Assuming the expansion
$\phi=\phi_{0}\left(x,y\right)+\alpha\phi_{1}\left(x,y\right)+\mathcal{O}(\alpha^{2})$,
$F=F_{0}\left(y\right)+\alpha F_{1}\left(y\right)+\mathcal{O}(\alpha^{2})$,
$G=G_{0}\left(x\right)+\alpha G_{1}\left(x\right)+\mathcal{O}(\alpha^{2})$ and
$\Omega\left(x,y\right)=1+\alpha\omega\left(x,y\right)+\mathcal{O}(\alpha^{2})$.
In General Relativity, the ansatz (52), leads to the C-metric which
accommodates accelerating black holes (see Griffiths:2009dfa  for a modern
interpretation as well as a historical review), even in the presence of
minimally coupled, self interacting scalar fields Anabalon:2012ta . Here $A$,
stands for the acceleration and $\Omega\left(x,y\right)=1$ in General
Relativity in vacuum.

To the lowest order on the string tension $\alpha$, Einstein equations lead to

$F\left(y\right)=F_{0}\left(y\right)=f_{3}y^{3}+f_{2}y^{2}+f_{1}y+f_{0}\text{
and }G\left(x\right)=G_{0}\left(x\right)=f_{3}x^{3}-f_{2}x^{2}+f_{1}x-f_{0}\
,$ (54)

fulfilling $G\left(\xi\right)=-F\left(-\xi\right)$. Here $f_{i}$ with
$i=\\{0,...,3\\}$ are integration constants. The quadratic, the linear or the
$f_{0}$ term in the polynomials (54) can be removed by a simultaneous,
constant shift of the independent variables $(x,y)$, maintaining the form of
the metric (52). For future purposes, it is better to keep all the $f_{i}$ as
non-vanishing at the moment.

The field equations of the $alpha^{\prime}$-corrected theory (4), at linear
order in $\alpha$ are solved by

$\displaystyle F_{1}\left(y\right)$ $\displaystyle=$ $\displaystyle
d_{3}y^{3}+d_{2}y^{2}+d_{1}y+d_{0}\ ,$ (55) $\displaystyle G_{1}(x)$
$\displaystyle=$ $\displaystyle
f_{3}h_{1}x^{3}+3f_{3}h_{2}x^{2}+\frac{\left(3d_{3}f_{1}-3f_{1}f_{3}h_{1}-6f_{2}f_{3}h_{2}+3d_{1}f_{3}-2d_{2}f_{2}\right)}{3f_{3}}x$
(57)
$\displaystyle-\frac{\left(-6f_{0}f_{3}h_{1}-3f_{1}f_{3}h_{2}+3d_{0}f_{3}-d_{2}f_{1}+6d_{3}f_{0}\right)}{3f_{3}}\
,$ $\displaystyle\omega(x,y)$ $\displaystyle=$ $\displaystyle
2\phi_{1}(x,y)+\frac{3f_{3}j_{1}x+3\left(2f_{3}h_{1}+f_{3}j_{1}-2d_{3}\right)y-6f_{3}h_{2}-2d_{2}}{3\left(x+y\right)f_{3}}\
,$ (58)

leading to the following inhomogenoeus, PDE for $\phi_{1}\left(x,y\right)$

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(x+y\right)\left(G_{0}\left(x\right)\frac{\partial^{2}\phi_{1}}{\partial
x^{2}}+F_{0}\left(y\right)\frac{\partial^{2}\phi_{1}}{\partial y^{2}}\right)$
(60) $\displaystyle+(f_{3}x^{3}+3f_{3}x^{2}y-2f_{2}xy-
f_{1}x+f_{1}y+2f_{0})\frac{\partial\phi_{1}}{\partial x}$
$\displaystyle+(f_{3}y^{3}+3f_{3}xy^{2}+2f_{2}xy+f_{1}x-f_{1}y-2f_{0})\frac{\partial\phi_{1}}{\partial
y}-6A^{2}f_{3}^{2}\left(x+y\right)^{5}\ .$

Here the constants $(d_{i},f_{j},h_{k},j_{l})$ are new integration constants
that emerge from the integration of the field equations at linear order in
$\alpha$. Even though the equation (60) seems not to admit an analytic
solution, it can be solved as a power series in the acceleration $A$, around
$A=0$. In order to be able to take the limit $A=0$ in (52), it is useful to
perform the following change of coordinates (see Chapter 14 of
Griffiths:2009dfa ):

$x=-\cos\theta\text{, }y=\frac{1}{Ar}\text{, }t=A\tau\ ,$ (61)

and choosing

$f_{2}=-f_{0}=1\text{ and }\ f_{3}=-f_{1}=-2mA\ ,$ (62)

leads to the following parametrization for the C-metric in General Relativity

$ds_{0}^{2}=\frac{1}{\left(1-Ar\cos\theta\right)^{2}}\left(-Q_{0}\left(r\right)d\tau^{2}+\frac{dr^{2}}{Q_{0}\left(r\right)}+\frac{r^{2}d\theta^{2}}{P_{0}\left(\theta\right)}+P_{0}\left(\theta\right)r^{2}\sin^{2}\theta
d\varphi^{2}\right)\ ,$ (63)

with

$\displaystyle Q_{0}\left(r\right)$ $\displaystyle=$
$\displaystyle\left(1-\frac{2m}{r}\right)\left(1-A^{2}r^{2}\right)\ ,$ (64)
$\displaystyle P_{0}\left(\theta\right)$ $\displaystyle=$ $\displaystyle
1-2mA\cos\theta\ .$ (65)

In terms of $\left(r,\theta\right)$, and choosing the constants $f^{\prime}s$
as in (62), the equation (60) is integrated, order by order in the
acceleration $A$. For such purpose, it is convenient to choose

$\phi_{1}\left(r,\theta\right)=\left(1-Ar\cos\theta\right)H\left(r,\theta\right)\
,$ (66)

with

$H\left(r,\theta\right)=\sum_{i=0}H_{i}\left(r,\theta\right)A^{i}$

which leads to the following functions at the lowest orders

$\displaystyle H_{0}\left(r,\theta\right)$ $\displaystyle=$
$\displaystyle-\frac{4m}{3r^{3}}-\frac{1}{r^{2}}-\frac{1}{mr}\ ,$ (67)
$\displaystyle H_{1}\left(r,\theta\right)$ $\displaystyle=$
$\displaystyle\left(\frac{26m}{3r^{2}}-\frac{1}{r}\right)\cos\theta\ ,$ (68)
$\displaystyle H_{2}\left(r,\theta\right)$ $\displaystyle=$
$\displaystyle\frac{2m\left(2\sin^{2}\theta-23\right)}{3r}\ .$ (69)

Other solutions are possible, but they lead to logarithmic or divergent
behavior for the dilaton as $r\rightarrow\infty$.

In order to clarify the meaning of the plethora of integration constants which
remain arbitrary on the metric functions, it is useful to reconstruct the
full, corrected spacetime (52), in $\left(r,\theta\right)$ coordinates. The
change of coordinates (61), induces the presence of $A^{-1}$ terms in the
metric coming from the terms (55)-(58) written in terms of $(r,\theta)$, which
are removed by setting $d_{3}=0$. Imposing the absence of divergences at
$\theta=0$ and $\theta=\pi$ on the metric functions suffices to fix all the
remaining integration constants, but $j_{1}$, leading to
$d_{2}=d_{1}=d_{0}=h_{1}=h_{2}=0,$ which in consequence leads to vanishing
corrections of the function $F$ and $G$, namely

$F_{1}\left(r,\theta\right)=0,\ G_{1}(r,\theta)=0\ ,$ (70)

and to a conformal factor $\omega\left(r,\theta\right)$ given by

$\omega(r,\theta)=2\phi_{1}(r,\theta)\ ,$ (71)

where we have also set $j_{1}=0$ since a non-vanishing value of $j_{1}$ can be
absorbed on the dilaton’s additive, arbitrary constant $\phi_{0}$.

Putting all these ingredients together, lead to the corrected metric which is
given by

$ds^{2}=\frac{1+2\alpha\phi_{1}\left(r,\theta\right)}{\left(1-Ar\cos\theta\right)^{2}}\left(-Q_{0}\left(r\right)d\tau^{2}+\frac{dr^{2}}{Q_{0}\left(r\right)}+\frac{r^{2}d\theta^{2}}{P_{0}\left(\theta\right)}+P_{0}\left(\theta\right)r^{2}\sin^{2}\theta
d\varphi^{2}\right)\ .$ (72)

Here $\phi_{1}\left(r,\theta\right)$ is given by (66) and
$Q_{0}\left(r\right)$ and $P_{0}\left(\theta\right)$ given by (64) and (65),
respectively. One can check, that the metric (72), with
$\phi_{1}\left(r,\theta\right)$ in (66), solve the field equations of the
theory (4), disregarding terms of the form
$\mathcal{O}\left(\alpha^{2}\right)$ and $\mathcal{O}\left(\alpha
A^{3}\right)$, i.e. when evaluated on the corrected C-metric, the field
equations vanish up to

$E_{\mu\nu}=\mathcal{O}\left(\alpha^{2}\right)+\mathcal{O}\left(\alpha
A^{3}\right)\ .$ (73)

It is very interesting to notice that the regularity conditions lead us to
move the whole effect of the $\alpha$-correction to the conformal factor. The
solution can be found to higher orders on the acceleration by performing the
integration of the PDE (60), at the desired order on $A$, ofter moving to
$\left(\tau,r,\theta\right)$ coordinates via (61), in such a manner that the
limit of vanishing acceleration is regular.

##  VII Mapping to the Einstein frame

Recently in Cano3 , the authors constructed the dimensional reduction of the
Heterotic String on a flat torus, to dimension four, and constructed rotating
solutions, perturbativelly in the rotation parameter, including the first
$\alpha^{\prime}$-correction, in the Einstein frame. It is interesting to
compare the setup we have considered here, defined by the action (4), with the
one of reference Cano3 , where disregarding the contribution of the
$B_{\mu\nu}$-field, leads to an action of the form

$I\left[g^{\prime}_{\mu\nu},\tilde{\phi}\right]=\int
d^{4}x\sqrt{-g^{\prime}}\left(R^{\prime}-\frac{1}{4}\nabla_{\mu}\tilde{\phi}\nabla_{\nu}\tilde{\phi}g^{\prime\mu\nu}+\alpha
e^{-\tilde{\phi}}\left(R_{\ \ \ \rho\sigma}^{\prime\mu\nu}R_{\ \ \
\mu\nu}^{\prime\rho\sigma}-4R_{\ \sigma}^{\prime\nu}R_{\
\nu}^{\prime\sigma}+R^{\prime 2}\right)\right)\ .$ (74)

Considering a Weyl transformation of the form

$\displaystyle g_{\mu\nu}$ $\displaystyle\mapsto$ $\displaystyle
g_{\mu\nu}^{\prime}=e^{\Phi}g_{\mu\nu}\ ,$ (75)

where $\Phi$ is some scalar function on the spacetime, the transformation of
the quadratic scalars constructed with the Riemann tensor are given by

$\displaystyle R_{\ \ \ \rho\sigma}^{\prime\mu\nu}R_{\ \ \
\mu\nu}^{\prime\rho\sigma}$ $\displaystyle=$ $\displaystyle
e^{-2\Phi}\left[R_{\ \ \ \rho\sigma}^{\mu\nu}R_{\ \ \
\mu\nu}^{\rho\sigma}-4R_{\ \sigma}^{\nu}\nabla_{\nu}\Phi^{\sigma}+2R_{\
\rho}^{\nu}\Phi^{\rho}\Phi_{\nu}-R\Phi^{\lambda}\Phi_{\lambda}\right.$
$\displaystyle+D_{2}\nabla_{\sigma}\Phi^{\nu}\nabla_{\nu}\Phi^{\sigma}+\left(\square\Phi\right)^{2}-D_{2}\nabla_{\sigma}\Phi_{\nu}\Phi^{\sigma}\Phi^{\nu}+D_{2}\square\Phi\Phi^{\lambda}\Phi_{\lambda}\left.+\frac{1}{8}D_{2}D_{1}\left(\Phi^{\lambda}\Phi_{\lambda}\right)^{2}\right]\
,$ $\displaystyle R_{\ \sigma}^{\prime\nu}R_{\ \nu}^{\prime\sigma}$
$\displaystyle=$ $\displaystyle e^{-2\Phi}\left[R_{\ \sigma}^{\nu}R_{\
\nu}^{\sigma}-D_{2}R^{\nu\sigma}\nabla_{\nu}\Phi_{\sigma}-R\square\Phi+\frac{1}{2}D_{2}R^{\nu\sigma}\Phi_{\nu}\Phi_{\sigma}-\frac{1}{2}D_{2}R\Phi^{\lambda}\Phi_{\lambda}\right.$
$\displaystyle+\frac{1}{4}D_{2}^{2}\nabla_{\sigma}\Phi^{\nu}\nabla_{\nu}\Phi^{\sigma}+\frac{1}{4}\left(3D-4\right)\left(\square\Phi\right)^{2}+\frac{1}{16}D_{2}^{2}D_{1}\left(\Phi^{\lambda}\Phi_{\lambda}\right)^{2}$
$\displaystyle\left.-\frac{1}{4}D_{2}^{2}\nabla_{\sigma}\Phi_{\nu}\Phi^{\sigma}\Phi^{\nu}+\frac{1}{4}D_{2}\left(2D-3\right)\square\Phi\Phi^{\lambda}\Phi_{\lambda}\right]\
,$ $\displaystyle R^{\prime 2}$ $\displaystyle=$ $\displaystyle
e^{-2\Phi}\left[R^{2}-2D_{1}R\square\Phi-\frac{1}{2}D_{1}D_{2}R\left(\partial\Phi\right)^{2}+D_{1}^{2}\left(\square\Phi\right)^{2}\right.$
$\displaystyle\left.+\frac{1}{2}D_{1}^{2}D_{2}\square\Phi\left(\partial\Phi\right)^{2}+\frac{1}{16}D_{1}^{2}D_{2}^{2}\left(\partial\Phi\right)^{4}\right]$

where $\Phi_{\lambda}=\nabla_{\lambda}\Phi$ and $D_{p}:=(D-p)$. These
expressions lead to the following transformation of the Gauss-Bonnet density,
which we had actually worked out in arbitrary dimension $D$

$\displaystyle R_{\ \ \ \rho\sigma}^{\prime\mu\nu}R_{\ \ \
\mu\nu}^{\prime\rho\sigma}-4R_{\ \sigma}^{\prime\nu}R_{\
\nu}^{\prime\sigma}+R^{\prime 2}$ (76) $\displaystyle=$ $\displaystyle
e^{-2\Phi}\left[R_{\ \ \ \rho\sigma}^{\mu\nu}R_{\ \ \
\mu\nu}^{\rho\sigma}-4R_{\ \sigma}^{\nu}R_{\
\nu}^{\sigma}+R^{2}+4D_{3}G^{\nu\sigma}\nabla_{\nu}\Phi_{\sigma}-2D_{3}R^{\nu\sigma}\Phi_{\nu}\Phi_{\sigma}\right.$
$\displaystyle-\frac{1}{2}D_{4}D_{3}R\left(\partial\Phi\right)^{2}-D_{3}D_{2}\nabla_{\sigma}\Phi^{\nu}\nabla_{\nu}\Phi^{\sigma}+D_{3}D_{2}\nabla_{\sigma}\Phi_{\nu}\Phi^{\sigma}\Phi^{\nu}$
$\displaystyle+\left.D_{3}D_{2}\left(\square\Phi\right)^{2}+\frac{1}{2}D_{3}^{2}D_{2}\square\Phi\left(\partial\Phi\right)^{2}+\frac{1}{16}D_{4}D_{3}D_{2}D_{1}\left(\partial\Phi\right)^{4}\right]\
,$ $\displaystyle=$ $\displaystyle e^{-2\Phi}\left(\mathcal{G+P}\right)$

where

$\mathcal{G}=R_{\ \ \ \rho\sigma}^{\mu\nu}R_{\ \ \ \mu\nu}^{\rho\sigma}-4R_{\
\sigma}^{\nu}R_{\ \nu}^{\sigma}+R^{2}$ (77)

and $\mathcal{P}$ stands for the remaining terms. Replacing these expression
in (74) and choosing the scalar $\Phi$ as

$\Phi=-\frac{4}{D-2}\phi\ ,$ (78)

and identifying the scalar field in (74) $\tilde{\phi}$ as

$\tilde{\phi}=\frac{4}{D-2}\phi,$ (79)

and setting $D=4$, leads to

$\displaystyle
I\left[g^{\prime}_{\mu\nu}=e^{-2\phi}g_{\mu\nu},\tilde{\phi}=2\phi\right]$
(80) $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}e^{-2\phi}\left[R+4\left(\partial\phi\right)^{2}\right.+\alpha\left.\left(\mathcal{G}-32\left(\partial\phi\right)^{4}-16G^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi+24\square\phi\left(\partial\phi\right)^{2}\right)\right]$

where we have disregarded boundary terms. Notice that this action belongs to
the family of the most general String Theory actions
$\alpha^{\prime}$-corrected, which after field-redefinitions lead to second
order field equations, since each of the derivative terms for the scalar
sector belongs to the Horndeski family Horndeski:1974wa . Indeed, upon
comparison with eq. (2.6) of reference Maeda2011  one can read from the action
(80) that the coefficients $(\lambda,\mu,\nu)$ of reference Maeda2011  are
given by $\lambda=-32,\mu=-16$ and $\nu=24$, and they indeed fulfil the
consistency constraint $\lambda+2\left(\mu+\nu\right)+16=0$. The relation
among the relative coefficients of the higher derivative operators of the
scalar attests about the UV finiteness of the action. In consequence, using
the results of reference Maeda2011 , one can see that the action (74) of
reference Cano3  in the Einstein frame and our action (4) in the string frame
are related by a field redefinition, composed with a change of frame.

Therefore, our static and rotating solutions of Sections II and IV, correspond
to a change of frame of the solutions found in Callan  and Cano3 ,
respectively, composed with a field redefinition. On the other hand, the
solution corresponding to accelerating black holes we presented in Section VI,
is completely new.

##  VIII The $d$-dimensional solution

The analytic solution that we constructed in $d=4$ can be generalized to
arbitrary dimension $d$ in a similar manner, although, as we will see, the
form of the general case is a bit more involved.

Consider the ansatz

$ds^{2}=-\left(1+\alpha
N_{1}\left(r\right)\right)^{2}\left(1-\left(\frac{\mu}{r}\right)^{d-3}+\alpha
f_{1}\left(r\right)\right)dt^{2}+\frac{dr^{2}}{1-\left(\frac{\mu}{r}\right)^{d-3}+\alpha
f_{1}\left(r\right)}+r^{2}d\Omega_{d-2}^{2}\ ,$

where now $d\Omega_{d-2}^{2}$ is the constant-curvature metric on the unit
$(d-2)$-sphere. By plugging this ansatz in the field equations for $d$
generic, we find the following general solution

$\displaystyle N_{1}(r)$
$\displaystyle=-\frac{C_{3}}{\mu^{2}f_{0}(r)}\left(\frac{\mu}{r}\right)^{d-3}+\frac{(d-3)(d-2)}{(d-1)\mu^{2}}\left(\frac{\mu}{r}\right)^{d-1}\left[F(r)+\frac{1}{f_{0}(r)}\left(\left(\frac{\mu}{r}\right)^{d-3}d-1\right)\right]$
$\displaystyle+\frac{C_{3}}{(d-3)\mu^{2}}\log{f_{0}(r)}+\frac{C_{2}}{\mu^{2}}\
,$ $\displaystyle f_{1}(r)$
$\displaystyle=-\frac{(d-3)}{(d-1)\mu^{2}}\left(\frac{\mu}{r}\right)^{2d-4}\left[(d-3)(d-2)F(r)+2(2d-3)\right]+\frac{C_{1}}{\mu^{2}}\left(\frac{\mu}{r}\right)^{d-3}$
$\displaystyle-\frac{C_{3}}{\mu^{2}}\left(\frac{\mu}{r}\right)^{d-3}\log{f_{0}(r)}\
,$

and

$\displaystyle\phi_{1}(r)$
$\displaystyle=\frac{\left(d-3\right)\left(d-2\right)^{2}}{2\left(d-1\right)r^{2}}\left(\frac{\mu}{r}\right)^{d-3}\left(F(r)-1\right)+\frac{(d-2)C_{3}}{2(d-3)\mu^{2}}\log{f_{0}(r)}+\frac{C_{4}}{\mu^{2}}$

where $f_{0}\left(r\right)=1-\left(\frac{\mu}{r}\right)^{d-3}$ and where
$F(r)$ is given in terms of the hypergeometric function,

$F\left(r\right)\,=\,{}_{2}F_{1}\left(1,\frac{d-1}{d-3},2\,\frac{d-2}{d-3},\left(\frac{\mu}{r}\right)^{d-3}\right)\,;$

$C_{1},\,C_{2},\,C_{3}$, and $C_{4}$ are integration constants, analogous to
the constants $A$, $B$, $C$, and $D$ of the $d=4$ case; for example, one can
identify $C_{3}=\mu^{2}B+2$, $C_{4}=\mu^{2}\bar{\phi}_{0}$, and so on. Some of
these constants can be fixed as in the 4-dimensional case, i.e. by rescaling
time coordinate $t$, shifting the zero-mode of $\phi$, neglecting
$\mathcal{O}(\alpha^{2})$ remnants, imposing a globally flat asymptotic
behavior as $r\rightarrow\infty$, and requiring regularity at the event
horizon. One can easily check that the 4-dimensional solution studied in the
previous sections is recovered in the case $d=4$. To see this, it is
convenient to consider the relation

${}_{2}F_{1}(1,3,4,z)=-\frac{3}{2z^{3}}\left[z(z+2)+2\log(1-z)\right]\,,$ (81)

with $z=1-f_{0}(r)$. The presence of logarithmic terms $\log
f_{0}(r)=\log(1-z)$ in the functions $N_{1}(r)$, $f_{1}(r)$ and $\phi_{1}(r)$
is related to the fact that the third argument of the hypergeometric function,
$c=2\frac{(d-2)}{(d-3)}$, turns out to be an integer for some dimensions
($d=4,5$). The logarithm, which in any case tends to zero at large $r$,
disappears if one chooses $C_{3}$ appropriately.

##  IX Conclusions

Summarizing, the solutions we have presented in this paper describe static,
spherically symmetric configurations in the graviton-dilaton sector of the
$d$-dimensional low-energy stringy effective action (4). This includes square-
curvature terms and a non-vanishing dilaton coupling. We have used the freedom
of field redefinitions to recast the action in a form that leads to second
order field equations, while still working in the string frame. The set of
solutions includes asymptotically flat black holes with regular event
horizons, which behave as thermodynamic objects, just like expected. As a
working example, we first focused on the the 4-dimensional case, which is
given by (28)-(30). We derived the corrections to the thermodynamic variables
introduced by the higher-curvature effects; we computed the Bekenstein-Hawking
entropy (27), the Hawking temperature (32), and the mass formula (42)
including the $\mathcal{O}(\alpha)$ effects. The computation of the Noether
charges was shown to be in exact agreement with the first law of black hole
mechanics as derived from the Wald entropy formula; the Euclidean action
formalism also reproduces these results. We also obtained the correction to
the C-metric, which contains accelerating black holes. We have show that
regularity conditions imply that the whole modifications is contained within
the conformal factor of the spacetime. We also integrated the equations of
motion for the stationary, non-static solution in the slowly rotating
approximation. This yields stringy corrections to the Kerr geometry in four
dimensions. Although, in contrast to the 4-dimensional case, the field
equations in arbitrary dimension $d$ are more involved, we showed that in the
static case they can still be solved explicitly in terms of hypergeometric
functions. Static and rotating black holes in these theories have already been
considered in the literature, when the theory is expressed in a different
fashion, which is possible due to the freedom of field redefinition.
Considering such freedom, we found the precise relation between our setup and
the frames previously considered on the literature. In particular, we mapped
our theory to the Einstein frame, including the higher curvature corrections,
and we showed the equivalence of our setup and that of reference Cano3 , where
rotating solutions were already presented. These solutions may serve as
working examples to investigate higher-curvature stringy effects in a concrete
setup.

## Acknowledgements

We thank Cristóbal Corral and Nicolás Grandi for their enlightening comments.
We also thank the anonymous referee who pointed out the possibility of mapping
our theory to a frame where rotating solutions have already been constructed
in Ref. [66], which led us to include Secs. VI and VII. F. A. is thankful for
funding via FONDECYT (Fondo Nacional de Desarrollo Científico y Tecnológico)
REGULAR 1201280. The work of M. C. is partially supported by Mexico’s National
Council of Science and Technology (CONACyT) Grant A1-S-22886, DGAPA-UNAM Grant
IN107520, and a PASPA fellowship. The work of G. G. was supported by CONICET
(Consejo Nacional de Investigaciones Científicas y Técnicas) and ANPCyT
(Agencia Nacional de Promoción Científica y Tecnológica) through Grants 755
PIP1109-2017 and PICT-2019-00303. The work of M. O. is partially funded by
Beca ANID (Agencia Nacional de Investigación y Desarrollo) de Doctorado Grant
21222264 and FONDECYT Grant 1221504. J. O. thanks the support of proyecto
FONDECYT REGULAR 1221504, Proyecto de Cooperación Internacional 2019/13231-7
FAPESP/ANID, and Dirección de Postgrado, Universidad de Concepción, through
Grant UCO1866.

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