Physics Letters B 747 (2015) 158–163
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Hawking–Page phase transition in new massive gravity
Shao-Jun Zhang
Instituto de Física, Universidade de São Paulo, C.P. 66318, 05315-970, São Paulo, SP, Brazil
a r t i c l e i n f o a b s t r a c t
Article history:
Received
25 April 2015
Received
in revised form 22 May 2015
Accepted
26 May 2015
Available
online 29 May 2015
Editor: M.
Cveti
ˇ
c
We consider Hawking–Page phase transition between the BTZ black hole with M ≥ 0and the thermal
soliton with M =−1in new massive gravity. By comparing the on-shell free energies, we can see that
there exists a critical temperature. The thermal soliton is more probable than the black hole below the
critical temperature while the black hole is more probable than the thermal soliton above the critical
temperature. By consistently constructing the off-shell free energies taking into account the conical
singularity, we show that there exist infinite non-equilibrium states connecting the BTZ black hole and
the thermal soliton, so that they provide a picture of continuous evolution of the phase transition.
© 2015 The Author. 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
Now it is well known that a black hole is like a thermodynam-
ical
system, not only it has entropy but also temperature [1–4].
In past decades, extensive efforts have been paid on the study
of thermodynamical properties of black holes. And various phase
transitions in various black holes have been found. The particularly
interesting
one, now named Hawking–Page transition [5], states
that there exists a phase transition between an AdS–Schwarzschild
black hole and a pure AdS vacuum in four dimensions. Above
a critical temperature, the AdS–Schwarzschild black hole is more
probable than the AdS vacuum, while below the critical tempera-
ture
the AdS vacuum is more probable than the AdS–Schwarzschild
black hole. This work has been extended successfully to cases of
higher dimensions in Ref. [6], and is explained to be holographic
dual to confinement/deconfinement transition in dual gauge the-
ory.
Such a phase transition can be read off from comparing their
on-shell free energies.
We
can also study the Hawking–Page transition using off-shell
free energy of the black hole, and it is more natural to see the
tunneling picture of the phase transition in this method. In the off-
shell
free energy, the temperature is taken to be arbitrary rather
than the Hawking temperature of the black hole. And then the
other quantities, i.e. the mass of the black hole, which is deter-
mined
by the temperature originally in on-shell free energy, now
can be treated as an arbitrary parameter for given temperature.
So the off-shell free energy will be a continuous function of the
continuous mass parameter, from positive value to zero which cor-
E-mail address: sjzhang84@hotmail.com.
responds to the vacuum, for given temperature. The extremum of
the off-shell free energy is an equilibrium state which is the black
hole, while the others correspond to non-equilibrium states which
can be seen as immediate states during phase transition. Then the
tunneling picture becomes clear that the black hole is connected
with the vacuum via infinite immediate states.
However,
the situation becomes more subtle in (2 + 1)-dimen-
sional
BTZ (Bañados–Teitelboim–Zanelli) black hole system [7],
where there exist two distinct solutions. One is the BTZ black
hole with mass M ≥ 0, and the other is the global AdS
3
soliton
with M < −1. There are various works studying the Hawking–
Page
phase transition in this system using on-shell method, as well
as off-shell method taking conical singularity into account [8–16].
Then a natural question arises: Because there exists a mass gap
between the BTZ black hole and the soliton, can we still get a pic-
ture
of continuous evolution of the phase transition between them
using off-shell method? In Ref. [17], the authors confirm this pic-
ture
by treating both the BTZ black hole and the soliton off-shell.
Both off-shell free energies will become continuous functions of
the mass, with M ≥ 0and M < 0 respectively, and both are con-
nected
at the point M = 0. Then for a given temperature, not only
there exist infinite non-equilibrium states with M ≥ 0, but also ex-
ist
infinite non-equilibrium states with M < 0. The two extrema of
the two off-shell free energies correspond to the BTZ black hole
and the thermal soliton, respectively. And now the tunneling pic-
ture
becomes clear that the BTZ black hole and the thermal soliton
are connected via immediate states.
On
the other hand, recently, a new kind of three dimensional
gravity theory was proposed, which is now known as new massive
gravity (NMG) [18,19]. In NMG, beyond the usual Einstein–Hilbert
action, particular higher-curvature terms are added. Then propa-
http://dx.doi.org/10.1016/j.physletb.2015.05.065
0370-2693/
© 2015 The Author. 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
.