Holographic Thermalization and Generalized Vaidya-AdS Solutions in Massive Gravity

We investigate the effect of massive graviton on the holographic thermalization process. Before doing this, we first find out the generalized Vaidya-AdS solutions in the de Rham-Gabadadze-Tolley (dRGT) massive gravity by directly solving the gravitational equations. Then, we study the thermodynamics of these Vaidya-AdS solutions by using the Minsner-Sharp energy and unified first law, which also shows that the massive gravity is in a thermodynamic equilibrium state. Moreover, we adopt the two-point correlation function at equal time to explore the thermalization process in the dual field theory, and to see how the graviton mass parameter affects this process from the viewpoint of AdS/CFT correspondence. Our results show that the graviton mass parameter will increase the holographic thermalization process.

down in this process, while the lattice QCD is not well-suited in dealing with the real time physics or Lorentzian correlation functions [1,2]. Therefore, other theoretical tools should be found out to describe this thermalization process.
A significant progress to deal with the strongly coupled system has been made in the last decade. The AdS/CFT correspondence has provided deep insights into the mapping between the strongly coupled field theory and the dynamics of a classical gravitational theory in the bulk, i.e. a gravitational spacetime with higher dimension [5][6][7]. Moreover, the AdS/CFT correspondence has been recognized as a useful tool to deal with the strongly coupled systems [2,[8][9][10][11][12]. Therefore, for the thermalization process of strongly coupled QGP, one can also construct a proper model in the bulk gravity to investigate this process [13], which is usually termed as the holographic thermalization process. Indeed, until now there have been many models proposed to study the holographic thermalization process, such as [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. Among them, a crucial model has been proposed [25,26] by using the equal time two-point correlation function (besides space-like Wilson loop and entanglement entropy) as a thermalization probe to investigate the thermalization behavior. From the viewpoint of AdS/CFT correspondence, this is equal to probing the evolution of a shell collapsing into an AdS black brane spacetime, which can be described as a Vaidya-like solution in the bulk gravity. There have been many generalizations of this model to other cases .
Many clues have been shown that the effect of massive graviton in the bulk gravity can be considered as the effect from the lattice in the dual field theory on the boundary, i.e. deducing the dissipation of momentum [55,56]. Therefore, the effect from the lattice during the thermalization process of QGP may be investigated from the bulk gravity with massive graviton, i.e. massive gravity. In fact, as an extension of the Einstein's general gravity, the massive gravity is a natural generalization [57]. However, it should be noted that this generalization is difficult, since the massive gravity usually has the instability problem of the Boulware-Deser ghost [58]. For more details, one can refer to the reviews of massive gravity [59,60]. Recently, the so called de Rham-Gabadadze-Tolley (dRGT) massive gravity has been proposed [60][61][62], which is a nonlinear massive gravity theory and has been found to be ghost-free [63,64]. Note that, the dRGT massive gravity is thought to be the only healthy theory in Poincaré invariant setups, and there have been many researches in dRGT massive gravity [56,[65][66][67][68][69][70][71][72][73][74][75][76][77][78].
In this paper, we will focus on the dRGT massive gravity, and first find out the generalized Vaidya-AdS solutions in this massive gravity by directly solving the gravitational equations. It is easily seen that these Vaidya-AdS solutions are consistent with the results in the previous work in the static spacetime [66]. In addition, we also investigate the thermodynamics of these Vaidya-AdS solutions by using the generalized Misner-Sharp energy in the (3+1)-dimensional dRGT massive gravity and unified first law [67,79]. Besides the discovery of the thermodynamical first law of these Vaidya-AdS solutions, we also find that the massive gravity is in its thermodynamic equilibrium state, which is consistent with the result considered in the FRW universe [67]. Finally, we investigate the holographic thermalization of the boundary field theory which can be holographically modeled by a massive shell collapsing into the Vaidya-AdS black brane. In particular, the thermalization is captured by the equal-time two-point correlation functions, which can be mapped to the bulk by computing the length of the geodesics between these separated points [25,26]. Our numerical results show that the graviton mass parameter in dRGT massive gravity can increase the holographic thermalization process, which indicates that the inhomogeneity of the boundary field theory will render the thermalization faster. This paper is organized as follows. In section II, we find out the generalized Vaidya-AdS solutions in the (3+1)-dimensional dRGT massive gravity; We adopt the Misner-Sharp energy and unified first law to investigate the thermodynamics of these generalized Vaidya-AdS solutions in section III; In section IV, we investigate the effect of the graviton mass parameter on the holographic thermalization process; Finally, we draw the conclusions and discussions in section V.
where l is the radius of the AdS spacetime, m 2 is the graviton mass parameter, and f is a fixed symmetric tensor, which is usually called the reference metric, c i are constants, and U i are symmetric polynomials of the eigenvalues of the (n + 2) × (n + 2) matrix K µ ν ≡ √ g µα f αν : The square root in K means ( √ A) µ ν ( √ A) ν λ = A µ λ and [K] = K µ µ = g µα f αµ (to extract the roots of the components one by one and then to make summation). The equations of motion turns out to be where In this section, we mainly focus on obtaining the generalized Vaidya-AdS solutions with a two dimensional maximally symmetric inner space in the (3+1)-dimensional dRGT massive gravity.
Therefore, the general metric ansatz can be where γ ij is the metric on a two-dimensional constant curvature space N with its sectional curvature k = ±1, 0, and the two-dimensional spacetime T spanned by the coordinates (v, r) possesses the metric as h ab . The energy momentum tensor for the radiation matter in the spacetime is given by where µ is the energy density and l a = (1, 0, 0, 0) in coordinates (v, r, x i ). In our case, we can also take the reference metric as the following in [65,66] with c 0 being a positive constant. Thus the symmetric polynomials become Therefore, the field equation in (3) can be simplified in our case as where Λ = − 3 l 2 , and For the metric ansatz in (5), the components of the field equation (8) can be explicitly expressed where a prime/overdot denotes the derivative with respect to r/v. Note that, the components G i j are not independent, since it can be found that they can be linearly expressed in terms of , and hence G i j = 0 do not yield independent equations. Therefore, from the above equation in (10) and (12), we can easily obtain the generalized Note that, our solutions (14) can be consistent with those in some previous work like [66]. Since if M (v) is independent of v, i.e. a constant, and hence f (v, r) can be written as f (r), then after making the transformation in the metric ansatz (5) we will easily find that the solutions (14) are just the same as the static solutions in (3+1)dimensional spacetime found in [66] .

III. THERMODYNAMICS OF THE GENERALIZED VAIDYA-ADS SOLUTIONS
In this section, we will investigate the thermodynamics of the above generalized Vaidya-AdS solutions by using the unified first law and Misner-Sharp energy. According to the unified first law [79], similar to the case of Einstein gravity, one can usually cast the equations of gravitational field (3) in a (3+1)-dimensional spacetime with a two dimensional maximally symmetric inner space into the form where A = V k r 2 and V = V k r 3 /3 are area and volume of the 2-dimensional space with radius r. W is called work density defined as W = −h ab T ab /2, while Ψ a is the energy supply vector, In addition, T ab is the projection of the four-dimensional stress energy T µν of matter into h ab , and E ef f is signed as the generalized Misner-Sharp mass in the modified gravity if it exists.
Note that, we have obtained the generalized Misner-Sharp mass in 4-dimensional dRGT massive gravity in [67] Therefore, after substituting the explicit forms of the generalized Vaidya-AdS solutions in (14), we can obtain the following quantities From which, we can easily check that the unified first law (16) is indeed satisfied for our solutions (14), and the generalized Misner-Sharp mass indeed exists in the 4-dimensional dRGT massive gravity. Note that, the generalized Misner-Sharp mass does not always exist for the modified gravity, i.e. the f (R) gravity in [80,81].
Next, we will investigate the thermodynamics of the generalized Vaidya-AdS solutions (14) On the other hand, the temperature of generalized Vaidya-AdS solutions can be considered as T = κ 2π . Here the surface gravity κ defined on the apparent horizon is κ = D a D a r = [79,[82][83][84], and D a is the covariant derivative associated with the metric h ab . In addition, the entropy of generalized Vaidya-AdS solutions is . Therefore, we can obtain Note that, from the equation of the apparent horizon f (r A , v) = 0, we can deduce a simple relation Obviously, after using this simple relation, we can easily find that the usual Clausius relation δQ = T dS holds on the apparent horizon for the generalized Vaidya-AdS solutions (14). Therefore, the unified first law in (16) on the apparent can be rewritten as which is just the first law of thermodynamics for the generalized Vaidya-AdS solutions. In addition, according to [86], the usual Clausius relation δQ = T dS held in our case also indicates that the dRGT massive gravity is an equilibrium state. It should be emphasized that the usual Clausius relation δQ = T dS does not always hold on the apparent horizon. For example, the usual Clausius relation does not hold for the f (R) gravity, which can be the effect from the nonequilibrium thermodynamics of space-time [80,85,86]. It should also be pointed out that the usual Clausius relation held or not held only depends on the explicit theory of gravity, i.e. like the case for the formula of entropy S(A) independent of the explicit solutions. Indeed, the usual Clausius relation δQ = T dS held in our case is consistent with the investigation in [67] by taking the FRW universe into account.

IV. HOLOGRAPHIC THERMALIZATION IN MASSIVE GRAVITY
In this section, we will focus on the holographic thermalization in the dRGT massive gravity with the 4-dimensional bulk spacetime, in which the dual gravitational solution is investigated under the above Vaidya-AdS solution with k = 0, where i = 1, 2 representing the codimension 2 space, z = l r and H(v, z) can be related to the mass of a collapsing black brane, which is usually set as a smooth function Holographic thermalization has been studied from various aspects, such as colliding shock waves, gravitational collapsing, holographic (quantum)quenches, holographic entanglement entropy and etc., see reviews [2,87]. In this section we will focus on the gravitational collapsing aspect and adopt the two-point correlation function at equal time to explore the thermalization process in the dual field theory, and then see how the graviton mass parameter affects this process. According to the AdS/CFT correspondence, the on-shell equal time two-point correlation function can be holographically approximated as [26] in which the conformal dimension ∆ of scalar operator O is assumed to be large enough, and L is the bulk geodesic length between the points (t 0 , x i ) and (t 0 , x i ) on the AdS boundary. Usually the geodesic length L is divergent due to the UV contribution from the AdS boundary, therefore, one should eliminate the divergence by adding a counter-term in order to obtain the renormalized geodesic length which is δL = L + 2 ln z 0 [25,26], where 2 ln z 0 comes from the contribution of a pure AdS boundary, and z 0 is a UV cut-off satisfying the boundary conditions in which d is the spatial separation between the two points lies entirely in the x 1 direction while t 0 is the time of the thermalization probe moving from the shell to the boundary, which will be recognized as thermalization time later.
In the following context, we would like to rename x 1 as x for simplicity and adopt it to parameterize the geodesic. Considering the parity symmetry in x direction x → −x, the proper length of the geodesic is given by In order to compute the length of the geodesic, we need to minimize the length (26) and solve the two coupled Euler-Lagrangian equations of motion for z(x) and v(x) respectively. We reach In order to solve the above equations, we impose the following boundary conditions of z(x) and v(x) at x = 0, where z (0) = v (0) = 0 can be deduced from the symmetry of the geodesic.
One can further simplify the EoMs (27) by virtue of the 'Hamiltonian' methods [88]. If regarding the integrand √ Π/z in (26) as the 'Lagrangian' L, we can define the 'Hamiltonian' H as Therefore, substituting (30) into the EoMs (27) we can substantially simplify the equations as So finally we obtain the second order ordinary differential equations for v(x) and z(x), which can be solved numerically if we impose the boundary conditions in (25) and (28).
In the following we will present the numerical results of the above differential equations. In the numerics we fixed the parameters c 0 = c 1 = 1, c 2 = −1/2 in order to render the background thermodynamically stable [56,66]; Besides, we set the shell thickness v 0 = 0.01 and the UV cutoff at z 0 = 0.01, respectively. In addition, we rescaled the horizon location to be z = 1 in order to more easily compare the positions of the horizon and the shell. In Fig.1   In Table.I we present the data of the thermalization times t 0 for various graviton masses and initial times. One finds that for a fixed initial time v * , the thermalization time t 0 decreases as the graviton mass increases, which indicates that the dual boundary field theory thermalizes and saturates into the equilibrium state faster. As we learned from [55], graviton mass is related to the inhomogeneity of the boundary field theory. Therefore, from the Table.I we can infer that on the boundary field theory, the greater the inhomogeneity is, the faster the system would saturate into the equilibrium state.
In Fig.2, we plot the renormalized geodesic length for different graviton masses and different separation of the points on the boundary. In particular we define a renormalized dimensionless geodesic length as δL−δL E , in which δL = δL/d and δL E is the value of the geodesic length arriving at equilibrium state. The brown, green and black lines in the Fig.2 correspond to graviton masses m 2 = 1, 0.5, 0.0001 respectively. Therefore, δL − δL E = 0 indicates that the system saturates an equilibrium thermal state. Hence, it is readily to see that as graviton mass becomes bigger, the corresponding boundary field theory saturates into equilibrium faster, which is consistent with the conclusion drawn from Table.I. Once again this indicates that greater inhomogeneity in the boundary field theory would make the system faster to enter the equilibrium state. From the panels (a) and (b) in Fig.2, we can also deduce that different separations of the points on the boundary will not change the conclusions above.

V. CONCLUSION AND DISCUSSION
In this paper, after obtaining the generalized Vaidya-AdS solutions by directly solving the equations of fields in the dRGT massive gravity, we have investigated the thermodynamics of these generalized Vaidya-AdS solutions. Besides the first law of thermodynamics obtained by using the unified first law and generalized Misner-Sharp mass, we also find that the usual Clausius relation δQ = T dS holds in our case, which indicates that the dRGT massive gravity is a equilibrium state. This result is consistent with that by taking the FRW universe into account. Moreover, we further investigate the holographic thermalization by considering a massive shell collapsing in the generalized Vaidya-AdS black brane spacetime, while the two-point correlation function at equal time has been chosen as a thermalization probe to investigate the thermalization behavior of the dual field on the boundary. On the other hand, according to the AdS/CFT correspondence, the two-point correlation function at equal time can be related to the length of the space-like geodesics in the bulk between the points (t 0 , x i ) and (t 0 , x i ) on the AdS boundary. Therefore, after some numerical calculations, our final result is that the graviton mass parameter can increase the holographic thermalization process.
Note that, some clues have been shown that the effect from the massive graviton in the bulk can be considered as the effect from the lattice in the dual field according to the applications of AdS/CFT in condensed matter, i.e. AdS/CMT. Therefore, an interesting question related to our results is that what is the dual physical meaning of the effect from the massive graviton for the field on the AdS boundary? In addition, another interesting question is about the relationship between the graviton mass and the two-point correlation function at equal time on the AdS boundary.
From our results, we can qualitatively find that the graviton mass affects the length of the bulk geodesic, and hence the two-point correlation function at equal time through (24), which finally deduce the thermalization time shorter. Note that, recently there are also some investigations on the inhomogeneous holographic thermalization [89,90], which will be also an interesting issue to be further studied. These investigations may give more insights to the information about the real thermalization process of QGP. On the other hand, it should be pointed out that other choices of probes like Wilson loops or entanglement entropy are also possible if the bulk spacetime is of dimension D > 3, i.e., the holographic thermalization process takes place in the space with dimension D > 1. Moreover, it is well known that in the Einstein gravity the entanglement entropy is the probe that thermalizes later and therefore sets the thermalization time of the field theory, thus an interesting future direction would be to know whether the dRGT massive gravity also follows the same pattern as Einstein gravity, namely that probes of codimension 2 are the ones that take longer to thermalize. In addition, Y.P Hu thanks a lot for the support from the Sino-Dutch scholarship programme under the CSC scholarship.