#  Regular $(2+1)$-dimensional spatially homogeneous
$\alpha^{\prime}$-corrected black holes in string theory

F. Naderi 111e-mail<EMAIL_ADDRESS>, A. Rezaei-Aghdam
222Corresponding author. e-mail<EMAIL_ADDRESS>
Department of Physics, Faculty of Basic Sciences, Azarbaijan Shahid Madani
University
53714-161, Tabriz, Iran

###### Abstract

We present the solution of string effective action on spatially homogeneous
$(2+1)$-dimensional spacetime with the source given by dilaton and
antisymmetric $B$-field in the presence of central charge deficit term
$\Lambda$. We consider the two-dimensional space part to be Weyl-related to a
surface of arbitrary negative constant Gaussian curvature with symmetries of
two-dimensional Lie algebra. It is shown that the geometry is a Lobachevsky-
type geometry that can be reduced to hyperbolic type if the negative Gaussian
curvature of the surface equals $-\frac{1}{2}$. At leading order string
effective action, we obtained a black hole solution whose line element is
Weyl-related to this homogeneous spacetime with arbitrary negative Gaussian
curvature. The solution can be reduced to the BTZ black hole by choosing a
special set of parameters. Also, it is shown that the central charge of the
CFT coincides with that of the AdS spacetime related to the Chern-Simons model
with gauge group $SO(2,2)$. The solutions appear to be in the high curvature
limit $R\alpha^{\prime}\gtrsim 1$, with emphasis on including the higher order
$\alpha^{\prime}$ corrections. Considering the two-loop (first order
$\alpha^{\prime}$) $\beta$-function equations of $\sigma$-model, we present
the $\alpha^{\prime}$-corrected black hole solutions.

##  1 Introduction

###  1.1 General considerations

Investigating the $(2+1)$-dimensional gravity theories, initiated by [1], have
attracted attention especially as a toy model of quantum gravity, motivated by
the background provided in [2, 3, 4], to survey the classical and quantum
dynamics of point sources, in addition to representation of Chern-Simons
theory for $(2+1)$-dimensional gravity [5, 6, 7]. Classical and quantum
solutions for gravity theories in this dimension have been widely
investigated, for instance, in [8, 9, 10, 11, 12, 13, 14, 15].

The first exact black hole solution for $(2+1)$-dimensional gravity with a
negative cosmological constant, called BTZ black hole solution, was first
obtained in [16, 17]. A modified BTZ black hole solution was then obtained to
a $(2+1)$-dimensional string theory with a matter source given by anti-
symmetric $B$-field with the contribution of the central charge deficit
$\Lambda$ [18]. BTZ black hole solutions have been widely improved,
generalized, and investigated from different physical viewpoints, to extend
the quantum theory of gravity, gauge field theory, and string theory, along
with improving the knowledge of gravitational interaction in lower dimensional
manifolds [10, 14, 19, 20, 21, 22, 23]. Also, the role of BTZ black hole in
making deformed graphene, which is a particular $2$-dimensional real system
[24], as a tabletop for QFT in curved spacetimes to study the measurable
effects, like Hawking-Unruh effect has been pointed out in [25]. An important
role, in this case, was played by surfaces of constant negative Gaussian
curvature, whose embedding into ${R}^{3}$ gives rise to essential
singularities, as a result of the Hilbert theorem [26, 27]. In this category,
the line element of $(2+1)$-dimensional spacetimes constructed by Beltrami,
hyperbolic, and elliptic pseudospheres have been shown to be Weyl-related to
the line element of Rindler, de Sitter, and non-rotating BTZ black holes.

In this work, we extend the above catalog with a new $(2+1)$-dimensional class
of black hole solutions in string theory, whose spatial part is Weyl-related
to a surface of negative constant Gaussian curvature with the symmetries of
two-dimensional Lie algebra, admitting a homogeneous metric. The homogeneous
spacetimes, which possess the symmetry of spatial homogeneity and are defined
based on the simply-transitive Lie groups classification [28], have been used
to construct cosmological and black hole solutions in $(4+1)$ dimensions [29,
30], $(3+1)$ dimensions [31, 32, 33, 34, 35], and $(2+1)$ dimensions [12].
Here, we show that the considered homogeneous surface, being an especial case
of Lobachevsky geometries, can be reduced to a hyperbolic surface if the
Gaussian curvature is set to $-\frac{1}{2}$. Considering a general case with
an arbitrary negative Gaussian curvature, we first find black hole solutions
for the leading order of string effective action on a $(2+1)$-dimensional
spacetime, which is conformal related to this homogeneous spacetime. The
solutions include a constant dilaton, field strength tensor of $B$-field, and
the central charge deficit term $\Lambda$, which appears as a negative
cosmological constant. We will also consider the AdS/CFT correspondence for
the obtained solutions at the boundary to obtain the central charge of the
CFT. It turned out that the string frame metric is in the high curvature
limit, i.e. $R\alpha^{\prime}\gtrsim 1$, where the $\alpha^{\prime}$ is square
of string length, $\alpha^{\prime}=\lambda_{s}^{2}/2\pi$. At this limit, the
role of higher-order $\alpha^{\prime}$ corrections to the string effective
action becomes significant, which is widely believed to regularize the
curvature singularity [36]. Although considering all orders in
$\alpha^{\prime}$ is recommended at the high curvature regime, we limit the
calculations to the first order of $\alpha^{\prime}$, hoping to present a
glimpse of the consequences that could be obtained considering all
$\alpha^{\prime}$ orders. Considering only the first order of corrections, the
regularizing effects of the $\alpha^{\prime}$ corrections have been already
investigated in [37, 38, 39, 40, 41], while the higher orders of
$\alpha^{\prime}$ corrections usually result in the order by order correcting
the lower order solutions. Similar to what we have done in [34, 35],
implementing a perturbative series expansion in $\alpha^{\prime}$ on the
background field for string effective action that includes the correction of
Gauss-Bonnet type, we find the $\alpha^{\prime}$ correction to the obtained
black hole solutions.

In the following, we add some introductory remarks on two-loop
$\beta$-functions of $\sigma$-model, which are equivalent to the
$\alpha^{\prime}$-corrected string effective action equations of motion,
followed by a brief review on geometries with constant Gaussian curvature.

###  1.2 String effective action and two-loop $\beta$-functions

We are going to construct black hole solutions to string effective action,
whose equations of motion are equivalent to the $\beta$-function equations of
$\sigma$-model, which assuring the conformal invariance of the $\sigma$-model,
are, on the other hand, equivalent to the field equations of the associated
gravity theory [42]. For a $\sigma$-model with background fields of metric
$g$, dilation field $\phi$, and antisymmetric $B$-field, the two-loop
$\beta$-functions (order $\alpha^{\prime}$) have been calculated in [43, 44],
where the $\beta$-function equations for the metric are

$\displaystyle\frac{1}{\alpha^{\prime}}{\beta}^{g}_{\mu\nu}=$
$\displaystyle{R}_{{\mu}{\nu}}-\frac{1}{4}{H}^{2}_{{\mu}{\nu}}-\nabla_{{\mu}}\nabla_{{\nu}}{\phi}+\frac{\alpha^{\prime}}{2}\big{[}R_{\mu\alpha\beta\gamma}R_{\nu}^{~{}\alpha\beta\gamma}-\frac{3}{2}R_{(\mu}^{~{}~{}~{}\alpha\beta\gamma}H_{\nu)\alpha\lambda}H_{\beta\gamma}^{~{}~{}\lambda}-\frac{1}{2}R^{\alpha\beta\rho\sigma}H_{\mu\alpha\beta}H_{\nu\rho\sigma}+\frac{1}{8}(H^{4})_{\mu\nu}$
(1)
$\displaystyle-\frac{f}{2}\big{(}R_{\mu\alpha\beta\nu}(H^{2})^{\alpha\beta}+2\,R_{(\mu}^{~{}~{}\alpha\beta\gamma}H_{\nu)\alpha\lambda}H_{\beta\gamma}^{~{}~{}\lambda}+R^{\alpha\beta\rho\sigma}H_{\mu\alpha\beta}H^{\nu\rho\sigma}-\nabla_{\lambda}H_{\mu\alpha\beta}\nabla^{\lambda}H_{\nu}^{~{}\alpha\beta}\big{)}$
$\displaystyle-\frac{1}{12}\nabla_{\mu}\nabla_{\nu}H^{2}+\frac{1}{4}\nabla_{\lambda}H_{\mu\alpha\beta}\nabla^{\lambda}H_{\nu}^{~{}\alpha\beta}+\frac{1}{12}\nabla_{\mu}H_{\alpha\beta\gamma}\nabla_{\nu}H^{\alpha\beta\gamma}+\frac{1}{8}H_{\mu\alpha\lambda}H_{\nu\beta}^{~{}~{}\lambda}(H^{2})^{\alpha\beta}\big{]},$

where
$H^{4}=H_{\mu\nu\lambda}H^{\nu\rho\kappa}H_{\rho\sigma}^{~{}~{}\lambda}H^{\sigma\mu}_{~{}~{}\kappa}$,
$H_{\mu\nu}^{2}=H_{\mu\rho\sigma}H_{\nu}^{\rho\sigma}$ and $H$ is the field
strength of $B$-field defined by
$H_{\mu\nu\rho}=3\partial_{[\mu}B_{\nu\rho]}$. The $f$ parameter stands for
the renormalization scheme (RS) dependence in $\beta$-functions and the
corresponding schemes to $f=1$ and $f=-1$, called $R^{2}$ and Gauss-Bonnet
schemes, have been particularly discussed in [43]. Solutions of various RS
$\beta$-function equations are different, but still equivalent because they
belong to various definitions of physical metric, dilaton, and $B$-field. We
are going to focus on the Gauss-Bonnet scheme, where the $\beta$-function of
$B$-field is given by [43]

$\displaystyle\frac{1}{\alpha^{\prime}}{\beta}^{B}_{\mu\nu}(f$
$\displaystyle=-1)=\hat{R}_{[{\mu}{\nu}]}+\frac{\alpha^{\prime}}{4}\big{(}2\hat{R}^{\alpha\beta\gamma}_{~{}~{}~{}~{}[\nu}\hat{R}_{\mu]\alpha\beta\gamma}-\hat{R}^{\beta\gamma\alpha}_{~{}~{}~{}~{}[\nu}\hat{R}_{\mu]\alpha\beta\gamma}+\hat{R}_{\alpha[\mu\nu]\beta}H^{\alpha\beta}-\frac{1}{12}H_{\mu\nu}^{~{}~{}\rho}\nabla_{\rho}H^{2}\big{)},$
(2)

in which the $\hat{R}^{\rho}_{\mu\nu\sigma}$ is the Riemann tensor of the
generalized connection with torsion
${\hat{\Gamma}}^{\rho}_{\mu\nu}={{\Gamma}}^{\rho}_{\mu\nu}-\frac{1}{2}H^{\rho}_{\mu\nu}$.333
${\hat{R}}^{\kappa}_{~{}\lambda\mu\nu}=R^{\kappa}_{~{}\lambda\mu\nu}-\frac{1}{2}\nabla_{\mu}H^{\kappa}_{\nu\lambda}-\frac{1}{2}\nabla_{\nu}H^{\kappa}_{\mu\lambda}+\frac{1}{4}H_{~{}\nu\lambda}^{\gamma}H_{~{}\mu\gamma}^{\kappa}-\frac{1}{4}H_{~{}\mu\lambda}^{\gamma}H^{\kappa}_{~{}\nu\gamma}.$
For dilation field, the averaged $\beta$-function, which is given in terms of
metric and dilation $\beta$-functions as
$\tilde{\beta}^{{\phi}}={\beta}^{{\phi}}-\frac{1}{4}{\beta}^{g}_{\mu\nu}g^{\mu\nu}$,
is given by [43]

$\displaystyle\begin{split}\frac{1}{\alpha^{\prime}}\tilde{\beta}^{{\phi}}&=-{R}+\frac{1}{12}{H}^{2}+2\nabla_{{\mu}}\nabla^{{\mu}}{\phi}+(\partial_{{\mu}}{\phi})^{2}-\Lambda-\frac{\alpha^{\prime}}{4}(R^{2}_{\mu\nu\rho\lambda}-\frac{1}{2}R^{\alpha\beta\rho\sigma}H_{\alpha\beta\lambda}H_{\rho\sigma}^{~{}~{}\lambda}+\frac{1}{24}H^{4}-\frac{1}{8}(H_{\mu\nu}^{~{}~{}2})^{2}),\end{split}$
(3)

in which $\Lambda$ indicates the central charge deficit of $D$-dimensional
bosonic theory, given by [42]

$\displaystyle\Lambda=\frac{2\,(26-D)}{3\alpha^{\prime}}.$ (4)

The $\beta$-function equation (3) can be obtained by variation of the
following string effective action with respect to the dilaton field [43]

$\displaystyle\begin{split}S=-\frac{1}{2\kappa_{D}^{2}}\int&d^{D}x\sqrt{g}e^{\phi}(R-\frac{1}{12}H^{2}+(\nabla\phi)^{2}+\Lambda+\frac{\alpha^{\prime}}{4}(R^{2}_{\mu\nu\rho\lambda}-\frac{1}{2}R^{\alpha\beta\rho\sigma}H_{\alpha\beta\lambda}H_{\rho\sigma}^{~{}~{}\lambda}+\frac{1}{24}H^{4}-\frac{1}{8}(H_{\mu\nu}^{~{}~{}2})^{2})),\end{split}$
(5)

in which $\kappa_{D}^{2}=8\pi G_{D}=\lambda_{s}^{D-2}{\rm
e}^{-\phi}=\lambda_{p}^{D-2}$, where $\lambda_{p}$ is the Planck length and
$G_{D}$ is the $D$-dimensional gravitational Newton constant.

The string frame metric $g_{\mu\nu}$ is related to the Einstein-frame metric
$\tilde{g}_{\mu\nu}$ in $D$-dimensional spacetime by

$\tilde{g}_{\mu\nu}={\rm e}^{\frac{2}{D-2}\phi}g_{\mu\nu}.$ (6)

To provide the Einstein-frame string effective action, a remained intrinsic
ambiguity to a given order $\alpha^{\prime}$ in the effective action was
considered in [43]. This ambiguity is the result of the invariance of the
$S$-matrix under a set of field redefinitions [45], which leads to a class of
physically equivalent effective actions parametrized by $8$ essential
coefficients [46]. The Gauss-Bonnet scheme has been used in [43] to provide a
set of these $8$ coefficients. In $3$ dimensions the effective action is given
by

$\displaystyle\begin{aligned} S=&-\frac{1}{2\kappa_{3}^{2}}\int
d^{3}x\sqrt{\tilde{g}}\bigg{(}\tilde{R}-\frac{1}{12}\,{\rm
e}^{{4}{}\phi}H^{2}-(\tilde{\nabla}\phi)^{2}+\Lambda{\rm
e}^{-2\phi}+\frac{\alpha^{\prime}e^{{2}{}\phi}}{4}\bigg{[}\tilde{R}^{2}_{\mu\nu\rho\lambda}-4\,\tilde{R}_{\mu\nu}^{2}+\tilde{R}^{2}-\left(\partial\phi\right)^{4}\\\
&+{\rm
e}^{{4}\phi}\left(-\frac{1}{2}\tilde{R}^{\alpha\beta\rho\sigma}H_{\alpha\beta\lambda}H_{\rho\sigma}^{~{}~{}\lambda}+H^{2}_{\mu\nu}\tilde{\nabla}^{\mu}\phi\tilde{\nabla}^{\nu}\phi-\frac{1}{6}H^{2}(\tilde{\nabla}\phi)^{2}\right)+{\rm
e}^{{8}{}\phi}\left(\frac{1}{24}H^{4}+\frac{1}{8}(H_{\mu\nu}^{~{}~{}2})^{2}-\frac{1}{16}(H^{2})^{2}\right)\bigg{]}\bigg{)},\end{aligned}$
(7)

in which $\tilde{\nabla}$ indicates the covariant derivative with respect to
$\tilde{g}_{\mu\nu}$.

The $\alpha^{\prime}$ corrections to the string effective action (5) are
significant when the curvature is at the high limit $R\alpha^{\prime}\gtrsim
1$. Actually, the string effective action is known to include two kinds of
corrections: the stingy type $\alpha^{\prime}$-expansion at high curvature
regime and the quantum nature loop expansion in string coupling at strong
string coupling limit $g_{s}\equiv{\rm{e}^{-\phi}}>1$ [37]. As long as the
$g_{s}$ is sufficiently weak in the high curvature regime, the loop
corrections are allowed to be neglected, and the $\alpha^{\prime}$ corrections
are enough to be included in the effective action [37].

###  1.3 Lobachevsky geometry

The surfaces of constant Gaussian curvature have attracted attention in the
classic studies of differential geometry [26, 47]. Particularly, these
spacetimes have been considered as real terrestrial laboratories in studying
the measurable effects, like the Hawking-Unruh effect [25]. As a consequence
of a theorem proven by Hilbert [26, 27, 48], embedding of these surfaces into
$R^{3}$ gives rise to essential singularities. Although no single good
parametrization exists for all surfaces, the surfaces of revolution can be
described by one such parametrization, namely the canonical
parametrization.444The surfaces of revolution are the swapped surfaces by a
curve for instance in the plane $(x,z)$ rotated around the $z$-axis by a full
angle. These surfaces can be parameterized in ${R}^{3}$ by [25]

$x(\rho,v)=R(\rho)\cos v\;,\;y(\rho,v)=R(\rho)\sin
v\;,\;z(\rho)=\pm\int^{\rho}\sqrt{1-{R^{\prime}}^{2}(\bar{\rho})}d\bar{\rho}\;,$
(8)

where $R(\rho)$ identifies the type of surface, $\rho$ is the meridian
coordinate, whose range is fixed by request of real $z(\rho)$, $v$ is the
parallel coordinate (angle) ranging in $[0,2\pi]$, and prime denotes
derivative with respect to $\bar{\rho}$. The parametrization (8) leads to the
following embedded line element

$dl^{2}\equiv dx^{2}+dy^{2}+dz^{2}=d\rho^{2}+{R}^{2}(\rho)dv^{2}.$ (9)

The Gaussian curvature of this line element is ${\cal
K}=-\frac{R^{\prime\prime}(\rho)}{R(\rho)}$ [47], which can be solved as a
differential equation for positive and negative constant ${\cal K}$, giving
the following cases

$R(\rho)=c\cos(\sqrt{{k}}\,\rho)\quad{\rm for}\quad{\cal K}=k\;,$ (10)
$R(\rho)=c_{1}\sinh(\sqrt{k}\,\rho)+c_{2}\cosh(\sqrt{k}\,\rho)\quad{\rm
for}\quad{\cal K}=-k\;,$ (11)

where $k$ (positive), $c,c_{1}$, and $c_{2}$ are real constants.

In the positive constant curvature case (10), three cases are usually
distinguished by $c=k^{-\frac{1}{2}}$, $c>k^{-\frac{1}{2}}$,
$c<k^{-\frac{1}{2}}$, which are describing a sphere of radius
$k^{-\frac{1}{2}}$, and two applicable surfaces to sphere via a redefinition
of $v\to(c\sqrt{k})v$. On the other hand, for negative constant curvature
class (11), called the Lobachevsky surfaces [25], the three special cases of
Beltrami, elliptic, and hyperbolic pseudospheres have attracted attention,
which are described as follows:

$\bullet$ The Beltrami pseudosphere, whose associated spacetime is known to be
conformal to the Rindler spacetime [25], is defined by (11) with
$c_{1}=c_{2}\equiv c$, and

$R(\rho)=c\;e^{\sqrt{k}\,\rho},\quad{\rm where}\quad
R(\rho)\in[0,k^{-\frac{1}{2}}]\Leftrightarrow\rho\in[-\infty,-k^{-\frac{1}{2}}\ln(c\sqrt{k})],$
(12)

in which $c$ is required to be a real positive number.

$\bullet$ With $c_{2}=0$, $c_{1}\equiv c$, the equation (11) describes the
elliptic pseudosphere by

$R(\rho)=c\;\sinh(\sqrt{k}\,\rho),\quad{\rm where}\quad
R(\rho)\in[0,k^{-\frac{1}{2}}\cos\vartheta]\Leftrightarrow\rho\in[0,{\rm
arcsinh}\cot\vartheta],$ (13)

in which $\sin(\vartheta)=c\sqrt{k}$. The spacetime whose space part is an
Elliptic surface has been shown to be Weyl to de-Sitter spacetime [49].

$\bullet$ The $c_{1}=0$, $c_{2}\equiv c$ case leads to the hyperbolic
pseudosphere with

$R(\rho)=c\;\cosh(\sqrt{k}\,\rho),\quad{\rm where}\quad
R(\rho)\in[c,\sqrt{k^{-1}+c^{2}}]\Leftrightarrow\rho\in[-{\rm
arccosh}(\sqrt{1+\frac{1}{k\,c^{2}}}),{\rm
arccosh}(\sqrt{1+\frac{1}{k\,c^{2}}})],$ (14)

in which $c$ is required to be a real positive number. The interesting
characteristic of hyperbolic pseudosphere is that the line element of the non-
rotating BTZ black hole is Weyl-related to the line element of the hyperbolic
spacetime [16], in such a way that [50]

$\begin{split}ds^{2}_{BTZ}&=-\left(\frac{r^{2}}{l^{2}}-m\right)\,dt^{2}+\frac{dr^{2}}{\left(\frac{r^{2}}{l^{2}}-m\right)}+r^{2}d\phi^{2}\\\
&=\frac{l^{2}m}{\sinh^{2}\left(\sqrt{m}\rho\right)}\left(-dt^{2}+d\rho^{2}+l^{2}\cosh^{2}\left(\sqrt{m}\rho\right)d\phi^{2}\right),\end{split}$
(15)

where the $r$ is given in terms of $\rho$ by $r=\sqrt{m}l\coth(\sqrt{m}\rho)$.

To find a black hole solutions for string effective action with a line element
Weyl-related to a homogeneous spacetime with Lobachevsky geometry, the paper
is organized as follows: Section 2 presents the characteristics of the
considered spacetime in detail and then the derivation of the black hole
solutions. After introducing the geometry of the $(2+1)$-dimensional
spacetime, the black hole solutions obtained by solving the one-loop order of
$\beta$-functions, i.e. the zeroth order $\alpha^{\prime}$, are provided in
subsection 2.1. The thermodynamic characteristics of these solutions are
investigated in section 3. The CFT central charge for the solution is
calculated in section 4. Then, section 5 presents the
$\alpha^{\prime}$-corrected black hole solutions constructed based on the
solutions of section 2, by solving the two-loop order $\beta$-function
equations in the Gauss-Bonnet RS. Finally, some concluding remarks are
presented in section 6.

##  2 Black hole solution of string effective action on spatially homogeneous
$(2+1)$-dimensional spacetime

On the $(2+1)$-dimensional spacetime, whose the $t$-constant hypersurface is
given by a homogeneous space corresponding to the $2$-dimensional Lie group
with real two-dimensional Lie algebra $[T_{1},T_{2}]=T_{2}$, the following
metric ansatz can be considered [12]

$\displaystyle ds^{2}=-dt^{2}+g_{ij}\sigma^{i}\sigma^{j},$ (16)

where $g_{ij}$ are constants and $\\{\sigma^{i},~{}i=1,2\\}$ are left
invariant basis $1$-forms on the Lie group, obeying
$\sigma^{2}=-\frac{1}{2}\sigma^{1}\wedge\sigma^{2}$,
$\sigma=(g^{-1}dg)^{i}T_{i}$, and $\sigma^{i}=(g^{-1}dg)^{i}$, where $g={\rm
e}^{x_{1}T_{1}}{\rm e}^{x_{2}T_{2}}$. The coordinate and non-coordinate basis
are related by

$\displaystyle\sigma^{1}=dx_{1}+x_{2}dx_{2},\quad\sigma^{2}=dx_{2}.$ (17)

Accordingly, the metric (16) recasts the following form

$\displaystyle
ds^{2}=-dt^{2}+(g_{11}+2g_{12}\,x_{2}+g_{22}\,x_{2}^{2})\,dx_{1}^{2}+2\left(g_{12}+g_{22}x_{2}\right)\,dx_{1}dx_{2}+g_{22}\,dx_{2}^{2},$
(18)

whose Gaussian curvature is constant, given by

$\displaystyle{\cal{K}}=-\frac{g_{11}g_{22}-g_{12}^{2}}{g_{22}}.$ (19)

Now, assuming ${\cal{K}}<0$ and defining

$\displaystyle k\equiv-{\cal{K}},$ (20)

the new coordinates $(\rho,\varphi)$ can be introduced by the following
coordinate redefinition

$\displaystyle\begin{split}x_{1}&=-L\,c_{1}\varphi-\sqrt{k}c_{1}\rho-\ln\left(2\,k\,{{\rm
e}^{-{{2}{\sqrt{k}}}c_{1}\rho}}+1\right)+\ln(k)+\frac{3}{2}\ln(2),\\\
x_{2}&={\frac{1}{{4g_{22}}\,k}}\left(\sqrt{{2\,g_{22}}}\left(2\,{{\rm
e}^{-\sqrt{k}c_{1}\rho}}k-{{\rm
e}^{\sqrt{k}c_{1}\rho}}\right)-4\,{g_{12}}\,k\right),\end{split}$ (21)

where $L$ and $c_{1}$ are real constants. Then, the line element (18) recasts
the following form

$\displaystyle ds^{2}\equiv
g_{\mu\nu}^{R}dx^{\mu}dx^{\nu}=-dt^{2}+c_{1}^{2}\left(d\rho^{2}+R^{2}(\rho)d\varphi^{2}\right),$
(22)

with

$\displaystyle\begin{split}R(\rho)&=\frac{\sqrt{2}L}{4\,k}\left(2\,k\,{{\rm
e}^{-\sqrt{k}c_{1}{\rho}}}+{{\rm
e}^{{{{\sqrt{k}}c_{1}\rho}}}}\right)=\frac{\sqrt{2}L}{2\,k}\cosh_{2k}(\sqrt{k}c_{1}\rho).\\\
\end{split}$ (23)

The $\cosh_{2k}$ function is known as the deformed hyperbolic function
introduced for the first time in [51, 52], in solving Schrodinger equation
with deformed potential.

In the case that the embedding of the space part of the above line element
into ${R}^{3}$ is of interest, according to (8), the $\rho$ coordinate must
range in

$\rho\in\left[{\frac{1}{\sqrt{k}c_{1}}\ln\left({\frac{\sqrt{2k}\,(a-1)}{Lc_{1}}}\right)},\frac{1}{\sqrt{k}c_{1}}\ln\left({\frac{\sqrt{2k}\,(a+1)}{Lc_{1}}}\right)\right],$
(24)

in which $a=\sqrt{{L}^{2}{{c_{1}}}^{2}+1}$, and hence

$R(\rho)\in\left[{\frac{L\sqrt{2}\left(2\,k+1\right)}{4k}},\frac{a}{\sqrt{k}c_{1}}\right].$
(25)

These maximal circles, which appeared as a consequence of the Hilbert theorem
[26, 27, 48], denote the non-removable singularities, i.e. the singular
boundaries for the considered embedded surface, which are called the Hilbert
horizons [25].

The obtained line element (22) with the $2$-dimensional space possessing Lie-
symmetry can be considered as an especial class of metric (9) with $R(\rho)$
given by (11). It is worth noting that, although the Beltrami and elliptic
type metrics can not be recovered from (22), the value of $k=\frac{1}{2}$ can
reduce the metric (22) to a hyperbolic type case, described in (14).

Now, motivated by the conformal relation of the BTZ black hole solution to the
spacetime with hyperbolic space geometry [50], we are going to find a black
hole solution, if exists, whose line element is conformally related to the
metric (22). In this regard, we start with the metric
$g_{\mu\nu}=h(\rho)g_{\mu\nu}^{R}$, i.e.

$\displaystyle
ds^{2}=h(\rho)g_{\mu\nu}^{R}dx^{\mu}dx^{\nu}=h(\rho)\left(-dt^{2}+d\rho^{2}+R^{2}(\rho)d\varphi^{2}\right),$
(26)

in which $R(\rho)$ is given by (23) and $h(\rho)$ is an arbitrary function.
The Gaussian curvature-related parameter $k$ (20) will be assumed to be an
arbitrary positive constant.

To construct the black hole solutions, we consider the $\beta$-function
equations (1)-(3) with contributions of dilation, antisymmetric $B$-field, and
central charge deficit $\Lambda$, which plays the role of a dilation potential
in the string effective action (5). We start with finding the solutions to the
$\beta$-function equations at one loop order, i.e. $\alpha^{\prime}=0$. Then,
the solutions to the two-loop order of $\beta$-function equations are
considered.

###  2.1 $(2+1)$-dimensional black hole solutions of one-loop $\beta$-function
equations

First, we consider the $\beta$-function equations (1)-(3) at the one-loop
order, i.e. in the absence of the $\alpha^{\prime}$ corrections. Considering
(26), with the $R(\rho)$ given by (23), as the string frame metric along with
the following form of field strength tensor of $B$-field

$\displaystyle H=\frac{1}{3!}E(\rho)\,dt\wedge d\rho\wedge d\varphi,$ (27)

the $(t,t)$, $(\rho,\rho)$, and $(\varphi,\varphi)$ components of metric
$\beta$-function (1), $(t,\varphi)$ component $B$-field $\beta$-functions (2),
and dilaton $\beta$-functions (3) yield following coupled set of differential
equations

$\displaystyle 2{\ln h^{\prime\prime}}+{\ln
h^{\prime}}\,\left({\ln(h\,R^{2})^{\prime}}+2\,\phi^{\prime}\right)-2{{{E}^{2}}{{h}^{-2}{R}^{-2}}}=0,$
(28) $\displaystyle 2{\ln R^{\prime\prime}}+2{\ln
h^{\prime\prime}}+2\phi^{\prime\prime}+{\ln
R^{\prime}}\,{\ln(h\,R^{2})^{\prime}}-{\ln
h^{\prime}}\,\phi^{\prime}-{{{E}^{2}}{{h}^{-2}{R}^{-2}}}=0,$ (29)
$\displaystyle 2{\ln R^{\prime\prime}}+{\ln h^{\prime\prime}}+2{{\ln
R^{\prime}}}^{2}+2\left({\ln h^{\prime}}+2\phi^{\prime}\right){\ln
R^{\prime}}+\frac{1}{2}\,{{\ln h^{\prime}}}^{2}+{\ln
h^{\prime}}\,\phi^{\prime}-\,{{{E}^{2}}{{h}^{-2}{R}^{-2}}}=0,$ (30)
$\displaystyle 3\,{\ln h^{\prime}}+2\,{\ln
R^{\prime}}-2\,\phi^{\prime}-2\,{\ln E^{\prime}}=0,$ (31)
$\displaystyle\begin{aligned} 4\,\phi^{\prime\prime}+4\,{\ln
R^{\prime\prime}}+4\,{\ln
h^{\prime\prime}}+2{\phi^{\prime}}^{2}+2{\ln(h\,R^{2})^{\prime}}\phi^{\prime}+4\,{{\ln
R^{\prime}}}^{2}+2\,{\ln h^{\prime}}\,{\ln R^{\prime}}+{{\ln
h^{\prime}}}^{2}-2\Lambda\,h-{{{E}^{2}}{{h}^{-2}{R}^{-2}}}=0.\end{aligned}$
(32)

Here, the prime stands for derivative with respect to $\rho$. Combining
equations (28)-(32) one can obtain the following equation

$\displaystyle\begin{aligned}
2\phi^{\prime\prime}+2{\phi^{\prime}}^{2}+\left(2\,{\ln R^{\prime}}+{\ln
h^{\prime}}\right)\phi^{\prime}-2\Lambda
h+{{{E}^{2}}{{h}^{-2}{R}^{-2}}}=0.\end{aligned}$ (33)

Solving these equations, we obtain the following set of solution

$\displaystyle\phi(\rho)={\phi_{0}},$ (34) $\displaystyle
h(\rho)=\frac{32k^{2}}{\Lambda}\left({{\rm
e}^{{{\sqrt{k}}c_{1}\rho}}}-2\,k{{\rm
e}^{{{{-\sqrt{k}}c_{1}\rho}}}}\right)^{-2},$ (35) $\displaystyle
E(\rho)=\frac{32\,b\,k^{2}c_{1}^{3}}{\Lambda^{\frac{3}{2}}{\rm
e}^{\phi_{0}}}\frac{\left(2k{{\rm e}^{{-{{\sqrt{k}}c_{1}\rho}}}}+{{\rm
e}^{{{\sqrt{k}}c_{1}\rho}}}\right)}{\left(2k{{\rm
e}^{{{{-\sqrt{k}}c_{1}\rho}}}}-{{\rm e}^{{{\sqrt{k}}c_{1}\rho}}}\right)^{3}},$
(36)

where $\phi_{0}$, and $b$ are real constant. Then, (33) lead the following
constraint between the constants of the solution

$\displaystyle\Lambda\,L^{2}-b^{2}{\rm e}^{-2\phi_{0}}=0.$ (37)

Accordingly, the metric (26) recasts the following form as the solution of the
one-loop $\beta$-function equations

$\displaystyle ds^{2}=\frac{32c_{1}^{2}k^{2}}{\Lambda\left(2\,k\,{{\rm
e}^{{{-\sqrt{k}}c_{1}\rho}}}-{{\rm
e}^{{{{\sqrt{k}}c_{1}\rho}}}}\right)^{2}}\left(-dt^{2}+d\rho^{2}+\frac{b^{2}{\rm
e}^{-2\phi_{0}}}{32\Lambda k^{2}}\left(2\,k\,{{\rm
e}^{{{-\sqrt{k}}c_{1}\rho}}}+{{\rm
e}^{{{{\sqrt{k}}c_{1}\rho}}}}\right)^{2}d\varphi^{2}\right).$ (38)

In the special case where the Gaussian curvature equals $-\frac{1}{2}$, i.e.
$k=\frac{1}{2}$, the metric (38) can be compared to (15) that represents the
line element to which the BTZ solution is conformally related. In this case,
(38) reads

$\displaystyle
ds^{2}=\frac{2c_{1}^{2}}{\Lambda\sinh^{2}\left(\frac{\sqrt{2}}{2}c_{1}\rho\right)}\left(-dt^{2}+d\rho^{2}+\frac{b^{2}{\rm
e}^{-2\phi_{0}}}{2\Lambda}\cosh^{2}\left(\frac{\sqrt{2}}{2}c_{1}\rho\right)d\varphi^{2}\right).$
(39)

Although the dependence on the Gaussian curvature constant is absent in (39),
the remaining constant $c_{1}$ can be considered to be related to the BTZ mass
by $c_{1}=\sqrt{2m}$. The conformal factor in (39) diverges at $\rho=0$.
However, it can be excluded from the $\rho$ coordinate range in (14) while
embedding it into $R^{3}$, considering $\rho\in(-{\rm
arccosh}(\sqrt{1+\frac{1}{k\,c^{2}}}),0)$, which leads to $r>0$ in the BTZ
metric (15).

Here, we continue with the general solution (38) to obtain the black hole
metric, applying the following redefinition

$\displaystyle-\frac{\sqrt{32}c_{1}k}{\sqrt{\Lambda}\left(2\,k\,{{\rm
e}^{{{-\sqrt{k}}c_{1}\rho}}}-{{\rm
e}^{{{{\sqrt{k}}c_{1}\rho}}}}\right)}d\rho=dr.$ (40)

Since the metric (38) has diverging conformal factor at

$\displaystyle\rho_{0}=\frac{\ln(2k)}{\sqrt{k}c_{1}},$ (41)

to have a well-defined coordinate redefinition, we can restrict the range of
this coordinate to $\rho<\rho_{0}$ to have $r>0$.555If the embedding into
$R^{3}$ is of interest, the $\rho_{0}$ belongs to the $\rho$-coordinate range
of (24). Then, (40) gives the $\rho$ coordinate in terms of new $r$ coordinate
as follows

$\displaystyle\rho={\frac{1}{2\sqrt{k}{c_{1}}}\ln\left({\frac{\left(r-c_{2}\right)\Lambda+8\,\sqrt{k}{c_{1}}}{2\left(r-c_{2}\right)\Lambda\,k}}\right)},$
(42)

in which $c_{2}$ is an integrating constant.

Now, applying the conformal transformation (6) on (38) gives the not-yet
finalized form of the Einstein-frame black hole metric in terms of $r$ as
follows

$\displaystyle ds^{2}_{E}=\tilde{g}_{\mu\nu}^{R}dx^{\mu}dx^{\nu}={{\rm
e}^{2\,\phi_{0}}}(-F_{0}(r)dt^{2}+F_{0}^{-1}(r)dr^{2}+W^{2}(r)d\varphi^{2}),$
(43)

where defining
$B(r)=\frac{\Lambda}{4}\left(r-{c_{2}}+8\,\sqrt{k}c_{1}\right)$, we have

$\displaystyle F_{0}(r)=B(r)\left(r-{c_{2}}\right),$ (44) $\displaystyle
W^{2}(r)={{\rm
e}^{-2\phi_{0}}}{\frac{b^{2}}{{\Lambda}^{2}k}}\left(B(r)-\sqrt{k}{c_{1}}\right)^{2}.$
(45)

Also, the field strength tensor (27) is

$\displaystyle H(r)=-\frac{1}{3!}\sqrt{\Lambda}\,W(r)\,dt\wedge dr\wedge
d\varphi.$ (46)

##  3 Thermodynamic behavior of the $(2+1)$-dimensional black hole solutions

Since the only physical quantity characterizing the obtained black hole
solution is its mass, one physical integrating constant needs to be present in
the solution. Noting (43)-(46), $b$ can be absorbed in a redefinition of the
$\varphi$ coordinate. To decide on the $c_{1}$ and $c_{2}$, we investigate the
thermodynamic quantities and satisfaction of the first law of black hole
thermodynamics.

The solution is associated with the horizons located at
$r_{h_{1}}=c_{2}-{\frac{8{\sqrt{k}}}{\Lambda}}c_{1}$ and $r_{h_{2}}=c_{2}$,
where $r_{h_{1}}<r_{h_{2}}$. The Hawking temperature can be derived from the
Euclidean regularity methods [53], by666The same result can be obtained via
the surface gravity definition
$\displaystyle\kappa^{2}_{h}=-\frac{1}{2}{(\nabla_{\mu}{\xi}_{\nu})\left(\nabla^{\mu}{\xi}^{\nu}\right)}|_{r=r_{H}}=\tilde{g}^{rr}\left(\sqrt{\tilde{g}_{tt}}\right)^{\prime}_{r=r_{H}}=k\,c_{1}^{2},$
(47) where $\xi$ is the Killing vector normal to the horizon such that
$T=\kappa_{h}/2\pi$.

$\displaystyle\begin{aligned}
T=&\frac{\sqrt{(\tilde{g}_{rr}^{-1})^{\prime}\tilde{g}_{tt}^{\prime}}|_{r=r_{h_{2}}}}{4\pi},\end{aligned}$
(48)

which gives the temperature of the leading order solution (43) by

$\displaystyle\begin{aligned} T_{0}=\frac{\sqrt{k}c_{1}}{2\pi}.\end{aligned}$
(49)

Furthermore, the black hole entropy can be obtained using the Wald’s formula
[54]

$\displaystyle\begin{aligned}
S=-2\pi\int_{h}d\varphi\sqrt{\tilde{\sigma}}\frac{\delta\cal{L}}{\delta\tilde{R}_{\mu\nu\rho\sigma}}\epsilon_{\mu\nu}\epsilon_{\rho\sigma},\end{aligned}$
(50)

where the $h$ denotes evaluating the integral on the horizon, $\tilde{\sigma}$
is the determinant of the metric of the $1$-dimensional boundary in the
Einstein-frame, and $\epsilon_{\mu\nu}$ is the binormal to the horizon whose
normalization is chosen as $\epsilon_{\mu\nu}\epsilon^{\mu\nu}=-2$. At the
leading order of Einstein-frame string effective action (7), i.e.
$\alpha^{\prime}=0$, we obtain

$\displaystyle\begin{aligned}
S_{0}=\frac{2\pi\,b\,c_{1}}{\kappa_{3}^{2}\Lambda}.\end{aligned}$ (51)

The solution (43) is not asymptotically flat and similar to BTZ black holes,
whose asymptotic Ricci scalar is constant, the Einstein-frame Ricci scalar is
a constant given by $\tilde{R}=-\frac{3}{2}{\rm e}^{-2\phi_{0}}\Lambda$. The
conserved mass of this non-asymptotically flat solution can be calculated via
the quasi-local formalism by [55]

$\displaystyle
m=\frac{1}{\kappa_{3}^{2}}\int_{B}d\varphi\sqrt{\tilde{\sigma}}n^{a}\tau_{ab}\xi^{b},$
(52)

in which $\kappa_{3}$ is the $3$-dimensional Newton constant, $B$ is the
$1$-dimensional boundary, and $n^{a}$ is the time-like unit normal vector to
the boundary $B$. Also, noting that with the constant dilation (34), the
central charge deficit term in the leading order of string effective action
(7) reduces to the negative cosmological constant term, the quasi-local stress
tensor is given by [56]

$\displaystyle{\tau_{ab}}=K_{ab}-\tilde{h}_{ab}K-\frac{1}{l}\tilde{h}_{ab},$
(53)

where $K_{ab}$ is the extrinsic curvature of the $2$-dimensional boundary
$\partial{\cal{M}}$ with induced metric $\tilde{h}_{ab}$, $K$ is the trace of
$K_{ab}$, and on this $(2+1)$-dimensional spacetime $l=\frac{2{\rm
e}^{\phi_{0}}}{\sqrt{\Lambda}}$.

The calculation of the mass is provided in the appendix A, where it is shown
that (52) leads to the mass expression (95), which yields the mass of the
zeroth order $\alpha^{\prime}$ solutions by

$\displaystyle
m_{0}={\frac{{b}\,c_{2}}{32\sqrt{k}}}\left(-c_{2}\Lambda+8\,\sqrt{k}{c_{1}}\right)+{\frac{{b}\,\sqrt{2}{{\rm
e}^{{\phi_{0}}}}}{32\Lambda\,\sqrt{k}}}\left(c_{2}\Lambda-4\,\sqrt{k}c_{1}\right)^{2}.$
(54)

As mentioned before, only one constant that characterizes the mass of the
black hole is required to be present in the solution. Noting (49), (51), and
(54), if we consider $c_{2}$ to be functionally related to $c_{1}$, i.e.
$c_{2}(c_{1})$,777The functional relation between the appeared constant in the
solutions has been for instance pointed out in AdS context in [57], where in
the Einstein-Scalar models with the scalar filed of the form
$\phi=\frac{\alpha}{r}+\frac{\beta}{r}$, the integrability of energy in
Hamiltonian formalism, which contains $\delta Q_{\phi}=\int\beta\delta\alpha
d\Omega+\dots$ term, requires a functional relation between the $\alpha$ and
$\beta$ coefficients in the asymptotic expansion of the scalar field. then the
satisfaction of the first law of thermodynamics $dm_{0}=T_{0}\,dS_{0}$ fixes
this relation as

$\displaystyle c_{2}=\frac{4c_{1}\sqrt{k}}{\Lambda},$ (55)

where the mass of these black hole solutions, recasts the following form

$\displaystyle m_{0}={\frac{{b}\sqrt{k}}{2\Lambda}}c_{1}^{2}.$ (56)

Using (55) and (56) in (43), the final form of the black hole solutions for
one-loop $\beta$-function equation in the Einstein-frame reads

$\displaystyle ds^{2}_{E}={{\rm
e}^{2\,\phi_{0}}}\left(-\left(\frac{1}{4}\Lambda\,{r}^{2}-\frac{8\sqrt{k}}{b}m_{0}\right)dt^{2}+\left(\frac{1}{4}\Lambda\,{r}^{2}-\frac{8\sqrt{k}}{b}m_{0}\right)^{-1}dr^{2}+{\frac{{b^{2}}\,{{\rm
e}^{-2\phi_{0}}}}{16\,{k}}}r^{2}d\varphi^{2}\right),$ (57)

while the antisymmetric $B$-field associated with the filed strength tensor
(46) becomes

$\displaystyle B_{t\varphi}=-{\frac{{b}\sqrt{\Lambda}}{2\sqrt{k}}}\,{{\rm
e}^{-\phi_{0}}}r^{2}.$ (58)

The obtained metric has constant curvature invariants without any essential
singularity.

To see the thermal stability of the solutions, one can consider the specific
heat capacity at constant pressure, given by

$\displaystyle C=T\left(\frac{\partial S}{\partial
T}\right)={\frac{2\,\pi\,{b}\,{c_{1}}}{\Lambda}},$ (59)

which is positive for $c_{1}>0$, indicating the thermodynamic stability of the
solutions.

The $c_{2}$ expression (55) leaves only the $r_{h_{2}}$ as the positive radius
of event horizon, where noting (56)888The other radius reads
$r_{h_{1}}=-\frac{4}{\Lambda}\sqrt{k}{{{c_{1}}}{}}.$

$\displaystyle r_{h_{2}}=\frac{4}{\Lambda}\sqrt{k}{{{c_{1}}}{}}.$ (60)

It is worth putting the event horizons in contact with the notion of Hilbert
horizon of the homogeneous spacetime described by metric (22). As we have
seen, the singular boundaries for the surface appear while embedding the
surface into ${R}^{3}$, where the $\rho$ coordinate is restricted to be in the
range of (24). Using the redefinition (40),999This redefinition gives the $r$
coordinate in terms of $\rho$ as
$r={\frac{{4}\,\sqrt{k}\,c_{1}\left(2\,k{{\rm
e}^{-2\,\rho\,\sqrt{k}{c_{1}}}}+1\right)}{\Lambda\,\left(2\,k{{\rm
e}^{-2\,\rho\,\sqrt{k}\,{c_{1}}}}-1\right)}}.$ the two radius of Hilbert
horizons defined by the $\rho$-coordinate bounds in embedding into $R^{3}$,
are

$\displaystyle
r_{Hh_{1}}=-\frac{4}{\Lambda}\sqrt{k}{{{c_{1}}}{}}-{\frac{4\,{L}^{2}{{c_{1}}}^{3}\sqrt{k}}{\left(\sqrt{{L}^{2}{{c_{1}}}^{2}+1}+1\right)\Lambda}},\quad
r_{Hh_{2}}=-\frac{4}{\Lambda}\sqrt{k}{{{c_{1}}}{}}+{\frac{4\,{L}^{2}{{c_{1}}}^{3}\sqrt{k}}{\left(\sqrt{{L}^{2}{{c_{1}}}^{2}+1}-1\right)\Lambda}}$
(61)

where, as fixed by (37), $L=b{\rm e}^{\phi_{0}}\Lambda^{-\frac{1}{2}}$. The
$r_{Hh_{1}}$ and $r_{Hh_{2}}$ are associated to the upper and lower bounds of
$\rho$ in (40), respectively. It should be also noted that the coordinate
redefinition (24) added an extra limit on the $\rho$ coordinate by restricting
it to $\rho<\rho_{0}$. Hence, if the embedding of the $2$-dimensional surface
into $R^{3}$ is considered in the solutions, the $\rho$ coordinate must belong
to the range
$\rho\in\left[{\frac{1}{\sqrt{k}c_{1}}\ln\left({\frac{\sqrt{2}\,(a-1)}{L\sqrt{k}c_{1}}}\right)},\rho_{0}\right]$,
which is equivalent to the $r$-coordinate that starting at $r_{Hh_{2}}$ goes
to infinity as $\rho$ approaches $\rho_{0}$. The $r_{Hh_{1}}<0$, not belonging
to this range of $\rho$, is not of our interest. Now, assuming the $L$ to take
the form of $L=(c_{1})^{-1}\sqrt{p-1}$, or equivalently
$b={\frac{\sqrt{\Lambda({p}-1)}{}}{{{\rm e}^{-\phi 0}}{c_{1}}}}$, in which $p$
is a constant, the radius of event horizon (60) and Hilbert horizon (61) are
related generally by

$\displaystyle r_{Hh_{2}}-r_{h_{2}}=\frac{4\,kc_{1}}{\Lambda}(\sqrt{p}-1).$
(62)

It shows that, similar to the case when the space part of BTZ solution is
embedded in $R^{3}$ [50], the geometry ends before reaching the event horizon
of the black hole. Also, (62) shows that the two kinds of horizons can not
coincide unless $c_{1}=0$, for which the entropy of the black hole vanishes.

Compared to the BTZ solution to string effective action, obtained in [18],
which was constructed with $\phi_{0}=0$ and including no integrating constant
associated with the $B$-field $\beta$-function equation, the obtained solution
(57) contains the constants $\phi_{0}$ and $b$ along with the constant $k$,
related to the Gaussian curvature of the homogeneous space. Although a
redefinition in the coordinates can eliminate the $b$, $\phi_{0}$, and $k$
dependent factor in the third term of the line element (57), it can not remove
the factor accompanied with the mass $m_{0}$ that contains these constants.
Hence, the solution (57) can be considered as a more general form of BTZ-type
solution of string effective action. If one sets $b=8\sqrt{k}$, with
$\phi_{0}=\ln(2)$ the BTZ black hole can be recovered from (57), where the
dependence of the metric on the Gaussian curvature constant $k$ becomes
absent.

In other words, we arrived at a BTZ-type solution starting from the spacetime
conformal related to the line element whose spatial part is a homogeneous
space with arbitrary negative Gaussian curvature ${\cal K}=-k$. The conformal
relation of BTZ solution to negative curvature hyperbolic space has been
addressed in [25, 50]. As mentioned before, hyperbolic space can be recovered
here by setting $k=\frac{1}{2}$. Here, we found a more general Lobachevsky-
type geometry with arbitrary Gaussian curvature that has a conformal relation
to the BTZ-type solution of string theory.

##  4 Central charge of the $CFT$ at boundary

In this section, we consider the AdS/CFT correspondence for the model at the
boundary and obtain the CFT central charge $c$, which can be calculated using
the following formula [11]

$\frac{1}{2\kappa_{3}^{2}}Tr(\tau^{ab})=-\frac{c}{24\pi}\mathcal{R},$ (63)

where $\tau^{ab}$ is the regularized stress energy tensor defined by (53) and
$\mathcal{R}$ is scalar curvature of the boundary surface. Considering metric
(57), by defining

$F_{0}(r)=\frac{1}{4}\Lambda\,{r}^{2}-\frac{8\sqrt{k}}{b}m_{0},$ (64)

and $C\equiv{\frac{{b^{2}}\,{{\rm e}^{-2\phi_{0}}}}{16\,{k}}}$, components of
$\tau^{ab}$ are given by

$\begin{split}\tau_{00}=\frac{\sqrt{C}F_{0}(r)}{\sqrt{-\det(\tilde{h})}}-\frac{\tilde{h}_{00}}{l},\quad\quad\tau^{22}=-\frac{\Lambda
r^{2}}{4\sqrt{-C\det(\tilde{h})}}-\frac{\tilde{h}_{22}}{l},\end{split}$ (65)

where $\tilde{h}^{ab}$ denotes the inverse of boundary metric. Using the
following identity

$2G_{ij}n^{i}n^{j}=\mathcal{R}-K_{ab}K^{ab}+K^{2},$ (66)

$\mathcal{R}$ can be obtained, where $G_{ij}$ is the Einstein tensor, and
$n^{i}=-\frac{1}{\sqrt{g_{rr}}}\delta^{i,r}$ is the unit outward pointing
normal vector to the boundary $\partial M_{r}$. For the geometry (57) we have

$G_{ij}n^{i}n^{j}=\frac{\Lambda}{4{\rm e}^{2\phi_{0}}},$ (67)
$\theta_{ab}=\frac{1}{2\sqrt{g_{rr}}}\partial_{r}\tilde{h}_{ab}.$ (68)

Based on Fefferman-Graham expansion [58], the boundary metric and its inverse
can be rewritten in the following forms for metric (57)

$\tilde{h}_{ab}=r^{2}\tilde{h}^{(0)}_{ab}+\tilde{h}^{(2)}_{ab}+O(\frac{1}{r^{2}}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tilde{h}^{(0)}=diag(-\frac{\Lambda{\rm
e}^{2\phi_{0}}}{4},C),\quad\tilde{h}^{(1)}=diag(\frac{8\sqrt{k}}{b}m_{0}{\rm
e}^{2\phi_{0}},0)$ (69)
$\tilde{h}^{ab}=\frac{1}{r^{2}}(\tilde{h}^{(0)})^{ab}-\frac{1}{r^{4}}(\tilde{h}^{(2)})^{ab}+\dots~{}.$
(70)

Then, we have

$K_{ab}=r\sqrt{F_{0}(r)}{{\rm e}^{\phi_{0}}}\tilde{h}^{(0)}_{ab}+\dots~{}.$
(71) $K=\tilde{h}^{ab}K_{ab}=\frac{2\sqrt{F_{0}(r)}{{\rm
e}^{-\phi_{0}}}}{r}-\frac{\sqrt{F_{0}(r)}{{\rm
e}^{-\phi_{0}}}}{r^{3}}\tilde{h}^{(0)^{ab}}\tilde{h}^{(2)}_{ab}+\cdot\cdot\cdot~{}.$
(72)

Now, using (66), the scalar curvature of boundary at infinity
$(r\rightarrow\infty)$ is given by

$\mathcal{R}=\frac{4}{l^{2}}+\dots~{}.$ (73)

Furthermore, for the determinant of the metric (69) we have

$\frac{1}{\sqrt{-\tilde{h}}}=\frac{1}{\sqrt{-\det(r^{2}\tilde{h}^{(0)}_{ab})}~{}\Big{(}1+\frac{1}{r^{2}}\tilde{h}^{(0)^{ab}}\tilde{h}^{(2)}_{ab}+\cdot\cdot\cdot\Big{)}^{\frac{1}{2}}}=\frac{2{\rm
e}^{-\phi_{0}}}{\sqrt{C\Lambda}r^{2}}~{}\Big{(}1-\frac{1}{2r^{2}}\tilde{h}^{(0)^{ab}}\tilde{h}^{(2)}_{ab}+\cdot\cdot\cdot\Big{)},$
(74)

Then, we find

$\begin{split}Tr(\tau^{ab})&=\tilde{h}^{ab}\tau_{ab}=\tilde{h}^{00}\tau_{00}+\tilde{h}^{22}\tau_{22}\\\
&=-\frac{{\Lambda}r^{2}-\frac{16\sqrt{k}}{b}m}{\sqrt{\Lambda}{\rm
e}^{\phi_{0}}r^{2}}~{}\Big{(}1-\frac{1}{2r^{2}}\tilde{h}^{(0)^{ab}}\tilde{h}^{(2)}_{ab}+\cdot\cdot\cdot\Big{)}-\frac{2}{l}\\\
&=-\frac{4}{l}+\dots~{}.\end{split}$ (75)

in which (69) has been used. Now, (63) leads to the central charge

$c=\frac{3l}{2G},$ (76)

which has the same form of the central charges associated to the Hilbert-
Einstein action from Chern-Simons theory with gauge group $SO(2,2)$ obtained
in [59], and the $(2+1)$ dimensional gravity from Maxwell and semi-simple
extension of the Poincarè gauge symmetric model obtained in [11].

##  5 $\alpha^{\prime}$-corrected $(2+1)$-dimensional black hole solutions for
two-loop $\beta$-function equations in Gauss-Bonnet RS

The string frame Ricci scalar of the obtained solution for the one-loop
$\beta$-function equations is $R=-\frac{3\Lambda}{2}$. Hence, noting (4), the
solutions are at the high curvature limit $R\alpha^{\prime}\gtrsim 1$, where
the $\alpha^{\prime}$ corrections to the string effective action become
significant. Here, aiming at investigating only the consequences of including
the $\alpha^{\prime}$ corrections and neglecting the quantum loop corrections,
the string coupling is assumed to be weak, i.e. $g_{s}={\rm e}^{-\phi_{0}}\ll
1$. Then, the two-loop $\beta$-function equations (1), (2), and (3),
equivalent to the equations of motion of Gauss-Bonnet
$\alpha^{\prime}$-corrected string effective action (7), are considered to be
solved to find the $\alpha^{\prime}$-corrections to the obtained black hole
solution. In doing so, we start with a string frame metric of the form

$\displaystyle
ds^{2}=-F(r)dt^{2}+{\frac{1}{F(r)}}{{dr}}^{2}+W^{2}(r)d\varphi^{2},$ (77)

and the $H$ filed strength (27), where $W^{2}(r)={\frac{{b^{2}}\,{{\rm
e}^{-2\phi_{0}}}}{16\,{k}}}r^{2}$. Then, considering the two-loop
$\beta$-functions (1)-(3) at Gauss-Bonnet RS, the $(t,t)$, $(\rho,\rho)$ and
$(\varphi,\varphi)$ components of metric $\beta$-function, $(t,\varphi)$
component of $B$-field $\beta$-function, and the dilaton $\beta$-function
reduce to the following coupled set of equations

$\displaystyle\begin{aligned} &{\ln F^{\prime\prime}}+{\ln
F^{\prime}}\,\left({\ln(FW)^{\prime}}+\phi^{\prime}\right)+{\frac{{E}^{2}}{F{W}^{2}}}+\frac{\alpha^{\prime}}{2}\bigg{[}\frac{{E}^{2}}{{W}^{2}}\bigg{(}-2\,{\ln
F^{\prime\prime}}+{\ln
F^{\prime}}\,\left({\ln\left({E}{F^{-2}W^{-3}}\right)^{\prime}}\right)\bigg{)}\\\
&-\frac{1}{F}\bigg{(}{{\ln F^{\prime\prime}}}^{2}+{{\ln
F^{\prime}}}^{2}\left(2\,{\ln F^{\prime\prime}}+{{\ln F^{\prime}}}^{2}+{{\ln
W^{\prime}}}^{2}\right)\bigg{)}+{\frac{3{E}^{4}}{2F{W}^{4}}}\bigg{]}=0,\end{aligned}$
(78)

$\displaystyle\begin{aligned} &2\phi^{\prime\prime}+{\ln
F^{\prime\prime}}+2\,{\ln W^{\prime\prime}}+{{\ln F^{\prime}}}^{2}+{\ln
W^{\prime}}{\ln(FW^{2})^{\prime}}+\phi^{\prime}\ln
F^{\prime}+{\frac{E}{F{W}^{2}}}\\\
&+\frac{\alpha^{\prime}}{2}\bigg{[}\frac{{E}^{2}}{{W}^{2}}\big{(}-{\ln
E^{\prime}}\,{\ln\left(FE^{3}W^{-6}\right)^{\prime}}+{\ln
W^{\prime}}\,{\ln\left(W\,F^{3}\right)^{\prime}}+2\,{{\ln
F^{\prime}}}^{2}-2\,{\ln E^{\prime\prime}}+6\,{\ln W^{\prime\prime}}+2\,{\ln
F^{\prime\prime}}\big{)}\\\ &+{F}\bigg{(}{{\ln F^{\prime}}}^{4}+\left({{\ln
W^{\prime}}}^{2}+2\,{\ln F^{\prime\prime}}\right){{\ln
F^{\prime}}}^{2}+\left(4\,{{\ln W^{\prime}}}^{3}+4\,{\ln W^{\prime}}\,{\ln
W^{\prime\prime}}\right){\ln F^{\prime}}+4\,{{\ln W^{\prime}}}^{4}\\\
&+8\,{{\ln W^{\prime}}}^{2}{\ln W^{\prime\prime}}+{{\ln
F^{\prime\prime}}}^{2}+4\,{{\ln
W^{\prime\prime}}}^{2}\bigg{)}-{\frac{3{E}^{4}}{2{W}^{4}F}}\bigg{]}=0,\end{aligned}$
(79)

$\displaystyle\begin{aligned} &2\,{\ln W^{\prime\prime}}+2\,{\ln
W^{\prime}}\,\left({\ln(FW)^{\prime}}+\phi^{\prime}\right)+{\frac{{E}^{2}}{F{W}^{2}}}-{\alpha^{\prime}}{}\biggl{[}{\frac{{E}^{2}}{{W}^{2}}\left(2\,{\ln
W^{\prime\prime}}+{\ln
W^{\prime}}\,{\ln\left({F^{2}W^{3}E^{-1}}\right)^{\prime}}\right)}\\\
&+2\,F\left(\frac{1}{2}{{\ln F^{\prime}}}^{2}{{\ln W^{\prime}}}^{2}+{\ln
W^{\prime}}\,\left({{\ln W^{\prime}}}^{2}+{\ln W^{\prime\prime}}\right){\ln
F^{\prime}}+\left({{\ln W^{\prime}}}^{2}+{\ln
W^{\prime\prime}}\right)^{2}\right)-{\frac{3{E}^{4}}{4{W}^{4}F}}\biggr{]}=0,\end{aligned}$
(80) $\displaystyle\begin{aligned}
&{\phi^{\prime\prime}}+\left({\ln(FW)^{\prime}}-\frac{4}{5}{\phi^{\prime}}\right){\phi^{\prime}}+{\frac{\Lambda}{5F}}+{\frac{{E}^{2}}{5{W}^{2}F}}-\frac{\alpha^{\prime}}{20}\biggl{[}{W}^{2}F{E}^{2}\bigg{(}-{\ln
E^{\prime}}\,{\ln\left({E^{3}F^{2}}{W^{-4}}\right)^{\prime}}\\\ &+{\ln
F^{\prime}}\,\left(5\,{\ln F^{\prime}}+12\,{\ln W^{\prime}}\right)+9\,{{\ln
W^{\prime}}}^{2}-2\,{\ln E^{\prime\prime}}+5\,{\ln F^{\prime\prime}}+12\,{\ln
W^{\prime\prime}}\bigg{)}-{\frac{25}{4}}{E}^{4}\\\
&+3{W}^{4}{F}^{2}\bigg{(}\left({{\ln F^{\prime}}}^{2}+{\ln
F^{\prime\prime}}\right)^{2}+4\,\left({{\ln W^{\prime}}}^{2}+{\ln
W^{\prime\prime}}\right)^{2}+2\,{\ln W^{\prime}}\,{\ln F^{\prime}}\,\left({\ln
W^{\prime}}\,{\ln(FW^{2})^{\prime}}+2\,{\ln
W^{\prime\prime}}\right)\bigg{)}\biggr{]}=0,\end{aligned}$ (81)
$\displaystyle\begin{aligned} &{\ln W^{\prime}}-{\phi^{\prime}}-{\ln
E^{\prime}}+\frac{\alpha^{\prime}}{16}{{\rm
e}^{-\phi}}\ln(EW^{-1})^{\prime}\biggl{[}\left({\ln W^{\prime}}\,{\ln
F^{\prime}}+2\,{{\ln W^{\prime}}}^{2}+2\,{\ln
W^{\prime\prime}}\right)F-{{\left({{\ln F^{\prime}}}^{2}+{\ln
F^{\prime\prime}}\right){F}^{2}}{{W}^{-2}}}\\\ &-{({-{{\ln
F^{\prime}}}^{2}+{\ln W^{\prime}}\,{\ln F^{\prime}}+2\,{{\ln
W^{\prime}}}^{2}-{\ln F^{\prime\prime}}+2\,{\ln
W^{\prime\prime}}}){{W}^{-2}}}\biggr{]}=0,\end{aligned}$ (82)

where the prime stands for derivation with respect to $r$.

Solving the set of equations of (78)-(82) to obtain the $\alpha^{\prime}$
corrections to the one-loop solution (57), by respecting the geometry of the
spacetime, to which the solutions have had Weyl relation, we find the
following solutions

$\displaystyle
ds^{2}=-F_{0}(r)(1+\alpha^{\prime}c_{3})dt^{2}+{\frac{1+\alpha^{\prime}c_{4}}{F_{0}(r)}}{{dr}}^{2}+{\frac{{b^{2}}\,{{\rm
e}^{-2\phi_{0}}}}{16\,{k}}}r^{2}d\varphi^{2}.$ (83) $\displaystyle
E(r)=-{\frac{\sqrt{\Lambda}{b}}{4\,\sqrt{k}}}{{\rm e}^{-\phi_{0}}}r,$ (84)

in which $F_{0}(r)$ is given by (64) and

$\displaystyle\begin{aligned}
c_{3}&={\frac{\Lambda}{\lambda(92\,\eta-17)}}\left(-46\,\eta-26+\sqrt{2116\,{\eta}^{2}-10304\,\eta+3022}\right),\\\
c_{4}&=\Lambda\eta,\end{aligned}$ (85)

where we have defined $\lambda=\alpha^{\prime}\Lambda$, which according to (4)
equals $\frac{46}{3}$. Also, $\eta\sim 0.08991143357$.101010$\eta$ is the
solution for equation
$58790944\,{\eta}^{4}-5628560\,{\eta}^{3}-349094\,{\eta}^{2}+62952\,\eta-2589=0$.

To calculate the $\alpha^{\prime}$-corrected mass in the quasi-local formalism
using (52), the boundary terms related to the $\alpha^{\prime}$ corrections in
the string effective action need to be considered to have a well-defined
variational principle. For the action (5), only the Gauss-Bonnet term is
associated with a non-vanishing boundary term,111111The ${\rm
e}^{{6}\phi}\tilde{R}^{\alpha\beta\rho\sigma}H_{\alpha\beta\lambda}H_{\rho\sigma}^{~{}~{}\lambda}$
in the effective action (5) needs a boundary term of type
$\int_{\partial{\cal M}}d^{2}x\,n_{\mu}{\rm
e}^{{6}\phi}h^{\sigma\rho}\nabla_{\sigma}\delta
g_{\nu\lambda}H^{\gamma\lambda}_{~{}~{}\rho}H_{\gamma}^{~{}\mu\nu},$ which
vanishes because if $\delta g_{\nu\lambda}$ assumed to be constant on the
boundary, then $h^{\sigma\nu}\nabla_{\sigma}\delta g_{\mu\nu}$ denoting the
projected covariant derivative to the boundary with $h^{\mu\nu}$ must vanish.
which leads to a correction term to the quasi-local stress tensor $\tau_{ab}$
(53), given by [60]

$\displaystyle\tau_{ab}=K_{ab}-\tilde{h}_{ab}K+\frac{1}{l}\tilde{h}_{ab}+\frac{\alpha^{\prime}{\rm
e}^{{2\phi}{}}}{2}\left(Q_{ab}-Q\,h_{ab}\right),$ (86)

written for $(2+1)$-dimensional spacetime, where

$\displaystyle
Q_{ab}=2KK_{ac}K_{b}^{~{}c}-2K_{ac}K^{cd}K_{db}+K_{ab}\left(K_{cd}K^{cd}-K^{2}\right)+2K\tilde{R}_{ab}+\tilde{R}K_{ab}-2K^{cd}\tilde{R}_{cadb}-4\tilde{R}_{ac}K_{b}^{~{}c}.$
(87)

The Riemann tensor of the boundary metric $\tilde{h}_{ab}$ has no non-zero
components. Also, the extrinsic curvature $K_{ab}$ calculated for the metric
(77) in Einstein-frame, obtained via conformal transformation (6), is given by

$\displaystyle K_{ab}={{{{\rm
e}^{\phi_{0}}}\sqrt{F}}{}}\left[\begin{array}[]{cc}-\frac{1}{2}F^{\prime}&0\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr 0&WW^{\prime}\end{array}\right],$ (90)

which leads to $Q_{ab}=0$.121212In fact, the boundary term for Gauss-Bonnet
action is given by  $I_{ct}=-\frac{1}{8\pi
G}\int_{\partial{\cal{M}}}d^{d-1}x\sqrt{\gamma}\frac{d-3}{2(d-4)}\left(\sqrt{\frac{\alpha^{\prime}}{2}(d-3)(d-4)\Lambda{\rm
e}^{\phi}+1}-1\right)L_{GB},$ which actually vanishes when $d=3$. Then, the
similar procedure provided in appendix A can be used to derive the
$\alpha^{\prime}$-corrected mass. The temperature can be calculated via the
definition (48). Furthermore, the $\alpha^{\prime}$-corrected black hole
entropy can be calculated using the Wald’s formula (50)

$\displaystyle\begin{aligned}
S&=-\frac{\pi}{2\kappa_{3}^{2}}\int_{H}dx\,\sqrt{\tilde{\sigma}}\epsilon_{\mu\nu}\epsilon_{\rho\sigma}\bigg{(}\left(1+\frac{1}{2}{\alpha^{\prime}{\rm
e}^{{2\phi}}}\tilde{R}\right)\left(\tilde{g}^{\mu\rho}g^{\nu\sigma}-\tilde{g}^{\mu\sigma}\tilde{g}^{\nu\rho}\right)-\frac{1}{4}\alpha^{\prime}{\rm
e}^{{6\phi}}H^{\mu\nu}_{~{}~{}\xi}H^{\nu\sigma\xi}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\alpha^{\prime}{\rm
e}^{{2\phi}{}}\left(R^{\mu\nu\rho\sigma}-\tilde{g}^{\mu\rho}\tilde{R}^{\nu\sigma}-\tilde{g}^{\nu\sigma}\tilde{R}^{\mu\rho}+\tilde{g}^{\mu\sigma}\tilde{R}^{\nu\rho}+\tilde{g}^{\nu\rho}\tilde{R}^{\mu\sigma}\right)\bigg{)}\\\
&=\frac{\pi}{2\,\kappa_{3}^{2}}\int_{H}dx\sqrt{\tilde{\sigma}}\left(4+\alpha^{\prime}E^{2}W^{-2}\right)\\\
&=\frac{2\pi}{\kappa_{3}^{2}}A_{h}\left(1+\frac{\lambda}{4}\right).\end{aligned}$
(91)

The mass, entropy, and temperature are then given as follows

$\displaystyle
m=\sqrt{\frac{1+\alpha^{\prime}c_{3}}{1+\alpha^{\prime}c_{4}}}m_{0},\quad
T=\sqrt{\frac{1+\alpha^{\prime}c_{3}}{1+\alpha^{\prime}c_{4}}}T_{0},\quad
S=\left(1+\frac{\alpha^{\prime}\Lambda}{4}\right)S_{0},$ (92)

in which $m_{0}$, $T_{0}$, and $S_{0}$ are given by (56), (49), and (51). When
$\alpha^{\prime}\rightarrow 0$, the thermodynamic quantities of
$\alpha^{\prime}$-corrected solutions (92) reduce to those of the leading
order solution.

##  6 Conclusion

We have constructed $(2+1)$-dimensional black hole solutions for string
effective actions at leading order and the first order of $\alpha^{\prime}$,
involving the contributions of dilaton, anti-symmetric $B$-field, and central
charge deficit $\Lambda$ that appeared in the role of a negative cosmological
constant. The basic idea of this work was considering the $(2+1)$-dimensional
spacetime whose two-dimensional space part has the symmetries of two-
dimensional Lie algebra, admitting a homogeneous metric. The homogeneous
surface has been shown to be of constant negative Gaussian curvature type
surfaces. It can be reduced to the hyperbolic geometry case if the Gaussian
curvature is $-\frac{1}{2}$. Investigating the embedding of the homogeneous
surface on $R^{3}$, we found the boundaries of the spacetime.

Inspired by the point that the line element of BTZ black hole is Weyl-related
to the line element of a spacetime with hyperbolic space geometry [50], we
found black hole solutions for leading order of string effective action on the
$(2+1)$-dimensional spacetime that is Weyl-related to the considered
homogeneous spacetime. Calculating the Wald entropy, temperature, and quasi-
local mass, the thermodynamic properties of the solutions have been
investigated. Generally, the leading order solutions and their thermodynamic
quantities are given in terms of the constants, including the central charge
deficit $\Lambda$, the Gaussian scalar curvature dependent parameter $k$,
constant dilaton $\phi_{0}$, the $B$-field $\beta$-function equation’s
integrating constant $b$, and the two integrating constant $c_{1}$ and
$c_{2}$. According to the physical discussion presented in [61], $\phi_{0}$ is
not a physical parameter. The $c_{2}$, on the other hand, has been found to be
functionally related to $c_{1}$ to satisfy the first law of thermodynamics.
But, compared to the BTZ solution, besides the mass and $\Lambda$, the
obtained line element at the leading order of string effective action contains
extra constants including the $k$, $\phi_{0}$, and $b$, which can not be
generally eliminated by redefinition of the coordinates. However, the solution
can recast the form of the BTZ black hole by choosing a particular set of
constants. By calculating the central charge of the CFT at the boundary for
the obtained solutions, expressed in terms of the Newton constant and AdS
radius $l$, it turned out that the central charge has the same form as in the
AdS spacetime related to the Chern-Simons model with gauge group $SO(2,2)$
[59], and $(2+1)$-dimensional gravity with Maxwell and semi-simple extension
of the Poincarè gauge symmetric model [11].

The presence of the central charge deficit $\Lambda$, which is of order
$\alpha^{\prime-1}$, resulted in the high curvature of the leading order
solutions with $R\alpha^{\prime}\gtrsim 1$. This characteristic underlines the
requirement of considering the $\alpha^{\prime}$ corrections in the effective
action or equivalently the higher order $\beta$-function equations. In these
cases, the whole $\alpha^{\prime}$ correction series is prescribed by the
conformal invariance condition to be taken into account. Nevertheless, aimed
at providing a pattern that may be obtained in the presence of
$\alpha^{\prime}$ corrections, we limited our calculation to the first order
$\alpha^{\prime}$ (two-loop) $\beta$-function equations as the equations of
motion of the $\alpha^{\prime}$-corrected Gauss-Bonnet string effective
action, which is valid at the weak string coupling limit. The mass, entropy,
and temperature calculated for the $\alpha^{\prime}$-corrected solutions
showed that the first order $\alpha^{\prime}$-corrections rescaled these
thermodynamic quantities compared to those of the leading order solutions.

Similar to the BTZ black hole, the solutions at both orders $\alpha^{\prime}$
are regular with no curvature singularities. Furthermore, deriving the heat
capacity, it has been shown that the solutions have thermodynamic stability.
Also, considering the singular boundaries of the negative Gaussian curvature
pseudosphere, interpreted as the Hilbert horizons, in relation to the event
horizons of the obtained black hole it turned out that, similar to the BTZ
black hole, the embedded space part of the homogeneous spacetime into $R^{3}$
may end before reaching the event horizons. It has been seen that these two
kinds of horizons can not generally coincide, unless when the mass of the
black hole vanishes. Also, it should be stressed that the metric is different
from those considered in the so-called deformed hyperbolic black holes [62],
whose two-dimensional metric curvature invariants are not constant.

There are several venues worth exploring regarding the solutions presented
here. The family of surfaces of constant negative Gaussian curvature has been
one of the most fruitful classes of surfaces considered in the approaches that
address the interrelation between various branches of physics. Particularly,
they have been used in investigating the measurable effects of QFT in curved
spacetime describing curved graphene, as a $2$-dimensional real system [25,
50, 63, 64, 65]. Their results concern the cases of Beltrami, elliptic, and
hyperbolic pseudosphere. In this context, the new solutions provided in this
work can be considered for the sake of studying the measurable effects, like
the Hawking-Unruh effect, while the QFT and condensed matter concepts are
taken into account.

## Acknowledgment

This research was supported by Azarbaijan Shahid Madani University under Grant
No. $1402/231-2$.

## Appendix A

In this appendix, we present the calculation for quasi-local mass. With
Killing vector $\bar{\xi}^{\mu}=\delta^{\mu}_{t}$, (52) leads to the following
mass expression for the leading order solution (43)

$\displaystyle
m_{0}=\frac{1}{\kappa^{2}}\int_{B}d\varphi\sqrt{\tilde{\sigma}}\sqrt{F_{0}(r)}\,\varepsilon,$
(93)

where, similar to the quasi-local mass of BTZ black holes calculated in [66,
67], $\varepsilon=K-\varepsilon_{0}$ in which $\varepsilon_{0}$ denoting the
zero of the energy, is a function of background metric and

$\displaystyle K=-{W^{\prime}(r)\sqrt{F_{0}(r)}}+\frac{{\rm
e}^{\phi_{0}}}{2}\sqrt{2\Lambda}W(r).$ (94)

Then, the conserved mass is provided by

$\displaystyle
m_{0}=\frac{b}{8\sqrt{k}}\sqrt{F_{0}(r)}\,\left(\sqrt{F_{00}(r)}-\sqrt{F_{0}(r)}+\sqrt{\Lambda}{{\rm
e}^{2\phi_{0}}}(W_{00}(r)-W(r))\right),$ (95)

in which $F_{0}(r)$ and $W(r)$ are given by (44) and (45), respectively, and
$F_{00}(r)$ and $W_{0}(r)$ are the background metric functions, given by

$\displaystyle{F_{00}(r)}=\frac{1}{4}{{r}\left(\Lambda\,{r}-2\,c_{2}\Lambda+8\,\sqrt{k}c_{1}\right)},\quad
W_{0}(r)=\frac{{{\rm
e}^{-\phi_{0}}}}{4}{\frac{{b}}{\sqrt{k\,\Lambda}}}({r\,(-2\,\Lambda\,c_{2}+\Lambda\,r+8\,\sqrt{k}{c_{1}})}).$
(96)

Accordingly, (95) yields the mass

$\displaystyle
m_{0}={\frac{{b}\,c_{2}}{32\sqrt{k}}}\left(-c_{2}\Lambda+8\,\sqrt{k}{c_{1}}\right)-{\frac{{b}\,\sqrt{2}{{\rm
e}^{{\phi_{0}}}}}{32\Lambda\,\sqrt{k}}}\left(c_{2}\Lambda-4\,\sqrt{k}c_{1}\right)^{2}.$
(97)

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