11institutetext: Instituto de Ciencias Exactas y Naturales (ICEN), Universidad
Arturo Prat,
Avenida Arturo Prat Chacón 2120, 1110939, Iquique, Chile22institutetext:
Science Institute, University of Iceland,
Dunhaga 3, 107 Reykjavík, Iceland

# Thermodynamics of the three-dimensional black hole with torsion

Luis Avilés 2 Diego Hidalgo 1 Omar Valdivia<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The stationary black hole solution of a Chern-Simons model based on the semi-
simple extension of the Poincaré gauge group is studied. The solution
resembles the metric properties of the BTZ geometry but contains, in addition,
non-vanishing torsion. The global structure of spacetime is characterized by
three conserved charges: two associated with the mass and angular momentum and
one extra constant triggered by spacetime torsion. Consequently, we show that
the entropy deviates from the standard Bekenstein-Hawking value and discuss
the implications of torsional charges in the context of black hole
thermodynamics.

††preprint:

##  1 Introduction

Thermodynamic properties of black holes are expected to give us important
clues in understanding the quantum nature of gravity PhysRevD.7.2333 ;
Hawking:1972aa ; Bardeen:1973aa . The development of black hole thermodynamics
has involved many significant results, such as the intimate connection among
the laws of gravitation with the laws of thermodynamics Jacobson:1995ab ;
Padmanabhan:2009vy , intrinsic relationships between black hole entropy and
Noether charges associated with diffeomorphism symmetry Wald:1993nt , and
thermal phase transitions due to the presence of a negative cosmological
constant Hawking:1983aa . In this regard, the discovery of the three-
dimensional black hole by Bañados, Teitelboim, and Zanelli (BTZ)
Banados:1992wn ; Banados:1992gq  was of particular importance since it allowed
us to investigate in a more straightforward way the nature of black holes and
their relationship to basic foundations of quantum physics. Further
developments of three-dimensional black holes can be found in Carlip:1995qv ;
Martinez:1996gn ; Carlip:1998wz ; Cardenas:2014kaa ; Perez:2015kea ;
Erices:2017nta  and references therein.

It is well known that three-dimensional gravity with a negative cosmological
constant can be formulated as a Chern-Simons (CS) theory for the anti-de
Sitter group $SO(2,2)$ ACHUCARRO198689 ; WITTEN198846 . The gauge field takes
values in the Lie algebra $\mathfrak{so}(2,2)$ and is written in terms of the
dreibein $e^{a}(x)$ and the spin connection $\omega^{a}(x)$, as independent
degrees of freedom. The equations of motion imply constant (Riemannian)
curvature and vanishing torsion, so the theory is purely metric on-shell.
However, there are interesting trends motivated by gauge-theoretic arguments
for considering gravity models based on Riemann-Cartan geometry containing
both metric and affine properties of spacetime as generic ingredients of the
gravitational dynamics (see, for instance, RevModPhys.48.393 ; Hehl:1994ue ;
Blagojevic:2013xpa , and references therein). In three-dimensions, Riemann-
Cartan gravity is studied in MIELKE1991399 . Further development along these
lines led to several interesting results, such as a stationary black hole
solution Garcia:2003nm , asymptotic symmetries, thermodynamic properties, and
microscopic description of the entropy Blagojevic:2006jk ; Blagojevic:2003uc ;
Blagojevic:2006hh . An important observation in this context is that entropy
differs from the Bekenstein-Hawking result by an additional contribution,
which depends on the presence of spacetime torsion.

In this work, we study the thermodynamic properties of another black hole
solution with torsion, originally reported in Hoseinzadeh:2014bla . The theory
behind this solution is a three-dimensional CS model genuinely invariant under
the semi-simple extension of the Poincaré gauge symmetry Soroka:2006aj . This
semi-simple enlargement is equivalent to an S-expansion procedure of the AdS
algebra in any dimension and is often referred to as AdS-Lorentz (AdSL) Lie
algebra Izaurieta:2006zz ; Diaz:2012zza ; Fierro:2014lka . The AdSL group was
introduced to describe the symmetries of a particle moving in AdS spacetime in
the presence of a constant electromagnetic field Gomis:2009dm . Moreover, AdSL
symmetry is also related to the so-called Maxwell group Schrader:1972zd
through an Inönü-Wigner contraction. In the context of CS theories of gravity,
expanded symmetry algebras have many interesting properties. For instance, any
expanded algebra may include new generators, increasing the gauge symmetry of
the particular theory. In such cases, it also means including compensating
gauge fields apart from the dreibein and spin connection. Interestingly, in
three-dimensions, this extra gauge field codifies the affine structure of
geometry so that gauge invariance is preserved.111There are other interesting
examples of black hole solutions for theories with extended symmetries, such
as higher spin systems Blencowe:1988gj ; Bergshoeff:1989ns ; Vasiliev:1995dn ;
Perez:2012cf .  Diaz:2012zza ; Salgado:2014qqa .

Here, we start by highlighting the fact that three-dimensional AdSL gravity
naturally contains torsion. Moreover, we show that the stationary solution
Hoseinzadeh:2014bla  is characterized by the standard constants of motion
$M,J,$ of Einstein’s gravity, and a new constant $b$ related to the presence
of spacetime torsion. The presence of this constant generalizes the “exotic”
thermodynamics analysis of Townsend and Zhang Townsend:2013ela  by including
the torsion in both the global charge and entropy formulas. Specifically, we
present the modifications in entropy caused by torsion and the additional term
in the first law, which comes from the new global charge. For completeness, we
display the Smarr formula with the effects of torsion taken into
consideration. For this purpose, we employ the methods used in the CS
formalism proposed in Carlip:1994gc ; Henneaux:2013dra ; Bunster:2014mua .

This paper is organized as follows. In Section 2.1, we review the AdSL
symmetry leading properties and the construction of a CS principle action. In
Section 2.2, we explain the stationary solution of this theory and review the
BMS-like gauge for the field content that allows solving the field equations.
In Section 2.3, we review the corresponding asymptotic symmetry and the
derivation of the asymptotic gauge connection useful for the computation of
the entropy. The study of the computation of the global charges, entropy,
first law, and Smarr formula is developed in Section 3. Finally, Section 4 is
devoted to discussing the results and future directions.

##  2 Three-dimensional AdS-Lorentz gravity

This section briefly reviews the formulation of a CS theory of gravity quasi
invariant under the AdSL group Diaz:2012zza . Some subtleties in deriving the
black hole solution of this gravity theory are highlighted.

###  2.1 Formulation of AdS-Lorentz CS gravity

In $(2+1)$-spacetime dimensions, the AdSL algebra is generated by spacetime
rotations $J_{a}$, spacetime translations $P_{a}$, and a new type of
generators $Z_{a}$, originally introduced in Schrader:1972zd ; Bacry:1970aa .
Commutation relations among these generators are given by

$\left[J_{a},J_{b}\right]=\epsilon_{abc}J^{c}\,,$ |  |  $\left[Z_{a},Z_{b}\right]=\epsilon_{abc}Z^{c}\,$,
---|---|---
$\left[J_{a},P_{b}\right]=\epsilon_{abc}P^{c}\,,$ |  |  $\left[J_{a},Z_{b}\right]=\epsilon_{abc}Z^{c}$ ,
$\left[P_{a},P_{b}\right]=\epsilon_{abc}Z^{c}$ , |  |  $\left[Z_{a},P_{b}\right]=\epsilon_{abc}P^{c}$ ,
(1)

where $\epsilon_{abc}$ is the Levi-Civita symbol, such that
$\epsilon_{012}=1,\,\epsilon^{012}=-1$, and $a=0,1,2$ is a Lorentz index,
raised and lowered with the Minkowski metric
$\eta_{ab}=\text{diag}(-1,+1,+1)$. As presented in (1), is clear that
generators $Z^{a}$ complete the semi-simple extension of the Poincaré algebra
$\mathfrak{iso}(1,2)$ generated by rotations $J_{a}$ and translations $P_{a}$.
Another observation is that (1) can also be written as the direct sum
$\mathfrak{so}(2,2)\oplus\mathfrak{so}(2,1)$, reason why it is usually
referred to as AdSL algebra. In fact, defining

$\tilde{J}_{a}=Z_{a}\,,\hskip 28.45274pt\tilde{P}_{a}=P_{a}\,,\hskip
28.45274pt\tilde{Z}_{a}=J_{a}-Z_{a}\,,$ (2)

the direct sum structure of AdS and Lorentz algebras is made explicit (see
Diaz:2012zza ; Salgado:2014qqa ). This last observation also implies that (1)
is equivalent to the direct sum of three copies of the Lorentz algebra.
Indeed, three commuting sets of $\mathfrak{so}(2,1)$ generators

$\left[J_{a}^{+},\,J_{b}^{+}\right]=\epsilon_{abc}(J^{+})^{c}\,,\hskip
34.14322pt\left[J_{a}^{-},\,J_{b}^{-}\right]=\epsilon_{abc}(J^{-})^{c}\,,\hskip
34.14322pt\left[\hat{J}_{a},\,\hat{J}_{b}\right]=\epsilon_{abc}\hat{J}^{c}\,,$
(3)

reproduce (1) by means of redefinitions

$Z_{a}=J_{a}^{+}+J_{a}^{-}\,,\hskip
34.14322ptP_{a}=J_{a}^{+}-J_{a}^{-}\,,\hskip
34.14322ptJ_{a}=\hat{J}_{a}+J_{a}^{+}+J_{a}^{-}\,.$ (4)

The most general invariant tensor associated to (1) has the following
components

$\displaystyle\left\langle J_{a}\,,J_{b}\right\rangle=\alpha_{0}\eta_{ab}\,,$
$\displaystyle\left\langle P_{a}\,,P_{b}\right\rangle=\alpha_{2}\eta_{ab}\,,$
(5a) $\displaystyle\left\langle
J_{a}\,,P_{b}\right\rangle=\alpha_{1}\eta_{ab}\,,$
$\displaystyle\left\langle Z_{a}\,,Z_{b}\right\rangle=\alpha_{2}\eta_{ab}\,,$
(5b) $\displaystyle\left\langle
J_{a}\,,Z_{b}\right\rangle=\alpha_{2}\eta_{ab}\,,$
$\displaystyle\left\langle Z_{a}\,,P_{b}\right\rangle=\alpha_{1}\eta_{ab}\,,$
(5c)

where $\alpha_{0},$ $\,\alpha_{1},\text{ and }\alpha_{2}$ are real arbitrary
constants. This bilinear form is non-degenerate if $\alpha_{1}\neq\alpha_{2}$
and $\alpha_{2}\neq\alpha_{0}$. The Cartan sub-algebra is a three-dimensional
space which suggest the presence of three independent Casimir operators
constructed from these generators Diaz:2012zza ; Fierro:2014lka .

A quasi-invariant gravity theory with AdSL symmetry can be constructed by
evaluating the CS action

$I\left[A\right]=\frac{k}{4\pi}\int_{\scriptscriptstyle\mathcal{M}}\left\langle
A\wedge\mathrm{d}A+\frac{2}{3}\,A\wedge A\wedge A\right\rangle\,,$ (6)

for the connection one-form $A=A_{\mu}\mathop{}\\!\mathrm{d}x^{\mu}$, taking
values in the algebra (1) spanned by $\\{P_{a},J_{a},Z_{a}\\}$

$A=\frac{1}{\ell}e^{a}P_{a}+\omega^{a}J_{a}+\sigma^{a}Z_{a}\,.$ (7)

As usual, $e^{a}(x)$ denote the one-form dreibein, $\omega^{a}(x)$ is the one-
form spin connection, and the one-form $\sigma^{a}(x)$ is the compensating
gauge field associated with the non-abelian generator $Z_{a}$. Here, $\ell$
denotes the AdS radius and $k$ is the CS level related to the Newton’s
constant $G$ according to $k=\frac{\ell}{4G_{N}}$. In terms of the gauge field
components, the action takes the form222Henceforth, the wedge product $\wedge$
between differential forms is understood, i.e.
$\omega^{a}\,e^{b}=\omega^{a}\wedge e^{b}=-e^{b}\wedge\omega^{a}.$

$I_{\scriptscriptstyle\mathrm{AdSL}}[e,\omega,\sigma]=\frac{k}{4\pi}\int_{\scriptscriptstyle\mathcal{M}}\left(L_{\text{EC}}+L_{\text{exotic}}+L_{\sigma}\right)\,,$
(8)

with

$\displaystyle L_{\mathrm{EC}}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{1}}{\ell}\left(2R_{a}e^{a}+\frac{1}{3\ell^{2}}\epsilon^{abc}e_{a}e_{b}e_{c}\right)\,,$
(9) $\displaystyle L_{\mathrm{exotic}}$ $\displaystyle=$
$\displaystyle\alpha_{0}\left(\omega^{a}\mathop{}\\!\mathrm{d}\omega_{a}+\frac{1}{3}\epsilon^{abc}\omega_{a}\omega_{b}\omega_{c}\right)+\frac{\alpha_{2}}{\ell^{2}}T^{a}e_{a}\,,$
(10) $\displaystyle L_{\sigma}$ $\displaystyle=$
$\displaystyle\frac{2\alpha_{1}}{\ell}F^{a}e_{a}+\alpha_{2}\left(2R^{a}\sigma_{a}+\frac{1}{3}\epsilon^{abc}\sigma_{a}\sigma_{b}\sigma_{c}+\mathrm{D}_{\omega}\sigma^{a}\sigma_{a}+\frac{1}{\ell^{2}}\epsilon^{abc}e_{a}\sigma_{b}e_{c}\right)\,.$
(11)

The two-form curvatures associated to the gauge fields $\omega^{a}$, $e^{a}$
and $\sigma^{a}$ are defined by

$R^{a}=\mathop{}\\!\mathrm{d}\omega^{a}-\frac{1}{2}\epsilon^{abc}\omega_{b}\omega_{c}\,,\hskip
28.45274ptT^{a}=\mathrm{D}_{\omega}e^{a}\,,\hskip
28.45274ptF^{a}=\mathrm{D}_{\omega}\sigma^{a}+\frac{1}{2}\epsilon^{abc}\sigma_{b}\sigma_{c}\,,$
(12)

where the covariant derivative is defined with respect to the full spin
connection as
$\mathrm{D}_{\omega}e^{a}\equiv\mathrm{d}e^{a}-\epsilon^{abc}\omega_{b}e_{c}$.
The total Lagrangian in (8) consists of three pieces: the three-form
$L_{\text{EC}}$ is the first order generalization of the Einstein-Hilbert term
with a negative cosmological constant. The piece $L_{\text{exotic}}$ contains
the exotic Lorentz-Chern-Simons term DESER1982372  and a torsional
contribution whose exterior derivative is locally related to the Nieh-Yan
topological invariant Nieh:1981ww ; Chandia:1997jf . The Lagrangian three-form
$L_{\sigma}$ is the extra contribution coming from the presence of the gauge
field $\sigma^{a}$.

Functional variation of (8) leads to the field equations

$\displaystyle\delta\omega^{a}$ $\displaystyle:$ $\displaystyle\qquad
0=\alpha_{0}R_{a}+\frac{\alpha_{1}}{\ell}\left(T_{a}+\epsilon_{abc}\sigma^{b}e^{c}\right)+\alpha_{2}\left(F_{a}+\frac{1}{2\ell^{2}}\,\epsilon_{abc}e^{b}e^{c}\right)\,,$
$\displaystyle\vskip 6.0pt plus 2.0pt minus 2.0pt\delta e^{a}$
$\displaystyle:$ $\displaystyle\qquad
0=\frac{\alpha_{1}}{\ell}\left(R_{a}+F_{a}+\frac{1}{2\ell^{2}}\,\epsilon_{abc}e^{b}e^{c}\right)+\frac{\alpha_{2}}{\ell^{2}}\left(T_{a}+\epsilon_{abc}\sigma^{b}e^{c}\right)\,,$
(13) $\displaystyle\vskip 6.0pt plus 2.0pt minus 2.0pt\delta\sigma^{a}$
$\displaystyle:$ $\displaystyle\qquad
0=\frac{\alpha_{1}}{\ell}\,\left(T_{a}+\epsilon_{abc}\sigma^{b}e^{c}\right)+\alpha_{2}\left(R_{a}+F_{a}+\frac{1}{2\ell^{2}}\,\epsilon_{abc}e^{b}e^{c}\right)\,,$

so in principle the space of solutions contains different branches controlled
by the possible values of the coupling constants $\alpha_{0}$, $\alpha_{1}$
and $\alpha_{2}$. However, requiring a non-vanishing determinant of (13), the
equations of motion reduce to

$\displaystyle R_{a}$ $\displaystyle=$ $\displaystyle 0\,,$ (14a)
$\displaystyle T_{a}+\epsilon_{abc}\sigma^{b}e^{c}$ $\displaystyle=$
$\displaystyle 0\,,$ (14b) $\displaystyle
F_{a}+\frac{1}{2\ell^{2}}\,\epsilon_{abc}e^{b}e^{c}$ $\displaystyle=$
$\displaystyle 0\,.$ (14c)

This is, in fact, the generic case where the field equations completely
determine the evolution of the system. Notice that the full dynamics of the
theory is captured by the $\alpha_{2}$ term in the action (8), regardless of
the choices for $\alpha_{0}$ and $\alpha_{1}$. Otherwise, $\omega^{a}$ would
be a redundant gauge field of the theory, and therefore the bilinear form (5)
would be degenerate. Another important observation is that, in this particular
dimension, the gauge field $\sigma^{a}$ is a source for torsion. This can be
seen explicitly in Eq.(14b) by splitting the Lorentz connection $\omega^{a}$
in terms of its torsion-free $(\mathring{\omega}^{a})$ and contorsion part
$(\kappa^{a})$ as $\omega^{a}=\mathring{\omega}^{a}+\kappa^{a}$ and then one
intermediately arrives to $\sigma^{a}=-\kappa^{a}$. This last result makes the
theory particularly appealing, for instance, when studying black hole
solutions and their thermodynamics, as we will see later.

###  2.2 Stationary black hole solution

Three-dimensional AdSL gravity admits a stationary black hole solution with
rotation Hoseinzadeh:2014bla . The line element is given by333Henceforth, we
will use the convention $8G_{\scriptscriptstyle N}=1$ with
$G_{\scriptscriptstyle N}$ being the Newton’s constant.

$\mathop{}\\!\mathrm{d}s^{2}=-N^{2}\mathop{}\\!\mathrm{d}t^{2}+\frac{dr^{2}}{N^{2}}+r^{2}\left(\mathop{}\\!\mathrm{d}\varphi+N_{\varphi}\mathop{}\\!\mathrm{d}t\right)^{2}\,,$
(15)

with the $r$-dependent functions

$N^{2}=-M+\frac{J^{2}}{4r^{2}}+\frac{r^{2}}{\ell^{2}}\,,\qquad
N_{\varphi}=-\frac{J}{2r^{2}}\,,$ (16)

and $M$ and $J$ are integration constants. This means the BTZ tetrad444From
now on, we will use bar notation both for the diagonal Minkowski metric
$\overline{\eta}=\text{diag}(-1,1,1)$ and gauge fields associated.

$\displaystyle\bar{e}^{0}$ $\displaystyle=N\mathop{}\\!\mathrm{d}t\,,$
$\displaystyle\bar{e}^{1}$ $\displaystyle=N^{-1}\mathop{}\\!\mathrm{d}r\,,$
(17) $\displaystyle\bar{e}^{2}$
$\displaystyle=r\left(\mathop{}\\!\mathrm{d}\varphi+N_{\varphi}\mathop{}\\!\mathrm{d}t\right)\,,$

is a solution while the torsionless part of the spin connection
$\mathring{\bar{\omega}}^{a}$ is solved from
$\mathrm{D}_{\mathring{\omega}}{e}^{a}=0$

$\displaystyle\mathring{\bar{\omega}}^{0}$
$\displaystyle=N\mathop{}\\!\mathrm{d}\varphi\,,$
$\displaystyle\mathring{\bar{\omega}}^{1}$
$\displaystyle=-\frac{N_{\varphi}}{N}\mathop{}\\!\mathrm{d}r\,,$ (18)
$\displaystyle\mathring{\bar{\omega}}^{2}$
$\displaystyle=r\left(\frac{\mathop{}\\!\mathrm{d}t}{\ell^{2}}+N_{\varphi}\mathop{}\\!\mathrm{d}\varphi\right)\,.$

Expressions (17) and (18) determine completely the metric part of the
solution. On the other hand, inserting $\sigma^{a}=-\kappa^{a}$ and (17)-(18)
into (14c) one finds the following components for the contorsion tensor

$\displaystyle\bar{\kappa}^{0}$
$\displaystyle=aC\mathop{}\\!\mathrm{d}t+B\mathop{}\\!\mathrm{d}r+(C-N)\mathop{}\\!\mathrm{d}\varphi\,,\vskip
6.0pt plus 2.0pt minus 2.0pt$ $\displaystyle\bar{\kappa}^{1}$
$\displaystyle=aF\mathop{}\\!\mathrm{d}t+\left(E+\frac{N_{\varphi}}{N}\right)\mathop{}\\!\mathrm{d}r+F\mathop{}\\!\mathrm{d}\varphi\,\,,$
(19) $\displaystyle\bar{\kappa}^{2}$
$\displaystyle=\left(aI-\frac{r}{\ell^{2}}\right)\mathop{}\\!\mathrm{d}t+H\mathop{}\\!\mathrm{d}r+\left(I-rN_{\varphi}\right)\mathop{}\\!\mathrm{d}\varphi\,,$

where we have defined the $r$-dependent functions

$\displaystyle B$ $\displaystyle\equiv$
$\displaystyle\frac{1}{I}\frac{\mathop{}\\!\mathrm{d}F}{\mathop{}\\!\mathrm{d}r}+\frac{HC}{I}\,,\vskip
6.0pt plus 2.0pt minus 2.0pt$ $\displaystyle C$ $\displaystyle\equiv$
$\displaystyle\sqrt{F^{2}+I^{2}+b\,},\vskip 6.0pt plus 2.0pt minus 2.0pt$ (20)
$\displaystyle E$ $\displaystyle\equiv$
$\displaystyle\frac{1}{C}\left(\frac{\mathop{}\\!\mathrm{d}I}{\mathop{}\\!\mathrm{d}r}+\frac{F}{I}\frac{\mathop{}\\!\mathrm{d}F}{\mathop{}\\!\mathrm{d}r}\right)+\frac{FH}{I}\,.$

Therefore, the solution depends on three arbitrary functions $I=I(r)$,
$F=F(r)$ and $H=H(r)$ and two additional integration constants $a$ and $b$.

Let us write down this solution in the BMS gauge. To this end, we introduce
the off-diagonal Minkowski metric

$\eta_{ab}=\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\
0&0&1\end{array}\right)\,,$ (21)

and solve field equations in the direct sum basis (2). By doing so, the
torsionless fields $\tilde{e}^{a}$ and $\tilde{\omega}^{a}$ can be set to obey
standard pure gravity boundary conditions, and the gauge field
$\tilde{\sigma}^{a}$ is simply a flat Lorentz connection Concha:2018jjj . In
the BMS gauge Barnich:2013yka ; Barnich:2012aw , the solution is parametrized
by the local coordinates $x^{\mu}=(u,r,\phi)$, where $u=t-f(r)$ and
$\phi=\varphi+g(r)$ correspond to the retarded time coordinate and new angular
coordinate, respectively, with

$\frac{\mathop{}\\!\mathrm{d}f(r)}{\mathop{}\\!\mathrm{d}r}=N^{-2}\,,~{}~{}~{}~{}\frac{\mathop{}\\!\mathrm{d}g(r)}{\mathop{}\\!\mathrm{d}r}=-\frac{N_{\varphi}}{N^{2}}\,.$
(22)

The boundary is located at $r=const.$ With this new basis and change of
variables, the metric (15) reads

$\mathop{}\\!\mathrm{d}s^{2}=\eta_{ab}\tilde{e}^{a}\tilde{e}^{b}=2\tilde{e}^{0}\tilde{e}^{1}+\left(\tilde{e}^{2}\right)^{2}\,,$
(23)

and the dreibein can be chosen as

$\displaystyle\tilde{e}^{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2\ell^{2}}\left(\ell^{2}\,\mathcal{M}(u,\phi)-r^{2}\right)\mathop{}\\!\mathrm{d}u-\mathop{}\\!\mathrm{d}r+\frac{1}{2}\mathcal{N}(u,\phi)\mathop{}\\!\mathrm{d}\phi\,,$
(24a) $\displaystyle\tilde{e}^{1}$ $\displaystyle=$
$\displaystyle\mathop{}\\!\mathrm{d}u\,,$ (24b) $\displaystyle\tilde{e}^{2}$
$\displaystyle=$ $\displaystyle r\,\mathop{}\\!\mathrm{d}\phi\,.$ (24c)

Here, $\mathcal{M}$ and $\mathcal{N}$ are arbitrary functions to be fixed by
the Einstein field equations. By considering the AdSL algebra as the direct
sum $\mathfrak{so}(2,2)\oplus\mathfrak{so}(2,1)$, the spin connection
$\tilde{\omega}^{a}$ is torsionless and therefore is given by

$\displaystyle\tilde{\omega}^{0}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2\ell^{2}}\mathcal{N}(u,\phi)\mathop{}\\!\mathrm{d}u+\frac{1}{2\ell^{2}}\left(\ell^{2}\,\mathcal{M}(u,\phi)-r^{2}\right)\mathop{}\\!\mathrm{d}\phi\,,$
(25a) $\displaystyle\tilde{\omega}^{1}$ $\displaystyle=$
$\displaystyle\mathop{}\\!\mathrm{d}\phi\,,$ (25b)
$\displaystyle\tilde{\omega}^{2}$ $\displaystyle=$
$\displaystyle\frac{r}{\ell^{2}}\mathop{}\\!\mathrm{d}u\,.$ (25c)

Finally, the gauge field $\tilde{\sigma}^{a}$ acquires the form

$\displaystyle\tilde{\sigma}^{0}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2\ell^{2}}\left(\ell^{2}\,\mathcal{M}(u,\phi)-\mathcal{R}(u,\phi)\right)\mathop{}\\!\mathrm{d}\phi\,,$
(26a) $\displaystyle\tilde{\sigma}^{1}$ $\displaystyle=$
$\displaystyle\mathop{}\\!\mathrm{d}\phi\,,$ (26b)
$\displaystyle\tilde{\sigma}^{2}$ $\displaystyle=$ $\displaystyle 0\,.$ (26c)

with $\mathcal{R}$ another arbitrary function. By inserting this BMS gauge
solution into the field equations (14), we get (see, for instance,
Barnich:2013yka )

$\mathcal{\dot{M}}(u,\phi)\ =\
\frac{1}{\ell^{2}}\mathcal{N}^{\prime}(u,\phi)\;,\quad\quad\mathcal{\dot{N}}(u,\phi)\
=\ \mathcal{M}^{\prime}(u,\phi)\;,$ (27)

where dot and prime denote derivatives with respect to the coordinates $u$ and
$\phi$, respectively, whose solutions are

$\mathcal{M}\ =\ \frac{\mathcal{L}^{+}+\mathcal{L}^{-}}{\ell}\,,\hskip
28.45274pt\mathcal{N}\ =\ \mathcal{L}^{+}-\mathcal{L}^{-}\;,$ (28)

with $\mathcal{L}^{\pm}=\mathcal{L}^{\pm}(x^{\pm})$ and
$x^{\pm}=\phi\pm\frac{1}{\ell}u$. The field equation associated with the gauge
field $\tilde{\sigma}^{a}$ leads to the extra condition

$\dot{\mathcal{R}}(u,\phi)=\mathcal{N}^{\prime}(u,\phi)\,,$ (29)

which yields

$\mathcal{R}=\ell\,\left(\mathcal{L}^{+}+\mathcal{L}^{-}-2\mathcal{L}\right)\,,\quad\quad\mathcal{L}=\mathcal{L}(\phi).$
(30)

Now, the transformation in (2) induces the following relationship among the
gauge fields of the BMS and the original AdSL basis

$e^{a}=\tilde{e}^{a}\,,\hskip 28.45274pt\omega^{a}=\tilde{\sigma}^{a}\,,\hskip
28.45274pt\sigma^{a}=\tilde{\omega}^{a}-\tilde{\sigma}^{a}\,.$ (31)

Hence, the solution for AdSL CS gravity is given by

$\displaystyle e^{0}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2\ell^{2}}\left(\ell^{2}\,\mathcal{M}-r^{2}\right)\mathop{}\\!\mathrm{d}u-\mathop{}\\!\mathrm{d}r+\dfrac{1}{2}\mathcal{N}\mathop{}\\!\mathrm{d}\phi\,,\hskip
14.22636pte^{1}=\mathop{}\\!\mathrm{d}u\,,\quad
e^{2}=r\mathop{}\\!\mathrm{d}\phi\,,$ (32) $\displaystyle\omega^{0}$
$\displaystyle=$
$\displaystyle\dfrac{1}{2\ell^{2}}\left(\ell^{2}\,\mathcal{M}-\mathcal{R}\right)\mathop{}\\!\mathrm{d}\phi\,,\hskip
85.35826pt\omega^{1}=\mathop{}\\!\mathrm{d}\phi\,,\hskip
11.38092pt\omega^{2}=0\,,$ (33) $\displaystyle\kappa^{0}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{2\ell^{2}}\mathcal{N}\mathop{}\\!\mathrm{d}u+\dfrac{1}{2\ell^{2}}\left(r^{2}-\mathcal{R}\right)\mathop{}\\!\mathrm{d}\phi\,,\hskip
39.83368pt\kappa^{1}=0\,,\hskip
17.07182pt\kappa^{2}=-\dfrac{r}{\ell^{2}}\mathop{}\\!\mathrm{d}u\,.$ (34)

It is important to remark that the integration constant $a$ for this gauge
vanishes. Moreover, it is direct to show that enforcing the functions
$\mathcal{M}$, $\mathcal{N}$ and $\mathcal{R}$ to be the constants

$\mathcal{M}\left(u,\phi\right)=M\,,\hskip
28.45274pt\mathcal{N}\left(u,\phi\right)=-J\,,\hskip
28.45274pt\mathcal{R}\left(u,\phi\right)=\ell^{2}\left(b+M\right)\,,$ (35)

and the arbitrary functions to be fixed by

$I(r)=rN_{\varphi}\,,\hskip
28.45274ptH(r)=\frac{rN^{2}_{\varphi}}{N^{2}}\,+\,\frac{1}{N}\frac{\mathop{}\\!\mathrm{d}N}{\mathop{}\\!\mathrm{d}r}\,,\hskip
28.45274ptF(r)=\frac{b+r^{2}N^{2}_{\varphi}-N^{2}}{2N}\,,$ (36)

we connect the results of the BMS basis (23) with the ones written in the AdSL
basis of the stationary solution (15). In conclusion, the BMS gauge basis
fixes the form of the arbitrary functions $I,H,F$ in terms of the BTZ
geometric data. Then, the phase space of this family of solutions is spanned
by the integration constants $M,J$ and $b$.

###  2.3 Asymptotic dynamics

The thermodynamic properties of a black hole can be ascertained by assessing
the Euclidean action associated with the black hole solution. The
thermodynamic functions mostly get contributions from the boundary terms.
Moreover, the extensive thermodynamic quantities, the global charges defined
at infinity, depend on asymptotic symmetries. Therefore, it is essential to
understand the behavior of dynamic fields in the asymptotic region.

In order to calculate the thermodynamics of the BTZ type solution (15), we
have to consider suitable fall-off conditions for the gauge field at infinity.
The gauge connection $A$ associated with the gauge fields (32) written in the
$\\{J_{a},P_{a},Z_{a}\\}$ basis is

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{1}{2\ell^{2}}\,\left(\mathcal{N}\left(u,\phi\right)\mathop{}\\!\mathrm{d}u+\left(\mathcal{R}\left(u,\phi\right)-r^{2}\right)\mathop{}\\!\mathrm{d}\phi\right)Z_{0}+\frac{r}{\ell^{2}}\mathop{}\\!\mathrm{d}uZ_{2}$
(37)
$\displaystyle+\,\left(\frac{1}{2\ell^{2}}\left(\ell^{2}\mathcal{M}\left(u,\phi\right)-r^{2}\right)\mathop{}\\!\mathrm{d}u-\mathop{}\\!\mathrm{d}r+\frac{1}{2}\mathcal{N}\left(u,\phi\right)\mathop{}\\!\mathrm{d}\phi\right)\frac{P_{0}}{\ell}\,+\,\frac{P_{1}}{\ell}\mathop{}\\!\mathrm{d}u\,+\,r\,\mathop{}\\!\mathrm{d}\phi
P_{2}$
$\displaystyle+\,\left(\frac{1}{2\ell^{2}}\left(\ell^{2}\,\mathcal{M}\left(u,\phi\right)-\mathcal{R}\left(u,\phi\right)\right)\mathop{}\\!\mathrm{d}\phi\right)J_{0}+\mathop{}\\!\mathrm{d}\phi
J_{1}\,.$

It is possible to remove the radial coordinate $r$ by performing an
appropriate gauge transformation on (37). Indeed, doing

$A=\theta^{-1}\mathop{}\\!\mathrm{d}\theta+\theta^{-1}a\,\theta\,,$ (38)

with the group element $\theta=e^{-\frac{r}{\ell}P_{0}}$. Then, the asymptotic
gauge connection is

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{1}{2\ell^{2}}\,\left(\mathcal{N}\left(u,\phi\right)\mathop{}\\!\mathrm{d}u\,+\,\mathcal{R}\left(u,\phi\right)\mathop{}\\!\mathrm{d}\phi\right)Z_{0}\,+\,\frac{1}{2\ell}\,\left(\mathcal{M}\left(u,\phi\right)\mathop{}\\!\mathrm{d}u+\mathcal{N}\left(u,\phi\right)\mathop{}\\!\mathrm{d}\phi\right)P_{0}$
(39) $\displaystyle+\
\frac{\mathop{}\\!\mathrm{d}u}{\ell}P_{1}+\frac{1}{2\ell^{2}}\left(\ell^{2}\,\mathcal{M}\left(u,\phi\right)-\mathcal{R}\left(u,\phi\right)\right)\mathop{}\\!\mathrm{d}\phi
J_{0}+\mathop{}\\!\mathrm{d}\phi J_{1}\,.$

Since we are interested in the thermodynamic properties of the solution (32),
it is convenient to rewrite (39) in the basis
$\\{J_{a}^{+},J_{a}^{-},\hat{J}_{a}\\}$, then

$a=\left(a^{+}_{u}+a^{-}_{u}\right)\frac{\mathop{}\\!\mathrm{d}u}{\ell}\ +\
\left(a_{\phi}^{+}+a_{\phi}^{-}+\hat{a}_{\phi}\right)\mathop{}\\!\mathrm{d}\phi\,,$
(40)

where

$\displaystyle a_{\phi}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{\ell}\mathcal{L}^{\pm}(u,\phi)J_{0}^{\pm}+J_{1}^{\pm}\,,$
(41) $\displaystyle\hat{a}_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{\ell}\mathcal{L}(\phi)\hat{J}_{0}+\hat{J}_{1}\,,$ (42)

and

$\mathcal{L}^{\pm}(u,\phi)=\frac{1}{2}\left(\ell\,\mathcal{M}(u,\phi)\,\pm\,\mathcal{N}(u,\phi)\right)\,,\hskip
28.45274pt\mathcal{L}=\frac{1}{2\ell}\left(\ell^{2}\,\mathcal{M}(u,\phi)-\mathcal{R}(u,\phi)\right)\,.$
(43)

The asymptotic gauge fields $a_{u}^{+}$ and $a_{u}^{-}$ will be specified
below. On the other hand, the asymptotic form of the connection (37) is
preserved under gauge transformations of the form
$\delta_{\lambda}a=\mathrm{d}\lambda+\left[a,\lambda\right]$, where $\lambda$
is zero-form a gauge parameter

$\Lambda=\theta^{-1}\lambda\theta\,,\qquad\lambda=\frac{1}{\ell}\varepsilon^{a}\left(u,\phi\right)P_{a}+\chi^{a}\left(u,\phi\right)J_{a}+\gamma^{a}\left(u,\phi\right)Z_{a}\,.$
(44)

Note that $\lambda=\lambda(u,\phi)$ depends only on spacetime coordinates $u$
and $\phi$ because we gauged away its $r$-dependence. The functions
$\varepsilon^{a},\chi^{a}$, and $\gamma^{a}$ are the symmetry parameters along
the generators of the AdSL algebra. Taking the connection (39) and the
$r$-independent gauge parameter (44), one can show its components can be
solved in terms of three arbitrary functions, i.e. $\lambda=\lambda(Y,f,h)$,
with $Y=Y(\phi)$, $f=f(u,\phi)$, and $h=h(u,\phi)$. Explicitly,

$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2\ell^{2}}\left(\ell^{2}\,\mathcal{M}-\mathcal{R}\right)Y-Y^{\prime\prime}\right)J_{0}+YJ_{1}-Y^{\prime}J_{2}+\frac{1}{2\ell}\left(\mathcal{M}f+\mathcal{N}Y+\mathcal{N}h-2f^{\prime\prime}\right)P_{0}$
(45) $\displaystyle+\
\frac{f}{\ell}P_{1}-\frac{f^{\prime}}{\ell}P_{2}+\frac{1}{2\ell^{2}}\left(\ell^{2}\,\mathcal{M}h+\mathcal{N}f+\mathcal{R}Y-2\ell^{2}h^{\prime\prime}\right)Z_{0}+hZ_{1}-h^{\prime}Z_{2}\,,$

whenever the functions $\mathcal{M},\mathcal{N}$ and $\mathcal{R}$ transform
according to Concha:2018jjj

$\displaystyle\delta_{\lambda}\mathcal{M}$
$\displaystyle=\mathcal{M}^{\prime}Y+2\mathcal{M}Y^{\prime}-2Y^{\prime\prime\prime}-2h^{\prime\prime\prime}+h\mathcal{M}^{\prime}+2\mathcal{M}h^{\prime}+\frac{2\mathcal{N}f^{\prime}}{\ell^{2}}+\frac{f\mathcal{N}^{\prime}}{\ell^{2}}\,,$
(46a) $\displaystyle\delta_{\lambda}\mathcal{N}$
$\displaystyle=\mathcal{M}^{\prime}f+2\mathcal{M}f^{\prime}-2f^{\prime\prime\prime}+\mathcal{N}^{\prime}Y+2\mathcal{N}Y^{\prime}+2\mathcal{N}h^{\prime}+h\mathcal{N}^{\prime}\,,$
(46b) $\displaystyle\vskip 6.0pt plus 2.0pt minus
2.0pt\delta_{\lambda}\mathcal{R}$
$\displaystyle=\ell^{2}\left(\mathcal{M}^{\prime}h+2\mathcal{M}h^{\prime}-2h^{\prime\prime\prime}\right)+\mathcal{N}^{\prime}f+2\mathcal{N}f^{\prime}+\mathcal{R}^{\prime}Y+2\mathcal{R}Y^{\prime}\,.$
(46c)

We will now determine the asymptotic form of the gauge fields along time
evolution. By considering one of the components of the field equations, we
find

$F_{iu}=0\hskip 22.76228pt\Longrightarrow\hskip
22.76228pt\dot{A_{i}}=\partial_{i}A_{u}+[A_{i},\,A_{u}]\,,$ (47)

with $i=1,2$ denoting the spatial coordinates. From here, one can recognize
the structure of a gauge transformation equation for the gauge field $A_{i}$.
Therefore, we can interpret this equation as the time evolution of $A_{i}$ as
a gauge transformation generated by the parameter $A_{u}$. To preserve the
asymptotic symmetries, the Lagrange multiplier must be
$A_{u}=\theta^{-1}a_{u}\theta$, with

$a_{u}=\lambda\left(\mu,\xi,\rho\right)\,,$ (48)

where we have included the chemical potentials $\mu$, $\xi$ and $\rho$,
following the analysis presented in Henneaux:2013dra . Therefore, the time
evolution of the gauge fields in the asymptotic region is given by the
following set of differential equations

$\displaystyle\dot{\mathcal{M}}$ $\displaystyle=$
$\displaystyle\mathcal{M}^{\prime}\mu+2\mathcal{M}\mu^{\prime}+\mathcal{M}^{\prime}\rho+2\mathcal{M}\rho^{\prime}+\frac{\mathcal{N}^{\prime}\xi}{\ell^{2}}+\frac{2\mathcal{N}\xi^{\prime}}{\ell^{2}}-2\mu^{\prime\prime\prime}-\rho^{\prime\prime\prime}\,,$
(49a) $\displaystyle\dot{\mathcal{N}}$ $\displaystyle=$
$\displaystyle\mathcal{M}^{\prime}\xi+2\mathcal{M}\xi^{\prime}+\mathcal{N}^{\prime}\mu+2\mathcal{N}\mu^{\prime}+\mathcal{N}^{\prime}\rho+2\mathcal{N}\rho^{\prime}-2\xi^{\prime\prime\prime}\,,$
(49b) $\displaystyle\dot{\mathcal{R}}$ $\displaystyle=$
$\displaystyle\ell^{2}\mathcal{M}^{\prime}\rho+2\ell^{2}\mathcal{M}\rho^{\prime}+\mathcal{N}^{\prime}\xi+2\mathcal{N}\xi^{\prime}+\mathcal{R}^{\prime}\mu+2\mathcal{R}\mu^{\prime}-2\ell^{2}\rho^{\prime\prime\prime}\,.$
(49c)

It is clear from these equations that configurations with constant values of
$\mathcal{M},\mathcal{N}$ and $\mathcal{R}$, as well as their corresponding
chemical potentials $\mu$, $\xi$ and $\rho$, solve the field equations (49).
Considering (48), the gauge connection (39) takes the form

$a=a_{u}\mathop{}\\!\mathrm{d}u\ +\
\left(\frac{1}{2}\left(\ell^{2}\,\mathcal{M}-\mathcal{R}\right)J_{0}+J_{1}+\frac{\mathcal{N}}{2\ell}P_{0}+\frac{\mathcal{R}}{2\ell^{2}}Z_{0}\right)\mathop{}\\!\mathrm{d}\phi\,,$
(50)

with

$\displaystyle a_{u}(\mu,\xi,\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{2\ell^{2}}\left(\ell^{2}\,\mathcal{M}-\mathcal{R}\right)\mu
J_{0}+\mu
J_{1}+\frac{1}{2\ell}\left(\mathcal{M}\xi+\mathcal{N}\mu+\mathcal{N}\rho\right)P_{0}+\frac{\xi}{\ell}P_{1}$
(51) $\displaystyle+\
\frac{1}{2\ell^{2}}\left(\ell^{2}\mathcal{M}\rho+\mathcal{N}\xi+\mathcal{R}\mu\right)Z_{0}+\rho
Z_{1}\,.$

Writing these results above in the basis
$\\{J_{a}^{+},J_{a}^{-},\hat{J}_{a}\\}$, the time evolution of the gauge
fields is given by

$\displaystyle\dot{\mathcal{L}}^{\pm}$ $\displaystyle=$
$\displaystyle\mathcal{L}^{\pm\prime}\nu_{\pm}+\mathcal{L}^{\pm\prime}\nu+2\mathcal{L}_{\pm}\nu_{\pm}^{\prime}+2\mathcal{L}_{\pm}\nu^{\prime}-\ell\left(\nu_{\pm}^{\prime\prime\prime}+\nu^{\prime\prime\prime}\right)\,,$
(52) $\displaystyle\dot{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\mathcal{L}^{\prime}\nu+2\mathcal{L}\nu^{\prime}-\ell\nu^{\prime\prime\prime}\,,$
(53)

where

$\nu_{\pm}=\rho\ \pm\ \frac{\xi}{l}\,,\hskip 28.45274pt\nu=\mu\,,$ (54)

are the chemical potentials in this basis. Finally, in the rest frame, the
asymptotic gauge connection (50) reads

$a=\left(a^{+}_{u}+a^{-}_{u}+\hat{a}_{u}\right)\mathop{}\\!\mathrm{d}u\ +\
\left(a_{\phi}^{+}+a_{\phi}^{-}+\hat{a}_{\phi}\right)\mathop{}\\!\mathrm{d}\phi\,,$
(55)

with

$\displaystyle a_{u}^{\pm}$ $\displaystyle=$
$\displaystyle\pm\left(\nu_{\pm}+\nu\right)\left(\frac{1}{\ell}\mathcal{L}^{\pm}J_{0}^{\pm}+J_{1}^{\pm}\right)\,,$
(56) $\displaystyle\hat{a}_{u}$ $\displaystyle=$
$\displaystyle\nu\left(\frac{\mathcal{L}}{\ell}\hat{J}_{0}+\hat{J}_{1}\right)\,,$
(57)

where $a_{\phi}^{\pm}$ and $\hat{a}_{\phi}$ are given in Eqs. (41) and (42).
In the following section, we will compute the black hole entropy following the
known formulae of the three-dimensional CS theory Bunster:2014mua .

##  3 Thermodynamics

We now study the thermodynamics of the stationary solution (32). For this
purpose, we follow for a large part the treatment presented in Carlip:1994gc ;
Henneaux:2013dra ; Bunster:2014mua . In general, black hole entropy is
determined by establishing regularity conditions and global charges. In a CS
gauge theory of gravity, regularity conditions are achieved by requiring
trivial holonomies along contractible cycles.

###  3.1 Entropy

To set regularity conditions easily, we will use the results obtained on the
basis $\\{J_{a}^{+},J_{a}^{-},\hat{J}_{a}\\}$. The connection one-form takes
values in the three copies of the Lorentz algebra such that

$A=A^{+}+A^{-}+\hat{A}\,,$ (58)

with

$A^{\pm}=\left(\omega^{a}\pm\frac{e^{a}}{\ell}+\sigma^{a}\right)J^{\pm}_{a}\,,\hskip
28.45274pt\hat{A}=\omega^{a}\hat{J}_{a}\,.$ (59)

For this gauge connection, the CS action (8) is rewritten as

$I_{\mathrm{CS}}\left[A\right]\ =\
I_{\mathrm{CS}}\left[A^{+}\right]+I_{\mathrm{CS}}\left[A^{-}\right]+I_{\mathrm{CS}}\left[\hat{A}\right]\,,$
(60)

where we have considered the following non-vanishing components of the
invariant tensor

$\left\langle
J_{a}^{\pm},\,J_{b}^{\pm}\right\rangle\,=\,\frac{1}{2}\left(\alpha_{2}\pm\alpha_{1}\right)\eta_{ab}\,,\hskip
28.45274pt\left\langle\hat{J}_{a},\,\hat{J}_{b}\right\rangle\,=\,\left(\alpha_{0}-\alpha_{2}\right)\eta_{ab}\,.$
(61)

In the microcanonical ensemble, the entropy can then be obtained from the
following expression Bunster:2014mua

$S=k\,\left(\left\langle a_{u}^{+}\,,a_{\phi}^{+}\right\rangle+\left\langle
a_{u}^{-}\,,a_{\phi}^{-}\right\rangle+\left\langle\hat{a}_{u}\,,\hat{a}_{\phi}\right\rangle\right)\Bigg{|}_{\text{on-
shell}}\,.$ (62)

Following the nomenclature of Bunster:2014mua , without loss of generality, we
shall make the assumption that we are in a “rest frame” such that the
functions $\\{\mathcal{L}^{\pm},\mathcal{L}\\}$ and $\\{{\nu^{\pm},\nu\\}}$
are constants at $r\to\infty$, i.e.

$\mathcal{L}^{\pm}\left(u,\phi\right)\xrightarrow[r\to\infty]{}\frac{\mathcal{L}_{0}^{\pm}}{2\pi}\,,\hskip
14.22636pt\mathcal{L}(u,\phi)\xrightarrow[r\to\infty]{}\frac{\mathcal{L}_{0}}{2\pi}\,,\hskip
14.22636pt\nu^{\pm}(u,\phi)\xrightarrow[r\to\infty]{}\frac{\nu_{0}^{\pm}}{2\pi}\,,\hskip
14.22636pt\nu(u,\phi)\xrightarrow[r\to\infty]{}\frac{\nu_{0}}{2\pi}\,.$ (63)

In addition, we conveniently rescale the fields
$\\{\mathcal{L}^{\pm},\mathcal{L}\\}$, such that

$\displaystyle a_{\phi}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{k}\mathcal{L}^{\pm}J_{0}^{\pm}+J_{1}^{\pm}\,,$ (64a)
$\displaystyle\hat{a}_{\phi}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{k}\mathcal{L}\hat{J}_{0}+\hat{J}_{1}\,,$ (64b)
$\displaystyle a_{u}^{\pm}$ $\displaystyle=$
$\displaystyle\left(\nu_{\pm}+\nu\right)\left(\frac{4\pi}{k}\mathcal{L}^{\pm}J_{0}^{\pm}+J_{1}^{\pm}\right)\,,$
(64c) $\displaystyle\hat{a}_{u}$ $\displaystyle=$
$\displaystyle\nu\left(\frac{4\pi}{k}\mathcal{L}\hat{J}_{0}+\hat{J}_{1}\right)\,.$
(64d)

From these results and considering the formulas (61)-(62), we deduce the
general expression for the entropy

$S=4\pi\left(2\alpha_{0}\nu\mathcal{L}+\alpha_{1}\left((\nu_{+}+\nu)\mathcal{L}^{+}+(\nu_{-}+\nu)\mathcal{L}^{-}\right)+\alpha_{2}\left((\nu_{+}+\nu)\mathcal{L}^{+}-(\nu_{-}+\nu)\mathcal{L}^{-}-2\nu\mathcal{L}\right)\right)\,.$
(65)

In order to fix the chemical potentials, we require the holonomies $H$ along
the thermal circle to be trivial, namely

$H=e^{\mathrm{i}a_{u}(r_{+})}\Bigg{|}_{\text{on-shell}}=-\mathbb{1}\,,$ (66)

where $\mathbb{1}$ is the identity generator of the AdSL algebra and $r_{+}$
is the largest root of $N(r)=0$. Then, for each temporal gauge connection (64)
satisfying the regularity condition (66), we obtain a set of equations that
allows us to find $\nu_{\pm}$ and $\nu$. According to this condition, the
eingenvalues of $\mathrm{i}a_{u}$ are given by $\pm\,\mathrm{i}\pi$.
Therefore,

$\displaystyle\text{tr}\left[\left(a_{u}^{+}\right)^{2}\right]$
$\displaystyle=$ $\displaystyle 2\pi^{2}\,,$ (67a)
$\displaystyle\text{tr}\left[\left(a_{u}^{-}\right)^{2}\right]$
$\displaystyle=$ $\displaystyle 2\pi^{2}\,,$ (67b)
$\displaystyle\text{tr}\left[\left(\hat{a}_{u}\right)^{2}\right]$
$\displaystyle=$ $\displaystyle 2\pi^{2}\,.$ (67c)

On the other hand, because of the isomorphism
$\mathfrak{so}(2,1)\simeq\mathfrak{sl}(2,\mathbb{R})$, then we conveniently
choose

$\hat{J}_{0}^{\pm}\simeq j_{0}\,,\hskip 28.45274pt\hat{J}_{1}^{\pm}\simeq
j_{1}\,,$ (68)

where $j_{0}$ and $j_{1}$ are both $\mathfrak{sl}\left(2,\mathbb{R}\right)$
elements.555See, for instance, appendix of Donnay:2016iyk  This isomorphism
allows us to use the $\mathfrak{sl}\left(2,\mathbb{R}\right)$ matrix
representation to compute the holonomy (67). Therefore, from (67), $\nu_{\pm}$
and $\nu$ are given by

$\nu_{+}+\nu=\sqrt{\frac{\pi k}{2\mathcal{L}^{+}}}\,,\hskip
28.45274pt\nu_{-}+\nu=\sqrt{\frac{\pi k}{2\mathcal{L}^{-}}}\,,\hskip
28.45274pt\nu=\sqrt{\frac{\pi k}{2\mathcal{L}}}\,.$ (69)

Then, by considering (63) and (69) the entropy (62) turns out to be

$S=2\pi\sqrt{k}\,\left(\alpha_{1}\left(\sqrt{\mathcal{L}^{+}_{0}}+\sqrt{\mathcal{L}^{-}_{0}}\right)+\alpha_{2}\left(\sqrt{\mathcal{L}^{+}_{0}}-\sqrt{\mathcal{L}^{-}_{0}}\right)+2\left(\alpha_{0}-\alpha_{2}\right)\sqrt{\mathcal{L}_{0}}\right)\,.$
(70)

From the expression (43) we can rewrite (70) in terms of the integration
constants $M$, $J$ and $b$ as

$S=\pi\sqrt{2k}\left(\alpha_{1}\left(\sqrt{\ell M-J}-\sqrt{\ell M+J}\right)+\
\alpha_{2}\left(\sqrt{\ell M-J}+\sqrt{\ell M+J}\right)+4\sqrt{-\ell
b}\left(\alpha_{0}-\alpha_{2}\right)\right)\,.$ (71)

Notice that the term proportional to $\alpha_{2}$ gives the standard result of
the BTZ black hole. However, we now have the contribution of the exotic term
$\alpha_{1}$ and also from the integration constant $b$. At this level, we
cannot set $\alpha_{0}-\alpha_{2}=0$ because the bilinear form (5)
degenerates. One can also write the entropy in terms of the global charges
associated with the isometries of this new geometry. To this end, in the next
section, we compute the global charges and show the appearance of a new
conserved charge.

###  3.2 Global and gauge charges

Let us consider diffeomorphisms and gauge symmetries together, both
parameterized by $\epsilon=(\xi,\lambda)$, with $\xi$ a vector field and
$\lambda$ a Lie algebra valued gauge parameter. Thus, a general infinitesimal
symmetry transformation reads666We use the notation $\xi{\lrcorner}=i_{\xi}$
for the interior product, for instance
$\xi{\lrcorner}e^{a}=\xi^{\mu}e\indices{{}^{a}_{\mu}}$.

$\delta
A_{\epsilon}=\mathcal{L}_{\xi}A-\mathrm{D}_{A}\lambda^{\prime}=\xi{\lrcorner}F-\mathrm{D}_{A}\lambda\,,$
(72)

where we have defined the exterior covariant derivative
$\mathrm{D}_{A}(\cdot)\equiv\mathop{}\\!\mathrm{d}(\cdot)+[A,\,(\cdot)]$ and
the displaced parameter $\lambda=\lambda^{\prime}-\xi{\lrcorner}A$. Following
Frodden:2019ylc , to define a conserved charge, we demand the so-called exact
symmetry condition, which consists in standing the infinitesimal gauge
transformations (72) to zero and solve for the parameters
$\epsilon=(\xi,\lambda)$. For instance, the global charges associated with the
invariance under time displacements (mass) and rotations (angular momenta) are
computed using, respectively, the Killing vectors $\xi^{\mu}_{(u)}=(1,0,0)$
and $\xi^{\mu}_{(\phi)}=(0,0,-1)$, respectively. The formula to derive these
charges, with $u=const$, is given by

$\delta
Q\left[\lambda\right]=-\frac{k}{2\pi}\,\int\mathop{}\\!\mathrm{d}\phi\,\left\langle\lambda,\delta
a\right\rangle\,.$ (73)

where we considered the expression for the asymptotic gauge field $a$
(39).777It is worth noting here that the quasi-local treatment for the charge
conservation ensures, given a spacetime with exact symmetries, the
independence of the radius $r$.  By choosing $\lambda=-\xi{\lrcorner}a$, the
exact symmetry condition $\delta_{\epsilon}A=0$ is solved for the Killing
vectors $\xi^{\mu}_{(u)}$ and $\xi^{\mu}_{(\phi)}$, only. The charges
associated with these isometries become

$\displaystyle\delta Q\left[\xi^{\mu}_{(u)}\right]$ $\displaystyle\equiv$
$\displaystyle\delta m=\frac{k}{2\ell}\left(\alpha_{1}\delta
M-\frac{\alpha_{2}}{\ell}\delta J\right)\,,$ (74) $\displaystyle\delta
Q\left[\xi^{\mu}_{(\phi)}\right]$ $\displaystyle\equiv$ $\displaystyle\delta
j=\frac{k}{2\ell}\left(\left(\alpha_{0}-\alpha_{2}\right)\ell\,\delta
b+\alpha_{1}\delta J-\alpha_{2}\ell\delta M\right)\,.$ (75)

The integration along $\phi$ is trivial because the charges are coordinate
independent. Moreover, since the phase space variation is linear, this
integration is trivial as well. Setting $k\equiv 2\ell$, we get the finite
charges

$\displaystyle m$ $\displaystyle=$
$\displaystyle\alpha_{1}M-\frac{\alpha_{2}}{\ell}J\,,$ (76) $\displaystyle j$
$\displaystyle=$ $\displaystyle\left(\alpha_{0}-\alpha_{2}\right)\,\ell
b+\alpha_{1}J-\alpha_{2}\ell\,M\,.$ (77)

Notice that the rotating BTZ case corresponds to
$(\alpha_{0},\alpha_{1},\alpha_{2})=(0,1,0)$, and the case
$(\alpha_{0},\alpha_{1},\alpha_{2})=(0,0,-1)$ with $b=0$, corresponds to the
exotic BTZ charges, where a reversion of the mass and angular momentum is
observed. Therefore, here we find that global charges essentially generalize
the exotic case found by Townsend and Zhang Townsend:2013ela  and contain, in
addition, a torsional correction proportional to $b$ in $j$.

There is an extra charge associated with a gauge symmetry in this solution.
This charge is computed using the gauge parameter $\lambda$ satisfying the
exact symmetry condition for $\epsilon=(0,\lambda)$, so that

$\delta_{\lambda}a=\mathop{}\\!\mathrm{d}\lambda+\left[a,\lambda\right]=0\,,$
(78)

where $\lambda$ is the algebra valued parameter given in (44) and $a$ the
gauge connection appearing in (39). This exactness condition admits the
solution for $\varepsilon^{a}=0$ and $\chi^{a}=-\gamma^{a}$, we get

$\lambda_{\scriptscriptstyle
w}\equiv-\frac{b}{2}J_{0}+J_{1}+\frac{b}{2}Z_{0}-Z_{1}\,.$ (79)

Then, replacing (79) into (73), and integrating both in $\phi$ spacetime
coordinate and in phase space, we obtain the new gauge charge

$Q\left[\lambda_{\scriptscriptstyle w}\right]\equiv
w=\left(\alpha_{0}-\alpha_{2}\right)\,\ell b\,.$ (80)

This is a novel result of this work. Although AdSL symmetry has three Casimir
operators Diaz:2012zza , and this new charge is expected, the surprising
result here is how the torsion supports it. It could be interesting to study
the possible relationship between the constant electromagnetic field of the
AdSL particle analysis Bacry:1970aa ; Gomis:2009dm  and this new charge. We
leave this for a future work.

From these charges in hand, it is possible to write the entropy (71) in terms
of the global charges $m,j$ and $w$. We find

$S=2\pi\sqrt{\ell}\left(\sqrt{(\ell m+j-w)(\alpha_{1}-\alpha_{2})}+\sqrt{(\ell
m-j+w)(\alpha_{1}+\alpha_{2})}+4\sqrt{w(\alpha_{2}-\alpha_{0})}\right)\,,$
(81)

or in terms of the inner/outer horizons $r_{\pm}$ and the integration constant
$b$ becomes

$S\,=\,4\pi\left(\alpha_{1}r_{+}-\alpha_{2}r_{-}+2\left(\alpha_{0}-\alpha_{2}\right)\,\ell\sqrt{-b}\right)\,,$
(82)

noticing that the integration constant $b<0$ holds. Here we have assumed that
$\alpha_{0}>\alpha_{2}$. The case $\alpha_{0}<\alpha_{2}$ is permissible as
well, giving a minus sign in front of the last term. To recover the result
found in Townsend:2013ela , one should set
$\alpha_{2}\rightarrow-\tilde{\alpha}_{2}$, with $\tilde{\alpha}_{2}>0$ and
$b=0$.

This expression for the entropy differs from the standard and exotic formula
because of the term proportional to the new gauge charge $w$. Due to that,
this charge is related to the integration constant appearing in the solution
for $\sigma^{a}=-\kappa^{a}$ gauge field, this result indicates that spacetime
torsion modifies both the black hole entropy and the first law of
thermodynamics, as we will see below.

###  3.3 The first law and Smarr formula

By studying the thermodynamics of the black hole with torsion (35),
calculation of the entropy formula (71) revealed the existence of a new
extensive variable associated with the integration constant $b$. Therefore, we
can establish the first law of black hole mechanics for this family of black
hole solution spanned by the parameters $M,J$, and $b$. With all the global
charges at hand, the first law of thermodynamics reads

$\delta m=T\delta S+\Omega\,\delta j+\mu\delta w\,,$ (83)

where we have used the standard formulas to compute the extensive quantities

$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{(r_{+}^{2}-r_{-}^{2})}{2\pi\ell^{2}\,r_{+}}\,,$ (84)
$\displaystyle\Omega$ $\displaystyle=$ $\displaystyle\frac{r_{-}}{\ell
r_{+}}\,,$ (85) $\displaystyle\mu$ $\displaystyle=$
$\displaystyle\frac{2(r_{+}^{2}-r_{-}^{2})-\sqrt{|b|}\ell
r_{-}}{\sqrt{|b|}\ell^{2}\,r_{+}}=\frac{4\pi T}{\sqrt{|b|}}-\Omega\,.$ (86)

All of these quantities might be further identified with standard physical
ones: a temperature (Hawking thermal radiation) $T$, angular velocity of the
horizon $\Omega$, and a chemical potential $\mu$ associated with the gauge
charge $w$.

Finally, by considering the fact that the entropy is a homogeneous
thermodynamical function of degree $1/2$ for $m,j$ and $w$, one easily obtains
the Smarr formula for this black solution

$S=\frac{2}{T}\left(m-\Omega j-\mu w\right)\,.$ (87)

When $w=0$, this expression reduces to the Smarr formula for the BTZ black
hole solution. Lastly, using these state variables and the thermodynamic
functions shown, it is possible to find other quantities, such as the heat
capacity and fluctuation modes, that will be needed, for example, to study the
critical behavior and phase transition of this torsional black hole.

##  4 Conclusions

In this paper, we have examined the thermodynamics of BTZ-type black holes in
the presence of spacetime torsion. After a suitable gauge fixing, we show that
the solution is characterized by three integration constants: $M,J$ as in
standard GR plus a new constant $b$ related to torsion. The presence of this
constant generalizes the “exotic” thermodynamics analysis of Townsend and
Zhang by including spacetime torsion in both the global charge formulas and
entropy. The solution has been obtained from a CS action valued on a semi-
simple enlargement of the Poincaré gauge symmetry, namely the AdSL symmetry.
This enlargement includes new gauge fields that allow the notion of a torsion
different from zero. Further, we found a novel conserved quantity related to
torsion. This implies the presence of this new gauge charge in the laws of
black hole thermodynamics.

Investigating the thermodynamics in the $\ell\rightarrow\infty$ limit of the
torsional black hole would be interesting. This observation is motivated by
taking the flat limit of the stationary solutions in AdSL gravity, arriving at
the cosmological solution of Maxwell CS theory Concha:2018zeb . Moreover, it
is also interesting to analyze whether it is possible to recover the entropy
formula obtained through a microscopic description from the Cardy formula.
This is a work in progress. Including matter fields in the AdSL CS theory is
another important topic to discuss. Since fermionic fields interact with
torsion, this might be the most intriguing scenario to investigate. A more
comprehensive interpretation of the conserved torsional charge may be
attainable by doing so. Finally, thermodynamic phenomena such as phase
transitions and critical behavior are interesting additional topics to
investigate in this torsional black hole. These phenomena involving a
conserved torsional charge have not been described previously.

## Acknowledgments

We thank Eloy Ayón-Beato, Fabrizio Canfora, Cristóbal Corral, Branislav
Cvetkovic, Ernesto Frodden, Oscar Fuentealba, Joaquim Gomis, Fernando
Izaurieta, José Martín-García, Niels Obers, Alfredo Pérez, Lárus Thorlacius,
and Jorge Zanelli, for insightful discussions and comments. DH is supported by
the Icelandic Research Fund via the Grant of Excellence titled “Quantum Fields
and Quantum Geometry” and by the University of Iceland Research Fund. L.A. is
supported by Fondecyt grants 3220805. OV acknowledges to ICE-CSIC and Fondecyt
11200742.

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