New Asymptotically Lifshitz Black Holes in Hořava gravity
We study asymptotically Lifshitz solutions with critical exponent in Hořava gravity in three and four spacetime dimensions. For and , we find a novel class of numerical solutions with regular universal horizon, but are characterized by non-analytic behavior near infinity. In the interior, inside the universal horizon, the unit timelike vector field associated with the preferred time foliation exhibits oscillatory behavior, qualitatively similar to that found earlier in asymptotically flat solutions. For no solutions of this type appear to exist. We comment on potential applications to holographic Lifshitz dualities.
Over the past decade holography has become an important tool to describe strongly coupled condensed matter systems. Since many real world systems of interest are non-relativistic, there has been growing interest in constructing gravitational models that exhibit non-relativistic Galilean or Lifshitz symmetries. In this paper we will focus on dual Lifshitz theories, where boost invariance is broken and space and time scale in an anisotropic way . The dynamical exponent in general will differ from unity. A field theory with this type of symmetry is thought to describe quantum critical points, which characterize zero temperature phase transitions associated with quantum fluctuations subir ().
On the gravitational side, it was shown that Einstein gravity with various additional matter fields in the bulk can have solutions with asymptotic Lifshitz scaling Kachru:2008yh (); Taylor:2008tg (); Goldstein:2009cv (). In these cases the dual gravitational theory is of course relativistic, with local Lorentz invariance. The Lifshitz scaling emerges only as feature of a special class of solutions, supported by the exotic matter fields. A more natural, alternative dual description is in terms of a gravity theory with a preferred frame Griffin:2011xs (); Janiszewski:2012nb ().
One such theory is Einstein-aether theory AEtheory (), where in addition to the metric there is a dynamical unit timelike vector field acting as an “aether”. Einstein-aether theory has been studied extensively as an alternative theory of gravity (i.e. focusing on asymptotically flat and cosmological solutions), for a review see AEreview (). Another theory is Hořava-Lifshitz gravity, or Hořava gravity for short, which has attracted much attention as a possibly renormalizable quantum theory of gravity Horava:2009uw (). Hořava gravity can be thought of in the low energy limit as Einstein-aether gravity restricted to the case where the aether is hypersurface orthogonal, defining a preferred time.
In Griffin:2012qx () the authors showed that Hořava gravity (with negative cosmological constant) is a viable setting for non-relativistic Lifshitz holography and worked out some aspects of the holographic dictionary. In holography, thermal states in the field theory are dual to black hole solutions in the bulk gravitational theory. Therefore to study Lifshitz hydrodynamics we need to find black hole solutions and their associated thermodynamics. However, in a theory of gravity with a preferred frame the loss of relativistic causality leads to the existence of apparently multiple horizons, which clashes with the laws of black hole thermodynamics see e.g. Dubovsky:2006vk (); Eling:2007qd (); Jacobson:2008yc (). A potential resolution is to define a black hole via the existence of a “universal horizon”, a spacelike surface where even modes of arbitrary speed do not reach infinity Barausse:2011pu (); Blas:2011ni (). The universal horizon is defined in terms of the preferred time foliation, in particular it is the surface where the preferred time . Several works have shown that one can indeed define a notion of temperature and a First law for the universal horizon Berglund:2012bu (); Berglund:2012fk (); Mohd:2013zca (); Cropp:2013sea ().
Asymptotically Lifshitz black holes with regular universal horizons have previously been found in Hořava gravity. In particular, in the special case where there is an analytic black hole solution Janiszewski:2014iaa () in four dimensions. One can perturb this solution in the context of the fluid-gravity correspondence and read off some of the transport coefficients of the dual theory Eling:2014saa (); Davison:2016auk (). However, the generic cases are of the most interest. Recently, Basu:2016vyz () found a numerical solution in three dimensions for and special choice of coupling constants.
In this paper we show numerically that there exist a wide class of solutions with regular universal horizon for and in both three and four dimensions. Near infinity, these solutions have non-analytic behavior, where series expansions are in terms of non-integer powers. We also explore the interior and show there is oscillatory behavior in the aether field, just as in asymptotically flat cases. Curiously, for no solutions of this type appear to exist. The structure of this paper is as follows. In Section II we briefly review Einstein-aether theory and how hypersurface orthogonal solutions are also solutions of Hořava gravity. In Section III we discuss asymptotic Lifshitz solutions, which can have analytic and non-analytic behavior when . In Section IV, we discuss generic features of black hole solutions, including how one imposes regularity. In Section V we present our results. We conclude with a discussion and areas for future research.
Ii Einstein-aether and Hořava-Lifshitz gravity
The action for Einstein-aether theory with a negative cosmological constant is
where , with
This turns out to be the most general effective action for a timelike unit vector field at 2nd order in derivatives (a term proportional to Ricci tensor contracted with two aether vectors is a combination of the and terms above).
Varying this action with respect to the metric, vector field, and the Lagrange multiplier , one finds the following field equations
The aether stress tensor is given by
and . The aether field equation is
One can show that solutions of Einstein-aether theory where the aether is hypersurface orthogonal are also solutions of Hořava-Lifshitz gravity Jacobson:2010mx (). In this case the aether foliation defines a preferred time foliation in the spacetime. First note that when the aether field is hypersurface orthogonal, the twist
vanishes. One can show that
Since squared twist must also vanish, adding it to the action doesn’t affect the solutions. Due to the fact that this scalar twist is a combination of the , and terms, we can use the freedom to add to eliminate one of , or in the action. A convenient choice is to eliminate , therefore we take from this point forward.
Secondly, hypersurface orthogonality implies the co-vector is the gradient of a scalar
If we choose coordinates such that , where is the preferred foliation of time, then in this gauge the Einstein-aether action reduces to the generic form of the Hořava-Lifshitz action (e.g. Griffin:2012qx ())
Here is the extrinsic curvature of the preferred time slices, the spatial metric on the slices, the intrinsic Ricci scalar and is the lapse function, . The mapping between the constants is given by
In Einstein-aether theory there are five propagating degrees of freedom with spin-2, spin-1, and spin-0 helicities Jacobson:2004ts (). In Hořava-Lifshitz the spin-1 mode is non-propagating. The squared speeds of the remaining modes are given in general space dimension by Griffin:2012qx ()
Iii Lifshitz Asymptotic Solutions
We begin to characterize solutions by considering the following ansatz for the metric and aether in terms of Eddington-Finkelstein like coordinates
Here the aether field is hypersurface orthogonal (vanishing twist), so that solutions will satisfy both the Einstein-aether and Hořava-Lifshitz field equations. In practice, we will work with the Einstein-aether field equations. Plugging the ansatz into these equations leads to a complicated set of second order ordinary differential equations for the functions , , and . One can take solutions in this case and re-express them in a gauge where only has time component. In this frame, the solution will also satisfy the Hořava-Lifshitz field equations.
In holography, the coordinates are identified with the dual field theory, while is the bulk radial coordinate. Lifshitz scaling by definition is an anisotropic scaling of space and time in the field theory: . In the full bulk metric we require an invariance under this scaling, with . This leads to the asymptotic form as
When we have the standard AdS asymptotic metric. If we plug this generic form into the field equations, one finds the following consistency conditions for it to be a solution Griffin:2012qx (),
Asymptotically Lifshitz solutions were first studied in detail in Janiszewski:2014iaa () in four-dimensional spacetime () by expanding the field equations in a power series around . It was found that the behavior of solutions separates into two classes depending on if or . In the case there is a solution characterized by two free parameters and . The series solution for and truncates and it was ultimately shown that there is a full analytic solution for general given by
This solution depends only on . Note that for , .
In the case there appears to be no analytic solution for general . The behavior of asymptotic solutions was determined to be dependent on the analyticity properties of the solution ansatz. If we assume the functions , , and are of the form of their leading asymptotics (17) times an analytic function, then the resulting series solution depends generically only on one free parameter. On general grounds Eling:2006ec () there should be a two parameter family of solutions near infinity. This indicates this ansatz is a restrictive special case that does not reflect the general properties of the asymptotic solutions.
Instead Janiszewski:2014iaa () noted that one can allow for non-analytic behavior by imposing
Substituting into the field equations, it was found that
restricting attention to non-negative cases. In general is not an integer. In this way it was found that solutions are indeed characterized generically by two free parameters which are the coefficient of in the expansion for and the coefficient of in the expansion of . Note that in special cases where one can choose so that in (21) is an integer. In this particular case one has an analytic power series solution around infinity that depends on two free parameters.
Recently, Basu:2016vyz () studied asymptotic Lifshitz solutions for the three-dimensional case (). They imposed the following condition of analyticity on the power series solutions around infinity
where and is the timelike Killing vector field. In order to have a non-zero and finite ADM-like mass it was found that . The field equations imply that there is a restriction to a subset of solutions with particular . The relevant condition on the couplings is
where is an integer . The solution again depends on two free parameters. The advantage of these power series solutions is that one can define and read-off the total mass (per unit volume) of the black hole and then construct a First Law of thermodynamics based on the temperature and entropy of a universal horizon. However, in the following we will show in many cases that there are regular black holes with universal horizons that do not necessarily have analytic behavior near infinity.
Iv Regular Black Hole Solutions
Asymptotically Lifshitz solutions are generally characterized by two free parameters. However, there is no guarantee that these solutions are regular everywhere in the bulk interior. Singularities generically arise at the location of the horizon associated with the spin-0 mode Eling:2006ec (); Barausse:2011pu (). In the case, the speed of the spin-0 mode (14) is infinite, so the horizon in this case coincides with the universal horizon. The universal horizon is defined as the spacelike surface where is orthogonal to the aether, i.e. . For the global analytic solution, imposing regularity at the universal horizon amounts to one additional condition relating to , Janiszewski:2014iaa () i.e.
where is the location of the universal horizon. is proportional to the total mass per area of the black hole. Therefore one is left with parameter family of regular black hole solutions. An analogous solution in the case was later found in Sotiriou:2014gna ().
We now want to determine numerically whether other one parameter families of regular solutions exist when . Following Eling:2006ec (); Barausse:2011pu () the idea is to impose regularity at the spin-0 horizon and then integrate outwards to infinity, demanding the Lifshitz asymptotic conditions (17). Practically, to implement this we redefine
where and should approach a constant plus corrections near infinity, while goes to unity. Asymptotic values of and depend on a choice of initial conditions at the spin-0 horizon, which is a reflection of the scaling freedom in the time coordinate . Under this rescaling
but the combinations and are invariant. Thus asymptotically Lifshitz conditions are imposed universally by demanding these combinations of functions approach unity at infinity.
For simplicity, the next step is to fix so that the location of the spin-0 horizon coincides with the usual metric/Killing horizon in the bulk which is where . Fixing fixes and then this condition can be used to solve for
This choice can be made without loss of generality by making the following field redefinition of the metric and the aether for a theory with arbitrary
which maps into and has the effect of mapping arbitrary into .
To impose regularity at the horizon we consider a series solution ansatz
where is the location of the spin-0 horizon. Using the time rescaling freedom discussed above we can fix one value, say , at the horizon. Plugging into the field equations, we find that the values of , and are determined (in a very complicated form using Maple) in terms of and . Thus, there is a two parameter family of regular spin-0 horizon solutions in the Lifshitz case, which agrees with the earlier results in Eling:2006ec (). These local solutions of course will not generically be asymptotically Lifshitz.
V Asymptotically Lifshitz Black Holes for
To find regular Lifshitz black holes we use the shooting method. We fix a small value of and use the series solution above to find the initial radial values. Then we integrate the field equations, a set of ordinary differential equations, out to infinity. The initial values depend on and we tune this value until the combinations and approach unity at very large values of , say . If the procedure finds a solution we will have a fixed numerical value of , meaning that the asymptotic condition has eliminated an additional free parameter. The resulting solution will depend on one free parameter (similar to the Schwarzschild solution in GR, asymptotically flat Einstein-aether black holes in Eling:2006ec (); Barausse:2011pu () and finally the solution found in Janiszewski:2014iaa ()). To determine the solution in the interior, we then take the value of and use the series solution for small negative to provide the necessary initial data to integrate inwards.
v.1 Black Holes
First we study the three-dimensional Hořava-Lifshitz theory can focus first on the case. is likely the case of most relevance for physical systems since in field theory the symmetry group can be enhanced to include Galilean symmetries. For a range of up to about we find regular solutions. In the shooting process one can see that for a range of the values of and are for very large while for others they become . The tuning leads to the set of values displayed in Table 1 where the asymptotic conditions are met.
The behavior of the functions for is shown in Figure 1. We have chosen the standard normalization/gauge where and approach unity at infinity.
We performed a non-linear curve fit of this data to the asymptotic forms
While the solutions have a regular spin-0 horizon, for there to be a black hole in Hořava-Lifshitz gravity, modes of arbitrary speed must be trapped, so we require a universal horizon. A universal horizon is located where
vanishes. For is negative. A plot of versus in the interior is shown in Figure 2.
At , vanishes so there is indeed a universal horizon present. In addition, note that the function tends to oscillate, so additional interior universal horizons appear at for example and . This is the same type interior oscillation found in Eling:2006ec (); Barausse:2011pu () for asymptotically flat solutions in four-dimensions. To see this behavior more clearly, we plot the function versus in Figure 3.
Again this is qualitatively the same type of behavior noted previously in the asymptotically flat cases.
The only noticeable change in the functions with is that as its value increases, the size of the interior peak(s) in tends to decrease and the oscillation tends to smooth out. See Figure 4.
At and beyond no solutions appeared to exist generically. However, this particular value range may be an artifact of numerical errors or inaccuracies in our simple first order series solution around the horizon. Solutions may exist for somewhat larger values. Also for larger there may be special solutions present. In Basu:2016vyz (), a regular black hole solution was found numerically with , which satisfies (23) for . For this choice of , we indeed found a nearly regular solution, at a very fined tuned value of the shooting parameter. A singularity appeared at large , which seems likely to be due to numerical errors or an inaccuracy.
v.1.1 Cases- No Generic Solution?
Now we will consider cases with larger values of the dynamical exponent . First we examined . In this case for a range of the behavior of the solutions was different than in . As we tuned the value of the asymptotic values of and tended to remain very close to unity but always below it. For example, for for between 0.94 and 1.75 the value of varied from 0.9843 to 0.9705. Below the initial conditions become complex. There was never a case where the asymptotic values became greater than unity. We found the same type of behavior for and also . This indicates there are generically no regular asymptotically Lifshitz black holes for , but it would be useful to probe this regime in more detail.
Perhaps is another special value (like ) where solutions exist. To check this we considered the case where . Here we found again regular solutions for a range of , similar to the case. In Figure 5, we have plotted the solution , which shows similar behavior to the case (with less oscillation of )
v.2 Lifshitz Black Holes
Now we consider four-dimensional black hole solutions. The behavior in this case throughout is qualitatively similar to the three-dimensional case. First, for a range of (here up to about 0.4) we found regular solutions. An example is shown in Figure 6.
Performing a Non-linear curve fit of our data for to leads to a reasonable agreement between () and the value obtained in (21) ( here) required by the non-analytic expansion around infinity. In the interior, there are universal horizons associated with the same type of oscillations appearing in oscillations, and as increased the behavior of the functions was the same as in the case.
v.2.1 Cases- No Solution?
We again consider , following the same methods as in the three-dimensional case. As we tuned the value of the asymptotic values of and again tended to remain very close to unity but always below it. For example, for and for between 1.605 and 0.919 the value of varied from 0.999179 to 0.9995. Below the initial conditions become complex. We again conclude that no asymptotic Lifshitz solutions of this type exist.
Finally, we again found solutions for which are qualitatively similar to the solutions in . An example is plotted in Figure 7.
We found numerically a new class of asymptotically Lifshitz black hole solutions in three and four dimensional Hořava-Lifshitz gravity. This class of solutions is distinguished by the non-analytic behavior of the solution near infinity. In particular, for and , solutions exist for a range of . We also studied the properties of the solution inside the horizon and showed there is oscillatory behavior in the aether field, just as in the asymptotically flat case. Interestingly, for , there seem to be generically no solutions of this type, although particular solutions with special values of parameters are likely to exist.
In the future, it would be interesting to understand the nature of the non-analytic asymptotic solution in these cases. While a regular universal horizon exists, it is not clear whether one can define a corresponding ADM-like mass for these solutions that can used to study black hole thermodynamics, such as the First Law. Are there other distinguishing properties of solutions, such as stability or instability? From the point of view of holography, these black holes describe the thermal state of a dual field theory with . What does the nature of the asymptotic solution here tell us about the properties of the dual theory? Finally, for other holographic applications, these solutions (in four-dimensions) may also be useful as background one can perturb in the fluid-gravity setting, following Eling:2014saa (); Davison:2016auk (). For example, it would be interesting use these black holes to study the characteristics of fluid transport coefficients in cases where .
I would like to thank Jishnu Bhattacharyya and David Mattingly for valuable discussions. This research was supported by the European Research Council under the European Union’s Seventh Framework Programme (ERC Grant agreement 307955).
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