# General Relativity from Einstein-Gauss-Bonnet gravity

1∧Fabrizio Canfora, 2,⋆Adolfo Cisterna,
3,∗Sebastián Fuenzalida, 4,†Carla Henríquez-Báez, 4,‡Julio Oliva  1 Centro de
Estudios Científicos (CECs), Casilla 1469, Valdivia, Chile  2Sede Esmeralda,
Universidad de Tarapacá, Av. Luis Emilio Recabarren 2477, Iquique, Chile
3Departamento de Física, Universidad Técnica Federico Santa María, Casilla
110-V, Valparaíso, Chile  4Departamento de Física, Universidad de Concepción,
Casilla 160-C, Concepción, Chile

###### Abstract

In this work we show that Einstein gravity in four dimensions can be
consistently obtained from the compactification of a generic higher curvature
Lovelock theory in dimension $D=4+p$, being $p\geq 1$. The compactification is
performed on a direct product space
$\mathcal{M}_{D}=\mathcal{M}_{4}\times\mathcal{K}^{p}$, where
$\mathcal{K}^{p}$ is a Euclidean internal manifold of constant curvature. The
process is carried out in such a way that no fine tuning between the coupling
constants is needed. The compactification requires to dress the internal
manifold with the flux of suitable $p$-forms whose field strengths are
proportional to the volume form of the internal space. We explicitly
compactify Einstein-Gauss-Bonnet theory from dimension six to Einstein theory
in dimension four and sketch out a similar procedure for this compactification
to take place starting from dimension five. Several black string/p-branes
solutions are constructed, among which, a five dimensional asymptotically flat
black string composed of a Schwarzschild black hole on the brane is
particularly interesting. Finally, the thermodynamic of the solutions is
described and we find that the consistent compactification modifies the
entropy by including a constant term, which may induce a departure from the
usual behavior of the Hawking-Page phase transition. New scenarios are
possible in which large black holes dominate the canonical ensamble for all
temperatures above the minimal value.††<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
##  I Introduction

The original Kaluza-Klein scheme Kaluza ; Klein  provides, via dimensional
reduction to four dimensions, a consistent method to unify gravity and
electromagnetism by starting from a purely geometrical higher dimensional
theory Overduin:1998pn . In higher dimensions, it is assumed that only gravity
exists, described by a five dimensional Einstein-Hilbert action, that gives
rise to the electromagnetic force once the spacetime is properly compactified
to dimension four. Two fundamental assumptions are made: First, gravity is
described by Einstein General Relativity (GR) and second, the higher
dimensional spacetime is empty. Despite this conceptual appealing of the
Kaluza-Klein framework, some subtleties emerge given by two experimental
signatures that are in tension with the simplest approaches of
compactification: the small, positive value of the four dimensional
cosmological constant that we observe today and the highly accurate particle
physics experiments whose consistency requires strict bounds on the size of
the extra dimensions. In the cosmological context, this procedure has made it
possible to find solutions of the Friedmann-Robertson-Walker class Castillo-
Felisola:2016kpe .
In addition, simple compactifications over direct product spaces, namely
consistent compactifications with vanishing gauge and dilaton fields, where
the internal manifold is of constant curvature, usually suffer from
incompatibilities at the level of the field equations Duff:1986hr . A concrete
example to see this, is the compactification of Einstein theory on a $D=d+p$
dimensional spacetime of the form
$\mathcal{M}_{D}=\mathcal{M}_{d}\times\mathcal{K}^{p}$, where
$\mathcal{M}_{d}$ is a $d$-dimensional spacetime and $\mathcal{K}^{p}$ an
internal $p$-dimensional Euclidean manifold of constant curvature.
Compatibility of the field equations will imply a vanishing of the Ricci
tensor of the internal manifold, which if assumed of constant curvature,
implies that it must be locally flat and consequently, its isometry group can
only be Abelian. Isometries of the internal manifold govern the symmetry group
of the compactified interaction and in consequence within this ansatz of
cylindrycal compactification, GR it is only unifiable with electromagnetism.
On the other hand, to accommodate a small non-vanishing four dimensional
cosmological constant, it is then mandatory to include an internal manifold
with a non-vanishing curvature which forces the radius of the extra dimensions
to be large. In consequence, to properly compactify Einstein theory over
internal manifolds constructed with small dimensions, with or without a small
cosmological constant, it is necessary to either give up on the emptiness of
the higher dimensions or to generalize the geometric structure of the theory
by including well-possed higher curvature terms into the gravitational action.
Including matter fields in higher dimensions seems to go against the very idea
of geometrization of interactions behind Kaluza-Klein compactifications. As
stated in the 80’s, this would be still consistent if there is a guiding
principle that fixes the matter content in higher dimensions in a relatively
unique manner. Supergravity achieves this goal in eleven dimensions, since its
bosonic matter content in uniquely defined by a metric, and a fundamental
three-form, which cannot be further coupled to other matter multiplets
Duff:1986hr .
On the other hand, at a practical level, including higher dimensional matter
fields has been shown to be fruitful to compactify Einstein theory. It is
known that dressing the internal manifold with fundamental $(p-1)$-forms
$A_{[p-1]}$, with a field strength $F_{[p]}=dA_{[p-1]}$ proportional to the
volume form of such space, renders possible compactifications of Einstein
gravity Freund:1980xh ; RandjbarDaemi:1982hi . In fact, the $p$-form charge
allows for a non-trivial curvature of the internal manifold while at the same
time accommodating a small cosmological constant. This dressing protocol is
also widely applied in the construction of black holes with $p$-form fields
depending on the horizon coordinates which support a black hole horizon from
collapsing when deviating from bald solutions Bardoux:2012aw  (see also
Edery:2015wha , Edery:2018jyp )
In higher dimensions, higher curvature terms when combined properly in such a
way that the resulting theory of gravity respect general covariance and
possesses second order field equations, give rise to Lovelock theory, the
natural generalization of Einstein gravity in higher dimensions
Lovelock:1971yv . This theory corresponds to an infinite series of higher
curvature terms of order $k$, which are non-trivial when $k<D/2$. In this way,
on top of a cosmological term, the Ricci scalar is the first term of the
series and the only one that it is non-trivial in dimension four. The first
dynamical correction of Lovelock gravity to the Einstein-Hilbert action
appears in dimension five and is given by a precise combination of quadratic
curvature terms known as the Gauss-Bonnet density Garraffo:2008hu , which also
appears as an $\alpha^{\prime}$-correction to GR in string theory
Zwiebach:1985uq . It is natural to wonder, how this model, dubbed Einstein-
Gauss-Bonnet (EGB) gravity, compactifies on direct product spacetimes.
Einstein-Gauss-Bonnet gravity with a cosmological constant can be easily
compactified over an internal manifold of non-vanishing constant curvature of
dimension $p$ to a spacetime of dimension $d\geq 5$. The presence of the
Gauss-Bonnet density renders the compactification trivial but at the price of
both, tuning the cosmological constant with the Gauss-Bonnet coupling and
forcing the internal manifold to have an hyperbolic structure, i.e a negative
constant curvature. The compactified theory becomes ill-behaved, the Killing
vectors of the hyperbolic internal manifold are not globally defined and then
it is not possible to accommodate the unification of gravity with any other
interaction. On might be tempted to dress the internal manifold with
$p$-forms, however we have shown in Cisterna:2020kde  that this only
eliminates the tuning of the cosmological constant with the Gauss-Bonnet
coupling, but the internal manifold remains hyperbolic. To solve these
problems it is mandatory to include higher dimensional matter in the shape of
$p$-forms non-minimally coupled to the curvature tensor, keeping the second
order character of the theory as a guiding principle. Such an interaction,
first described by Horndeski in Horndeski:1976gi  and later generalized in
Feng:2015sbw  provides the only non-minimally coupled gauge invariant
electrodynamics with second order field equations that when going to flat
spacetime reduces to Maxwell equations. This model renders possible a generic
compactification of Einstein-Gauss-Bonnet gravity on an internal manifold of
positive constant curvature Cisterna:2020kde , a fact that is extendable to
any Lovelock theory. Regarding these compactifications, a natural concern
arises: Whether or not a given Lovelock theory can be compactified to Einstein
gravity in dimension four? In four dimensions Lovelock densities are either
topological or identically zero, nevertheless, after the compactification,
traces of these geometrical entities survive at the level of the field
equations, rendering the resulting theory not compatible. In Canfora:2008iu
it has been shown that Lovelock gravity can be compactified to Einstein theory
in dimension four as long as the theory at least includes the cubic term of
the Lovelock series, and in consequence with an internal manifold of dimension
$p\geq 3$. The price to pay for this spontaneous compactification, beyond the
fact that it exists starting from seven dimensions, is that the cosmological
constant and the Gauss-Bonnet and cubic the Lovelock coupling are related to
each other in a manner that does not lead to a symmetry enhancement, and
therefore, such relation it is not expected to survive quantum corrections.
In this paper we show how any Lovelock theory, in particular Einstein-Gauss-
Bonnet theory, can be compactified to four dimensional Einstein gravity
starting from five or six dimensions and without imposing any relation among
the model parameters, namely for a generic Lovelock theory. To accomplish this
we employ the same strategy of Cisterna:2020kde  and we show that there is no
need to consider any interaction of order higher than two in the curvature.
The article is organized as follows: Section II is devoted to introduce our
model and the corresponding field equations. In Section III we show how to use
our model to compactify Einstein-Gauss-Bonnet theory to Einstein gravity in
dimension four by considering a two-dimensional internal manifold of constant
curvature. In section IV we provide a method for this compactification to take
place starting from five dimensions. Section V is destined to study the
thermodynamic features of the corresponding anti-de Sitter black brane
solutions of section III and IV. We conclude in Section VI.

##  II Theory and field equations

We start by considering the Lovelock Lagrangian of order $k$ in $D$ dimensions

$\displaystyle\mathcal{L}_{Lovelock}[g]$
$\displaystyle=\sum^{[(D-1)/2]}_{k=0}\alpha_{k}\mathcal{L}^{k},$ (1)

where the $\alpha_{k}$ are the Lovelock couplings and $\mathcal{L}^{k}$ the
Euler densities defined by

$\displaystyle\mathcal{L}^{k}=\frac{1}{2^{k}}\delta^{A_{1}\cdots
A_{2k}}_{B_{1}\cdots B_{2k}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}\cdots
R^{B_{2k-1}B_{2k}}{}{}_{A_{2k-1}A_{2k}},$ (2)

being $\delta^{A_{1}\cdots A_{2k}}_{B_{1}\cdots B_{2k}}$ the anti-symmetric
generalized Kronecker delta. Each Euler density will contribute non-trivially
to the field equations for $k\leq\left[\frac{(D-1)}{2}\right]$, otherwise they
are either topological or identically zero. Notice that we have parameterized
the sum in (1) in such a way that in even dimension the topological Euler
density is not present.
We will perform compactifications of (1) from a $D=d+p$ dimensional direct
product spacetime $\mathcal{M}_{D}$ to a $d$-dimensional spacetime
$\mathcal{M}_{d}$, being the internal manifold $\mathcal{K}^{p}$ a Euclidean
$p$-dimensional space of constant curvature. As previously explained, in order
to perform the compactification to Einstein gravity in dimension four it is
necessary to include suitable non-minimal couplings between curvature tensors
and $p$-form fields. The corresponding theory Feng:2015sbw  is constructed in
complete analogy with the Lovelock Lagrangian, and it is built in terms of a
polynomial invariant made of curvature tensors and the field strength of the
corresponding $p$-form fields. We start by introducing the bi-linear
combination

$\displaystyle Z^{A_{1}\cdots A_{p}}{}_{B_{1}\cdots B_{p}}$
$\displaystyle:=F^{A_{1}\cdots A_{p}}F_{B_{1}\cdots B_{p}},$ (3)

where $F_{[p]}=dA_{[p-1]}$, so the new interaction can be defined as

$\displaystyle\mathcal{L}_{p-forms}\left[g,A_{[p-1]}\right]=\sum^{[(D-1)/p]}_{n=1}\sum^{[(D-np)/2]}_{k=0}\frac{\beta_{k}}{2^{k}(p!)^{n}}$
$\displaystyle\delta^{A_{1}\cdots A_{2k}C_{1}^{1}\cdots C_{1}^{p}\cdots
C_{n}^{1}\cdots C_{n}^{p}}_{B_{1}\cdots B_{2k}D_{1}^{1}\cdots D_{1}^{p}\cdots
D_{n}^{1}\cdots D_{n}^{p}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}\cdots
R^{B_{2k-1}B_{2k}}{}{}_{A_{2k-1}A_{2k}}$ $\displaystyle\times
Z^{D_{1}^{1}\cdots D_{1}^{p}}{}_{C_{1}^{1}\cdots C_{1}^{p}}\cdots
Z^{D_{n}^{1}\cdots D_{n}^{p}}{}_{C_{n}^{1}\cdots C_{n}^{p}},$ (4)

being $\beta_{k}$ coupling constants. The field equations are of second order
which can be proved using the Bianchi identity satisfied by the $p$-forms

$\displaystyle\nabla_{\left[A_{1}\right.}F_{\left.B_{1}\cdots
B_{p}\right]}=\nabla^{\left[A_{1}\right.}F^{\left.B_{1}\cdots B_{p}\right]}=0$
(5)

as well as the following identity on the $Z$ tensor

$\displaystyle\nabla_{\left[D_{1}\right.}\nabla^{\left[C_{1}\right.}Z^{\left.A_{1}\cdots
A_{p}\right]}{}_{\left.B_{1}\cdots B_{p}\right]}$
$\displaystyle=\nabla_{\left[D_{1}\right.}F^{\left[A_{1}\cdots
A_{p}\right.}\nabla^{\left.C_{1}\right]}F_{\left.B_{1}\cdots
B_{p}\right]}+p(-1)^{p}F^{\left[A_{1}\cdots
A_{p}\right.}R^{E}{}_{\left[B_{1}D_{1}\right.}{}^{\left.C_{1}\right]}F_{\left.B_{2}\cdots
B_{p}\right]E}.$ (6)

Note that the order $k$ in (4) can be set to zero and only $Z$ tensors might
appear in the theory. Such a model, dubbed quasitopological electromagnetism
has been recently explored in the context of spherically symmetric black holes
Liu:2019rib ; Cisterna:2020rkc ; Cano:2020qhy . As a further guiding
principle, we require the matter field equations to be linear on the $p$-form
field, and therefore we restrict to Lagrangians that are quadratic in
$F_{[p]}$ in (4) and therefore linear in $Z$ namely, we keep as non-vanishing
only the contributions coming from the term with $n=1$ in (4).
The metric field equations of (1) supplemented with (4) consequently take the
form

$\displaystyle\sum^{[(D-1)/2]}_{k=0}\alpha_{k}E^{\left(k\right)}_{AB}-\sum^{[(D-p)/2]}_{k=0}\beta_{k}T^{\left(k,1\right)}_{AB,p}$
$\displaystyle=0,$ (7)

where $E^{\left(k\right)}_{AB}$ is the Lovelock tensor of order $k$ in the
curvature,

$\displaystyle E^{\left(k\right)}_{AB}$
$\displaystyle=-\frac{1}{2^{k+1}}g_{\left(A\right|C}\delta^{CA_{1}\cdots
A_{2k}}_{\left|B\right)B_{1}\cdots
B_{2k}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}\cdots
R^{B_{2k-1}B_{2k}}{}{}_{A_{2k-1}A_{2k}}$ (8)

and $T^{\left(k,1\right)}_{AB,p}$ is the corresponding energy-momentum tensor
associated to (4),

$\displaystyle T^{\left(k,1\right)}_{AB,p}=$
$\displaystyle\frac{1}{2^{k+1}p!}g_{AB}\delta^{A_{1}\cdots A_{2k}C_{1}\cdots
C_{p}}_{B_{1}\cdots B_{2k}D_{1}\cdots
D_{p}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}\cdots
R^{B_{2k-1}B_{2k}}{}{}_{A_{2k-1}A_{2k}}Z^{D_{1}\cdots D_{p}}{}_{C_{1}\cdots
C_{p}}$ $\displaystyle-\frac{k}{2^{k}p!}\delta^{A_{1}A_{2}\cdots
A_{2k}C_{1}\cdots C_{p}}_{B_{1}\left(A\right|\cdots B_{2k}D_{1}\cdots
D_{p}}R^{B_{1}}{}_{\left|B\right)A_{1}A_{2}}R^{B_{3}B_{4}}{}{}_{A_{3}A_{4}}\cdots
R^{B_{2k-1}B_{2k}}{}{}_{A_{2k-1}A_{2k}}Z^{D_{1}\cdots D_{p}}{}_{C_{1}\cdots
C_{p}}$ $\displaystyle+\frac{2k}{2^{k}p!}\delta^{A_{1}\cdots A_{2k}C_{1}\cdots
C_{p}}_{\left(A\right|\cdots B_{2k}D_{1}\cdots
D_{p}}g_{A_{2}\left|B\right)}R^{B_{3}B_{4}}{}{}_{A_{3}A_{4}}\cdots
R^{B_{2k-1}B_{2k}}{}{}_{A_{2k-1}A_{2k}}\nabla_{A_{1}}F^{D_{1}\cdots
D_{p}}\nabla^{B_{2}}F_{C_{1}\cdots C_{p}}$
$\displaystyle+\frac{2pk}{2^{k}p!}\delta^{A_{1}\cdots A_{2k}C_{1}\cdots
C_{p}}_{\left(A\right|\cdots B_{2k}D_{1}\cdots
D_{p}}g_{A_{2}\left|B\right)}R^{B_{3}B_{4}}{}{}_{A_{3}A_{4}}\cdots
R^{B_{2k-1}B_{2k}}{}{}_{A_{2k-1}A_{2k}}R^{D_{1}}{}_{E}{}^{B_{2}}{}_{A_{1}}Z^{ED_{2}\cdots
D_{p}}{}_{C_{1}\cdots C_{p}}$
$\displaystyle-\frac{p}{2^{k}p!}\delta^{A_{1}\cdots A_{2k}C_{1}\cdots
C_{p}}_{B_{1}\cdots B_{2k}\left(A\right|\cdots
D_{p}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}\cdots
R^{B_{2k-1}B_{2k}}{}{}_{A_{2k-1}A_{2k}}Z_{\left|B\right)}{}^{D_{2}\cdots
D_{p}}{}_{C_{1}\cdots C_{p}},$ (9)

while for the gauge field we have

$\displaystyle\sum^{[(D-p)/2]}_{k=0}\frac{\beta_{k}}{2^{k-1}}\delta^{A_{1}\cdots
A_{2k}C_{1}\cdots C_{p}}_{B_{1}\cdots B_{2k}D_{1}\cdots
D_{p}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}\cdots
R^{B_{2k-1}B_{2k}}{}{}_{A_{2k-1}A_{2k}}\nabla^{D_{1}}F_{C_{1}\cdots C_{p}}=0.$
(10)

In the next section we will address the compactification of Einstein-Gauss-
Bonnet gravity from dimension $D=d+p$ to Einstein theory in dimension four,
paying particular attention to the six dimensional case.

##  III Compactifying Einstein-Gauss-Bonnet gravity to Einstein theory in
dimension four

Let us address now the compactification of Einstein-Gauss-Bonnet theory from
dimension $d+p$ to Einstein gravity in dimension four, for generic values of
the couplings. We will provide the general analysis for an arbitrary dimension
$d$ and set $d=4$ when required. For the compactification to exists, our
interaction (4) should contain all terms up to quadratic order in the Riemann
tensor. This imposes the following action principle

$\displaystyle I\left[g,A_{\left[p-1\right]}\right]=$
$\displaystyle\int\sqrt{-g}d^{d+p}x\left(R-2\Lambda+\frac{\alpha_{2}}{4}\delta^{A_{1}\cdots
A_{4}}_{B_{1}\cdots
B_{4}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}R^{B_{3}B_{4}}{}{}_{A_{3}A_{4}}-\frac{1}{2p}Z^{C_{1}\cdots
C_{p}}{}_{C_{1}\cdots C_{p}}\right.$
$\displaystyle\left.+\frac{\beta_{1}}{2p!}\delta^{A_{1}A_{2}C_{1}\cdots
C_{p}}_{B_{1}B_{2}D_{1}\cdots
D_{p}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}Z^{D_{1}\cdots D_{p}}{}_{C_{1}\cdots
C_{p}}\right.$ $\displaystyle\left.+\frac{\beta_{2}}{4p!}\delta^{A_{1}\cdots
A_{4}C_{1}\cdots C_{p}}_{B_{1}\cdots B_{4}D_{1}\cdots
D_{p}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}R^{B_{3}B_{4}}{}{}_{A_{3}A_{4}}Z^{D_{1}\cdots
D_{p}}{}_{C_{1}\cdots C_{p}}\right).$ (11)

Notice that when $d+p=6$, this is the most general combination of the
Lovelock-like form leading to a linear matter equation. Here the couplings
$\alpha_{2}$ and $\beta_{1}$ have mass dimension $-2$, while $\beta_{2}$ has
mass dimension $-4$. Varying the functional (11) with respect to the metric
field we obtain

$\displaystyle G_{AB}+\Lambda g_{AB}+\alpha_{2}H_{AB}$
$\displaystyle=T^{\left(0,1\right)}_{AB}+\beta_{1}T^{\left(1,1\right)}_{AB}+\beta_{2}T^{\left(2,1\right)}_{AB},$
(12)

where $G_{AB}$ and $H_{AB}$ are respectively the Einstein and Gauss-Bonnet
tensors

$\displaystyle G_{AB}$ $\displaystyle=R_{AB}-\frac{1}{2}g_{AB}R,$ (13)
$\displaystyle H_{AB}$
$\displaystyle=2RR_{AB}-4R_{AC}R^{C}{}_{B}-4R^{CD}R_{ACBD}+2R_{ACDE}R_{B}{}^{CDE}-\frac{1}{2}g_{AB}\mathcal{GB},$
(14)

with $\mathcal{GB}=R^{2}-4R_{AB}R^{AB}+R_{ABCD}R^{ABCD}$ corresponding to the
Gauss-Bonnet density. In addition, the first energy-momentum tensor
$T^{\left(0,1\right)}_{AB}$ gives the dynamics of the minimally coupled
contribution

$\displaystyle T^{\left(0,1\right)}_{AB,p}=$
$\displaystyle\frac{1}{2}Z_{B}{}{}^{C_{2}\cdots C_{p}}{}_{AC_{2}\cdots
C_{p}}-\frac{1}{4p}g_{AB}Z^{C_{1}\cdots C_{p}}{}_{C_{1}\cdots C_{p}},$ (15)

while the non-minimally coupled sectors contribute as

$\displaystyle T^{\left(1,1\right)}_{AB,p}=$
$\displaystyle\frac{1}{4p!}g_{AB}\delta^{A_{1}A_{2}C_{1}\cdots
C_{p}}_{B_{1}B_{2}D_{1}\cdots
D_{p}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}Z^{D_{1}\cdots D_{p}}{}_{C_{1}\cdots
C_{p}}-\frac{p}{2p!}\delta^{A_{1}A_{2}C_{1}\cdots
C_{p}}_{B_{1}B_{2}\left(A\right|\cdots
D_{p}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}Z_{\left|B\right)}{}^{D_{2}\cdots
D_{p}}{}_{C_{1}\cdots C_{p}}$
$\displaystyle-\frac{1}{2p!}\delta^{A_{1}A_{2}C_{1}\cdots
C_{p}}_{B_{1}\left(A\right|D_{1}\cdots
D_{p}}R^{B_{1}}{}_{\left|B\right)A_{1}A_{2}}Z^{D_{1}\cdots
D_{p}}{}_{C_{1}\cdots C_{p}}+\frac{1}{p!}\delta^{A_{1}A_{2}C_{1}\cdots
C_{p}}_{\left(A\right|B_{2}D_{1}\cdots
D_{p}}g_{A_{2}\left|B\right)}\nabla_{A_{1}}F^{D_{1}\cdots
D_{p}}\nabla^{B_{2}}F_{C_{1}\cdots C_{p}}$
$\displaystyle+\frac{p}{p!}\delta^{A_{1}A_{2}C_{1}\cdots
C_{p}}_{\left(A\right|B_{2}D_{1}\cdots
D_{p}}g_{A_{2}\left|B\right)}R^{D_{1}}{}_{E}{}^{B_{2}}{}_{A_{1}}Z^{ED_{2}\cdots
D_{p}}{}_{C_{1}\cdots C_{p}}$ (16)

and

$\displaystyle T^{\left(2,1\right)}_{AB,p}=$
$\displaystyle\frac{1}{8p!}g_{AB}\delta^{A_{1}\cdots A_{4}C_{1}\cdots
C_{p}}_{B_{1}\cdots B_{4}D_{1}\cdots
D_{p}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}R^{B_{3}B_{4}}{}{}_{A_{3}A_{4}}Z^{D_{1}\cdots
D_{p}}{}_{C_{1}\cdots C_{p}}$ $\displaystyle-\frac{p}{4p!}\delta^{A_{1}\cdots
A_{4}C_{1}\cdots C_{p}}_{B_{1}\cdots B_{4}\left(A\right|\cdots
D_{p}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}R^{B_{3}B_{4}}{}{}_{A_{3}A_{4}}Z_{\left|B\right)}{}^{D_{2}\cdots
D_{p}}{}_{C_{1}\cdots C_{p}}$
$\displaystyle-\frac{1}{2p!}\delta^{A_{1}A_{2}\cdots A_{4}C_{1}\cdots
C_{p}}_{B_{1}\left(A\right|\cdots B_{4}D_{1}\cdots
D_{p}}R^{B_{1}}{}_{\left|B\right)A_{1}A_{2}}R^{B_{3}B_{4}}{}{}_{A_{3}A_{4}}Z^{D_{1}\cdots
D_{p}}{}_{C_{1}\cdots C_{p}}$ $\displaystyle+\frac{1}{p!}\delta^{A_{1}\cdots
A_{4}C_{1}\cdots C_{p}}_{\left(A\right|\cdots B_{4}D_{1}\cdots
D_{p}}g_{A_{2}\left|B\right)}R^{B_{3}B_{4}}{}{}_{A_{3}A_{4}}\nabla_{A_{1}}F^{D_{1}\cdots
D_{p}}\nabla^{B_{2}}F_{C_{1}\cdots C_{p}}$
$\displaystyle+\frac{p}{p!}\delta^{A_{1}\cdots A_{4}C_{1}\cdots
C_{p}}_{\left(A\right|\cdots B_{4}D_{1}\cdots
D_{p}}g_{A_{2}\left|B\right)}R^{D_{1}}{}_{E}{}^{B_{2}}{}_{A_{1}}R^{B_{3}B_{4}}{}{}_{A_{3}A_{4}}Z^{ED_{2}\cdots
D_{p}}{}_{C_{1}\cdots C_{p}}.$ (17)

By virtue of the identities (5) and (6) we clearly see that the energy-
momentum tensors (16)-(17) are casted in a manifestly second-order fashion. On
the other hand, variations with respect to the gauge field deliver the
following second order Maxwell like equation

$\displaystyle(p-1)!\nabla^{D_{1}}F_{D_{1}\cdots
D_{p}}-\beta_{1}\delta^{A_{1}A_{2}C_{1}\cdots C_{p}}_{B_{1}B_{2}D_{1}\cdots
D_{p}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}\nabla^{D_{1}}F_{C_{1}\cdots C_{p}}$
$\displaystyle-\frac{\beta_{2}}{2}\delta^{A_{1}\cdots A_{4}C_{1}\cdots
C_{p}}_{B_{1}\cdots B_{4}D_{1}\cdots
D_{p}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}R^{B_{3}B_{4}}{}{}_{A_{3}A_{4}}\nabla^{D_{1}}F_{C_{1}\cdots
C_{p}}=0.$ (18)

In order to proceed with the compactification on
$\mathcal{M}_{D}=\mathcal{M}_{d}\times\mathcal{K}^{p}$, we consider the
following direct product metric

$\displaystyle ds^{2}$
$\displaystyle=g_{AB}dx^{A}dx^{B}=\tilde{g}_{\mu\nu}\left(y\right)dy^{\mu}dy^{\nu}+\hat{g}_{ij}\left(z\right)dz^{i}dz^{j}.$
(19)

Here $\tilde{g}_{\mu\nu}dy^{\mu}dy^{\nu}$ stands for the $d$-dimensional
spacetime manifold $\mathcal{M}_{d}$ while
$\hat{g}_{ij}\left(z\right)dz^{i}dz^{j}$ represents a $p$-dimensional
Euclidean manifold $\mathcal{K}^{p}$

$\displaystyle\hat{g}_{ij}\left(z\right)dz^{i}dz^{j}$
$\displaystyle=\frac{d\vec{z}\cdot
d\vec{z}}{\left(1+\frac{\gamma}{4}\sum^{p}_{j=1}z^{2}_{j}\right)^{2}}$ (20)

of constant curvature, i.e

$\hat{R}_{ijkl}=\gamma(\hat{g}_{ik}\hat{g}_{jl}-\hat{g}_{il}\hat{g}_{jk})$
(21)

with $\gamma$ defining the corresponding curvature radius
$R_{0}=|\gamma|^{-1}$. From now on quantities with a tilde are intrinsically
defined on the $d$-dimensional spacetime (the brane), while quantities with a
hat refer to the internal manifold of dimension $p$. In accordance with our
dressing approach, the $A_{p-1}$ gauge field lives on the internal manifold
and in consequence we take it to be proportional to the volume form of
$\mathcal{K}^{p}$

$\displaystyle\hat{F}_{i_{1}\cdots i_{p}}$
$\displaystyle=\frac{q_{m}}{\left(1+\frac{\gamma}{4}\sum^{p}_{j=1}z^{2}_{j}\right)^{p}}\hat{\epsilon}_{i_{1}\cdots
i_{p}},$ (22)

where $q_{m}$ plays the role of a generalized magnetic charge and
$\hat{\epsilon}_{i_{1}\cdots i_{p}}=\delta^{z_{1}\ldots z_{p}}_{i_{1}\ldots
i_{p}}$ is the antisymmetrized Kronecker delta. This choice immediately
provides a solution of the gauge field equation (18).
We are left then with the Einstein equations. As the spacetime is a direct
product of two manifolds, the metric field equations split into the field
equations on the brane and on the internal manifold, yielding

$\displaystyle\left(\alpha_{2}+\beta_{2}q^{2}_{m}p!\right)\tilde{H}_{\mu\nu}+\left[1+2\alpha_{2}\gamma
p\left(p-1\right)+\beta_{1}q^{2}_{m}p!\right]\tilde{G}_{\mu\nu}$
$\displaystyle+\left[-\frac{\gamma}{2}p\left(p-1\right)+\Lambda-\frac{\alpha_{2}}{2}\gamma^{2}p\left(p-1\right)\left(p-2\right)\left(p-3\right)+\frac{q^{2}_{m}}{4}\left(p-1\right)!\right]\tilde{g}_{\mu\nu}=0,$
(23)

and

$\displaystyle\left(-\frac{\alpha_{2}}{2}\hat{g}_{ij}+\frac{\beta_{2}}{2}q^{2}_{m}p!\hat{g}_{ij}\right)\tilde{\mathcal{GB}}_{d}+\left[-\frac{1}{2}\hat{g}_{ij}-\alpha_{2}\gamma\left(p-1\right)\left(p-2\right)\hat{g}_{ij}+\frac{\beta_{1}}{2}q^{2}_{m}p!\hat{g}_{ij}\right]\tilde{R}_{d}$
$\displaystyle+\left[-\frac{\gamma}{2}\left(p-1\right)\left(p-2\right)\hat{g}_{ij}+\Lambda\hat{g}_{ij}-\frac{\alpha_{2}}{2}\gamma^{2}\left(p-1\right)\left(p-2\right)\left(p-3\right)\left(p-4\right)\hat{g}_{ij}-\frac{q^{2}_{m}}{4}\left(p-1\right)!\hat{g}_{ij}\right]=0.$
(24)

When compactifying, most incompatibilities emerge from these field equations.
In consequence to further analyze a possible incompatibility we trace both
equations, obtaining

$\displaystyle\left(\alpha_{2}+\beta_{2}q^{2}_{m}p!\right)\left(4-d\right)\tilde{\mathcal{GB}}_{d}+\left[1+2\alpha_{2}\gamma
p\left(p-1\right)+\beta_{1}q^{2}_{m}p!\right]\left(2-d\right)\tilde{R}_{d}+\left[-\gamma
dp\left(p-1\right)+2d\Lambda\right.$
$\displaystyle\left.-\alpha_{2}\gamma^{2}dp\left(p-1\right)\left(p-2\right)\left(p-3\right)+\frac{q^{2}_{m}}{2}d\left(p-1\right)!\right]=0,$
(25)

and

$\displaystyle\left(-\alpha_{2}p+\beta_{2}q^{2}_{m}pp!\right)\tilde{\mathcal{GB}}_{d}+\left[-p-2\alpha_{2}\gamma
p\left(p-1\right)\left(p-2\right)+\beta_{1}q^{2}_{m}pp!\right]\tilde{R}_{d}+\left[-\gamma
p\left(p-1\right)\left(p-2\right)+2p\Lambda\right.$
$\displaystyle\left.-\alpha_{2}\gamma^{2}p\left(p-1\right)\left(p-2\right)\left(p-3\right)\left(p-4\right)-\frac{q^{2}_{m}}{2}p\left(p-1\right)!\right]=0,$
(26)

To demonstrate the net effect of the inclusion of our interaction (4) we
explicitly compactify from six dimensions to Einstein gravity in dimension
four, i.e $d=4$ and $p=2$. Both traces then become

$\displaystyle\left(2\beta_{1}q^{2}_{m}+4\gamma\alpha_{2}+1\right)\tilde{R}_{4}+\left(4\gamma-4\Lambda-q^{2}_{m}\right)$
$\displaystyle=0,$ (27)
$\displaystyle\left(4\beta_{2}q^{2}_{m}-2\alpha_{2}\right)\tilde{\mathcal{GB}}_{4}+\left(4\beta_{1}q^{2}_{m}-2\right)\tilde{R}_{4}+\left(4\Lambda-q^{2}_{m}\right)$
$\displaystyle=0,$ (28)

whose compatibility is ensured by the relations

$\displaystyle
q^{2}_{m}=\frac{\alpha_{2}}{2\beta_{2}},\quad\gamma=-\frac{1}{4}\frac{24\Lambda\alpha_{2}\beta_{1}\beta_{2}-8\Lambda\beta_{2}^{2}+\alpha_{2}^{2}\beta_{1}-3\alpha_{2}\beta_{2}}{\beta_{2}\left(8\Lambda\alpha_{2}\beta_{2}-\alpha_{2}^{2}-4\alpha_{2}\beta_{1}+4\beta_{2}\right)},$
(29)

that fix the magnetic charge, which is an integration constant, in terms of
the couplings $\alpha_{2}$ and $\beta_{2}$ and fix as well the
compactification radius $\gamma$. A few comments are in order. First, the
presence of $\beta_{2}$, that controls a $k=2$ order term in (4) is mandatory
otherwise, although $\mathcal{GB}_{4}$ is topological in $d=4$ it presence at
the level of the field equations renders impossible the compactification,
indeed it constraints the system in such a way that no general relativity
solutions are allowed (see also Kastor:2006vw  and Giribet:2006ec ). Second,
we observe that $\alpha_{2}$ and $\beta_{2}$ are related through the magnetic
charge, an integration constant that can take any value, and in consequence
there is no tuning among the couplings of the theory. This is in stark
contrast to what happens in Canfora:2008iu , where for the compactification to
exists a cubic Lovelock interaction must be included whose coupling
$\alpha_{3}$ is directly fixed in terms of $\alpha_{2}$ and $\Lambda$. Third,
this compactification can be performed from dimension six due to the fact that
all terms in (4) are not topological and in consequence contribute non-
trivially in the critical dimension, opposite to the cubic Lovelock
interaction which is non-topological starting from dimension seven. Finally,
it is interesting to notice that in a scenario in which the higher curvature
terms arise as higher curvature corrections proper of an effective field
theory, $\alpha_{2}\sim M^{-2}\sim\beta_{1}$ while $\beta_{2}\sim M^{-4}$,
where $M$ is the energy scale that defines the effective approach, namely
physical quantities must be expanded in the limit $M$ large. From the second
relation in (29), one can show that the compactification radius $R_{0}^{2}\sim
M^{-2}$, and therefore the perturbative scheme is indeed consistent with
having a perturbatively small, compact, extra dimension.
Therefore, after fixing the radius of compactification as well as the value of
the magnetic charge as in (29), one consistently obtains an Einstein equation
induced on the brane, which can be read from (23) leading to

$\frac{1}{16\pi
G_{\text{eff}}}\left(G_{\mu\nu}+\Lambda_{\text{eff}}g_{\mu\nu}\right)=0\ ,$
(30)

where we have defined the effective Newton and cosmological constants
respectively as

$\displaystyle G_{\text{eff}}$
$\displaystyle=\frac{1}{32\pi}\frac{\beta_{2}(8\Lambda\alpha_{2}\beta_{2}-\alpha_{2}^{2}-4\alpha_{2}\beta_{1}+4\beta_{2})}{(\beta_{2}-\alpha_{2}\beta_{1})(\alpha_{2}^{2}+2\alpha_{2}\beta_{1}+8\Lambda\alpha_{2}\beta_{2}+2\beta_{2})}\
,\
\Lambda_{\text{eff}}=\frac{(8\Lambda\beta_{2}-\alpha_{2})}{16(\beta_{2}-\alpha_{2}\beta_{1})}\
,$ (31)

The field equations are now solvable, for example, by the following
$6$-dimensional black 2-brane (static spherically symmetric case)

$ds^{2}=-\left(\frac{r^{2}}{l^{2}_{\text{eff}}}-\frac{\mu}{r}+K\right)dt^{2}+\frac{dr^{2}}{\left(\frac{r^{2}}{l^{2}_{\text{eff}}}-\frac{\mu}{r}+K\right)}+r^{2}d\Sigma^{2}_{K}+\frac{dz_{1}^{2}+dz_{2}^{2}}{[1+\frac{\gamma}{4}(z_{1}^{2}+z_{2}^{2})]^{2}}$
(32)

with $K$ the curvature of the transverse manifold $d\Sigma_{K}^{2}$, $\mu$ an
integration constant related with ADM mass and
$l_{\text{eff}}^{2}=-3/\Lambda_{\text{eff}}$ an effective (A)dS radius

$l_{\text{eff}}^{-2}=\frac{1}{48}\left(\frac{8\Lambda\beta_{2}-\alpha_{2}}{\alpha_{2}\beta_{1}-\beta_{2}}\right).$
(33)

The four dimensional brane is given then by a Schwarzschild (A)dS black hole
with an effective cosmological constant, i.e an Einstein solution. From the
latter equation, one can see that for perturbative, higher curvature
couplings, the effective four-dimensional cosmological constant turns out to
be large, which in the case of de Sitter solutions is in tension with a direct
application of this scenario to late time cosmology.

##  IV The scalar case: One dimensional internal manifold

In order to compactify Einstein-Gauss-Bonnet gravity starting from dimension
five it is necessary to consider a one dimensional internal manifold, and in
consequence the dressing must be given by $0$-forms, i.e scalar fields. This
kind of dressing, has been successfully applied in order to construct
homogenous anti-de Sitter black string/p-branes in general relativity and
Lovelock theories Cisterna:2017qrb ; Cisterna:2018jsx ; Cisterna:2018mww ;
Arratia:2020hoy ; Cisterna:2019scr . Analogously to the case we have
previously faced, in which the fundamental forms were proportional to the
volume form of the internal manifold, the scalars are required to be linear in
the internal manifold coordinate, providing in this way an immediate solution
to the corresponding Klein-Gordon equation. Our interaction (4) is modified in
such a manner that now the curvature tensors are coupled with a specific
combination of first derivatives of the scalars. We then consider

$\mathcal{L}_{scalar}[g,\phi]=-\frac{1}{2^{2k+1}}\delta^{CA1...A_{2k}}_{DB1...B_{2k}}R^{B1B2}{}{}_{A1A2}...R^{B_{2k-1}B_{2k}}{}{}_{A_{2k-1}A_{2k}}\nabla_{C}\phi^{i}\nabla^{D}\phi^{i}$
(34)

which is equivalent to a generalized non-minimal kinetic coupling controlled
by the Lovelock tensor of order $k$

$\mathcal{L}_{scalar}[g,\phi^{i}]=-\frac{1}{2^{2k+1}}E^{CD}_{(k)}\nabla_{C}\phi^{i}\nabla_{D}\phi^{i}$
(35)

In complete analogy with the previous section we truncate our theory up to
order $k=2$, then our action principle is written as

$\displaystyle I\left[g,\phi,\psi\right]=$
$\displaystyle\int\sqrt{-g}d^{5}x\left[R-2\Lambda+\frac{\alpha_{2}}{4}\delta^{A_{1}\cdots
A_{4}}_{B_{1}\cdots
B_{4}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}R^{B_{3}B_{4}}{}{}_{A_{3}A_{4}}-\frac{1}{2}g^{AB}\nabla_{A}\phi\nabla_{B}\phi\right.$
$\displaystyle\left.+\frac{\beta_{1}}{8}G^{AB}\nabla_{A}\phi\nabla_{B}\phi+\frac{\gamma_{1}}{64}H^{AB}\nabla_{A}\phi\nabla_{B}\phi-\frac{1}{2}g^{AB}\nabla_{A}\psi\nabla_{B}\psi+\frac{\beta_{2}}{8}G^{AB}\nabla_{A}\psi\nabla_{B}\psi\right.$
$\displaystyle\left.+\frac{\gamma_{2}}{64}H^{AB}\nabla_{A}\psi\nabla_{B}\psi\right]$
(36)

where, in order to avoid any fine tuning between the couplings, one has to
introduce at least two different scalar fields (as the computation here below
shows). This is not necessary in the previous six dimensional case because the
curvature of the internal manifold provides an extra scale that prevents the
appearance of such relations.
The resulting theory belongs to a sector of Horndeski gravity, the most
general scalar-tensor theory with second order field equations
Horndeski:1974wa . The kinetic coupling to the Einstein tensor corresponds to
a quadratic sector of Horndeski gravity in dimension four while the coupling
with the Gauss-Bonnet tensor appears naturally in dimension greater or equal
than five. Taking the variation with respect to the metric, we obtain

$\displaystyle G_{AB}+\Lambda g_{AB}+\alpha_{2}H_{AB}$
$\displaystyle=T^{\left(0\right)}_{\phi_{i}AB}+\beta_{i}T^{\left(1\right)}_{\phi_{i}AB}+\gamma_{i}T^{\left(2\right)}_{\phi_{i}AB},$
(37)

where

$\displaystyle T^{(0)}_{\phi_{i}AB}$
$\displaystyle=\frac{1}{2}\left(\partial_{A}\phi_{i}\partial_{B}\phi_{i}-\frac{1}{2}g_{AB}\partial_{C}\phi_{i}\partial^{C}\phi_{i}\right),$
(38) $\displaystyle T^{\left(1\right)}_{\phi_{i}AB}$
$\displaystyle=\frac{1}{2}\left(\frac{1}{2}\partial_{A}\phi_{i}\partial_{B}\phi_{i}R-2\partial_{C}\partial_{(A}\phi_{i}R_{B)}^{\,\,C}-\partial_{C}\phi_{i}\partial_{D}\phi_{i}R_{A\,\,\,B}^{\,\,\,C\,\,\,D}-\nabla_{A}\nabla^{C}\phi_{i}\nabla_{B}\nabla_{C}\phi_{i}+\nabla_{A}\nabla_{B}\phi_{i}\square\phi_{i}\right.$
$\displaystyle\left.+\frac{1}{2}G_{AB}\left(\partial\phi_{i}\right)^{2}-g_{AB}\left[-\frac{1}{2}\nabla^{C}\nabla^{D}\phi_{i}\nabla_{C}\nabla_{D}\phi_{i}+\frac{1}{2}\left(\square\phi_{i}\right)^{2}-\partial_{C}\phi_{i}\partial_{D}\phi_{i}R^{CD}\right]\right)\
,$ (39) $\displaystyle T^{(2)}_{\phi_{i}AB}=$
$\displaystyle-\frac{1}{64}g_{AB}\nabla_{C}\phi_{i}\nabla^{D}\phi_{i}\delta^{CA_{1}\cdots
A_{4}}_{DB_{1}\cdots
B_{4}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}R^{B_{3}B_{4}}{}{}_{A_{3}A_{4}}+\frac{1}{32}\nabla_{C}\phi_{i}\nabla_{\left(A\right|}\phi_{i}\delta^{CA_{1}\cdots
A_{4}}_{\left|B\right)B_{1}\cdots
B_{4}}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}R^{B_{3}B_{4}}{}{}_{A_{3}A_{4}}$
$\displaystyle+\frac{1}{16}\nabla_{C}\phi_{i}\nabla^{D}\phi_{i}\delta^{CA_{1}\cdots
A_{4}}_{DB_{1}\cdots\left(A\right|}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}R^{B_{3}}{}_{\left|B\right)A_{3}A_{4}}+\frac{1}{8}\nabla^{B_{3}}\nabla_{C}\phi_{i}\nabla_{A_{3}}\nabla^{D}\phi_{i}\delta^{CA_{1}\cdots
A_{4}}_{DB_{1}\cdots\left(B\right|}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}g_{\left|A\right)A_{4}}$
$\displaystyle+\frac{1}{16}\nabla_{C}\phi_{i}\nabla^{E}\phi_{i}\delta^{CA_{1}\cdots
A_{4}}_{DB_{1}\cdots\left(B\right|}R^{B_{1}B_{2}}{}{}_{A_{1}A_{2}}R_{A_{3}E}{}{}^{B_{3}D}g_{\left|A\right)A_{4}}\
.$ (40)

where $\phi_{i}$ stands for the scalar fields and $i=1,2$. On the other hand,
the scalar fields fulfil

$\displaystyle\left(g^{AB}-\frac{\beta_{i}}{4}G^{AB}-\frac{\gamma_{i}}{32}H^{AB}\right)\nabla_{A}\nabla_{B}\phi^{i}$
$\displaystyle=0.$ (41)

To start with, we take a five dimensional product space
$\mathcal{M}_{D}=\mathcal{M}_{4}\times\mathbb{R}$ whose metric along the lines
of (19) can be written as

$\displaystyle ds^{2}=d\tilde{s}_{4}^{2}+dz^{2}\ .$ (42)

As we said, each scalar field is linear in the internal manifold coordinate

$\phi(z)=\lambda_{0}z,\hskip 14.22636pt\psi(z)=\lambda_{1}z,$ (43)

being $\lambda_{0}$ and $\lambda_{1}$ integration constants, the scalar
charges, and they immediately solve each corresponding Klein-Gordon equation.
Computing the field equations on the brane and on the extra coordinate and
tracing them, we get

$\displaystyle\left(4\Lambda+\lambda_{0}^{2}+\lambda_{1}^{2}\right)-\left(1-\frac{\beta_{1}}{4}\lambda_{0}^{2}-\frac{\beta_{2}}{4}\lambda_{1}^{2}\right)\tilde{R}$
$\displaystyle=0$ (44)

and

$\displaystyle\left(-2\Lambda+\frac{\lambda_{0}^{2}}{2}+\frac{\lambda_{1}^{2}}{2}\right)+\left(1+\frac{\beta_{1}}{4}\lambda^{2}_{0}+\frac{\beta_{2}}{4}\lambda^{2}_{1}\right)\tilde{R}+\left(\alpha_{2}-\frac{\gamma_{1}}{8}\lambda^{2}_{0}-\frac{\gamma_{2}}{8}\lambda^{2}_{1}\right)\tilde{\mathcal{GB}}=0.$
(45)

Compatibility will then require

$\lambda_{0}^{2}=\frac{8\alpha_{2}-\gamma_{2}\lambda_{1}^{2}}{\gamma_{1}},\hskip
14.22636pt\Lambda=-\frac{1}{4}\frac{\left(\gamma_{1}\lambda_{1}^{2}-\gamma_{2}\lambda_{1}^{2}+8\alpha_{2}\right)\left(-\beta_{1}\gamma_{2}\lambda_{1}^{2}+\beta_{2}\gamma_{1}\lambda_{1}^{2}+8\alpha_{2}\beta_{1}+12\gamma_{1}\right)}{\gamma_{1}\left(-3\beta_{1}\gamma_{2}\lambda_{1}^{2}+3\beta_{2}\gamma_{1}\lambda_{1}^{2}+24\alpha_{2}\beta_{1}+4\gamma_{1}\right)}.$
(46)

For an spherically symmetric ansatz on the brane, solving Einstein equations
delivers

$ds^{2}=-\left(\frac{r^{2}}{l^{2}_{\text{eff}}}-\frac{\mu}{r}+K\right)dt^{2}+\frac{dr^{2}}{\left(\frac{r^{2}}{l^{2}_{\text{eff}}}-\frac{\mu}{r}+K\right)}+r^{2}d\Sigma_{K}^{2}+dz^{2}$
(47)

where the effective cosmological constant, by means of (46), is given by

$l^{-2}_{\text{eff}}=\frac{1}{6}\frac{\left(\gamma_{1}-\gamma_{2}\right)\lambda_{1}^{2}+4\Lambda\gamma_{1}+8\alpha_{2}}{\left(-\beta_{1}\gamma_{2}+\beta_{2}\gamma_{1}\right)\lambda_{1}^{2}+8\alpha_{2}\beta_{1}-2\gamma_{1}}\
.$ (48)

It is interesting to note that, from here, it is possible to obtain a
Schwarzschild black string, i.e a five dimensional black string solution to
(37) in which the four dimensional brane is given by a Schwarzschild black
hole. This might be done by considering the case in which
$\Lambda=\lambda_{0}=0$. For such a case, even the $\beta_{1}$ interaction
maybe neglected and the result is nothing else than the standard black string

$ds^{2}=-\left(K-\frac{\mu}{r}\right)dt^{2}+\frac{dr^{2}}{\left(K-\frac{\mu}{r}\right)}+r^{2}d\Sigma_{K}^{2}+dz^{2}$
(49)

Here we observe the evident relation between black strings and
compactifications on direct product spaces. The original five dimensional
Schwarzschild black string exists because general relativity admits
cylindrical compactifications of the form on
$\mathcal{M}_{D}=\mathcal{M}_{4}\times\mathbb{R}(\mathcal{S}^{1}$). This
solution is particularly interesting because in five dimensions these
configurations suffers from a long wavelength instability, the Gregory-
Laflamme instability Gregory:1993vy , revealing the weakness of the cosmic
censorship in higher dimensions Lehner:2010pn . One might conjecture that this
is related with the fact that the Schwarzschild black hole is a solution of
general relativity, and that when uplifting the solution to a five dimensional
black string we are still under the domain of the same theory, namely, solving
Einstein field equations in dimension five. However, in five dimensions
Einstein theory is not the most general gravitational theory we have at hand,
in fact, Einstein-Gauss-Bonnet gravity plays that role. By means of our
compactification procedure, we have been able to construct a black string in
five dimensional Einstein-Gauss-Bonnet gravity (47,49) which precisely
represent an Einstein solution on the four dimensional brane. In consequence
it would be appealing to study its mechanical stability at least in the linear
regime.

##  V Thermodynamic quantities

We address now the thermodynamic analysis of the $6$-dimensional black
$2$-brane solution (32)111The $5$-dimensional black string (47) follows the
same procedure.. The black $2$-brane temperature is obtained, as usual, by
requiring a smooth Euclidean continuation of the solution, then

$\displaystyle T$ $\displaystyle=\frac{r_{+}}{2\pi
l_{\text{eff}}^{2}}+\frac{\mu(r_{+})}{4\pi r_{+}^{2}}\ ,$ (50)

where $r=r_{+}$ denotes the location of the black hole horizon and the
effective AdS length is given in (33). On the other hand, by means of Wald’s
formula Wald:1993nt  we obtain that the entropy density is

$s=\frac{S}{\mathrm{Vol}\left[\mathcal{K}^{2}\right]}=\frac{\sigma
r_{+}^{2}}{4G_{\text{eff}}}+32\pi\sigma K\alpha_{2}\ ,$ (51)

with $G_{\text{eff}}$ defined in (31). Here $\mathrm{Vol}[\mathcal{K}^{2}]$
represents the volume of the $2$-dimensional internal manifold
$\mathcal{K}^{2}$, while $\sigma$ corresponds to the volume of the transverse
manifold of the four dimensional black hole on the brane. It is interesting to
notice that there are two types of contributions to the black hole entropy.
Since the effective theory that dictates the dynamics of the four dimensional
manifold is GR, the entropy of the black hole has a contribution that goes as
$r_{+}^{2}$, namely as the area of the event horizon. This contribution
acquires corrections from all the higher curvature matter couplings, and can
be written as $A/(4G_{\text{eff}})$. It is also worth pointing out that there
is an extra, universal constant contribution to the entropy, which comes from
the presence of the higher dimensional Gauss-Bonnet term (see also Cai:2001dz
).
Using the first law of black hole thermodynamic, $dm=Tds$, we obtain that the
constant $\mu$ is identified to the mass of the solution by

$m=\frac{\sigma
r_{+}(r_{+}^{2}+Kl_{\text{eff}}^{2})}{8G_{\text{eff}}l_{\text{eff}}^{2}\pi}=\frac{\mu\sigma}{8\pi
G_{\text{eff}}}\ ,$ (52)

where $r_{+}=r_{+}(\mu)$. We observe that there is no gap, i.e the mass
density goes to zero in the limit $r_{+}\rightarrow 0$. As expected, the
Kretschmann invariant for (32) as $r\rightarrow 0$ behaves as

$\displaystyle R_{ABCD}R^{ABCD}=\mathcal{O}(r^{-6}),$ (53)

revealing the existence of a curvature singularity located at the origin.It is
very interesting to see the effect of the constant term in the entropy,
induced by the higher curvature terms in higher dimensions, which lead to a
consistent dimensional reduction thanks to the non-minimally coupled Maxwell
field. Considering the temperature, entropy and mass, respectively given in
(50), (51) and (52), one finds the following expression for the Helmholtz free
energy

$\frac{F\left(r_{+}\right)}{\mathrm{Vol}\left[\mathcal{K}^{2}\right]}=\frac{r_{+}}{4G_{\text{eff}}}\left(1-\frac{r_{+}^{2}}{l_{\text{eff}}^{2}}\right)-32\pi\alpha_{2}\left(\frac{3r_{+}}{l_{\text{eff}}^{2}}+\frac{1}{r_{+}}\right)\
,$ (54)

where we have focused on the spherically symmetric case, namely, we have fixed
$\sigma=4\pi$ and $K=1$. Since the temperature is not modified with respect to
that of Schwarzschild-AdS, it takes a minimum value, at which the free energy
reduces to

$\frac{F\left(T_{\text{min}}\right)}{\mathrm{Vol}\left[\mathcal{K}^{2}\right]}=2\sqrt{3}l_{\text{eff}}\left(\frac{1}{36G_{\text{eff}}}-\frac{32\pi\alpha_{2}}{l_{\text{eff}}^{2}}\right)\
.$ (55)

Notice that $F(T_{\text{min}})$ can have either signs depending on the precise
values of the Gauss-Bonnet coupling, and the presence of the effective
cosmological and Newton constant, allows to modify the sign even when the
higher curvature terms are of a perturbative nature. It is also interesting to
notice that for arbitrarily small black holes the free energy goes to minus
infinity due to the presence of the Gauss-Bonnet coupling $\alpha_{2}$, in
stark contrast to the behavior of small black holes in GR whose free energy
asymptotically vanish. Nevertheless the presence of the additive term in the
entropy, sets a lower bound on the mass of the black holes and consequently a
lower bound on their radii. This standard bound comes from considering an
idealized, quasi-static process of black hole fusion where the initial black
holes have masses $M_{1}$ and $M_{2}$ and, disregarding the gravitational
radiation, the variation of the entropy reads

$\Delta S=8\pi G_{\text{eff}}M_{1}M_{2}-128\pi^{2}\alpha_{2}-\frac{64\pi
G_{\text{eff}}^{3}}{l_{\text{eff}}^{2}}M_{1}M_{2}(2M_{1}^{2}+3M_{1}M_{2}+2M_{2}^{2})+\mathcal{O}(l_{\text{eff}}^{-4})\
,$ (56)

where $\Delta S=S_{\text{final}}-S_{\text{initial}}$. Since the expression for
finite $l_{\text{eff}}$ is not very illuminating, we have taken the expansion
of for large $l_{\text{eff}}$. In the asymptotically flat case, the positivity
of $\Delta S$ clearly imposes a lower bound on the black hole masses.
Our discussion is based on the original interpretation of these type of terms
given in Jacobson:1993xs . According to this interpretation, the extra term in
the entropy for two far apart black holes must be added, leading to a
potential violation of the law of increasing entropy in a proccess of quasi-
static black hole coalesence for $\alpha_{2}>0$, unless the mass is bounded
from below. If $\alpha_{2}<0$ no such bound appears using this argument. It is
important to remark that when the GB term emerges as a higher curvature
correction in string theory, $\alpha_{2}$ is positive. Nevertheless, there are
two extra arguments supporting the Jacobson-Myers interpretation. Firstly,
this constant term (despite its temperature independence) is actually non-
vanishing only in the presence of the black hole horizon, since otherwise the
domain of integration is void. Secondly, there are many physical systems which
possess non-zero extensive entropy at zero temperature, as for instance the
antiferromagnetic Potts model with not too large q (q=2 for the Ising model).

Figure 1 depicts the free energy curves for different values of the
parameters. Since no other integration constant, beyond the mass is present,
the appropriate ensemble is the canonical ensemble and therefore the
statistically most favoured configuration is the one that minimizes the
Helmholtz free energy for a given fixed temperature. It is also important to
notice that the vacuum configuration, namely the solution with vanishing
$F(T)$ is equal to thermal AdS${}_{4}\times S^{2}$ with non-vanishing flux of
the Maxwell field on the two-dimensional, compactified internal sphere.
Remarkably, when $F(T_{\text{min}})$ is negative, there is no Hawking-Page
transition Hawking:1982dh  and the large black holes always dominate the
ensemble.

Figure 1: Free energy versus temperature for Schwarzschild-AdS in General
Relativity in four dimensions (left panel) and for the compactified black hole
solution Schwarzschild-AdS${}_{4}\times S^{2}$ (right panel). The compactified
solution corresponds to the values $\alpha_{2}=0.1,\ \beta_{1}=0.2,\
\beta_{2}=0.05$ and $\Lambda=-1$.

##  VI Further comments

We have been able to perform a dimensional reduction of EGB theory to
dimension four, giving raise to General Relativity, for arbitrary values of
the couplings. The consistency of the compactification relies on the presence
of non-minimally coupled, magnetically charged, $p$-forms. As a guiding
principle for the introduction of this matter field, we have restricted to the
family of couplings introduced in Feng:2015sbw , which mimic the structure of
Lovelock theories, leading to second order field equations. The Gauss-Bonnet
term, as well as other higher curvature couplings with matter fields naturally
emerge as perturbative correction of fundamental theories in effective field
theory scenarios (see e.g. Burgess:2007pt ). Whether or not the precise
couplings considered here emerge from a low energy limit of a fundamental
theory after considering the freedom of field redefinitions, is beyond our
present objectives, nevertheless we have shown that in such potential
scenario, the radius of compactification of the two-dimensional compact
manifold considered in Section III, is indeed proportional to the inverse of
the energy scale $M$, and therefore can be consistently considered as small.
The structure of the theories considered suggest that our results can be
extended to Lovelock theories beyond the EGB Lagrangian. One might even been
able to consider a given, arbitrary Lovelock theory in dimension $D$ and
obtain a consistent arbitrary Lovelock theory in dimension $d=D-p$ by
considering magnetically charged, non-minimally coupled $p$-forms in the
family (4). As mentioned above, in vacuum, such compactifications usually
require to introduce relations between the couplings, which are not compatible
with the interpretation of the higher curvature terms as perturbative
corrections (for solutions of Lovelock theories in $N+1$-dimensions see
Kastor:2017knv ). As one increases the dimension of the compact manifold,
releasing its geometry, it is natural to expect the presence of some higher
curvature constraints on its curvature. For the case of topological black
holes, it was shown in Dotti:2005rc  that a new constant characterising the
square of the Weyl tensor of the horizon appears on the lapse function and the
effects of such constant in the thermodynamics has recently been explored in
Hull:2021bry . A simple family of manifolds that would allow to go beyond
constant curvature internal spaces is the products of spheres. These manifolds
will allow to introduce further parameters in the compactifications which may
help when contrasting the four-dimensional obtained theory with experimental
evidence.
Finally, as explained in Section V, the consistent compactification we have
found, may lead to black holes in AdS${}_{4}\times S^{2}$ which are always
thermodynamically favoured with respect to the solitonic thermal background.
In addition, since the Hawking-Page phase transition plays a very important
role in holography Witten:1998zw , it would be appealing to explore some
holographic properties of the present compactification scenario.

##  VII Acknowlegdments

This work is partially funded by FONDECYT grants 1200022, 1210500 and 1181047.
J.O. also thanks the support of Proyecto de Cooperación Internacional
2019/13231-7 FAPESP/ANID. C.H acknowledges support from the National Agency
for Research and Development (ANID) /Scholarship Program/BECA DE DOCTORADO
NACIONAL/2017 - 21171394. S.F. thanks the support of Beca Doctorado USM. The
Centro de Estudios Científicos (CECs) is funded by the Chilean Government
through the Centers of Excellence Base Financing Program of ANID.

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