Warped $AdS_3$ black hole in minimal massive gravity with first order formalism

We obtain a black hole solution of minimal massive gravity theory with Maxwell and electromagnetic Chern-Simons terms using the first order formalism. This black hole solution can be translated into the spacelike warped $AdS_3$ black hole solution with some parameters' conditions changing their coordinates system into a Schwarzschild one. Applying the Wald formalism to this theory with the first order formalism, we also find out the entropy, mass, and angular momentum of this black hole solution satisfies the first law of black hole thermodynamics. Under the assumption that minimal massive gravity theory with suitable asymptotically warped $AdS_3$ boundary conditions is holographically dual to a two dimensional boundary conformal field theory, we find appropriate central charges by using the relations between entropy of this black hole and the Cardy formula in dual conformal field theory described by left and right moving central charges and temperatures.


Introduction
There are several three dimensional gravity theories those are suggested to explain various physical problems for example topologically massive gravity (TMG) [1,2], new massive gravity (NMG) [3], minimal massive gravity (MMG) [4] etc. Various 3-dimensional gravity theories can be represented by the Chern-Simons-like Lagrangian [5]. It is well known that an extension of general relativity in 3-dimensions is TMG which is composed of Einstein-Hilbert term, cosmological constant and gravitational Chern-Simons term, breaking parity symmetry. There exists a single massive spin 2-mode on the linearization of this theory. TMG theory also allows some black hole solutions with AdS 3 asymptotics [6,7,8,9,10].
In the viewpoint of AdS/CF T correspondence there exists a discrepancy between any 3-dimensional gravity theories with asymptotic AdS 3 geometry and its dual CFT on the boundary. Whenever the spin-2 graviton modes propagating on the bulk have positive energy, the central charge of a dual boundary CFT is negative, i.e. non-unitary CFT. This discrepancy is also related to a problem that asymptotic AdS 3 black hole solutions have negative mass values whenever bulk graviton modes have positive energy, what is called "bulk vs. boundary clash". An alternative method have been suggested to circumvent this discrepancy, that is "minimal massive gravity (MMG)" theory [4].
One of the purposes of this paper is to construct a black hole solution in MMG theory including Maxwell and electromagnetic Chern-Simons terms with the first order formalism. An intrinsically rotating black hole has been found to TMG theory including Maxwell and electromagnetic Chern-Simons term [9]. The authors of [9] have used a dimensional reduction procedure of [11] to search for a stationary rotating black hole solution. In this paper we shall use the first order formalism to find a black hole solution. MMG theory can be represented by a Lagrangian which is composed of that of TMG theory including the torsion term coupled with an auxiliary field h and another 'ehh' term with a dimensionless coupling constant α. Auxiliary field h has the same odd-parity and mass dimension with the spin connection ω. The MMG field equation can not be obtained from an action represented by the metric alone. Elimination of the auxiliary field h from the MMG action can not reduced to the action for the metric only, leading to the correct field equation through the variation of the action for the metric. Using linearization of the field equation, it is possible to exist parameter regions to evade "bulk vs. boundary clash" with the condition of positivity of the central charge [4]. Therefore we use the first order formalism to find a black hole solution persisting the Lagrangian form with auxiliary field h. The solution we have found is the space-like warped AdS 3 black hole under considering theory. Similar black hole solutions have been studied in [8,9,10,12,13,14] Another purpose of this paper is to calculate mass, angular momentum and entropy of the black hole to be founded under considering theory. Various methods to calculate entropy, mass and angular momentum as conserved charges have been developed and we have introduced and referred these methods to [15]. We use the Wald's formalism to obtain entropy, mass and angular momentum of black holes with the first order formalism. In this method we can define charge variation forms to calculate these physical quantities. Entropy of black holes can be obtained by an integration of the charge variation form at bifurcation surface H, i.e. event horizon. Mass and angular momentum of black holes can also be obtained by an integration of the charge variation form at spatial infinity respectively. This method becomes a good working tool in case of having some boundaries at infinity such as asymptotically Minkowski and AdS space-time. Applying this method to the warped AdS 3 black hole solution in our considering theory, we calculate entropy, mass and angular momentum of this black hole. These physical quantities are different with those of the warped AdS 3 black hole in TMG case. These values are satisfied with the first law of black hole thermodynamics and also represented by a Smarr relation. In [14] there have been investigations for the warped AdS 3 black hole in generalized minimal massive gravity (GMMG). It has been shown that the space-like warped AdS 3 black hole [16] is a solution of GMMG which theory has an additional higher curvature terms occurring in new massive gravity (NMG) [17]. The authors of [14] have calculated entropy, mass and angular momentum of this black hole using different method from ours.
Finally, we have investigated that the entropy of a black hole can be represented by descriptions of quantities of dual CFT side, if any, according to the prescription of AdS/CF T correspondence. By using the Cardy formula the entropy of a black hole can be described by central charges and temperatures of the dual CFT. If there exists a holographic dual CFT for the MMG theory, then we can find the corresponding central charges by using the Cardy formula with the thermodynamical quantities for the black hole. The warped AdS 3 vacua has been referred in [18]. This paper is organized as follows. In section 2, we survey the Lagrangian and construct equations of motion in MMG theory with Maxwell and electromagnetic Chern-Simons term by using the first order formalism. In section 3, we find a black hole solution to solve the field equations. In section 4, we briefly review the Wald formalism to investigate thermodynamic properties of the black hole. Using Wald formalism we define entropy, mass and angular momentum of a black hole with the first order formalism, and find these quantities for the black hole solution we have found. Comparing the entropy of the black hole with the Cardy formula we find central charges. In section 5, we summarize our results and add some comments. Appendices are attached to explain some formulae to compute some equations, entropy, mass and angular momentum of the warped AdS 3 black hole in MMG theory.

Minimal massive gravity theory with Maxwell and electromagnetic Chern-Simons term
In MMG theory the Lagrangian can be written in terms of Lorentz vector-valued one-form e a , dualized connection one-form ω a = 1/2 · ǫ abc ω bc and auxiliary field h a . From these the local Lorentz covariant torsion and curvature 2-forms can be defined by The Lagrangian 3-form for MMG theory is given by where σ is a sign and Λ 0 a cosmological constant. Lorentz indices a, b, c, · · · are suppressed and operation '·' and '×' represent contractions of η ab and ǫ abc with wedge products. The third term describes the 'Local Lorentz Chern-Simons' term with a mass parameter µ. The fourth term is introduced to avoid "bulk vs. boundary clash" with a massless parameter α. Because the 3dimensional Newton constant has inverse mass dimension, the Lagrangian 3-form (2.2) should have mass-squared dimension. So the cosmological constant Λ 0 has mass squared dimension. The dreibein e a can be assigned zero mass dimension and even parity. If we assign the same mass dimension and odd parity to auxiliary field h a and connection ω a , then the "Lorentz Chern-Simons" term is the only parity breaking term. We also include the Maxwell and electromagnetic Chern-Simons term in the Lagrangian 3-form with a massless dimension parameter µ E . In order to use the first order formalism, we change the above action with dreibein as follows Therefore the total Lagrangian with the gravitational constant can be represented by where κ, i.e. the gravitational constant 8πG is absorbed in L M C . To find equations of motion we consider the variation of the total Lagrangian 3-form with respect to e, h and ω. Then the variation for L tot is given by where So, we can get equations of motion as follows where the 2-form field T a associated with the matter part is given by (2.12) Now we can use (2.9) to make the torsion free condition then we should consider the shifted connection Ω = ω + αh. Using these new connection 1-forms we can re-express equations of motion as follows T (Ω) = 0 , (2.14) From the variation with respect to the gauge field A, The equation of motion for the Maxwell and electromagnetic Chern-Simons term is given by

Black hole solution
To find a black hole solution with stationary circular symmetry, the dimensional reduction method has been used in [6,7,8,9,19]. Following the procedure of this method, we can take a metric ansatz with two commuting Killing vectors ∂ t and ∂ φ as where x µ expresses two coordinates t, φ and R(ρ) 2 = − det λ. The function ζ(ρ) is introduced as a scale factor for arbitrary reparametrizations of the coordinate ρ. If we assume that the space-time has two commuting Killing vectors ∂ t and ∂ φ , the special linear group SL(2, R) of transformations in the Killing vector space is locally isomorphic to the Lorentz group SO(1, 2). Therefore the parametrization of the matrix λ µν can be described by The special linear transformation of λ ab corresponds to the Lorentz transformation of a vector X = (T, X, Y ), and f (ρ) 2 = − det λ = X 2 = −T 2 + X 2 + Y 2 is the pseudo-norm in Minkowski space. We now take then (3.18) becomes (3.21) For the above metric, we can obtain dreibein as follows where we simply omit the representation of coordinate ρ from functions of ρ such as f , ζ, R and h. From now using this dreibein we will develop the first order formalism instead of the dimensional reduction method. To solve the equations of motion we should find the components of the electromagnetic field F ab = e µ a e ν b F µν which can be represented by where ' ′ ' means the differentiation with respect to ρ, and other components vanish. Also we can get the components of (2.5) Using dreibein (3.22) and the above components of the electromagnetic field we firstly consider the equation of motion for the electromagnetic field (2.17) to find a solution. Then we can obtain two equations These equations can be easily solved as For the simplest case we consider constants C 0 = C 2 = 0. Then we can find F a = 0. Applying these results to (3.24) we get If we take ζ = µ E , then the second equation can be simply reduced to To solve the above equation let us take functions as some polynomials Substituting the above functions into (3.30) then we can find an equation of ρ which has cubic term as its maximum order. If we require the cubic term to be vanished, then we can choose T 1 = 0 which means that the electric field does not make any physical effect. The other coefficients of ρ terms give us three relations as follows Substituting functions (3.31) into (3.28) with ζ = µ E , we can get With these values we can solve the equations in (3.32) to obtain two coefficients If f 2 = 0, then the linear term of function f 2 = f 2 ρ 2 + f 1 ρ + f 0 can be set to zero (f 1 = 0) by a translation of ρ. Therefore we can simply represents three functions in the metric ansatz (3.21) as follows We now investigate the equations of motion (2.14), (2.15) and (2.16) to find a black hole solution. Firstly we consider torsion free condition (2.14) for the metric ansatz (3.21). It is convenient to use some relations derived from (3.35) while we calculate connection 1-forms, curvature 2-forms and auxiliary fields ; By substituting (3.22) into (2.14) we can find connection 1-forms where we regard ζ as a constant because we take ζ = µ E before. We can also find the curvature 2-forms from the definition (2.1) with the above shifted connection 1-forms Ω a , We are dealing with a simplest case (3.27) with C 0 = C 2 = 0, i.e. F a = 0. It means that the matter part of the equations of motion (2.15) and (2.16) vanished, T a = 0. So, we can find auxiliary fields h a from (2.15) as follows

39)
where parameters have been changed by ξ 2 = ζ 2 − 4αΛ 0 andμ = µ(1 + σα) 2 . Now we re-express the equation (2.16) as component forms By using the results of dreibein, shifted connections and auxiliary fields we can find five equations but two of these are the same one. So, the equations resulted from (3.40) are given by Adding (3.41) to (3.44) and then multiplying 2/µζ 2 , we can obtain where we use the first relation for functions in (3.35). Therefore we can find a constant (3.46) By using (3.35) and multiplying ξ 2 /4ζ 2 , Eq.(3.42) becomes (3.47) By using (3.35), (3.47) and mutiplying 2/µζ 2 , Eq. (3.43) becomes Finally we can obtain Substituting (3.47) into (3.46) we can also find the same result with (3.49). When α goes to zero, then (3.49) approaches These results can be identified with the same results (3.9) and (3.13) in [9]. Let c 2 = 1 and f 2 = β 2 for convenience, and set f 0 = −β 2 ρ 2 0 and h 0 = ω(1 − β 2 ). Then we can represent three functions (3.35) as follows By using these functions we can rewrite this black hole solution which is the same form of [9] and [19] with a parameter condition 0 < β 2 < 1 for causally regular black hole solution. If we change the coordinate system and take some parameters as follows where functions constituting the metric are defined by Since the parameter condition for the causally regular black hole solution is 0 < β < 1, we take ν > 1 for convenience. From (3.31) and (3.33) we can express the gauge field A as follows If we change coordinates and parameters following (3.54), then electromagnetic potential can be represented by The undetermined parameter φ 1 can be determined by considering the magnitude of the electromagnetic field, but this does not relevant to the black hole solution.

Thermodynamic properties of black holes
In this section we investigate thermodynamic properties of the warped AdS 3 black hole in MMG theory. In order to find these properties we need to find mass, angular momentum and entropy of the black hole. These physical quantities can be defined by the Wald formalism [20,21]. In [15] we have investigated the Wald formalism for the sake of calculation of entropy, mass and angular momentum of black holes with the first order formalism. Here we briefly survey the Wald formalism and definitions of entropy, mass and angular momentum of black holes.
If there exist a black hole solution with a local symmetry generated by a Killing vector ξ, then the entropy of the black hole is defined on a bifurcation surface, and the corresponding mass and angular momentum are defined well at spatial infinity. In order to find these definitions we firstly consider a diffeomorphism invariant theory described by a Lagrangian d-form L , where d indicates the space-time dimensions. The variation of a Lagrangian is induced by a field variation where ψ describes dynamical fields collectively. E ψ = 0 means equations of motion constructed by fields variation δψ and Θ is a (d − 1)-form "symplectic potential" constructed by dynamical fields ψ and variations of them.
Let us consider a vector field ξ on a space-time manifold and variations of fields ψ induced by a diffeomorphism generated by this vector, (4.60) Since we are considering a diffeomorphism invariant theory, the variation of the Lagrangian can be represented by the Lie derivative of the Lagrangian under this variation, The above formula means that the vector fields ξ on a space-time generate infinitesimal local symmetries. Applying this formula into (4.59), we can define a closed (d − 1)-form Noether current, under on-shell condition. So, this current can be described by an exact (d − 2)-form, where Q ξ is constructed from fields and their derivatives. In diffeomorphism covariant framework [20,21], the phase space is given by the projection of the field configuration space composed of the solutions of field equations with a corresponding symplectic form. The variation δ ξ ψ under on-shell condition describes the flow vector corresponding to the 1-parameter family of diffeomorphisms generated by ξ. Therefore the variation of the Hamiltonian conjugated to ξ is prescribed by the symplectic form which is defined by where the righthand side is the symplectic form which is defined by the integration of the symplectic current ω on the Cauchy surface Σ. If we take ξ as a symmetry vector field of all dynamical fields, i.e. £ ξ ψ = 0, and their variation δψ satisfies the linearized equation, then symplectic current is given by By using the variations of Noether current (4.62), (4.63) and Lagrangian (4.59), we can find the variation of the Hamiltonian as follows where the second integration should be performed on the boundary of the Cauchy surface Σ. Since ξ generate a symmetry of solutions of fields ψ, i.e. £ ξ ψ = 0, the symplectic current vanishes. Therefore δH ξ = 0 gives us a boundary relation If we consider a stationary black hole solution with a Killing vector ξ which vanishes on bifurcation surface H, then the above integration should be performed at interior boundary H, i.e. bifurcation surface, and at its outer boundary, i.e. spatial infinity of the Cauchy surface Σ. Therefore the above variational form identity (4.67) can be represented by If we assume that the Killing vector ξ specifies time translation and axial rotation with an angular velocity Ω H , i.e. where the Hawking temperature is given by T H = κ S /2π with the surface gravity κ S . Following the Wald formalism we can define mass and angular momentum of a black hole as follows [15] and the entropy of a black hole can also be defined by where the charge variation form δχ ξ is given by If we include the gravitational constant 1/8πG in the Lagrangian from the beginning, we can omit this constant from definitions (4.71) and (4.72). Now we investigate whether this definition for the black hole entropy is correct or not. Firstly we consider the Lagrangian for the Chern-Simons-like form [5] L CSL = 1 2 g rs a r · da s + 1 6 f rst a r · (a s × a t ) . This Lagrangian form is introduced to describe diverse gravity theories, i.e. TMG, NMG, MMG etc., which may include Chern-Simons term and some auxiliary fields. The notation a r means a collection of Lorentz vector valued 1-forms a ra µ dx µ , where r is a "flavor" index running 1 · · · N . In MMG theory case, flavor N represents fields of this theory, i.e. dreibein e, connection 1-form ω and auxiliary field h. g rs and f rst represent metric and coupling constants on the flavor space respectively. To find the charge variation form we firstly consider the variation of this Lagrangian form which is given by where the first term gives equations of motion and we can read the symplectic potential from the second term Θ(a, δa) = 1 2 g rs δa r · a s . (4.76) Following the Wald formalism we can obtain Noether charge with the on-shell condition. In order to find the charge variation form we consider the variation of Noether charge and interior product of the symplectic potential. Then the charge variation form for the Chern-Simons like Lagrangian (4.74) is given by Because Killing vector ξ vanishes on the bifurcation surface H, the interior product of the symplectic potential with this vector should be vanished, i.e. i ξ Θ = 0. Applying this condition to the variation of the Noether charge, then the variation of this charge is equal to the charge variation form (4.78). Therefore we can define the entropy of a black hole (4.72) with the charge variation form in the Chern-Simons-like gravity theories.
The calculation for BTG black hole entropy in TMG including gravitational Chern-Simons term has been performed in [22]. In that paper, the charge for the black hole entropy has been given by Q ′ ξ = Q ξ − C ξ where δC ξ = i ξ Θ + Σ ξ with a choice Σ ξ = 0. So, the variation δQ ′ ξ is the same definition with δχ ξ . Now we are able to calculate mass, angular momentum and entropy of a black hole with definitions (4.71) and (4.72). To find these physical quantities in MMG theory with Maxwell and electromagnetic Chern-Simons term, we consider the symplectic potential Θ coming from the total derivative term of the variation of the Lagrangian L tot . The symplectic potential from (2.7) is given by Considering the variation of the Lagrangian for a vector field ξ, Noether current is given by Noether current is closed when equations of motion are satisfied, so it can be represented by an exact form The symplectic potential (4.79) can be divided into two parts Θ grav (e, ω, h, δe, δω, δh) = −σδω · e + 1 2µ δω · ω + δe · h , (4.82) Combining two definitions (4.80) and (4.81) we can find conserved charges for gravitational and electromagnetic interactions respectively When we calculate Noether current (4.80), we should consider the Lagrangian L as two part. One is L grav which is related to the gravitational interaction, i.e. terms including e, ω and h, and the other is L em which is related to the gauge field A or F . Changing connection 1-forms ω into the shifted ones Ω = ω + αh with a condition e · h = 0, we can obtain By using above symplectic potential and conserved charge we can find the charge variation form for the gravitational interaction as follows and for the electromagnetic interaction where 2π comes from the solid angle in three-dimensions and it depend on the definition of charge.
As an example, if we consider only electric potential term, then −i ξ A part of (4.89) describes an electric potential and the integration of δF on the bifurcation surface H is corresponded with the variation of the electric charge.
In order to calculate these charge variation forms we firstly consider dreibein from the metric of the black hole (3.55) dr , e 2 = ℓR(dθ + N θ dt) . We can find connection 1-forms by applying dreibein into torsion free condition (2.14) , With these connection 1-forms we can find curvature 2-forms by using the definition R(Ω) = dΩ + 1 2 Ω × Ω .
where functions F (r) and G(r) are simply represented by (4.95) Applying above curvature 2-forms into Eq.(2.15) with T a = 0 we can also find auxiliary fields The electromagnetic field can be extracted from (3.58) with ζ = µ E = 2ν/ℓ The mass and angular momentum of black holes can be calculated from (4.88), and the electric charge of black holes can also be obtained from (4.89). In order to find this charge variation form we need to calculate non-vanishing interior products by the Killing vector ξ and angle θ parts of the variations of dreibein, connection 1-forms, auxiliary fields and electro-magnetic fields. All these calculations are represented in appendix B. Then the charge variation form for the black hole mass is given by Then we can obtain black hole mass form by using the first definition of (4.71) where we should change the gravitational constant G to Gℓ to give the correct mass dimension. As α goes to zero with parameter conditions σ = 1 and µ = 3ν/ℓ, the above result becomes the mass of the warped AdS 3 black hole in TMG [10,15,23].
As we have already seen before, Eq. (2.16) can be reduced to (3.40) with a condition T a = 0. So, this equation can be solved with all the results (4.91), (4.92) and (4.96). More details are explained in appendix A. Considering Eq.(A.10) we can take for a special case which satisfy conditions (A.9) and (A.11). The second part of (4.100) vanishes with these parameters (4.101). So the mass of the warped AdS 3 black hole in MMG theory can be reduced as follows To find the angular momentum of the black hole we need to calculate the charge variation form (4.88) with a Killing vector ξ = ∂ ∂θ . But this charge variation form δχ ξ [ ∂ ∂θ ] can be simply represented by a total variation of this charge form As α goes to zero, the above charge form is reduced to the one of TMG theory and then gives us correct angular momentum for the warped AdS 3 black hole in TMG with parameter conditions σ = 1 and µ = 3ν/ℓ [10,15]. From functions (3.56) we can calculate each functions in square bracket. The value of the angular momentum should be calculated at spatial infinity, so we expand the above charge form (4.103) as a function of r. This charge form includes r 2 , r and constant terms. Other terms including negative power of r vanish as r goes to infinity. These calculations are summarized in appendix C. It can be shown that coefficients of r 2 and r vanish by using the result of (4.101). So, the rest constant term gives a value associated to the charge form for the angular momentum of the black hole after some tedious calculations Using the definition of the angular momentum (4.71) with δχ grav [ ∂ ∂θ ], we can obtain the angular momentum of the warped AdS 3 black hole in minimal massive gravity To obtain the above correct result we need to consider the change of coordinates to be dimensionful. So, we change the gravitational constant G to Gℓ. This change makes the angular velocity to be dimensionful with 1/ℓ and rotational Killing vector with ℓ.
To find the entropy of the warped AdS 3 black hole in this theory, we firstly consider the calculation for the charge variation form from the definition (4.72 where Ω H is given by (B.6). The charge variation form for the entropy of the warped AdS 3 black hole is given by (C.7). By using parameter conditions (4.101), functions (3.56), (B.8), (B.9) with variations of functions in appendix B, we can rearrange the charge variation form for the entropy of the black hole. Then we can take the radial coordinate value r = r + at the event horizon H. From all of these substitutions we can rearrange the charge variation form as a formula of r + and r − with their variations δr + , δr − and δR + . So, the result is given by Using the variation form δR described by (B.10) at the event horizon r + of the black hole and 2R + = 2νr + − r + r − (ν 2 + 3) appropriately, we can rearrange the charge variation form with variations δr + and δr − . Summarizing the above terms then we can obtain the charge variation form as follows (4.108) The entropy of a black hole can be defined by where we change the Newton constant G into Gℓ to make parameters to be dimensionful. Using the Hawking temperature for this warped AdS 3 black hole we can obtain the entropy of the space-like warped AdS 3 black hole in this theory which is given by To calculate the variation for the electric charge part, we need to consider the charge variation form for the electromagnetic interaction (4.89). By using the gauge field representations (3.58) and (4.97), the result for the calculation of δχ em ξ vanishes. Therefore the contribution of Φ H δQ for the first law of black hole thermodynamics is disappeared. So entropy, mass and angular momentum of this black hole satisfies the thermodynamic first law of black hole δM = T H δS BH + Ω H δJ , (4.112) with the dimensionful angular velocity Also we obtain Smarr relation between mass, angular momentum and entropy of the black hole The same relation with (4.114) have been obtained in case of the warped AdS 3 black hole in TMG theory with Maxwell and electromagnetic Chern-Simons term in [9]. The definition for the charge variation form (4.106) lead us to the same correct results for the entropy of the BTZ black hole, the space-like warped AdS 3 black hole in TMG [22,24].
It has been shown that the space-like warped AdS 3 black hole can be regarded as the ground state with U (1) L × SL(2, R) R symmetry in TMG with ν > 1 [10]. This black hole solution is discrete quotient of warped AdS 3 space. So, it is conjectured that quantum TMG with suitable warped AdS 3 boundary conditions is holographically dual to a 2D boundary CFT with a suitable central charges. Under this conjecture we can define left and right moving energies in terms of left and right central charges and temperatures. They can be defined by which is obtained by considering quotient along the isometry ∂ θ . The black hole entropy can be described by as follows the Cardy formula. According to the result in [25], mass and angular momentum can be represented by left and right moving energies Even though we did not obtain the warped AdS 3 black hole solution without matter term, we can refer (3.55) as a solution for MMG theory with matter term contribution T a = 0. So the above conjectural results can also be applied with this AdS 3 black hole in MMG theory. Therefore mass and angular momentum can be represented with β 2 = (ν 2 + 3)/4ν 2 in (3.54) as follows (4.120) Comparing the above formulae with (4.102) and (4.105), then we can obtain left and right moving central charges as follows It has been investigated that the variation of the bulk action including Chern-Simons term is related to the gravitational anomaly for the boundary CFT theory [26]. The bulk variation, i.e. general coordinate variation or local Lorentz transformation, can be described by a boundary integration with a coefficient of Chern-Simons term. The gravitational anomaly for the CFT can also be described by the same integration with a coefficient related to the difference between left and right moving central charges. In the context of AdS/CF T correspondence the comparison between two values make a relation c L − c R = 96πη where η means the coefficient of Chern-Simons term. In [10] they have found that the difference between two central charges of TMG in warped AdS 3 matches the coefficient of gravitational anomaly with η = −1/32πGµ and µ = 3ν/ℓ. From (4.121) the difference between central charges of MMG in warped AdS 3 is given by If we use two conditions µ = 3ν/ℓ(1+α) and α = 2ν 2 −3 from (4.101), then η = −2(ν 2 −1)ℓ/96πGν gives us the same value as (4.123). As α goes to zero, we obtain (4.122) again. So, we may suppose that the calculation of a general coordinate transformation for the action of MMG theory should be the boundary integration with coefficient η = −(1 + α)ℓ/96πGν.

Conclusion
In this paper we have constructed a space-like warped AdS 3 black hole solution in MMG theory with Maxwell and electro-magnetic Chern-Simons terms. In order to find black hole solution we solve equations of motion (2.14), (2.15) and (2.16) using first order formalism. This black hole solution can be interpreted into a space-like warped AdS 3 black hole by changing coordinates system into Schwarzschild one with ν > 1 such as referred to the case in TMG theory [9,10].
According to Wald's procedure with first order formalism, we can define mass and angular momentum of a black hole as variational forms which is related to the integration of charge variation form δχ ξ at spatial infinity (4.71) [15]. Mass of the space-like warped AdS 3 black hole can be given by (4.100) with a parameter α. As α goes to 0 with µ = 3ν/ℓ, mass of this black hole (4.100) becomes the mass value of the same black hole in TMG. Angular momentum of this black hole can be represented as an integral of the charge form (C.6) at spatial infinity with parameter α. In the same manner we can also obtain the same value of the charge form in TMG which is calculated in (B.10) of [15] as α approaches to 0. Solving equations of motion (2.16), we can obtain three parameter conditions (A.9), (A.10) and (A.11). We can find special parameter values (4.101) satisfying these parameter conditions. With these special parameter values (4.101) we have obtained mass and angular momentum of this black hole (4.102) and (4.105).
The entropy of the black hole can also be defined by an integral of the charge variation form (4.72) at the bifurcation surface H. It is well-known that the entropy of a black hole can be calculated as a conserved charge. But if we consider a theory including non-tensorial terms like a gravitational Chern-Simons term, the definition for the entropy of a black hole should be modified [22]. In this paper we have defined the entropy of a black hole as an integral of the charge variation form (4.72). It can be identified that the calculation of the entropy for the warped AdS 3 black hole using definition (4.72) leads to the correct entropy value appeared in [10]. In this paper we have calculated the entropy of the warped AdS 3 black hole (4.111) in MMG theory. This result gives us the correct relation for the first law of black hole thermodynamics (4.112). And we also have obtained Smarr relation (4.114) which describes finite relations between physical values of black holes.
The authors of [27] have constructed the quasi-local conserved charge for the Chern-Simons-like theories of gravity [5] with first order formalism. To find this charge they have used the off-shell ADT method [28,29,30,31] and the field variation with the Lorentz-Lie derivative The Lorentz-Lie derivative is defined by whereλ a = 1/2 · ǫ a bc λ bc is a generator of the local Lorentz transformation and £ ξ describes the Lie derivative for a vector ξ. The purpose to introduce this derivative is to circumvent the divergence of the connection 1-form ω a on the bifurcation surface H, even though the interior product of a Killing vector ξ for the connection becomes finite [32]. In our approach, the variation of the field variables for a vector field ζ can be described by where · · · means the term proportional to equations of motion [4,33] and ϕ diff [ζ] is a generator of a diffeomorphism associated to a vector field ζ, which comes from the sum of primary constraints derived by a Hamiltonian analysis [4]. Therefore it is enough to adapt the Wald formalism for the diffeomorphism invariant Lagrangian as we derive the charge variation form to define physical quantities, i.e. entropy, electric charge, mass and angular momentum of a black hole [15].
There has been a conjecture that TMG theory with suitable asymptotically stretched AdS 3 boundary conditions ν > 1 is holographically dual to a 2-dimensional boundary CFT with appropriate left and right moving central charges [10]. Applying this conjecture into this black hole solution, we have found left and right moving central charges in MMG theory respectively. In [10] it has been shown that the difference between left and right moving central charges matches the coefficient of the diffeomorphism anomaly with a relation c L − c R = 96πη. Considering MMG theory, we can find (4.123) with η = −(1 + α)ℓ/96πGν. So, the difference between central charges in this theory seems to appear as a shift by a parameter α in comparison with TMG case.
Subtracting (A.2) from (A.5) describes a relation From (4.95) functions F (r) and G(r) can be represented by simple form Using (A.6) and (A.7), we get two relations Now we substitute all these solutions to Eq.(3.40) which comes from (2.16), then we have three conditions as follows − α 4μ If we take special parameters, then the above three equations for parameter conditions (A.9), (A.10) and (A.11) are satisfied by these parameters (A.12).

B Some calculations for charge variation forms
The variation of non-vanishing angle θ part of all relevant fields for calculation of the charge variation form δχ ξ are given by Non-vanishing interior products of dreibein, connection 1-form for Killing vector ξ = ∂ ∂t are given by Non-vanishing interior products of auxiliary fields h a and electro-magnetic fields F for Killing vectors ξ = ∂ ∂t are given by Non-vanishing interior products of relevant fields for Killing vector ξ = ∂ ∂θ are given by In order to calculate the charge variation form for the black hole entropy, we should get nonvanishing interior products with Killing vector ξ = ∂ ∂t + Ω H ∂ ∂θ . The results are as follows i ξ e 0 = N , i ξ e 2 = ℓR(N θ + Ω H ) , i ξ F = 0 , where the angular velocity of the horizon Ω H is defined by Ω H = dθ dt r + = −N θ (r + ) = − 2 (2νr + − r + r − (ν 2 + 3)) = − 1 R + . (B.6) As r approaches to the event horizon r + , All terms including Ω H + N θ vanish.
From the above formulae and (3.56) we can get some relations These functions are helpful to calculate the charge variation form for the mass of the black hole.