Lovelock-Lifshitz Black Holes
In this paper, we investigate the existence of Lifshitz solutions in Lovelock gravity, both in vacuum and in the presence of a massive vector field. We show that the Lovelock terms can support the Lifshitz solution provided the constants of the theory are suitably chosen. We obtain an exact black hole solution with Lifshitz asymptotics of any scaling parameter in both Gauss-Bonnet and in pure 3rd order Lovelock gravity. If matter is added in the form of a massive vector field, we also show that Lifshitz solutions in Lovelock gravity exist; these can be regarded as corrections to Einstein gravity coupled to this form of matter. For this form of matter we numerically obtain a broad range of charged black hole solutions with Lifshitz asymptotics, for either sign of the cosmological constant. We find that these asymptotic Lifshitz solutions are more sensitive to corrections induced by Lovelock gravity than are their asymptotic AdS counterparts. We also consider the thermodynamics of the black hole solutions and show that the temperature of large black holes with curved horizons is proportional to where is the critical exponent; this relationship holds for black branes of any size. As is the case for asymptotic AdS black holes, we find that an extreme black hole exists only for the case of horizons with negative curvature. We also find that these Lovelock-Lifshitz black holes have no unstable phase, in contrast to the Lovelock-AdS case. We also present a class of rotating Lovelock-Lifshitz black holes with Ricci-flat horizons.
Candidates for gravity duals of non-relativistic scale invariant theories have recently attracted a great deal of attention for several reasons. One is that some condensed matter systems realized in laboratories at their critical points are described by non-relativistic conformal field theories. Furthermore, the AdS/CFT correspondence [1, 2] describing the duality between strongly coupled conformal field theory and gravity continues to find more applications in other branches of physics such as QCD quark-gluon plasmas , atomic physics, and condensed matter physics [4, 5, 6, 7, 8, 9]. This has led to a study of gravity-gauge duality in a much broader context than its original AdS/CFT formulation, extending to non-relativistic and Lifshitz field theories, with the aim of gaining more knowledge of the strong coupling behavior of these other physical systems. Non-relativistic conformal symmetry contains the scaling symmetry
where z is the dynamical exponent. This transformation exhibits anisotropic scale invariant behavior. Actually, for condensed matter applications, one typically is interested in considering anisotropy between different spatial dimensions. For this scaling symmetry is the familiar conformal symmetry. Such a non-relativistic scale invariance (1.1) can be exhibited by either a Galilean-invariant theory or a Lifshitz-invariant theory. From a holographic point of view, this suggests the following (asymptotic) form for the spacetime metric
that obeys the scale invariance
noted previously in a braneworld context .
A four-dimensional anisotropic scale invariant background using an action involving a two form and a three form field with a Chern-Simons coupling
can be engineered to yield solutions with this asymptotic behavior , where and . Such an action was argued to be rather generic in string theory, although no explicit brane realization or embedding into ten-dimensions was given. For these matter fields, lots of effort has been expended in extending this solution to the case of asymptotic Lifshitz solutions. One of the first analytic examples was reported in Ref.  for a sort of higher-dimensional dilaton gravity without restricting the value of the dynamical exponent . An exact topological black hole solution with hyperbolic horizon which happens to be asymptotically Lifshitz with was found in ; further solutions with and with spherical topology were subsequently obtained . However in general such asymptotic Lifshitz black holes must be investigated numerically [13, 14, 15, 16]. Other possibilities for the matter needed to support such a background have been investigated . Asymptotic Lifshitz solutions in the vacuum of higher-derivative gravity theories (with curvature-squared terms in the action) have been investigated [17, 18, 19, 20]; the higher curvature terms with suitable coupling constant play the role of the desired matter.
Recently gravity theories including higher powers of the curvature, particularly curvature-cubed interactions, have attracted increased attention . This is because, in the context of AdS/CFT correspondence, corrections from higher powers of the curvature must be considered on the gravity side of the correspondence in order to scrutinize CFTs with different values for their central charges. Amongst the cornucopia of higher-curvature gravity theories, Lovelock gravity theories play a special role in that the number of metric derivatives in any field equation is never larger than 2. Third-order Lovelock gravity is supersymmetric, and therefore one can define superconformal field theories via the AdS/CFT correspondence [22, 23]. Furthermore quasi-topological gravity including curvature-cubed interactions, while not supersymmetric, can be considered to be dual to some non-supersymmetric but conformal gauge theory in the limit of a large number of colours .
In this paper, we consider the existence of Lifshitz solutions in third order Lovelock gravity both in vacuum and in the presence of a massive vector field. Since the higher curvature terms appear to play the role of some kind of matter field, it is natural to ask whether they can support a Lifshitz solution in vacuum or not. We find that the answer is yes, albeit under restricted circumstances. We also search for asymptotic Lifshitz black holes in the presence of a massive vector field, whose action is given via a dualization of the action (1.4). The solutions we find – both analytically and numerically – can be regarded as higher-curvature modifications to those obtained from Einsteinian gravity coupled to matter [13, 14, 16].
The outline of our paper is as follows. We give a brief review of the field equations of third order Lovelock gravity in the presence of massive vector field in Sec. 2. In Sec. 3 we present the -dimensional exact Lifshitz solutions in vacuum and in the presence of a massive vector field. In Sec. 4 we obtain the series solutions to the field equations near the horizon, while the series solutions at large will be given the Appendix. We then obtain numerical solutions to these equations. The thermodynamics of these Lovelock-Lifshitz black holes will be given in Sec. 5. Section 6 will be devoted to the rotating Lovelock-Lifshitz solutions. We finish our paper with some concluding remarks.
2 Field equations
The fundamental assumptions in standard general relativity are the requirements of general covariance and 2nd-order differential field equations for the metric. Based on the same principles, the Lovelock Lagrangian is a very general Lagrangian in classical gravity that produces second order field equations for the metric for arbitrary powers of the curvature . The action of third order Lovelock gravity in the presence of an Abelian massive vector field may be written as
where , is the cosmological constant, and are Gauss-Bonnet and third order Lovelock coefficients, is the Einstein-Hilbert Lagrangian, is the Gauss-Bonnet Lagrangian, and
is the third order Lovelock Lagrangian. We assume that the Gauss-Bonnet coefficient, which has the dimension of (length), is positive as in the heterotic string theory .
In Lovelock gravity only terms with order less than (where is the integer part of ) contribute to the field equations, the rest being total derivatives in the action. For 3rd-order Lovelock gravity we therefore consider -dimensional spacetimes with (though in situations where we set our solutions will be valid for ). Varying the action with respect to the metric tensor and gauge field the equations of gravitation and gauge fields are
is the energy-momentum tensor of gauge field, is just the Einstein tensor, and and are given as:
The metric of an -dimensional asymptotically Lifshitz static and radially symmetric spacetime may be written as:
where the functions and must go to as goes to infinity. In Eq. (2.5) is the metric of an -dimensional hypersurface with constant curvature and volume . We can write
though it should be noted that our solutions are valid whenever describes any Einstein space.
3 Lifshitz Solutions
3.1 Vacuum Solutions
We first investigate the possibility of having -dimensional Lifshitz solutions
in Lovelock gravity in the absence of matter. In order to have an asymptotically Lifshitz solution in 3rd-order Lovelock gravity, the following constraints on the cosmological constant and third order Lovelock coefficient
hold for an arbitrary value of , as is easily obtained via straightforward calculation.
Note that for the constraints (3.2) become
and so asymptotically Lifshitz solutions exist in the vacuum of Gauss-Bonnet gravity () as well. The cosmological constant here is half that of an AdS spacetime. If we set then a Lifshitz solution exists in third order Lovelock gravity provided
Thus, one may have Lifshitz solutions in Lovelock gravity without matter, demonstrating that its higher curvature terms can, for the proper choice of Lovelock coefficients, have the desired effect that matter fields induce.
The preceding analysis was for . For we find for any value of that
furnishes an exact Lifshitz solution to Lovelock gravity without matter, provided one of the above conditions (3.2), (3.3) is satisfied, with the function undetermined by the field equations (though boundary conditions constrain it to asymptote to 1). We can choose . This is a naked singularity for but is an asymptotically Lifshitz black hole for . The arbitrariness of is due to a degeneracy of the field equations; if either of (3.2) or (3.3) hold then the field equations are each proportional to the factor .
This degeneracy of the field equations has been noted previously in 5-dimensional Einstein-Gauss-Bonnet gravity with a cosmological constant , where (converting notation appropriately) condition (3.3) was obtained. Here we see that a more general degeneracy occurs in 3rd order Lovelock gravity. We shall see that this degeneracy is lifted when matter is present. We find that no other exact solutions to the field equations exist for these symmetries and asymptotic behavior.
3.2 Matter Solutions
In order to have asymptotically Lifshitz solutions in 3rd-order Lovelock gravity in the presence of matter, the following constraint
must hold for the cosmological constant and charge where we define
for simplicity and we use this definition throughout the paper. Since we obtain the constraint
which in turn yields
provided . Equation (3.10) shows that for the cosmological constant is negative. For , the cosmological constant can be positive provided
where the last inequality comes from the condition (3.9).
Note that for Einstein gravity () and the above conditions reduce to those which are given in  for . In Gauss-Bonnet gravity (), these conditions become
where now must be larger than . If , then the charge becomes zero and the conditions (3.11) reduce to the conditions (3.3) as expected. More generally, if , then vanishes and the conditions (3.7) reduce to the conditions (3.2) as expected.
4 Asymptotic Lifshitz Black holes
In this section we seek black hole solutions in 3rd-order Lovelock gravity that are asymptotically Lifshitz. We therefore consider the field equations (2.8)-(2.10) with the conditions (3.7), but with the more general ansatz (2.5) and (2.7).
4.1 Series solutions near the horizon
We now consider the near-horizon behavior of such solutions. Requiring that and go to zero linearly, that is
and inserting these expansions into the equations of motion arising from Eqs. (2.8)-(2.9) with the conditions (3.7), and solving for the various coefficients, we find that . This is consistent with the fact that the flux should go to a constant at the horizon. Also, one may note that by scaling time we can adjust the constant by an overall multiplicative factor (note the use of in the expansion of as well, which is due to in the one-form ). We find the following constraint on the st order constants
which means that not all boundary conditions are allowed. Note that here the coefficient is arbitrary, and should be chosen suitable in order to explore the numerical solutions. The higher order coefficients may be found easily, but their expressions are very lengthy and so we don’t write them here for reasons of economy.
One may also investigate the behavior of the metric functions at large . We relegate this to the Appendix, where we show that the powers of for are in general non-integer.
4.2 Numeric Solutions
and find the first derivatives of , and as
Now having a system of 4 first order ordinary differential equations, one may explore the numeric solutions by choosing suitable initial conditions for , , and , where is the radius of horizon. This can be done by using the method explained in .
First, we apply the numerical method for which can be solved exactly, and compare the numerical solution to the exact one. Indeed, for , the charge vanishes, and the problem reduces to the case of black holes in third order Lovelock gravity with specific choices of Lovelock coefficients. In this case, we obtain the exact solutions of Lovelock gravity , which have . The diagrams of the functions and versus for with and have been shown in Fig 4.1. In this figure the exact solutions have also been shown. As one can see from the plots, the numerically obtained functions and are equal and both of them are exactly lie on the exact solution within limits of numerical precision. We regard this as a good test of our methods. Note that for and (the solution without matter) , the metric functions increase from zero at to at . However for and , the metric functions grow to be larger than unity at intermediate values of and asymptote to 1 at infinity.
Now, we consider the solutions for , for which matter exists, where we must solve the system of equations numerically. Figure 4.2 shows the function as a function of for a small black hole of Einstein, Gauss-Bonnet and Lovelock gravity with and . As one can see from this figure, while for Einstein, Gauss-Bonnet and Lovelock with (solution without matter) are almost the same, they are different for for the same values of and . Furthermore, for , the metric function rapidly increases to values larger than 1, eventually asymptoting to as , while for this function monotonically increases from zero at to at . Thus, it appears that asymptotic Lifshitz solutions are more sensitive than are their (AdS) counterparts to the corrections induced by Lovelock gravity. This feature also occurs for large black holes with (see Fig. 4.3). Note that for , the function in Gauss-Bonnet gravity is larger than that in third order Lovelock gravity. This is due to the fact that the slope of the function decreases as increases (see the terms in with the factor in Eq. (4.5) with , which are negative). Figures 4.4-4.5 show that this fact also occurs for the small and large black holes with . For the case of , one can see from Figs 4.6 and 4.7 that both the asymptotic AdS and Lifshitz black holes are different for various order of Lovelock gravities, although the difference between the metric function for various order of Lovelock gravity becomes more relevant as increases.
We find that the dependence on of is not as significant as for . For this and reasons of economy, we plot only the function in one case (see Fig. 4.8). We also find that is monotonically increasing for topological black holes with , whereas (as noted previously) has a maximum larger than at several times .
We pause to comment on the relative importance of the various terms in the Lovelock action. While there is no essential problem in considering all terms to be similar in magnitude, we obtain some insight into Lovelock gravity by treating it as an approximation (in powers of the curvature) to a full quantum theory of gravity. For simplicity we shall consider solutions obtained in this paper for , which asymptote to the Lifshitz background (3.1). In this case the th order Lagrangian of Lovelock gravity for the metric (2.5) may be written as:
where . The above expression for the Lagrangian is correct for any value of , though in this paper we have considered only . The Lagrangian (4.6) can be evaluated on the solutions at large by the use of the expansions of the metric functions which are given in the Appendix. It is a matter of calculation to show that the ratio at large is
and so .
5 Thermodynamics of black holes
The entropy of a black hole in Lovelock gravity is 
where the integration is done on the -dimensional spacelike hypersurface of the Killing horizon with induced metric (whose determinant is ), and is the th order Lovelock Lagrangian of . It is a matter of calculation to show that the entropy of a black hole per unit volume of the horizon in third order Lovelock gravity is
This reduces to the area law of entropy for .
One can obtain the temperature of the event horizon by using standard Wick-rotation methods, yielding the result
For the case of asymptotic Lifshitz black hole with (identified) hyperbolic horizon, the temperature of the exact solution (3.5) with horizon radius in vacuum is
The temperature of more general Lovelock-Lifshitz black holes can be calculated numerically. First, we review the thermodynamics of asymptotic AdS black holes () to compare it with the asymptotic Lifshitz black holes. As one can see from figure 5.1, the logarithm of the temperature of black holes versus the logarithm of the entropy for large black holes is linear, while for small black holes this occurs only in the case of . For , one encounters with an extreme black hole in Einstein (blue dashed line) and Lovelock gravity (blue solid line). However, the horizon radius of the extreme black hole of Lovelock gravity is smaller than that of Einstein gravity. Note that the slopes of the graphs of versus for small black holes of Einstein and Lovelock gravity with are negative, and therefore they are unstable. The unstable phase in Lovelock gravity occurs for a smaller radius black hole than in Einstein theory.
Plotting versus for the case of asymptotic Lifshitz black holes with , we see from Fig. 5.2 that the temperature of black holes in Lovelock gravity with given entropy is smaller than the temperature of black holes of Einstein gravity with the same entropy. Conversely, at a given temperature the entropy of the Lovelock black holes – extremal and non-extremal – is larger than for the Einstein ones for both and . The horizon radii of extreme black holes in both Einstein and Lovelock gravity for with are smaller than their counterparts, and for a given smaller in Lovelock gravity than in Einstein gravity.
Numerical calculations show that there is no unstable phase for black holes for .