#### Mass and angular momentum of black holes in 3D gravity theories with first order formalism

Eur. Phys. J. C
Mass and angular momentum of black holes in 3D gravity theories with first order formalism
Soonkeon Nam 0
Jong-Dae Park 0
0 Department of Physics and Research Institute of Basic Science, Kyung Hee University , Kyungheedae-ro 26, Dongdaemun-gu, Seoul 02447 , South Korea
We apply the Wald formalism to obtain masses and angular momenta of black holes in three dimensional gravity theories using the first order formalism. Wald formalism suggests that the entropy of a black hole can be defined by an integration of a conserved charge on the bifurcation horizon, and mass and angular momentum of a black hole as an integration of some charge variation form at spatial infinity. The action of three dimensional gravity theories can be represented by a form including some auxiliary fields. As well-known examples we have calculated masses and angular momenta of some black holes in topologically massive gravity and new massive gravity theories using the first order formalism. We have also calculated mass and angular momentum of BTZ black hole and new type black hole in minimal massive gravity theory with the action represented by the first order formalism. We have also calculated the entropy and central charges of new type black hole. According to Ad S/C F T correspondence we suggest that the left and right moving temperatures should be equal to the Hawking temperature in the case of new type black hole in minimal massive gravity.
1 Introduction
For last few decades there have been paid lots of attention to
three dimensional gravity theories. Studying for three
dimensional gravity theories provides an arena to explain some
conceptual feature of the realistic four dimensional general
relativity and some fundamental issues of quantum gravity.
In general three dimensional spacetime of the Einstein
gravity theory has no propagating degrees of freedom [1]. There
only exists a black hole solution, i.e. BTZ black hole, with the
negative cosmological constant [2,3]. A well known
modification of Einstein’s gravity theory in three dimensions is
the topologically massive gravity theory (TMG) which
consists of Einstein–Hilbert term, cosmological constant and
the gravitational Chern–Simons term [4–6], breaking parity
symmetry with a new mass scale parameter. The
linearization of this theory describes the existence of a single massive
graviton mode. This theory also allows some black hole
solutions having Ad S asymptotics [7,8]. There have been many
investigations for this TMG theory from the viewpoint of
Ad S/C F T correspondence [9–12].
A few years ago there was a new proposition, new
massive gravity (NMG) theory, which is composed of Ricci scalar
with the cosmological constant and some specific
combination of Ricci scalar square and Ricci tensor square [13]. The
original aim of the introduction of higher curvature terms in
NMG theory is to present the non-linear completion of the
Pauli–Fierz theory for massive spin 2 fields. This NMG
theory has the parity symmetry and two propagating massive
graviton modes contrary to TMG theory. There have been
found some black hole solutions such as BTZ, warped AdS
and new type black hole in NMG theory with a negative
cosmological constant [14–18]. It is natural to consider the
existence of the holographically dual conformal field theory
(CFT) on the boundary if we could get an AdS solution in
some gravity theory [19–22]. In the viewpoint of Ad S/C F T
correspondence there was an attempt to extend this NMG
theory to a theory having higher than square curvature terms to
be consistent with the holographic c-theorem [23–25].
In the context of Ad S/C F T correspondence there exists
an inconsistent problem between any three dimensional
gravity theories having asymptotic AdS geometry and its dual
CFT on the boundary. The central charge of a dual boundary
CFT becomes negative whenever the spin-2 graviton modes
propagating on the bulk have positive energy, implying the
dual CFT’s non-unitarity. It is closely related to a problem
that the asymptotic Ad S3 black hole solution have negative
mass value whenever the bulk graviton modes have
positive energy, so-called “bulk vs. boundary clash”. In order to
circumvent this inconsistency there were some suggestions
like the Boulware–Deser ghost [26], “Zwei Dreibein Gravity
(ZDG)” as a viable alternative to NMG [27,28]. Recently
the new “minimal massive gravity (MMG)” theory has been
suggested to resolve this inconsistency and to represent an
alternative to TMG [29]. The field equation of MMG theory
is distinguished from the TMG’s one, including the
additional symmetric curvature-squared terms while preserving
the single bulk graviton mode state. In the action level this
MMG theory can be represented by a “Chern–Simons-like
formulation” [28]. The action of the MMG theory include the
torsion term coupled with an auxiliary field h which have the
same odd-parity and dimension of mass with the spin
connection ω, and another parity-even ‘ehh’ term considering
h-squared term with a dimensionless parameter α. For more
details see some references and therein [29–33].
In spite of its difficulties there have been many studies
to obtain mass and angular momentum of black holes on
the curved background in three dimensional gravity
theories. Arnowitt–Deser–Misner (ADM) have suggested a
wellknown method to give some conserved charges in
general relativity, which describes an surface integral with
linearized metric at infinity in asymptotically flat spacetime
[34]. Another method to obtain mass and angular
momentum is to consider the integration of the stress–energy
tensors with their counterterms on the asymptotic Ad S
boundary surface using Brown–York formalism [11,35]. Another
formalism has evolved to an extended formalism, so-called
Abott–Deser–Tekin (ADT) formalism, including higher
curvature gravity theories [36–38]. Wald has proposed a method
how to calculate conserved quantities using Noether charge
and symplectic potential on the covariant phase space
establishing the first law of black hole thermodynamics in any
covariant gravity theory [39–42]. The ADT charge is given
by a surface integral of an antisymmetric ADT potential at
spatial infinity, which is related to a conserved current that is
described by a linearized field equation around an associated
constant background contracted with a background Killing
vector. There was a different suggestion to obtain mass and
angular momentum in asymptotically Ad S spacetime using
Ad S/C F T correspondence [43]. This explains that
according to Ad S/C F T correspondence mass and angular
momentum can be obtained by some combination of E L and ER
which express left and right energies of dual CFT. These
energies can be represented by terms of left and right central
charges and their temperatures respectively. An interesting
method to find quasi-local conserved charges can be
represented using the relation between off-shell ADT potential and
linearized Noether potential [44,45]. There are many works
to obtain mass and angular momentum of black holes on Ad S
background using methods mentioned above [43–53]. Also
there have been some studies about conserved charges,
central charges and the behavior of the correlation functions of
dual CFT by using the holographic renormalization method
[54,55].
One purpose of this paper is an attempt to obtain mass
and angular momentum of black holes using the first order
formalism. As we consider MMG theory, the field equation
of this theory is composed of general TMG equation with a
parameter γ and an additional symmetric tensor Jμν which
comprises symmetric squared Shouten tensors in order to
evade “bulk vs. boundary clash”. The MMG field equation
is given by
1 γ
σ¯ Gμν + Λ¯0gμν + μ Cμν + μ2 Jμν = 0,
(
1
)
with some shifted parameters σ¯ , Λ¯0 and a non-zero
dimensionless parameter γ as a function of parameter α [29]. Since
parameters σ and Λ0 are no longer sign and cosmological
constant respectively, they should be replaced by σ¯ and Λ¯0.
This equation cannot be obtained from an action for the
metric alone by back-substituting the equation of the auxiliary
fields h into the MMG action. So, dealing with field
equations including auxiliary fields seems more correct and
consistent way. Most of conserved charges are described by a
surface integral having some tensors which are induced by
the variation of a metric as its integrand. Using the first order
formalism with field equations including auxiliary fields, we
can obtain the charge variation which is represented by
nontensorial form structures as we can see below (
20
). The
conserved charges and central charges in MMG theory have been
calculated by Tekin using ADT method [56]. The entropy of
BTZ black hole in MMG theory have been obtained by Setare
and Adami [57] with the first order formalism according to
Tachikawa’s method [58]. Also the conserved charges in
generalized minimal massive gravity (GMMG) theory [28,59]
have been obtained by Setare and Adami with the same
method [60]. There were some studies of the properties of
the linearized equation and holographic renormalization in
MMG theory [61,62]. Nowadays there was a study about
black hole entropy as the horizon Noether charge for
diffeomorphism and local Lorentz symmetry [63]. It has also
been performed some studies for the quasi-local conserved
charges by considering the Lorentz diffeomorphism invariant
gravity theories [64–67].
In this paper we use the Wald’s method to obtain mass
and angular momentum of black holes with the first order
formalism. In this method mass and angular momentum is
just defined by an integration of the variational form at
spatial infinity (22). Even though it is not an exact derivation of
the charge variation form with the first order formalism, it
is enough to get mass and angular momentum of the three
dimensional gravity theories. This method seems to work
well in cases of having some boundaries at infinity such as
asymptotically Minkowski and Ad S spacetime. We consider
well-known three dimensional gravity theories and deal with
field equations including auxiliary fields using the first order
formalism. We calculate well-known results of mass and
angular momentum of black holes as some examples. Next
we calculate mass and angular momentum of BTZ black hole
and new type black hole of MMG theory as a new result.
Another purpose of this paper is to find the thermodynamic
relation of new type black hole in MMG theory. The action of
the three dimensional gravity theories can be represented by
an integral of a Lagrangian three-form L constructed as a sum
of the wedge products of N “flavors” of Lorentz vector valued
one-form fields, {ar |r = 1, . . . , N }, such as the dreibein
ea , the spin-connection ωa and some proper auxiliary fields
ha , f a etc. to get a set of equivalent first-order equations. In
this formulation the “Chern–Simons-like” Lagrangian takes
the form [28,33]
1 1
LC SL = 2 grs ar · das + 6 frst ar · (as × at ),
(
2
)
where grs is a symmetric invertible constant flavor metric
and frst defines a totally symmetric coupling constants on
the flavor space. The dot and cross represent the wedge
product including the contraction of Lorentz vectors with ηab and
abc respectively. In order to get the boundary central charges
of new type black hole in MMG theory, we investigate the
Poisson brackets of the primary constraint functions by using
the Hamiltonian analysis [28,29]. Furthermore, we calculate
the entropy of new type black hole using the charge
variation form including the gravitational Chern–Simons term
[58] with the first order formalism. According to the
prescription of the Ad S/C F T , the entropy of a black hole can
be interpreted in terms of the quantities of the dual CFT side
through the Cardy formula [68,69]. It is also well known that
the conserved charges of the bulk gravity are related with the
energies EL and ER of the dual CFT which is represented in
terms of left and right moving central charges and square of
temperatures. Comparing the entropy and mass of new type
black hole with those of the Cardy formula and the charge
relations in accordance with Ad S/C F T correspondence, we
should suggest that the left and right moving temperatures of
the dual CFT are equal to the Hawking temperature of new
type black hole.
This paper is organized as follows. In Sect. 2, we briefly
review about Wald’s method to define entropy, mass and
angular momentum of a black hole [39–42]. In Sect. 3, as
some examples we compute masses and angular momenta
of some black holes in TMG and NMG theories with the
charge variation form obtained by using the first order
formalism. In Sect. 4, we obtain mass and angular momentum
of BTZ black hole and mass of new type black hole in MMG
theory as some new results. In Sect. 5, we calculate the
central charges and entropy of new type black hole. From these
results, we obtain Smarr relation between mass and entropy
of new type black hole. In Sect. 6, We summarize our results
and add some comments. Appendices are attached to the last
to explain some useful formulae to calculate mass and
angular momentum of the warped Ad S black hole in TMG theory.
2 Brief review of the Wald formalism
In this section we briefly survey the Wald formalism which
has established entropy, mass and angular momentum of
black holes. According to this formalism, black hole entropy
is the integral of the diffeomorphism Noether charge
associated with the horizon-generating Killing field which vanishes
on the Killing horizon [39]. If there exists a black hole
solution with a Killing vector ξ which generates a local symmetry
of the solution, then the corresponding canonical mass and
angular momentum of the solution are well defined at spatial
infinity.
Consider a diffeomorphism invariant theory defined by a
Lagrangian n-form L, where n is the spacetime dimension.
The variation δ L is induced by a field variation δφ
δ L = Eφ δφ + dΘ(φ, δφ).
where φ means the dynamical fields. The Eφ describes the
field equation Eφ = 0 which is constructed from the
dynamical variables φ and their first derivatives, and (n − 1)-form
Θ is the symplectic potential which is constructed by the
dynamical fields and their first variations. The (n − 1)-form
symplectic current is defined by the anti-symmetrized field
variation of Θ,
ω(φ, δφ1, δφ2) = δ1Θ(φ, δ2φ) − δ2Θ(φ, δ1φ).
Then the symplectic form can be defined by an integration
over a Cauchy surface Σ in globally hyperbolic spacetime
Ω(φ, δ1φ, δ2φ) =
δξ L = £ξ L = diξ L .
(
7
)
linearized equation. then the symplectic current is given by
Because the above equation is a total derivative, it shows that
the vector fields ξ on a spacetime generate infinitesimal local
symmetries. According to the Eq. (
3
) under this variation a
Noether current (n − 1)-form Jξ , which is defined by
ω(φ, δφ, £ξ φ) = δΘ(φ, £ξ φ) − £ξ Θ(φ, δφ)
= δ Jξ − diξ Θ(φ, δφ) = δd Qξ − diξ Θ.
Substituting the above formula into (
13
) then the variation of
the Hamiltonian becomes
Jξ = Θ(φ, £ξ φ) − iξ L ,
can be associated to each vector ξ a . Applying exterior
derivative to this current gives
d Jξ = dΘ(φ, £ξ φ) − diξ L = −Eφ £ξ φ.
Therefore the current Jξ is closed when equations of motion
are satisfied, i.e. Eφ = 0. It means that the Noether current
can be represented by the exact form,
Jξ = d Qξ ,
where (n − 2)-form Qξ is constructed from the fields and
derivatives that are appearing in Lagrangian with ξ .
In order to derive the first law of black hole mechanics for
the perturbations of a black hole in an arbitrary
diffeormorphism covariant theory, we investigate an identity at first.
Consider φ to be any solution of the equations of motion.
Let δφ be an arbitrary variation of the dynamical field off
the solution φ. Then we survey the variation of the Noether
current
δ Jξ = δΘ(φ, £ξ φ) − iξ δ L .
Here we put ξ to be an arbitrary fixed vector field in this
variation, i.e. δξ = 0. With (
3
),
iξ δ L = iξ (Eφ δφ + dΘ(φ, δφ))
= £ξ Θ(φ, δφ) − diξ Θ(φ, δφ),
where we apply the equations of motion, Eφ = 0. Therefore
the relation (
11
) becomes
δ Jξ = δΘ(φ, £ξ φ) − £ξ Θ(φ, δφ) + diξ Θ(φ, δφ).
(
12
)
The phase space is the space of solutions to the field
equation in the covariant framework. The variation δξ φ satisfying
equations of motion describes the flow vector of the phase
space corresponding to the 1-parameter family of
diffeomorphisms generated by ξ . Then the variation of the Hamiltonian
Hξ conjugate to ξ is related to the symplectic form (
5
)
(
8
)
(
9
)
(
10
)
(
11
)
∂Σ
where the integral over ∂Σ . Because of £ξ φ = 0 the
symplectic current vanishes. So, Eq. (
13
) implies δ Hξ = 0. Therefore
the last line of the above formula becomes
Now consider a stationary black hole solution with a Killing
field ξ which generates a Killing horizon and vanishes on a
bifurcation surface H. If we choose the hypersurface Σ to
have its outer boundary at spatial infinity and interior
boundary at H, then the variational identity can be expressed with
two boundary terms
δ Qξ =
δ Qξ − iξ Θ.
H
∞
If we assume that the asymptotic symmetries have been
specified by the time translational Killing field and axial
rotational one with the horizon angular velocity ΩH, i.e.
∂ ∂
ξ = ∂t + ΩH ∂φ
then the outer boundary integral of (
16
) can be defined as
the total energy and the angular momentum. Comparing (
16
)
with the first law of thermodynamics
the left hand side gives the black hole entropy as the form
TH δS = δE − ΩHδJ ,
Sent =
2π
κ
H
Qξ .
δχξ = δ Qξ − iξ Θ,
If we re-express the charge variation as a form
(
14
)
(
15
)
(
16
)
(
17
)
(
18
)
(
19
)
(
20
)
(
21
)
δ Hξ =
where Σ is a Cauchy surface. If ξ is a symmetry of all
dynamical fields, i.e. £ξ φ = 0, and their variation δφ satisfy the
δE =
∞
∂
δχξ ∂t , δJ = −
∞
∂
δχξ ∂φ .
(
13
)
the right hand side of (
16
) gives the suitable definition of the
total energy and the angular momentum up to constant, i.e.
In this paper we intend to deal with three dimensional gravity
theories. So we define the variation of the mass and
angular momentum of a black hole in three dimensional gravity
theories as the form
1
δM = − 8π G
∞
∂ 1
δχξ ∂t , δJ = 8π G
∞
∂
δχξ ∂φ ,
(22)
In the action Lorentz indices a, b, c, . . . are suppressed, and
contractions of ηab and abc with wedge products are
represented by the sign ‘·’ and ‘×’ respectively. The third term of
the action with the factor 1/μ describes the ‘Local Lorentz
Chern–Simons’ term. The auxiliary field h is a Lagrange
multiplier for the torsion-free constraint and has the same
parity and dimension of ω.
The variation of the action is given by
where 1/8π G is a constant for three dimensional gravity
theories.
δ L = δe ·
3 Masses and angular momenta of black holes in three
dimensional gravity theories: examples
3.1 Topologically massive gravity
We consider diffeomorphism invariant Lagrangians in three
dimensional gravity theories with the first order
orthonormal frame. In these cases the Lagrangian can be written in
terms of local Lorentz vector-valued 1-form frame fields ea
and connection 1-forms ωab. The spacetime metric tensor is
denoted by the relation
gμν = ηabeμaeνb,
1
ωa = 2 abcωbc.
where ηab is the Minkowski metric. The connection 1-form
ωab can be expressed by a dualised form ωa as
(23)
(24)
Both ea and ωa are considered as independent variables to be
varied separately in the action. The action is represented by
the integral of the Lagrangian 3-form L which can be
constructed from wedge products of the frame fields ea and
connections ωa . Firstly we consider three-dimensional
gravitational Chern–Simons theory or Topologically Massive
Gravity theory (TMG). The Lagrangian form of this theory (TMG)
is given by
Λ0
L = −σ e · R + 6 e · e × e
where D is the Lorentz covariant exterior derivative. From
the above variation we obtain equations of motion as follows
Λ0
− σ R + 2 e × e + Dh = 0,
1
− σ De + μ R + e × h = 0,
T (ω) = De = 0,
and symplectic potential
1
Θ = −σ δω · e + 2μ δω · ω + δe · h.
Following the Wald’s formalism, we can find Noether current
using (
8
),
jξ = d Qξ = Θ(φ, £ξ φ) − iξ L
1
= d − σ iξ ω · e + 2μ iξ ω · ω + iξ e · h .
So, from the above equation we can read the Noether charge
1
Qξ = −σ iξ ω · e + 2μ iξ ω · ω + iξ e · h.
From (
16
) we can calculate the following variation form
δχξ = δ Qξ − iξ Θ = −σ (iξ ω · δe + δω · iξ e)
1
+ μ iξ ω · δω + iξ e · δh + δe · iξ h.
Now we consider a general metric form
(26)
dr 2
ds2 = − f (r )2dt 2 + f (r )2 + r 2(dφ + N (r )dt )2.
(27)
(28)
(29)
(30)
(31)
(32)
(33)
In order to apply the first order orthonormal frame formalism
to this theory we need to take 1-form frame fields as follows,
e0 = f (r )dt, e1 = fd(rr ) , e2 = r (dφ + N (r )dt ). (34)
We firstly consider the third equation of (28) which means the
torsion free condition in (26). For convenience we express
functions f (r ) and N (r ) as the abbreviated form without r .
Then we can find connection 1-forms ωa ,
ω0 = 21 r N e0 + rf e2,
ω1 = 21 r N e1,
ω2 = − 21 r N e2 + f e0,
where ‘ ’ denotes the differentiation of a function with respect
to r . It is easy to find curvature 2-forms and auxiliary field
ha by substituting (34), (35) into the definition of curvature
2-form (26) and second equation of (28).
For the first example we now consider BTZ black hole.
The metric of BTZ black hole with Λ0 = − 12 is given by
the metric form (33) with functions
Now in order to compute mass and angular momentum of
a black hole we consider the charge variation (
20
). Using
the above formula (37), we can rephrase the charge variation
form (32) as follows
δχξ = −σ iξ ω · δe − σ δω · iξ e
1 1
+ μ iξ ω · δω + μ 2 iξ e · δe.
The mass and angular momentum of a black hole is defined
by (22) on the boundary, i.e. spatial infinity. To compute the
charge variation form for mass and angular momentum of
a black hole we examine the interior products and
variations of frame fields and connection 1-forms. Since we are
now dealing with the variation of Hamiltonian (
14
) at spatial
boundaries on the Cauchy surface, we only need to consider
dφ component to compute the charge variation form δχξ .
The variation forms of connection 1-forms to be related to
dφ component are given by
δω0 = δ f dφ, δω2 = − 21 r 2δ N dφ.
There are no variation forms of the frame fields related to dφ
component. Therefore the charge variation for a black hole
mass in TMG becomes
1
− μ
The variation of frame fields ea and connections ωa should
be performed by the coordinate of the horizons r+ and r−
since these horizon coordinates behave as physical
quantities. In other words, variation means the difference between
solutions and background. Then from the definition of (22)
we can get the variation formula for the mass,
1
δM = − 8π G
∞
Integrating and considering the total variation of the right
hand side of the above formula, we can get the mass of BTZ
black hole in TMG theory,
which is the same result in [7,8].
In order to get the angular momentum we need to consider
the asymptotic rotational symmetry, i.e. taking the Killing
vector as ξ = ∂∂φ . Following the same procedure with the
case of the mass the charge variation (38) with this rotation
Killing vector at spatial infinity is given by
Using the above functions (36), the auxiliary fields ha can be
simply expressed as
r 2 2 r+r− ,
M = σ + + r− + 4Gμ 3
8G 2
(35)
(36)
(37)
(38)
(39)
(41)
(42)
dφ.
(43)
(44)
Substituting this form into the definition (22) for the angular
momentum formula, then we obtain
r r r +2 + r −2 .
J = σ + − + 8Gμ 2
4G
These results are also the same with [7,8]. The charge
variation (43) can be re-expressed by a total variation form. So,
we can represent the charge form χξ as
This value should be computed at spatial infinity. The part
of this value with negative power of r vanishes as r goes
to infinity. r 2 term does not give any contribution to (43)
because there is no r+ and r− in this coefficient and this term
only means AdS background. So, considering the constant
term is enough to compute the charge form χξ . Then we can
obtain the same result (45) with the charge definition (22)
and (46).
Now we investigate the space-like warped Ad S3 black
hole solution as the second example [43,53]. The metric of
the space-like warped black hole solution is given by
4dr 2
ds2 = −N (r )2dt 2 + 4R(r )2 N (r )2
+ 2 R(r )2(dθ + N θ (r )dt )2,
where
r
R(r )2 = 4
N (r )2 =
N θ (r ) =
3(ν2 − 1)r + (ν2 + 3)(r+ + r−)
−4ν r+r−(ν2 + 3) ,
2(ν2 + 3)(r − r+)(r − r−) ,
4R(r )2
2νr −
r+r−(ν2 + 3) .
2 R(r )2
From the metric we can easily read off the 1-form frame fields
e0 = N (r )dt, e1 = 2 R(r2)dNr (r ) ,
e2 =
R(r )(dθ + N θ (r )dt ),
and 1-form connections can be calculated using the third
equation of (28), i.e. torsion-free conditions, as follows
ω0 =
ω1 =
ω2 = −
∂ ,
In order to find the black hole mass we consider ξ = ∂t
then non-vanishing interior products and variations of the
frame fields, connections and auxiliary fields can be
represented by Appendix B. Then the charge variation δχξ for the
mass of the warped Ad S3 black hole is given by
3(ν2 − 1)
2
2r+r−
where we replace the gravitational constant G with G in the
definition (22). The metric for the warped Ad S3 black hole
(47) is written as the form with dimensionless coordinates.
Only the cosmological constant has a length dimension.
So, we should change these coordinates to be dimensionful
to give the correct result. Therefore integrating the above
formula we can get
M =
(ν2 + 3)
24G
1
r+ + r− − ν
1
+ 2μ
ν2
2 +
Since the above formula can be represented by the total
variation form, we simplify this form to become a charge form
2 +
When r goes to infinity the charge form (
54
) can be expressed
as a polynomial of r with r 2 term as its highest order. The
coefficients of r 2 and r vanish to leave the constant term
only. Substituting all functions into the above form with σ =
1, μ1 = 3ν , we obtain
∂
χξ ∂θ
= −
1
r+ + r− − ν
(r+ − r−)2 dθ .
From the definition of the angular momentum Eq. (22), we
can get the angular momentum of the warped Ad S3 black
hole
J = −
ν(ν2 + 3)
96G
1
r+ + r− − ν
To obtain the above result we also consider the change of the
coordinate to be dimensionful. So, we change the
gravitational constant G to G . Because these coordinates changes
the angular velocity’s dimension, so in order to have correct
dimension we need to change the angular velocity with 1/
and rotational Killing vector with . Then we can obtain the
correct angular momentum (
56
) which is the same result with
[43].
3.2 New Massive Gravity The Lagrangian of New Massive Gravity (NMG) theory can be represented by the first order form
δ L = d
1
− σ δω · e − m2 δω · f + δe · h
1
+ δh · De − m2 δ f · (R + e × f )
Firstly, we investigate BTZ black hole solution in NMG
theory. BTZ black hole is a solution of NMG theory with a
cosmological constant Λ0 which appears (
64
) below. So, we
can use the same metric form with (33), functions (36) and
the same frame 1-form fields (34). Using second and third
equations of motion of (
59
), we can get two auxiliary fields
With auxiliary fields solution (
63
) a parameter condition can
be appeared by solving the fourth equation of (
59
)
(
58
)
(
60
)
(
61
)
(
62
)
(
63
)
(
64
)
f a
= 212 ea , ha = 0.
(
53
)
where h and f are auxiliary fields and m is a mass parameter
[70,71]. The variation of NMG Lagrangian is given by
L N MG = − σ e·R+ Λ60 e·e×e+h·De− m12 f · R+ 21 e× f ,
(
57
)
We can re-express the symplectic potential Θ and the Noether
charge Qξ by inserting auxiliary fields (
63
) into (
62
) and then
calculate the charge variation as follows
1
δχξ = − σ + 2m2 2 (iξ ω · δe + iξ e · δω).
Performing the interior products and variations of frame
1form fields and connection 1-forms with ξ = ∂∂t we obtain
Therefore we can obtain the mass of BTZ black hole in NMG
theory,
M =
r 2 2 1
+ + r− σ + 2m2 2 .
8G 2
To find the angular momentum we consider the Killing vector
ξ = ∂∂φ . Then as r goes to infinity, the charge variation
becomes
Applying the above formula to the definition of the angular
momentum (22), we can get
Therefore we can obtain the angular momentum of the BTZ
black hole in NMG theory,
r r 1
+ − σ + 2m2 2 .
J = 4G
Now we investigate the new type black hole solution which
appears as another solution in NMG theory [14]. The metric
form of this black hole is given by
dr 2
ds2 = − f (r )2dt 2 + f (r )2 + r 2dφ2,
where
(
65
)
This non-rotating new type black hole solution is represented
by the general form (33) with N (r ) = 0. Solving equations
of motion in (
59
), then we can find
(
67
)
(
68
)
(
69
)
(
70
)
(
71
)
e0 = f (r )dt, e1 = fd(rr ) , e2 = r dφ,
ω0 = rf e2, ω1 = 0, ω2 = f e0.
Solving the second and third equations of motion in (
59
) with
(
71
), we can simply determine the auxiliary fields f a
and all auxiliary fields ha vanish.
So, the charge variation (
62
) with the condition, ha = 0,
becomes
δχξ = −σ (iξ ω · δe + iξ e · δω)
1
− m2 (iξ ω · δ f + iξ f · δω)v
For the computation of the charge variation we need to get the
variations of frame fields, connection 1-forms and auxiliary
fields. The non-vanishing dφ components of useful variations
are given by
δω0 = δ f dφ, δ f 2 = δ( f f )dφ.
All variations of the frame fields related to dφ component
vanish. Because we are dealing with the non-rotating new
type black hole, we only consider the computation of the
black hole mass. Performing the calculation of (75) for the
Killing vector ξ = ∂∂t with the function (
71
), we can get the
charge variation
From above conditions we should take a relation σ =
2m2 2 = 1, then the other condition gives a relation Λ0 =
− 212 . So we can rearrange the charge variation such as
Then the variation of the mass of the black hole with the
definition (22) is given by
1
δM = − 8π G
∞
This result has been computed in [49,50,55].
4 Masses and angular momenta of black holes in
Minimal Massive Gravity theory
The Lagrangian of Minimal Massive Gravity (MMG) theory
is given by
α
LMMG = LTMG + 2 e · h × h
Λ0
= −σ e · R + 6 e · e × e + h · T (ω)
where the gravitational Chern–Simons term and some
additional term with auxiliary fields ha are included [29]. In order
to get equations of motion we investigate the variation of the
MMG Lagrangian, then the variation becomes
α
δ L M MG = δ L T MG + 2 δ(e · h × h)
= δe ·
− σ R(ω) + Λ20 e × e + D(ω)h + α2 h × h
1
+ δω · μ
R(ω) − σ T (ω)
+ e × h
+ δh · (T (ω) + αe × h)
+ d
T (ω) + αe × h = 0, R(ω) + μe × h − σ μT (ω) = 0,
− σ R(ω) + Λ20 e × e + D(ω)h + α2 h × h = 0. (82)
We can also read the symplectic potential Θ from the
variation of the Lagrangian and calculate Noether charge using
(
8
),
1
Qξ = − σ iξ ω · e + 2μ iξ ω · ω + iξ e · h.
The first equation of (82) does not guarantee the torsion free
condition in this theory. So we should make these equations
torsion free through shifting connections. Shifting
connections ω to new dual spin-connections Ω = ω + αh, then
equations of motion (82) become
T (Ω) = 0,
αΛ0
R(Ω) + 2 e × e + μ(1 + σ α)2e × h = 0,
α Λ0
D(Ω)h − 2 h × h + σ μ(1 + σ α)e × h + 2 e × e = 0.
Assuming the frame 1-form fields ea is invertible and 1 +
σ α = 0 then we can find the field equations condition,
e · h = 0,
i.e. symmetric condition for hμν , using the following
identities,
D(Ω)T (Ω) ≡ R(Ω) × e,
D(Ω)R(Ω) = 0.
Noether charge and symplectic potential can be rearranged
using connection shifting Ω = ω + αh as follows
1
Qξ = −σ iξ Ω · e + 2μ iξ Ω · Ω + (1 + σ α)iξ e · h
α
− 2μ (iξ Ω · h + iξ h · Ω − αiξ h · h),
1
Θ = −σ δΩ · e + 2μ δΩ · Ω + (1 + σ α)δe · h
α
− 2μ (δΩ · h + δh · Ω − αδh · h).
To get the above shifted symplectic potential we used the
relation δh · e = δe · h from the condition (84).
4.1 BTZ black hole in MMG
From now we investigate BTZ black hole which is a solution
of MMG theory. The metric of this black hole is the same with
(33) and frame 1-form fields are also the same with (34). To
find the shifted connection 1-form Ω we use the first equation
of (82), i.e. T (Ω) = de + Ω × e = 0, then these connection
1-forms are given by the same form with (35). Solving the
second equation of motion in (83) with functions (36) we can
simply represent the auxiliary fields
ha
1
= − μ(1 + σ α)2
αΛ0
2
where we replace the constant part with λ for convenience.
Solving third equation of (83) we can find a parameter
condition
αλ2 + 2σ μ(1 + σ α)λ − Λ0 = 0.
α
+ λ 2μ
Using the above result ha = −λea , Noether charges and
symplectic potential can be reduced to
1
Qξ = −σ iξ Ω · e + 2μ iξ Ω · Ω − λ(1 + σ α)iξ e · e
(iξ Ω · e + iξ e · Ω + αλ iξ e · e),
1
Θ = −σ δΩ · e + 2μ δΩ · Ω − λ(1 + σ α)δe · e
α
+ λ 2μ (δΩ · e + δe · Ω + αλ δe · e).
Therefore we can represent the charge variation form δχξ
using above Noether charge and symplectic potential
α
δχξ = − σ − λ μ
1
(iξ Ω · δe + iξ e · δΩ) + μ iξ Ω · δΩ
2 α2
− 2λ(1 + σ α) − λ μ
iξ e · δe.
From now we follow the procedure for BTZ black hole
in TMG and NMG. The non-vanishing variation forms of
shifted connection 1-forms related to dφ component are
given by the same results with (39). All variations of the frame
fields related to dφ are vanished. Performing the calculation
(89) for the Killing vector ξ = ∂∂t , the charge variation for
BTZ black hole mass in MMG becomes
Applying this result to the definition (22), we can obtain the
variation of black hole mass
Substituting the λ value (86) into the above formula, then we
obtain the mass of BTZ black hole in MMG theory,
M =
r +2 + r −2
8G 2
α(1 − αΛ0 2)
σ + 2μ2 2(1 + σ α)2
r r
+ − . (92)
+ 4Gμ 3
TH = r2+2π−2rr+−2 .
To find the angular momentum of BTZ black hole in MMG
theory we consider the charge variation form for the Killing
vector ξ = ∂∂φ . Then the computation of the charge variation
form (89) is given by
Adapting the definition of (22) the variation form of the
angular momentum is given by
(93)
1
δJ = 8π G
Substituting the λ value (86) into the above formula then we
can obtain the angular momentum of BTZ black hole
r r
+ −
J = 4G
α(1 − αΛ0 2)
σ + 2μ2 2(1 + σ α)2
r +2 + r −2 .
+ 8Gμ 2
As already mentioned in case of BTZ black hole in TMG
theory, we can re-express the charge variation form (93) as a
total variation form. Therefore the charge form for the
angular momentum of the BTZ black hole in MMG theory can be
represented by
This result is the same with (46) except replacing the
coefficient σ with σ − λ α . Only the constant term contributes to
μ
the computation of the angular momentum. So, we can obtain
the angular momentum (95) of BTZ black hole in MMG
theory. The same results have been calculated by using ADT
formalism in [56].
From the metric form of BTZ black hole (33) and (36),
we can simply read the angular velocity as
With (97) and (98) the relation between the variation (91),
(94) and the variation of the entropy can give the first law of
black hole thermodynamics.
4.2 New type black hole in MMG
There exists a black hole solution in MMG at a special point,
i.e. “merger point” [31]. This black hole resembles new type
black hole solution in NMG theory. This solution is not
locally isometric to the Ad S vacuum but asymptotically Ad S
as r goes to infinity. There also exists a d S type solution but
we pay our attention to Ad S type solution. The black hole
solution has the same form of (
70
) with (
71
). So we can
take same forms of frame 1-form fields ea (
72
), connection
1-forms ωa replaced by Ωa (
73
).
Solving the second equation of motion in (83) with the
function (
71
), we can determine the auxiliary fields ha as
detailed forms
h0
h1
h2
1 1
= μ(1 + σ α)2 2 2 −
1 1
= μ(1 + σ α)2 2 2 −
1 1
= μ(1 + σ α)2 2 2
αΛ0
2
e0,
1 −
αΛ0 e1,
2
r+ + r−
r
The charge variation (
20
) with Noether charge and
symplectic potential (85) can be represented by a simple form
δχξ = −σ (iξ Ω · δe + iξ e · δΩ)
1
+ μ iξ Ω · δΩ + (1 + σ α)(iξ e · δh + iξ h · δe)
α
− μ (iξ Ω · δh + iξ h · δΩ − αiξ h · δh).
To compute this charge variation we need to find
nonvanishing variations of frame fields, connection 1-forms and
auxiliary fields. The dφ component of all variations of the
frame fields vanish. The non-vanishing variations are given
by the form
δΩ0 = δ f dφ, δh2 = μ(1 +1 σ α)2 δ( f f )dφ.
Because this black hole solution is non-rotating one, so we
only need to perform the calculation of the black hole mass.
To obtain the mass of this black hole we should consider
the time-like Killing vector ξ = ∂∂t . Then the non-vanishing
interior products of frame 1-forms, connection 1-forms and
auxiliary fields with this Killing vector are given by
(99)
(100)
(101)
iξ e0 = f, iξ Ω2 = f f ,
( f f )
iξ h0
2
1
= μ(1 + σ α)2
αΛ0
− 2
(102)
(103)
(104)
(105)
(106)
(107)
(108)
(109)
where λ is the same constant with (86) whenever we consider
the function f of (
71
). Inserting (101) and (102) into (100) we
can find the charge variation form δχξ [ ∂∂t ]. But this form of
charge variation can be simply reduced to the total variation
form, so we can obtain the charge form
∂
χξ ∂t
Therefore the coefficient of f 2 in (103) are the same that of
( f f )2. So, we can simplify the charge
∂
χξ ∂t
= 21 σ − λ μα ( f 2 − 2( f f )2)dφ
α 1
= − σ + 2 4 2 (r+ − r−)2dφ.
From the above result we can obtain the mass of new type
black hole in MMG
M = − 8π1G χξ ∂∂t = 16 G1 2 σ + α2 (r+ − r−)2
∞
α 2
= 32Gμ2 4(1 + σ α)2 (r+ − r−) .
This result is described by the square of the difference
between two horizons r+ and r−. It is the same form with that
of new type black hole in NMG theory except the parameter
shift σ + α/2.
5 Thermodynamic properties of the new type black hole
in minimal massive gravity
In this section we briefly explain to find the central charge of
new type black hole and calculate the entropy of this black
hole using the method of [58]. we compare these result with
the Cardy formula for black hole entropy. According to the
description of AdS/CFT correspondence, we can represent
the relation between the black hole mass and the left and
right moving energies [68,69]. This result of black hole mass
is the same that we have calculated the above (109). From
these result we can obtain first law of thermodynamics of
new type black hole. Then we suggest the relation between
the Hawking temperature and the left and right temperatures
which come from the left and right moving energies.
5.1 Boundary central charges
The method to find the boundary central charge of this MMG
theory is described in detail by using the Poisson bracket
algebra and Hamiltonian analysis [28,29,70,72]. The Lagrangian
of the three dimensional gravity theories can be represented
by the Chern–Simons- like form as follows [28],
1 1
LC SL = 2 grs ar · das + 6 frst ar · (as × at ).
The notation ar means a collection of Lorentz vector valued
1-forms aμra d x μ, where r is a “flavor” index running 1 · · · N .
Flavor N describes the fields of this MMG theory e, h and ω.
grs is the metric on the flavor space and frst is the coupling
constants. This description also includes other gravity
theories, i.e. TMG, NMG, ZDG, even MMG theory. Performing
the separation of space-time component
ara = a0ra dt + aira d xi ,
with writing 0i j = i j , the Lagrangian density becomes
L = − 21 i j grs air · a˙ j + a0r · φr ,
s
where the Lorentz vectors a0r are Lagrange multipliers for
primary constraints
φr =
i j grs ∂i asj + 21 frst ais × atj .
The Hamiltonian density can be obtained with these primary
constraints
H = − 21 i j grs air · ∂0a j − L = −a0r · φr .
s
(115)
(116)
(117)
(118)
(119)
In order to obtain the Poisson brackets of the primary
constraints the smeared functionals should be defined by
integrating the constraint function (113) against a test function
ξ r (x ) as follows
ϕ[ξ ] =
Σ
d2x ξar (x )φra (x ) + Q[ξ ],
where Σ is a space-like hypersurface and Q[ξ ] is a boundary
term to remove delta-function singularities in the brackets of
the constraints [28]. With this boundary term, ϕ is said to
be “differentiable” under a general variation of the fields.
In [28] the Poisson brackets of the constraint functions are
computed by using
{aira (x ), asjb(y)}P.B. = i j grs ηabδ(
2
)(x − y).
They are given by
{ϕ[ξ ], ϕ[η]}P.B. = ϕ[[ξ, η]] +
Σ
d2xξar ηbsPrasb
−
∂Σ
d xi ξ r · [grs ∂i ηs + frst (ais × ηt )],
(110)
where
(111)
(112)
(113)
[ξ, η]tc = f rs c a ηc,
t abξ r s
and
Prasb = f tq[r fs] pt ηabΔ pq + 2 f tr[s fq] pt (V ab) pq ,
Vapbq = i j aipaa qjb, Δ pq = i j aip · a qj .
The Poisson brackets of the first-class constraints with the
dynamical fields of the theory generate local Lorentz
transformations and diffeomorphisms. The second term of (117) is
related to the secondary constraint associated with the
consistency condition guaranteeing time independence of the
primary constraints. If we regard the test function ξar (x ) as the
gauge parameters of boundary condition preserving gauge
transformation, then the Poisson bracket algebra is
isomorphic to the Lie algebra of the asymptotic symmetries and
generally can be centrally extended [73].
The Chern–Simons-like model (110) is manifestly
invariant under diffeomorphism and local Lorentz transformations.
To understand these constraints which generate these
symmetries, it is convenient to investigate the Poisson brackets
of the gauge transformation with the dynamical variables of
the theory. The calculations can be performed using (113),
(115) and (116) as a form
(114)
{ϕ[ξ ], air } = ∂i ξ r + f rst ais × ξ t .
The constraint ϕω[ξ ω] generate a local Lorentz
transformation with ξ ω = ζ and ξ r = 0 for r = ω, as follows
β =
In (121), we have used a relation f rsω = grt ftsω = δrs where
a coupling constant frsω is given by grs . Diffeomorphisms
associated with a vector field ζ μ are generated by an
appropriate combinations of constraint functions
where £ζ describes the Lie derivative with respect to the
vector field ζ μ and · · · means the term proportional to equations
of motion. So, on-shell this gives the correct transformation
rule for the dynamical variables of the theory.
To find the boundary central charges from the Poisson
bracket algebra (117) we need to consider the two sets of
mutually commuting first class constraints
L±[ζ ] = ϕe[ζ μeμ] + ϕh [ζ μhμ] + a±ϕω[ζ μeμ],
with constant a±. Then the Poisson brackets should be
{L±[ξ ], L±[η]} = ∓
L±[[ξ, η]],
{L+[ξ ], L−[η]} = 0.
2
To calculate above commutators, we firstly need to determine
the non-zero components of the metric grs and the coupling
constants frst on the “flavor” space in MMG theory. They
are given by
1
geω = −σ, ghe = 1, gωω = μ ,
feωω = −σ, feee = Λ0, fhωe = 1,
1
fωωω = μ , fehh = α.
Using (117), the Poisson brackets of the primary constraints
can be determined as follows,
{φωa, φωb}P.B. = εabcφωc, {φωa, φeb}P.B. = εabcφec,
{φωa, φhb}P.B. = εabcφhc , {φea , φeb}P.B. = Λ0εabcφhc ,
{φha , φh }P.B. = αεabcφωc, {φea , φh }P.B. = αεabcφec
b b
+ μ(1 + σ α)εabcφωc + σ μ(1 + σ α)εabc hc
φ .
Let e¯ be the Ad S background dreibein. By using the
“merger point” conditions (104), (105), the auxiliary field
(99) becomes h¯ = βe¯ where the parameter β is given by
on the Ad S background. Also, for the new type black hole
case, we can find the Poisson brackets between smeared
functionals
{ϕe[ξ e], ϕe[ηe]}P.B. = Λ0ϕh [[ξ, η]],
{ϕh [ξ h ], ϕh [ηh ]}P.B. = β2αϕh [[ξ, η]],
{ϕh [ξ h ], ϕω[ηe]}P.B. = βϕh [[ξ, η]],
{ϕω[ξ e], ϕe[ηe]}P.B. = ϕe[[ξ, η]],
{ϕω[ξ e], ϕω[ηe]}P.B. = ϕω[[ξ, η]],
{ϕe[ξ e], ϕh [ηh ]}P.B. = βμ(1 + σ α)ϕω[[ξ, η]]
+ βαϕe[[ξ, η]] + βσ μ(1 + σ α)ϕh [[ξ, η]],
where ξ h = ξ μhμ = βξ e and [ξ, η] = ξ e × ηe. With the
parameter
Λ0 = −
Therefore we can use the above parametrization for a± and
two identities
a± = −σ μ(1 + σ α) ±
= −βα ±
a±2 + 2βμ(1 + σ α) = 2
{L±[ξ ], L±[η]} = ∓
to find out the Poisson bracket
2
L±[[ξ, η]]
σ μ2(1 + σ α)2
α
= ±
2
a±,
∓
2
+ ω¯ φ ±
1
2σ + α ± μ
2
∓ a± e¯φ
× ηe .
∂Σ
dφξ e · ∂φ ηe
After including the proper normalizations of first class
constraints, we can find the asymptotic symmetry algebra
consisting of two sets of Virasoro algebra with a central charge
3
cL,R = 2G
where G is the three dimensional Newton constant. We notice
that the central charges can not be reduced to the TMG
representations in the TMG limit α → 0, because there exist a
relation (106) between μ and α at “merger point” in MMG
theory. This is somewhat different case from the Ad S
background one since the solution of new type black hole in TMG
does not exist. The same result can also be obtained by using
the variation of the boundary charge Q±[ξ ]
d xi ges ξ e + ghs ξ h + a±gωs ξ e · δais
where s sums over e, h and ω. With some proper constants
the coefficient of the integral give the same central charges.
5.2 Thermodynamics
To find the thermodynamic relations for new type black hole
in MMG theory, we should use the previous result for the
mass in Sect. 4.2 and calculate the entropy of the black hole.
To obtain the black hole entropy of new type black hole, we
firstly consider the Lagrangian for the Chern–Simons-like
form (110). The variation for this Lagrangian is given by
δ LC SL = d
1
2 grs δar ·as
+δar · grs das + 21 frst as ×at ,
where the first term gives the symplectic potential
Θ(a, δa) = 21 grs δar · as ,
and the second term describes equations of motion.
Following the Wald’s formalism, we can find Noether charge
κ
TH = 2π =
r
+4π− r2− .
Qξ = 21 grs iξ ar · as ,
with the on-shell condition. Let us consider the variation of
Noether charge
δ Qξ = 21 (grs iξ δar · as + grs δar · iξ as ).
which calculation should be performed at the event horizon of
a black hole H and ΩH is the angular velocity at the horizon
when we consider a rotating black hole case. Therefore we
just consider the first variational charge form. By using (103),
(106) and (107), we can obtain the charge variation form
Substituting (148) and (149) into (147) and then integrate
the variational form, we can obtain the entropy of new type
black hole in MMG theory as follows
π
SB H = 2G
α
σ + 2
(r+ − r−) = 4Gπμα2(r2+(1−+r−σ)α)2 .
δ Qξ = grs iξ ar · δas ,
which is the same form of the charge variation form (
20
) on
the bifurcation horizon H, i.e., δ Qξ = δχξ . Therefore, we
can define the variation form of the entropy of a black hole
in the Chern–Simons-like gravity theory
κ 1
2π δSB H = − 8π G
H
δχξ .
The calculation for the black hole entropy including the
gravitational Chern–Simons term in the action has been
performed in [58]. The charge for the black hole entropy has
been defined by Qξ = Qξ − Cξ where δCξ = iξ Θ + Σξ
with a choice Σξ = 0. So, it is the same definition with
(146). Now we look for the entropy of new type black hole
in MMG theory. Then the entropy formula can be written by
the variational form
1 2π
δSB H = − 8π G · κ
H
The Killing vector ξ vanishes on the bifurcation horizon H,
so the interior product of the symplectic potential should also
be vanished, i.e., iξ Θ = 0. By using this condition, the first
term of the variation form (144) becomes the same one with
the second term. So, we can get
(145)
(146)
,
(147)
(148)
(149)
(150)
Considering the mass formula (109), the only physical
parameter is the mass M of the new type black hole and
can also be given by the difference between two horizons
r+ − r−. So we can obtain the first law of black hole
thermodynamics by using the variation of mass (109) and (150)
δM = TH δSB H .
According to the usual description of AdS/CFT
correspondence, the black hole entropy can be represented by the Cardy
formula which depicts the entropy of a dual CFT in terms of
central charges cL and cR at the temperatures TL and TR as
follows
π 2
3
SB H =
(cL TL + cR TR ).
In three dimensional gravity theories, the mass of an
asymptotically Ad S black hole can be obtained by
M = EL + ER ,
where the left and right moving energies of the dual CFT can
be defined as
EL =
π 2
6
cL TL2, ER =
cR TR2.
π 2
6
Here, we suggest that the left moving and right moving
temperatures should be equal to the Hawking temperature such
as
(151)
(152)
(153)
(154)
(155)
(156)
of these variations (22), we have obtained masses and
angular momenta of the asymptotically Ad S3 black holes in three
dimensional gravity theories with the first order formalism.
Following the Wald formalism, we have calculated
symplectic potential and conserved Noether charge which consist
of dreibeins, spin connections and auxiliary fields. To find
conserved charges such as mass and angular momentum, we
need to calculate the value (
20
). Firstly, we have reproduced
well-known results of the mass and angular momentum for
some black holes in TMG and NMG as some examples.
Secondly we have paid our attention to compute new results of
mass and angular momentum of BTZ and new type black
holes in MMG theory. These results are the same form with
one in TMG theory with a modified σ parameter which is a
function of α and Λ0. The mass of new type black hole in
MMG theory is also the same form with that of NMG theory.
Only the difference is a parameter σ + α/2 which include a
parameter α related to the auxiliary field h term in the action.
The MMG theory is an alternative modified theory to avoid
“bulk vs. boundary clash” [29]. MMG equation (
1
) can be
expressed by a simple modification of equation in TMG
theory including an auxiliary field term with a parameter α.
The elimination of the auxiliary fields from the MMG action
cannot make an action for the metric only. So in order to get
proper conserved charges such as mass and angular
momentum, we should make a certain formula including auxiliary
fields even though some results have obtained by using the
Eq. (
1
) in [60]. Definitions (22) are enough to find mass
and angular momentum in three dimensional gravity theories
while we are dealing with the integration at spatial infinity.
We also calculate the central charges of new type black
hole by investigation of the Poisson brackets of the constraint
functions coming from the “Chern–Simons-like” Lagrangian
form. We compute the entropy of new type black hole by
taking use of the method in [58] with the first-order formalism.
From these results, we can read the Smarr relation (156)
which is also derived from the first law of black hole
thermodynamics (151). The entropy and mass can be compare with
the Cardy formula (152) and energy definition of the dual
CFT (153). This comparison leads us to suggest a relation
(155) that the left and right moving temperatures should be
equal to the Hawking temperature. Then the Smarr relation
is satisfied under this suggestion.
Many constructions of the conserved charge for black
holes are introduced in a review paper [74]. By using the
linearization of the metric on the asymptotically flat
spacetime, the conserved charge has been constructed by Arnowitt,
Deser and Misner (ADM) [34]. This ADM formalism has
been extended to the covariant and higher curvature gravity
theories by Abbott, Deser and Tekin [36–38]. There was a
non-trivial generalization of the ADT charge formalism by
promoting to the off-shell level and including non-covariant
term like a gravitational Chern–Simons term [44,45]. In [74]
TL = TR = TH ,
1
M = 2 TH SB H .
then relations (152) and (153) are given by the same results
(150) and (109) with the central charges (139). Therefore the
Smarr formula is given by
From the variation of this formula with r+ − r− we can also
derive the first law of black hole thermodynamics (151).
6 Conclusion
In this paper we have investigated the variation form (22) of
the mass and angular momentum of black holes through the
Wald formalism. These variations are satisfied with the first
law of black hole thermodynamics (
18
). It have been shown
that the Lagrangian for diverse three dimensional gravity
theories can be expressed by using first order formalism which
comprise with auxiliary fields [28–30]. Using the definition
the quasi-local conserved charge for the Chern–Simons-like
theories of gravity (110) has been constructed with the first
order formalism. To find this charge the authors of [74] have
used the off-shell ADT method and the field variation with
the Lorentz–Lie derivative
With all functions of (48) and their derivatives we can
calculate curvature 2-forms. From now we abbreviate all
functions as their abridged form without coordinate r including
the above formulae. Substituting connections (
50
) into the
second equation in (26), curvature two-forms are given by
R0 = e1 ∧ e2
+ e1 ∧ e
R1 = e2 ∧ e0
−
R2 = e0 ∧ e1 3
,
·
.
2
(A.2)
Solving the second equation of (28) with torsion free
condition, then we obtain the auxiliary fields as follows
(157)
(158)
δξ ara = Lξ ara − δrωd λ˜a ,
where the definition of the Lorentz–Lie derivative L is given
by
Lξ ea = £ξ ea + λacec,
and λ˜a = 1/2 · abcλbc is a generator of the local Lorentz
transformation. This derivative has been introduced to avoid
the divergence of the spin connection on the event horizon,
even though the interior product between spin connection
and Killing vector becomes finite on the bifurcation surface
[63].
In our approach, the Wald formalism for the
diffeomorphism invariant Lagrangian has been adapted. Because the
variation of the field variables for a vector filed ξ can be
described by (123), the symplectic current ω(φ, δφ, £ξ φ) in
(
13
) vanishes when we take £ξ φ = 0. So, we define the
charge variational form as follow the path of the solution on
the Cauchy surface Σ .
Acknowledgements S. N. was supported by Basic Science Research
Program through the National Research Foundation of Korea (NRF)
funded by the Ministry of Education (no. 2013R1A1A2004538). J. D.
P. was supported by a Grant from the Kyung Hee University in 2009
(KHU-20110060) and Basic Science Research Program through the
National Research Foundation of Korea (NRF) funded by the Ministry
of Education (no. 2015R1D1A1A01061177).
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Some calculations for the curvature
twoforms of the warped Ad S3 black hole
In this appendix we summarize some useful functions and
relations to calculate curvature 2-forms of the warped Ad S3
black hole. This black hole solution is represented by some
functions given by (48). The second function of (48) can be
represented by the form
R N =
·
·
+
+
+
·
·
·
+
·
·
+
+
R
R = 2r +
N =
2N R
2
and their derivatives as follows
r+r−(ν2 + 3) .
(A.4)
From the above formulae and (48) we can get some relations
(ν2 + 3) 2r − r+ − r−
2 (r − r+)(r − r−)
1 3(ν2 − 1) r
− (r − r+)(r − r−) r + 4 R2
.
(A.5)
= −
3(ν2 − 1) 2νr
8 R2
r+r−(ν2 + 3)) ,
,
Now we use the previous results to calculate curvature
2forms (A.2). The calculations of the right hand side of (A.2)
are given by
+
·
2R N
2 −
2R N
2
−
+ 3
Substituting all the above results into (A.2) we can rearrange
curvature 2-forms for simple forms
1
· R2 2νr −
With (A.5) and (A.6) we can calculate the coefficients of the
dreibeins in (A.3), then the auxiliary fields ha are simply
given by
h0 = 21μ
h1 = 21μ
h2 = 21μ
ν2
ν2
ν2
2 −
2 −
2 +
Appendix B: Some useful formulae for the charge
variations of the warped Ad S3 black hole in Topologically
Massive Gravity
(r−δr+ + r+δr−),
−
To calculate the charge variation for the warped Ad S3
black hole we should make the variation of some functions
appeared in Eq. (
51
). The useful variations for (
51
) is as
follows
δh0
δh2 =
= − μ δ(G R)dθ ,
2μ
ν2
2 +
3(ν2 − 1)
2
δ R + μ δ(F R) dθ . (B.3)
To find the black hole mass variations we should find some
interior product of dreibeins, connection 1-forms and
auxiliary fields with ξ = ∂∂t . The non-vanishing interior products
of dreibeins and connections are given by
iξ ω2
= −
iξ h0
iξ h2
1
− R (r − r−)δr+ + (r − r+)δr−
.
In order to get the charge variation (
51
) and (
53
), we consider
the variations of frame fields, i.e. dreibein, connections
1forms and auxiliary fields. The charge variations are defined
by the integral over a certain Cauchy surface with constant
time and two boundaries at horizon surface and infinity. So,
we only need to consider the variation of the angle θ part,
i.e. dθ terms. The non-vanishing variations for dreibein (
49
)
and connection 1-forms (
50
) are as follows
δe2 = δ Rdθ , δω0 =
δω2 = −δ(R3 N θ )dθ .
The variations of auxiliary fields (A.9) are given by
2
δ(N R R )dθ ,
(B.1)
(B.2)
To find the black hole angular momentum variations with
ξ = ∂∂θ the non-vanishing interior products are given by
iξ e2 =
R, iξ ω0 =
2N R
2
R, iξ ω2
= −
R4 N θ = −
R
+R 2r +
(N R R )2 =
−
1
+ 2μ
R
+ 2r +
,
To compute the angular momentum (
54
) we need to use the
following some functions
Owing to the above formulae the charge variation for the
angular momentum (
54
) can be expressed by the following
form
ν2 + 3 4ν
C = 3(ν2 − 1) (r+ + r−) − 3(ν2 − 1) r+r−(ν2 + 3).
Using the above approximations we can re-express the charge
(B.9) with σ = 1 and μ1 = 3ν . So, rearranging the charge
with respect to r orders becomes
χξ
∂
∂θ =
where the coefficients of r 2 and r are vanished. Therefore it
gives the result (
55
).
(ν2 + 3(ν2 − 1))R2
2 3(ν2 − 1)(ν2 + 3)
+ μ 4 2
2
2
8
3(ν2 − 1)
3(ν2 − 1)
3(ν2 − 1) r
R
r
R
R
r
where
(B.9)
As r goes to the infinity the previous result can be expanded
as a series expansion of r . The useful asymptotic expansions
including R(r ) described by (48) are given by
1 3 2
1 − 2r C + 8r 2 C + · · · ,
1 1 2
1 + 2r C − 8r 2 C + · · · ,
3(ν2 − 1)
2
1 2
1 + 8r 2 C · · · ,
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