Thermodynamics of Lifshitz Black Holes

Transcription

1 Thermodynamics of Lifshitz Black Holes Hai-Shan Liu Zhejiang University of Based on work with Hong Lu and C.N.Pope, arxiv: PLB; arxiv:1402:5153 JHEP; arxiv:

2 Black Hole Thermodynamics Black hole is a thermodynamic system Schwartzchild black hole Reissner-Nordstrom black hole These black holes are asymptotic to AdS or Minkowski spacetimes, there are well-defined procedures to calculate these thermodynamical quantities.

3 Lifshitz vacuum Lifshitz spacetime Scaling symmetry For example, Schrodinger equation (z=2)

4 Lifshitz black hole Asymptotic Lifshitz black hole For Lifshitz black holes, T and S can be calculated by using the same standard methods. For asymptotic AdS or flat spacetime, there are various ways of calculating the mass, such as AMD, ADT mass. However, although Lifshitz geometry has been popular for years, there still doesn t exist a general method of defining its mass..

5 An example [Pang,JHEP 1001,116(2010)] Theory Solution(has only one parameter)

6 An example [Pang,JHEP 1001,116(2010)] Thermodynamics: Suppose the massive vector doesn t contribute to the first law, one may expect the first law looks like, we can compute temperature and entropy through standard procedure

7 An example [Pang,JHEP 1001,116(2010)] However, as said before, there is no method to calculate black hole mass, the massless vector field A is divergent as r goes to infinity, the electrical potential can not be well defined. There are two quantities we can not get, so we can not obtain the first law. Since the solution has only one parameter, and we know, a naïve thought is to integrate it out and interpret it as mass, just like what Pang did. However, we will show this was wrong.

8 An example [Pang,JHEP 1001,116(2010)] Even this doing gets the correct first law for such a degenerate case, we don t know if the massive vector contribute to the first law or how much part the massive vector contributes to the first law. All these difficulties force us turn to Wald.

9 Wald Formula In order to illustrate the Wald formula, we turn to a simple system, pure gravity We consider a general variation of the Lagrangian We can define an 1-form and its Hodge dual

10 Wald Formula We now specialize to a variation that is induced by an infinitesimal diffeomorphism One can show that denotes a contraction Then we can define a charge (on-shell)

11 Wald Formula Wald shows that the variation of the Hamiltonian with respect to the integration constants of a given solution is denotes two boundaries, one on the horizon and one at infinity. First law is given by

12 Wald Formula We consider spherically-symmetric solution The expressions turn out to be

13 Wald Formula At horizon

14 Wald Formula At infinity, we take Schwarzschild black hole in n dimention for example So first law is obtained

15 Comments As we can see from this simple example, the variation in horizon gives TdS is general. In fact this is Wald s entropy, which Wald expressed in a now well known formula However, there is no general formula for the variation at the asymptotic infinity.

16 Wald formula for a general class of theories We consider Einstein gravities in n dimensions that couples to a scalar and a vector We consider static solutions of general type

17 Wald formula for a general class of theories With the same procedure, we derive the Wald formula The result is insensible to non-derivative term in the Lagrangian. It is straightforward to generalize the results to multiple scalars and multiple vectors.

18 Application I Einstein-Maxwell-Dilaton theories The lagrangian is given by We consider N= 2 and, which admits a charged Lifshitz black hole[alishahiha, JHEP 1112, 036 (2011)]

19 Einstein-Maxwell-Dilaton theories The Wald formula is The variation over horizon gives TdS as usual, which we make a gauge choice so that Vector A vanishes on horizon. This implies that And the variation in infinity is given by

20 Einstein-Maxwell-Dilaton theories We can calculate the electrical charges We can see that is a fixed constant and it will not contribute to the first law. Hence the first law is given by with

21 Aplication II Einstein-Proca theory with a Maxwell field After introducing the Wald formula through a simple system, we turn back to the more complex one which we begin with. We consider static solutions of general type

22 Einstein-Proca theory with a Maxwell field Wald formula And

23 Einstein-Proca theory with a Maxwell field First law is given by The next step is to evaluate at asymptotic infinity, which we need obtain the falloffs of the fields. We first study the linearized field equations around the Lifshitz vacua, we use tilded fields to denote small perturbations.

24 Einstein-Proca theory with a Maxwell field Expand the equation of motion up to the linear order in tilded fields, we obtain a set of coupled linear differential equations, which can be solved exactly.

25 Einstein-Proca theory with a Maxwell field with

26 Einstein-Proca theory with a Maxwell field correspond to Maxwell field, correspond to graviton mode and correspond to massive vector To understand the graviton mode, we can turn to the metric component

27 Einstein-Proca theory with a Maxwell field In order to analyze the massive vector, we consider its equation of motion with electric solutions around Lifshitz vacuum. We can solve its equation and get

28 Einstein-Proca theory with a Maxwell field Once we got the falloff, we can obtain the first law through Wald formula and

29 Einstein-Proca theory with a Maxwell field We define Then we have

30 Einstein-Proca theory with a Maxwell field We define And The first law is given by

31 Einstein-Proca theory with a Maxwell field Back to Pang s solution Compared to linear mode

32 Einstein-Proca theory with a Maxwell field We can see this explicit example corresponds to, which means M = 0. Then what left is With T and S can be calculated by using the standard procedure, and we verify that the first law is indeed satisfied.

33 Application III Einstein-Proca theory with a non-dynamiacal sacalar field The theory is given by The Wald formula can be obtained by set

34 Einstein-Proca theory with a non-dynamiacal sacalar field An exact Lifshitz black hole solution was obtained for z=2,[balasubramanian and McGeevy, Phys.Rec.D 80, ]

35 Einstein-Proca theory with a non-dynamiacal sacalar field Substituting the solution to the Wald formula, The variation over horizon gives TdS, and in infinity gives, So the first law is give by with T and S are given by In this case, the naïve thought to integrate out TdS to get the mass turns out to be correct.

36 Summary We consider theory with static solutions of general type and get the Wald formula Only gravity part contributes on the horizon.

37 Summary The first law can be derived even without analytical solutions, what are needed are their asymptotic behaviors. We can pick up all the contributions to the first law The result is rather general, both in theory and solutions. If you find a solution, you can just put it into the Wald formula, and then comes out the first law.

38 New understanding of first law Every field in a two derivative gravity theory has two integration constants in the most general solution. The Wald formula gives the first-order differential relation between them. One of the two integration constants associated with graviton mode, like p, is constant related to arbitrary scaling of time, which we can set to 1, then the conjugate constant, say q, can be interpreted as mass.

39 Thank you!