Thermodynamic Stability of the Stationary Lifshitz Black Hole of New Massive Gravity

In this paper we investigate the thermodynamic properties of the stationary Lifshitz black hole solution of New Massive Gravity. We study the thermodynamic stability from local and global point of view. We also consider the space of equilibrium states for the solution within the framework of Thermodynamic Information Geometry. By investigating the proper thermodynamic metrics and their curvature invariants we find a set of restrictions on the parameter space and the critical points indicating phase transitions of the system. We confirm our findings by analytical analysis of the geodesics on the space of equilibrium states.


Introduction
Inconsistencies between classical gravity and other fundamental forces lead to the development of various modified theories of gravity in hope to resolve the long-standing problems. For this reason, models in three dimensions with black hole solutions become very attractive area of research. They can be considered as simpler toy models of gravity, especially in gauge/gravity correspondence, where such systems are dual to two-dimensional conformal field theories. Depending on how one approaches the problem there are number of ways to construct three-dimensional models of gravity.
In the first approach, one add topological Chern-Simons terms to the standard Einstein-Hilbert action. The resulting theory is known as Topologically Massive Gravity (TMG) proposed by Deser, Jackiw and Tempelton in [1,2]. Here, the propagating degree of freedom is a massive graviton. Among other exact solutions the theory also admits the famous Banados-Teitelboim-Zanelli (BTZ) black hole [3]. Since its proposal, TMG has been extensively studied in the literature and different features of the model has already been uncovered (see for example [4][5][6][7][8][9][10][11]). An extension of TMG with additional curvature squared term in the field equation has been proposed by Bergshoeff, Hohm and Townsend in [12]. It is called Minimal Massive Gravity (MMG). For certain range of the parameters MMG admits positive energy of the bulk graviton and positive central charges of the dual conformal field theory (CFT), thus avoiding the bulk-boundary unitarity problem arising in TMG.
In the second approach, the Einstein-Hilbert action is modified by higher-derivative correction terms, which give rise to the three-dimensional New Massive Gravity theory (NMG) [13]. In contrast to TMG, NMG is a parity preserving theory [13] and with certain constraints on the parameters, upon linearization about an AdS background, yields a unitary and ghost free theory [14]. Furthermore, a holographic renormalization study of NMG was conducted in [15], where in the context of AdS/CFT correspondence it was suggested that the dual CFT could be a logarithmic conformal field theory (LCFT). These properties of NMG motivate us to study its exact solutions, some of which can be found in [16][17][18][19][20][21][22][23].
An important aspect of black hole physics is their thermodynamics. Since the original works of Hawking, Beckenstein and others [24][25][26][27], one can study the thermal properties of gravitational systems in hope to uncover essential features of the underlying quantum dynamics. In this context, the so called Thermodynamic Information Geometry (TIG) can provide us with valuable insights. TIG takes advantage of powerful tools from differential geometry and mathematical statistics making it a very useful framework. This is due to the fact that geometry studies mutual relations between elements, such as distance and curvature.
Thermodynamic geometry was first introduced by Weinhold [28] in 1975 and later by Ruppeiner [29]. Weinhold showed that the laws of equilibrium thermodynamics can be represented in terms of an abstract metric space. This can be achieved by utilizing the Hessian of the internal energy with respect to the extensive parameters of the system and considering it as a Riemannian metric on the space of macro states. On the other hand, Ruppeiner developed his geometric approach within fluctuation theory, where one implements the entropy as a ther-modynamic potential. Here, one can use the Hessian of the entropy to find the probability for fluctuation between different macro states. Later it was discovered that both metric approaches are conformally related via the temperature being the conformal factor.
As it turns out, Hessian thermodynamics is not the only way to define a Riemannian metric on the equilibrium manifold. More general approach was proposed by Quevedo in [30], who considers Legendre invariant metrics. They preserve the physical properties of the system under different choices of thermodynamic potential. However, the price to pay is that there are infinitely many Legendre invariant metrics to choose from. For this reason, another two approaches were proposed. The first one is given by Hendi, Shahram, Panahiyan, Panah and Momennia (HPEM) in [31], where the authors consider thermodynamic metric with specific conformal function, which seems to resolve the problem of redundant singularities in Quevedo's approach. The second one is considered by Mirza and Mansoori (MM) in [32][33][34][35], which is based on conjugate thermodynamic potentials, specifically chosen to reflect the relevant thermodynamic properties of system under consideration. Some applications of these approaches to different gravitational systems can be found for example in [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. In order to identify the admissible thermodynamic metrics for a given black hole solution, a case by case study is required.
This paper is organized as follows. In Section 2 we present the stationary Lifshitz black hole solution of New Massive Gravity obtained in [22]. Here, we calculate the curvature invariants of the solution and identify the relevant physical singularities. We also find the location of the event horizon by investigating the Killing symmetries of the solution. In Section 3 we identify the parameter regions of local and global thermodynamic stability of the black hole solution in different ensembles. In Section 4 we study the problem of thermodynamic stability within the framework of Thermodynamic Information Geometry. We identify the admissible thermodynamic metrics and study their properties. In Section 5 we investigate geodesics on the space of equilibrium states, which correspond to the available thermodynamic metrics. This allows us to study the thermodynamic length (the shortest distance) between two macro states, which can be used to optimize the implementation of quasi-static protocols in a given ensemble. Finally, in Section 6 we give our concluding remarks.

Stationary Lifshitz black hole solution of NMG
The three-dimensional new massive gravity (NMG) was originally proposed in [12] as a paritypreserving and unitary solution to the problem of consistently extending the Fierz-Pauli field theory for a massive spin-2 particle to include interactions. Its action is given by [13,14] where Λ 0 is the cosmological constant, m is a mass parameter, and S µν is the Shouten tensor, The field equations for the metric can be derived by varying the action with respect to the metric tensor, thus where we have defined the tensor quantity Imposing specific choice of the parameters, namely one can obtain as a solution the static Lifshitz black hole [47] where x = ρ 2 / 2 . Now, boosting the metric (2.6) via where ω is a is a real constant with |ω| < 1, one arrives at the stationary Lifshitz metric 1 [22]: Without loss of generality, we will assume 0 ≤ ω < 1. Solving the equation g xx = 0 one finds two positive roots, namely the event horizon at x = M , and a central singularity at x = 0. The root x = 0 is a true singularity, due to the divergence of the Ricci curvature, Higher-order invariants also confirm this conclusion: (2.11) We know that g xx = 0 is not sufficient to locate the event horizon. For this purpose we can inspect the Karlhede's differential invariant, which is known to vanish on the horizon [48]. In our case it is zero on x = M . However, in order to be more rigorous, one is compelled to study the Killing symmetries of the solution. In this case, the standard rotating Killing vector is given by where K µ (t) = (1, 0, 0) T , K µ (θ) = (0, 0, 1) T , and Ω is the angular velocity on the horizon. If the surface x = M is a Killing horizon, then the Killing vector should become null on it, i.e. (2.14) There are two solutions to this equation, namely which, on the surface x = M , yield a constant angular velocity Therefore, the rigidity theorem is valid and x = M satisfies all requirements for an event horizon.
3 Local and global thermodynamic stability

Specific heats and local thermodynamic stability
The thermodynamics (TD) of the stationary Lifshitz black hole (2.8) is described by the energy E, the entropy S, and the angular momentum L, as found in [22] One can check that the first law of thermodynamics, is satisfied with T being the Hawking temperature and Ω being the angular velocity, , Ω = ω .
Throughout the paper the local coordinates on the equilibrium space of macro-states for the stationary Lifshitz black hole will be the intensive parameters T and Ω 2 . Considering 0 ≤ ω < 1 and > 0, one finds 0 ≤ Ω < 1.
one can immediately express the relevant extensive thermodynamic quantities: One can also consider the extensive parameters S and L as coordinates on the statistical manifold, which are related to the energy E via a cubic equation However, we prefer to work in (T, Ω) space, where the relations (3.6) between the relevant parameters are straightforward.
The specific heats of the black hole in (T, Ω) space are given by [34] C Ω (T, Ω) = T ∂S(T, Ω) ∂T and The Davies critical points are the set of divergences of C Ω,L , namely the spinodal where the metric (2.8) is also singular, thus one has to consider only cases where Ω = 1/ . The latter is already assured by Eq. (3.4). The second spinodal occurs when the specific heat changes its sign (C Ω = C L = 0). For C Ω this is possible only in the extremal case T = 0, which is not allowed by the principles of thermodynamics. For C L = 0, besides on T = 0, the change of sign occurs also on the curve On the other hand, local thermodynamic stability requires only positive specific heats C Ω,L > 0, which leads to (T > 0, 0 ≤ Ω < 1) for C Ω , and (T > 0, 0 ≤ √ 3 Ω < 1) for C L . Therefore, if we want for both specific heats to be positive, one has to impose This is the condition for local thermodynamic stability of the stationary Lifshitz black hole solution of NMG. However, one can also consider a weaker condition for values of the angular velocity in the range 1/( √ 3 ) ≤ Ω < . In this case, the black hole is locally TD stable only with respect to C Ω , but not C L .

Ensembles and global thermodynamic stability
While local thermodynamic stability identifies whether a certain phase in equilibrium is a local maximum of the total entropy, global TD stability is concerned with phases of the system corresponding to the global maximum. In canonical and grand-canonical ensembles local TD stability translates to positive specific heats, while global stability under thermal fluctuations translates in the concavity of Helmholtz and Gibbs free energies respectively.
In canonical ensemble the preferred phase of the system is the one that minimizes the Helmholtz free energy [49], The Helmholtz free energy has local extrema at Ω = √ 21/ (7 ) and Ω = 0, which are saddle curves for arbitrary values of the temperature T > 0. The free energy F and its derivatives are discontinuous on Ω = 1, thus indicating a phase transition. In order for the black hole to be in a global thermodynamic equilibrium, the following concavity condition must be satisfied This leads to the constraint which is less restrictive than the condition (3.11) for local TD stability.
In the grand-canonical ensemble the preferred phase of the system is the one that minimizes the Gibbs free energy, Considering T > 0, the local extremum is located at Ω = 0, which is a local maximum. The Gibbs free energy and its derivatives are also discontinuous at Ω = 1. The concavity condition in this case, assures global TD stability. This is always true within the range given in Eq. (3.4).
One notes that the conditions for global thermodynamic stability in both ensembles restricts distinctively the angular velocity Ω of the black hole, while including the condition for local TD stability from Eq. (3.11). The thermodynamic stability of the stationary Lifshitz black hole solution can be analyzed further by identifying the proper Riemannian metrics on the space of equilibrium states, which we show in the next section.

Thermodynamic geometry on the equilibrium manifold
In this section we investigate the thermodynamic stability of the system by several Riemannian metrics defined on (T, Ω) equilibrium space of the stationary Lifshitz black hole.

Hessian thermodynamic metrics
The simplest choice one can consider is the Ruppeiner metric defined by the Hessian of the entropy, where ∂ a denotes differentiation with respect to (T, Ω). Due to the probabilistic interpretation of Hessian metrics [50], one requires these metrics to be positive definite. This can be assured by imposing Sylvester's criterion, which states that all the principal minors of the metric tensor be strictly positive definite, i.e.
Unfortunately, these conditions cannot be simultaneously satisfied for g (R) ab , because g ΩΩ < 0 is always negative, thus there are no sub-regions in (T, Ω) space, where the metric tensor (4.1) is positive definite. The same is true if one considers the angular momentum L or the free Helmholtz energy F as thermodynamic potentials in (T, Ω) space. Therefore, we will not study these metrics here.
On the other hand, one can take advantage of Weinhold's approach utilizing the Hessian of the internal energy of the system instead of the entropy [51]. In this case, one finds Here, Sylvester's criterion (4.2) is applicable and further restricts the possible values of the angular velocity, namely where r ≈ 0.0342/ 2 is the only positive root of the following cubic equation The Weinhold scalar curvature yields It is singular at the root Ω = √ r, but the probability for fluctuating to a state on this spinodal or beyond is zero due to the fact that it is safely excluded by Sylvester's criterion. In other words, no geodesics (quasi-static processes) can pass through this spinodal within this approach. Moreover, the root √ r ≈ 0.185/ is within the region of local thermodynamic stability (3.11), suggesting Weinhold's metric as a viable metric in the (T, Ω) space.
We can now follow the standard interpretation [52], where the sign of the thermodynamic scalar curvature can be linked to the nature of the inter-particle interactions in composite thermodynamic systems. In this case, we find that R (W ) > 0 within the region given in Eq.
one finds that for large at fixed temperature the Weinhold thermodynamic curvature vanishes, thus the particle correlations become weak. The same is true for large temperatures at fixed . For states near the curve Ω = 1, the curvature R (W ) is also vanishing, thus we have a weakly coupled system near Ω = 1. In the non-rotating (static) case, Ω = 0, the strength of the interactions saturates at a value inversely proportional to the temperature T of the black hole. Therefore, one has weaker correlations (interactions) for larger temperatures and stronger interactions for smaller temperatures.
Another Hessian thermodynamic metric one can consider is the Hessian of the Gibbs free energy Imposing Sylvester's criterion one finds which precisely coincides with the range for local thermodynamic stability (3.11), thus one can take the Gibbs metric as a viable thermodynamic metric. The upper limit Ω = √ 3/(3 ) is where the inverse of the Gibbs metric becomes singular. The scalar curvature is given by which is positive in the range (4.9), specified by Sylvester's criterion. This suggests elliptic information geometry, which corresponds to repulsive inter-particle interactions. As in the Weinhold's case, the Gibbs curvature vanishes for high temperatures or large values of the gravitational parameter . In the static case R (G) is again inversely proportional to the temperature

New thermodynamic geometry
Although their convenient probabilistic interpretation Hessian thermodynamic metrics often fails to reproduce all relevant critical points. One way to avoid such problem is proposed in [32][33][34][35], where the authors take advantage of conjugate thermodynamic potentials to construct the proper Riemannian metrics on the space of equilibrium states of a given black hole system. Within the formalism of the New Thermodynamic Geometry (NTG) [33] one can find a positive definite metric on (T, Ω) space by using the Gibbs free energy as conjugate potential. For this purpose, let us take the differential from both sides of G = E − T S − ΩL, The metric with respect to G(T, Ω) is defined bỹ (4.14) Sylvester's criterion leads to the same restriction as in Eq. (4.9). However, most unexpectedly, this geometry is Ricci flat everywhere, where Y = 1 − 2 Ω 2 , which suggests free dual gauge theory. In order to make it more explicit we change coordinates to (S, Ω) space. Taking into account the Jacobian of the transformation, where andĝ = J t .g (G) .J, we end up with the metric on the cylinder In the last step we traded our coordinates for Further substitution by X = σ cos χ and Y = σ sin χ yields the desired flat two-dimensional Euclidean space. The latter means that the curve Ω = 1 is only a coordinate singularity in (T, Ω) space. It is most obvious, if one calculates the relevant thermodynamic quantities in (S, Ω) space 21) and the specific heats of the black hole which are regular everywhere. Nevertheless, the metricg ab comes from more general microscopic considerations. Consider a physical system in equilibrium with a large thermal reservoir. The configurational probability distribution is given by where y is the configuration (a set of random variables or a sample space), t is a time variable, β = 1/T is the inverse temperature of the environment (k B = 1), Z is the partition function, and H is the Hamiltonian of the system. The Hamiltonian is split into two parts -collective variables X a and their conjugate momenta λ a , i.e. βH = λ a (t)X a (y). The λ's are the experimentally controllable parameters of the system and define the accessible thermodynamic state space. In our case, one has λ a = (T, Ω) and X a = (S, L). If the partition function Z, which normalizes the probability distribution (4.24), is calculated in the so called "isobaric" 3 ensemble, then it can be directly related to the Gibbs potential G via [53] ln Z = −βG = ψ, (4.25) where ψ is the free entropy. In statistical quantum thermodynamics the first derivatives of the free entropy give the first moments of the collective variables [54] ∂ψ ∂λ a = − X a , (4.26) while the second derivative yields the covariance matrix (4.27) Substituting λ a = (T, Ω), X a = (S, L) and β = 1/T , we find that the covariance matrix (4.27) exactly corresponds to the thermodynamic metric from Eq. (4.14). Thus the Mirza-Mansoori (MM) approach has a direct relation to quantum statistics.

Thermodynamic length and quasi-static processes
In this section we consider geodesics on the equilibrium state space spanned by (T, Ω). The action for the thermodynamic geodesics is written by [54] L = where t is an affine parameter on the geodesics (not necessarily corresponding to time), λ a (t) = (T (t), Ω(t)) are the set of intensive thermodynamic parameters, and (t i , t f ) denote the initial and final states. We can vary the action to obtain the system of coupled geodesic equations where the dot is a derivative with respect to t. By definition the thermodynamic length L, between two equilibrium states at t i and t f respectively, is the on-shell value of the action (5.1) for the geodesic curve connecting those states. We can also define a related quantity, called the thermodynamic divergence of the path, which is a measure of the energy dissipation or entropy production for a transition between two equilibrium points at particular rates of change. In other words, J measures the efficiency of the quasi-static protocols and satisfies the following bound The latter follows from the Cauchy-Schwarz inequality for integrals and provides a formal definition of the degree of irreversibility of the process 4 (see [55] and references therein). We are now ready to begin our analysis of the thermodynamic geodesics in (T, Ω) parameter space. For the Gibbs metric (4.14) one finds the following coupled geodesic equations where The singular points in Eq. (5.7) correspond to the singularities of the metric (4.14) and its inverse. One way to find analytically a non-trivial solution is to consider a quasi-static process with a constant geodesic profile for the angular velocity, Ω(t) = Ω 0 = const. In this case, Eq. (5.5) is trivially satisfied, while Eq. (5.6) for the temperature becomes simpler For later convenience let us solve the more general equation where α is a real constant. This equation is equvalent to giving the first integral and consequently the general solution to (5.9): Given some initial conditions T (0) = T 0 andṪ (0) = T 0 , one finds In Eq. (5.8) one has α = −5/6, thus the temperature profile along the geodesics is (5.14) Here, we have assumed an initial macro-state T (0) = T 0 and an initial rate of temperature changeṪ (0) = T 0 . The thermodynamic lengthL (G) , between two macro-states at t = 0 and at t = τ , yieldsL For this specific choice of geodesics, the thermodynamic divergence J = L 2 saturates the equality, thus L 2 measures also the energy dissipation along the path. In the limit Ω 0 → 1/ the thermodynamic lengthL (G) becomes infinite. This means that one cannot use (T, Ω) coordinates for quasi-static evolution near this limit. In this case, a change of experimentally controllable parameters of the system, for example to (X, Y ) chart defined just below Eq. (4.19), is necessary to define meaningful thermodynamic states near Ω = 1. The price to pay is that we loose clarity of the physical process, due to the fact it is unclear how to treat (X, Y ) as thermodynamic quantities. When one considers a constant angular velocity geodesic profile for the Hessian-Gibbs metric (4.8) the system of geodesic equations (5.2) reduces to In this case, one has two options, namely Ω 0 = 0 or Ω 0 = 1/ . The first option, Ω(t) = Ω 0 = 0, leads to 3TT =Ṫ 2 , (5.17) with a solution given by Therefore, the corresponding TD length reduces to It is always finite, thus a quasi-static protocol is always possible on this path. The second choice, Ω 0 = 1/ , leads to non-singular geodesic equation with solution However, in the limit Ω → 1/ , the thermodynamic length diverges A non-trivial profile for the temperature T (t) can be obtained if we set Ω 0 = 0. In this case, the second equation (5.24) turns out to be the same as in the previous case (5.17) with the same solution (5.18) for T (t). However, the thermodynamic length of the path, connecting two macro states, is different By comparing the thermodynamic lengths in the considered cases above one can determine in which approach the system has greater probability of fluctuating from one macro state to another. In other words, which thermodynamic manifold leads to optimal implementation of quasi-static protocols in (T, Ω) space. The ratios of the computed TD lengths, compared at Ω 0 = 0, are independent of the initial rate of temperature change T 0 . One notices that for small initial temperatures, T 0 1, the "isobaric" lengthL (G) is greater than both Hessian-Gibbs (grand-canonical) L (G) and Weinhold's length L (W ) . In this case, it is less probable for the system to fluctuate into neighboring states, if it is realized in the "isobaric" ensemble. For large temperatures, T 0 1, the "isobaric" length is smaller and it becomes more efficient to fluctuate into neighboring states. Finally, Weinhold's length is always smaller than the Hessian-Gibbs length along the chosen geodesics, thus it is more efficient for the implementation of quasi-static protocols.

Conclusion
Investigating the thermodynamic properties of various black hole solutions in three dimensions plays an important role in revealing hidden relations between classical gravitational theories and quantum field theories in general. In this context, motivated by the remarkable dualities between gravitational (string) and gauge field theories [56], also known as the holographic principle, we study the thermodynamic properties of the stationary Lifshitz black hole solution of New Massive Gravity obtained in [22].
Our findings uncover the suitable Riemannian metrics on the space of equilibrium states of the black hole solution, together with several criteria for thermodynamic stability of the system. Our investigation has been conducted mostly within the framework of Thermodynamic Information Geometry, which takes advantage of differential geometry to study statistical features of various models.
The first set of restrictions (3.4) on the parameter space of the stationary Lifshitz black hole comes from its metric (2.8). The scalar curvature (2.9) and other higher-order invariants (2.10)-(2.12) are regular everywhere except at = 0. The curve Ω = 1 is also a regular one with respect to the curvature invariants, suggesting it is only a coordinate singularity. This is also true in the thermodynamic case, where all corresponding thermodynamic scalar curvatures vanish on this curve.
The second set of restrictions (3.11) comes from imposing local thermodynamic stability on the black hole. However, for values of the angular velocity in the range 1/( √ 3 ) ≤ Ω < , the black hole is locally TD stable only with respect to C Ω , but not C L . On the other hand, imposing global stability in the grand-canonical ensemble, we find the same restriction as in (3.4), while there is a different condition (3.14) on Ω in the canonical ensemble. Nevertheless, the restrictions in both ensembles include partially or entirely the condition (3.11) for local thermodynamic stability.
Further conditions comes from the admissible thermodynamic metrics on the (T, Ω) equilibrium state space of the black hole solution. Here, several approaches were considered. In Weinhold's case, one requires positive definite metric, which leads to condition (4.4), where the upper bound values of the angular velocity Ω are defined by a cubic equation (4.5). When one considers the Hessian of the Gibbs free energy one finds the condition (4.9), which stays within the local and global TD stability of the black hole. By defining a positive definite metric within the New Thermodynamic Geometry we found that the metric is in one to one correspondence with the covariance matrix from quantum thermodynamics. In our specific case, when utilizing the Gibbs free energy as conjugate thermodynamic potential, the correspondence is valid in the so called isothermal-isobaric ensemble. The resulting condition on the parameter space is the same as in (4.9). For clarity, we briefly state the results from all cases in Table 1.