Physics Letters B 801 (2020) 135180
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Entropy in three-dimensional general relativity: Kerr-AdS black hole
M. Blagojevi
´
c, B. Cvetkovi
´
c
∗
Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade-Zemun, Serbia
a r t i c l e i n f o a b s t r a c t
Article history:
Received
27 November 2019
Accepted
17 December 2019
Available
online 27 December 2019
Editor:
W. Haxton
Black hole thermodynamics of the Kerr-AdS spacetime in three-dimensional general relativity is analyzed
using a Hamiltonian approach. The values of the conserved charges and entropy, obtained by a proper
treatment of the AdS asymptotic conditions, are shown to satisfy the first law of black hole dynamics.
© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Asymptotic conditions play a crucial role in understanding basic
features of the black hole thermodynamics. In the case of asymp-
totically
flat black holes in general relativity (GR), the first law is
shown to hold for any stationary and axially symmetric black hole
with a bifurcate Killing horizon [1]. The situation with black holes
in an asymptotically anti-de Sitter (AdS) background is more in-
volved.
As discussed by Gibbons et al. [2]for Kerr-AdS black holes,
the relation between conserved charges, entropy and the angular
velocity on one side, and the first law on the other, is not fully
settled.
In
order to clearly understand thermodynamics of Kerr-AdS
black holes, one is naturally led to consider the corresponding
three-dimensional (3D) models, as they allow us to investigate the
problem in a technically much simpler context. A detailed study of
the subject can be found in Hawking et al. [3], where the authors
analyzed, inter alia, how different coordinate systems affect the
form of thermodynamic variables; see also [4,5]. However, some
aspects of their analysis of energy and the first law require further
consideration.
A
Hamiltonian approach to black hole entropy has been re-
cently
proposed in Ref. [6], and applied to asymptotically flat Kerr
black holes in Ref. [7]. In the present paper, we use the same
approach to analyze thermodynamic properties of the Kerr-AdS
black hole in three-dimensional GR (GR
3
). Our analysis shows
that a proper treatment of the AdS asymptotic conditions en-
sures
a consistency between the conserved charges, angular ve-
locity
and black hole entropy, expressed by the validity of the first
law.
*
Corresponding author.
E-mail
addresses: mb@ipb.ac.rs (M. Blagojevi
´
c), cbranislav@ipb.ac.rs
(B. Cvetkovi
´
c).
2. Hamiltonian approach to entropy
Although the Hamiltonian approach to entropy [6,7]is focused
on black holes in Poincaré gauge theory [8], where both the tor-
sion
and the curvature define the gravitational dynamics, it can be
equally well used in the realm of GR as a Riemannian theory of
gravity.
As
a preparation for such an approach in 3D spacetime, we in-
troduce
the first-order orthonormal frame formulation of GR
3
, in
which the coframe b
i
= b
i
μ
dx
μ
and the antisymmetric spin con-
nection
ω
ij
= ω
ij
μ
dx
μ
(1-forms) are independent dynamical vari-
ables.
The related field strengths are the torsion T
i
= db
i
+ω
ij
k
b
k
≡
∇
b
i
and the curvature R
ij
= dω
ij
+ ω
i
k
ω
kj
(2-forms), and metric is
defined by g = η
ij
b
i
⊗ b
j
, with η
ij
= (1, −1, −1). In the absence
of matter, the gravitational dynamics is defined by the Lagrangian
3-form
L
G
=−a
0
ε
ijk
b
i
R
jk
−
1
3
Λ
0
ε
ijk
b
i
b
j
b
k
, (2.1)
where a
0
= 1/16π (in units G = 1), Λ
0
is a cosmological con-
stant,
and the completely antisymmetric tensor ε
ijk
is normalized
by ε
012
= 1. Varying L
G
with respect to b
i
and ω
ij
, one obtains the
field equations of GR
3
in vacuum,
δb
i
:−a
0
ε
ijk
R
jk
− Λ
0
ε
ijk
b
j
b
k
= 0 ,
δ
ω
ij
− 2a
0
ε
ijk
∇b
k
= 0 . (2.2)
The second equation yields the condition for vanishing torsion,
∇b
k
= 0, whereupon the first equation takes the standard form of
Einstein equation with a cosmological constant.
Our
approach to black hole entropy is an extension of the
Hamiltonian treatment of conserved charges as boundary terms at
infinity. In the case of a black hole solution, we assume that the
https://doi.org/10.1016/j.physletb.2019.135180
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© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.