# Conserved charges of minimal massive gravity coupled to scalar field

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## Abstract

Recently, the theory of topologically massive gravity non-minimally coupled to a scalar field has been proposed, which comes from the Lorentz–Chern–Simons theory (JHEP 06, 113, 2015), a torsion-free theory. We extend this theory by adding an extra term which makes the torsion to be non-zero. We show that the BTZ spacetime is a particular solution to this theory in the case where the scalar field is constant. The quasi-local conserved charge is defined by the concept of the generalized off-shell ADT current. Also a general formula is found for the entropy of the stationary black hole solution in context of the considered theory. The obtained formulas are applied to the BTZ black hole solution in order to obtain the energy, the angular momentum and the entropy of this solution. The central extension term, the central charges and the eigenvalues of the Virasoro algebra generators for the BTZ black hole solution are thus obtained. The energy and the angular momentum of the BTZ black hole using the eigenvalues of the Virasoro algebra generators are calculated. Also, using the Cardy formula, the entropy of the BTZ black hole is found. It is found that the results obtained in two different ways exactly match, just as expected.

## 1 Introduction

We know that the pure Einstein–Hilbert gravity in three dimensions exhibits no propagating physical degrees of freedom [2, 3, 4, 5]. But adding the gravitational Chern–Simons term produces a propagating massive graviton [6]. The resulting theory is called the topologically massive gravity (TMG). Inclusion of a negative cosmological constant yields a cosmological topologically massive gravity (CTMG). In this case the theory exhibits both gravitons and black holes. Unfortunately, in this model a problem is found with the usual sign for the gravitational constant, which means that the massive excitations of CTMG carry negative energy. In the absence of a cosmological constant the sign of the gravitational constant can be changed but if \(\Lambda <0\) this will give a negative mass to the BTZ black hole, making the existence of a stable ground state doubtful in this model [7]. TMG has a bulk–boundary unitarity conflict. In other words either the bulk or the boundary theory is non-unitary so there is a clash between the positivity of the two Brown–Henneaux boundary charges *c* and the bulk energies [8]. Recently an interesting version of three-dimensional massive gravity was introduced by Bergshoeff et al. [9], dubbed Minimal Massive Gravity (MMG), which has the same minimal local structure as TMG. The MMG model has the same gravitational degree of freedom as the TMG has and the linearization of the metric field equations for the MMG yields a single propagating massive spin-2 field. It seems that the single massive degree of freedom of the MMG is unitary in the bulk and gives rise to an unitary CFT on the boundary. Following this paper some interesting work has been done on the MMG model [10, 11, 12, 13, 14, 15, 16, 17, 18].

The authors of [19] have introduced the Chiral Gravity (CG) by formulating the TMG at a special point in parameter space, where the curvature radius of \(AdS_3\) equals the inverse of the graviton mass \(\mu \). Recently the authors of [1] have proposed a generalization of chiral gravity. They have considered a Chern–Simons action for the spin connection in the presence of a scalar field and a constraint that enforces the spin connection to remain torsion-less. So the model includes the TMG and the CG as particular cases. Here we would like to extend the Lagrangian of this model so that it describes minimal massive gravity theory, non-minimally coupled to a scalar field. There are many papers about the coupling of the scalar field and gravity in three dimensions. For example, the contribution of the scalar fields to the conserved charges has been studied in the context of black hole solutions of 3D gravity with a scalar field in [20]. The authors in this paper have shown that although the generators of the asymptotic symmetries acquire a contribution from the scalar field, the asymptotic symmetry group remains the same as in pure gravity. In Ref. [21] the author has considered the three-dimensional gravity coupled to a scalar field, with special attention to black hole configurations. A finite action for three-dimensional gravity with a minimally coupled self-interacting scalar field has been constructed in [22]. In [23] a spinning hairy black hole in gravity, minimally coupled to a self-interacting real scalar field in three spacetime dimensions, has been presented. The authors in this paper have shown that the presence of a scalar field with a slower fall-off at infinity leads to anti-de Sitter asymptotic behavior, which differs from the one found by Brown and Henneaux but has the same symmetry group as in pure gravity.

In this paper we try to find a general formula for a quasi-local conserved charge for our model in a first order formalism. We use the formula obtained to find the energy, the angular momentum and the entropy of the BTZ black hole solution in the context of this theory. It is interesting that although the conserved charges (energy, angular momentum and entropy) of a BTZ black hole and also a Virasoro central charge depend on the coupling parameter \(\alpha \) in MMG [24], the contributions from the MMG interaction term in the energy, angular momentum entropy and also Virasoro central charge vanish for the solution we consider (the BTZ black hole solution in the presence of constant scalar field). It seems that for this special solution case, the coupling to the scalar field removes the dependence on \(\alpha \).

There are several approaches to obtain mass and angular momentum of black holes for different gravity theories [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. The authors of [28] have obtained the quasi-local conserved charges for black holes in any diffeomorphically invariant theory of gravity. By considering an appropriate variation of the metric they have established a one-to-one correspondence between the ADT approach and the linear Noether expressions. They have extended this work to a theory of gravity containing a gravitational Chern–Simons term in [29] and have computed the off-shell potential and quasi-local conserved charges of some black holes in TMG.

In the metric formalism of gravity for the covariant theories defined by a Lagrangian *n*-form *L* Wald has shown that the entropy of black holes is the Noether charge associated with the horizon-generating Killing vector field, evaluated at the bifurcation surface [25]. The presence of the purely gravitational Chern–Simons terms and mixed gauge gravitational ones gives rise to a non-covariant theory of gravity in the metric formalism. Tachikawa extended the Wald approach to include non-covariant theories [42]. Hence, regarding this extension, one can obtain the black hole entropy as a Noether charge in the context of non-covariant theories as well. Another way (apart from the Tachikawa method) to obtain the entropy of black holes in the context of such theories has been studied in Refs. [43, 44] in an appropriate way.

The remainder of this paper is organized as follows. In Sect. 2 we first briefly introduce the model of [1]. Then, by adding a convenient term to the Lagrangian of [1], this model is generalized, such that it describes a minimal massive gravity theory, non-minimally coupled to a scalar field. In contrast to the Lagrangian of [1] this model is not torsion-free. In Sect. 3 the equations of motion are obtained. It is shown that the new field *h* is not a symmetric tensor, in contrast to the usual minimal massive gravity. Also it is shown that the BTZ black hole spacetime solves the equations of motion. In Sect. 4 an expression for the conserved charges of the considered model is found, associated with the asymptotic Killing vector field \(\xi \) based on the quasi-local formalism. In Sect. 5 a stationary black hole solution of the minimal massive gravity coupled to a scalar field is considered. Then a general formula for entropy of such black hole solutions is found. In Sect. 6 the obtained formula is applied to conserved charges and the entropy on the BTZ black hole solution of the minimal massive gravity coupled to a scalar field model. The energy, angular momentum and entropy of these black holes are obtained. In Sect. 7, the central extension term is calculated and with this we find the central charges and the eigenvalues of the Virasoro algebra generators for the BTZ black hole solution. Also, we obtain again the energy and angular momentum of this black hole using the eigenvalues of the Virasoro algebra generators. Then the entropy of the BTZ black hole is found using the Cardy formula. In the final section the results are summarized.

## 2 Minimal massive gravity coupled to a scalar field

^{1}The spin connection can be decomposed in two independent parts,

*m*are two parameters that are introduced to adapt the cosmological constant and the mass parameter of the TMG term, respectively. The last term in the Lagrangian (9) makes this theory torsion free. The above Lagrangian describes a topologically massive gravity theory non-minimally coupled to a scalar field.

^{2}First of all, we consider the following redefinitions in the Lagrangian (9):

As will be discussed in Sect. 4, *e* and \(\omega \) both are invariant under a general coordinate transformation. Also, \(\varphi \) is a scalar field and therefore the Lagrangian (14) is invariant under a general coordinate transformation. On the other hand, *e* is invariant under a general Lorenz gauge transformation, \(e ^{a} \rightarrow \Lambda ^{a}_{ b} e^{b}\), but \(\omega \) is not, \(\omega \rightarrow \Lambda \omega \Lambda ^{-1}+ \Lambda \mathrm{d} \Lambda ^{-1}\). It is easy to check that the curvature 2-form and the torsion 2-form both are invariant under the general Lorenz gauge transformation. So the Lagrangian (14) is not invariant under the general Lorenz gauge transformation due to the presence of the topological Chern–Simons term (the third term in the Lagrangian). Thus, in order to obtain the conserved charges of such a theory, we will have to use an extension of the Tachikawa method [42], which is presented in [31, 50]. We assume that \(h^{a}_{\mu } \mathrm{d}x^{\mu }\) is a Lorentz vector-valued 1-form.

## 3 Equations of motion

*f*is an arbitrary Lorentz vector-valued 1-form. By combining Eqs. (26) and (27) we find

*h*

It can be easily seen from Eq. (31) that \(h^{a}\) is a Lorentz vector-valued 1-form. As we mentioned earlier, dreibein, curvature 2- form and torsion 2-form are invariant under general coordinate transformations (and under general Lorenz gauge transformation). Also, as \(D(\Omega )\) is an exterior covariant derivative with respect to \(\Omega \), the equations of motion (25)–(28) are covariant.

*l*is the AdS space radius. By assuming that \(\varphi \) is a constant, say \(\varphi = \varphi _{0}\), the BTZ black hole spacetime solves the equations of motion (25)–(28). So, by taking \(\varphi = \varphi _{0}\) for the BTZ black hole spacetime, Eq. (31) becomes

A few comments are necessary. Minimal massive gravity is an extension of topologically massive gravity. Topologically massive gravity is a torsion-free theory but minimal massive gravity is not. A term such as (13) is added to the TMG in order to construct minimal massive gravity. In Sect. 2 it was seen that the Lagrangian (9) describes a topologically massive gravity, non-minimally coupled to a scalar field. In an ordinary case we expect that minimal massive gravity, non-minimally coupled to a scalar field, can be constructed by adding a term like (13) to the Lagrangian (9). The equations of motion of the minimal massive gravity, non-minimally coupled to a scalar, are given by Eqs. (25)–(28). These equations are very complicated because of the complex form of the auxiliary field \(h^{a}\) (see Eq. (31)). Therefore writing the equations of motion in metric form is very difficult. Also, it is not useful.

## 4 Quasi-local conserved charges

In this section an expression to the conserved charges of the considered model associated with the asymptotic Killing vector field \(\xi \), based on quasi-local formalism for conserved charges, will be found [28, 29, 30, 31, 32].

*SO*(2, 1). In general, \(\lambda ^{a}_{ b}\) is independent of the dynamical fields of the considered model. It is a function of the spacetime coordinates and of the diffeomorphism generator \(\xi \). The total variation of the dreibein and the spin connection are defined by [50]

*e*and \(\omega \) both are invariant under a general coordinate transformation).

*G*denotes the Newtonian gravitational constant and \(\Sigma \) is a space-like codimension two surface. Also, integration over

*s*is just integration over an one-parameter path in the solution space and \(s=0\) and \(s=1\) correspond to the background solution and the solution of interest, respectively.

*e*and \(\omega \) is a dynamical field which is an arbitrary function of coordinates. So it is clear that Eqs. (61), (66) and (72) can be used to obtain charges associated with the solutions of the theory, which may have a non-constant scalar field.

## 5 General formula for entropy of black holes in minimal massive gravity coupled to scalar field

## 6 Application for BTZ black hole solution with \(\varphi = \varphi _{0}\)

## 7 Virasoro algebra and the central term

*SO*(2, 1) gauge groups [58]

*n*dependence, that is, \(n (n ^{2} -1)\), in the above expression it is sufficient that a shift is made from \(Q( \xi _{m} ^{\pm } )\) by a constant [59]. Thus by the substitution

*E*and the angular momentum

*j*of the BTZ black hole by the following equations, respectively:

## 8 Conclusion

In this paper, the theory of topologically massive gravity non-minimally coupled to a scalar field which comes from the Lorentz–Chern–Simons theory [1] is considered. The Lagrangian of this theory is given by Eq. (9). It is a torsion-free one. The theory is extended by adding an extra term (13) which makes the torsion non-zero. The Lagrangian of the extended theory is given by Eq. (14). In Sect. 3, we have obtained the equations of motion (25)–(28) of the extended theory in such a way that they are expressed in terms of the usual torsion-free spin connection (24). We have shown that the BTZ spacetime together with \(\varphi = \varphi _{0} \) solves the equations of motion (25)–(28) when one of Eq. (40) or (41) is satisfied. In Sect. 4, the off-shell ADT current is generalized and it is deduced that it is conserved for any asymptotically Killing vector field as well as a Killing vector field which is admitted by spacetime everywhere. Then the Poincaré lemma is used to define the generalized off-shell ADT charge (60). Consequently, we have defined the quasi-local conserved charge (61) for the considered theory. In Sect. 5 a general formula (79) is found for the entropy of the stationary black hole solution in the context of the considered theory. We have used the obtained formulas to calculate the energy (88), the angular momentum (90) and the entropy (93) of the BTZ black hole solution. These quantities satisfy the first law of black hole mechanics. In Sect. 7 we have obtained the central extension term (115) and then read off the central charges (120) and the eigenvalues of the Virasoro algebra generators (121) for the BTZ black hole solution. We have calculated the energy (122) and the angular momentum (123) of this black hole using the eigenvalues of the Virasoro algebra generators. Also, we have calculated the entropy of the BTZ black hole by using the Cardy formula (124). Comparing Eqs. (122)–(124) with Eqs. (88), (90) and (93) we found that, although they have been obtained in two different ways, they exactly match, as we expected.

We should mention that here we have considered a special example, a BTZ black hole together with \(\varphi =\varphi _{0}\) where \(\varphi _{0}\) is a constant. In this case, the equations of motion (26) and (27) imply that \(h^{a}\) is proportional to \(e^{a}\) (see Eq. (33)). Thus, on one hand, the contributions from the MMG interaction term in the energy, the angular momentum and the entropy become proportional to \(\alpha \beta \) (see Eqs. (66), (79)). On the other hand, the remaining equation of motion (25) requires that \(\alpha \beta =0\). Therefore, the contributions from the MMG interaction term in the energy, angular momentum and entropy vanish for the given example. In the model considered, we have one more equation of motion than in the ordinary MMG model. Herewith we can deduce that the coupling to the scalar field removes the dependence on \(\alpha \) for the given example. A similar argument holds for the Virasoro central charges.

We do not expect that the results obtained match with the MMG ones when we take a constant scalar limit. The reason for this comes from the fact that the Lagrangian (14) does not become the Lagrangian of the MMG [9] when the scalar field is a constant. Instead, it will reduce to the Lagrangian of the Mielke–Baekler model with torsion when the scalar field is considered as a constant (the Mielke–Baekler Lagrangian (15) has an extra term \(\theta _{T} e \cdot T (\omega )\) over the TMG. Subsequently, the Lagrangian (14), when the scalar field is a constant, differs from the Lagrangian of the MMG by a term of the form \(\theta _{T} e \cdot T (\omega )\)).

## Footnotes

- 1.
Here, we use Latin and Greek Letters to characterize Lorentz and coordinate indices, respectively. Also, we work in the Einstein–Cartan formalism. In this formalism the independent dynamical fields are the spin connection \(\omega ^{ab}\) and the dreibein 1-form \(e^{a}\).

- 2.
In order to generalize the Lagrangian (9), we consider the spin connection \(\tilde{\omega }^{ab}\) and the dreibein \(e^{a}\) as two independent dynamical fields on an equal footing.

## Notes

### Acknowledgements

M. R. Setare thanks Simon del Pino and Gaston Giribet for helpful comments.

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