On the static Lovelock black holes
Abstract
We consider static spherically symmetric Lovelock black holes and generalize the dimensionally continued black holes in such a way that they asymptotically for large go over to the ddimensional Schwarzschild black hole in dS/AdS spacetime. This means that the master algebraic polynomial is not degenerate but instead its derivative is degenerate. This family of solutions contains an interesting class of pure Lovelock black holes which are the th order Lovelock vacuum solutions having the remarkable property that their thermodynamical parameters have the universal character in terms of the event horizon radius. This is in fact a characterizing property of pure Lovelock theories. We also demonstrate the universality of the asymptotic Einstein limit for the Lovelock black holes in general.
pacs:
04.50.h, 04.20.Jb, 04.70.s, 97.60.LfI Introduction
With a view to understand gravity in higher dimensions, there has been extensive work on various generalizations of General Relativity in higher
dimensional spacetimes. The modifications should however be consistent with the following general features: (a) general covariance  the
Lagrangian must be a scalar density constructed from the Riemann curvature which yields a nontrivial equation of motion, (b) the equivalence
principle and (c) the equation of motion to be second order quasilinear. This uniquely identifies the LanczosLovelock Lagrangian (LLgravity)
which is a homogeneous polynomial in the Riemann curvature with specific coefficients where zeroth, linear and quadratic orders respectively
correspond to the cosmological constant, EinsteinHilbert and GaussBonnet terms lovelock . It is pertinent here to note that the
GaussBonnet term also arises in the one loop correction of the lowenergy effective action in string theory string . Not only string
theory but to physically realize high energy effects of gravity within the classical framework also asks for higher dimensions and the inclusion
of such higher order terms in the gravitational action. It is important that this is a purely classical motivation for higher dimensions for the
consideration of high energy effects dad . The requirement (c) ensures the unique physical evolution for a given initial value problem.
Further it is interesting that the LanczosLovelock Lagrangian is also characterized by several different considerations which include the
derivation of the equation of motion by the Bianchi derivative Dadhich:2008df and the equivalence of the metric and Palatini formulation
for arbitrary connection by the LeviCivita consistent truncation Dadhich:2010 . The LLgravity is therefore the most natural
generalization of the Einstein gravity in the strong gravitational regime where higher order curvature terms may become important and represent high
energy corrections.
In this work, we shall employ the study of properties of the static spherically symmetric Lovelock black holes for understanding gravitational
dynamics in higher dimensions. Spherically symmetric black hole solutions were first discovered for GaussBonnet extension of GR in five
dimensions boul deser ; wiltshire . There is a very extensive body of work on black holes in general Lovelock theories beginning with the
three classic papers wheeler ; whitt ; MS and followed by LovelockBH . For static spherically symmetric vacuum solutions, the equation
ultimately reduces to an algebraic equation involving an th order polynomial (henceforth called the master equation/polynomial). The problem
then reduces to simply solving this polynomial. As is well known there exists no standard method to solve it for . One of the obvious
strategies could be to assume that the polynomial is degenerate so that it is trivially solved. This is precisely what is done for the
dimensionally continued black hole solutions BTZ ; CTZ . It has been motivated by the considerations of the extension of the Euler density
to the next higher dimension as well as the embedding of Lorentz group into the larger AdS group and above all a unique
value of . This would mean that all Lovelock coefficients are not independent but are given in terms of the single one, . Thus
dimensional continuity prescription is nothing but the complete degeneracy of the master polynomial. This is all very fine but the solution does
not asymptotically go over to the corresponding Einstein solution for large . By the corresponding Einstein solution, we mean the dimensional Schwarzschild black hole in dS/AdS spacetime. Henceforth this is what would be implied by the Einstein solution or limit. As the higher order Lovelock terms are supposed to give
correction to the Einstein gravity, it is therefore pertinent that the solution should have the right Einstein limit far away from the source.
The question is, could the assumption of complete degeneracy be modified such that the solution asymptotically approaches the Einstein solution?
The natural generalization of degeneracy is the derivative degeneracy; i.e. the master polynomial is not degenerate but its first derivative is
completely degenerate. That is, it could be written as which is also trivially solvable. The solution would then tend to
the degeneracy case of dimensionally continued black holes BTZ ; CTZ for where is the horizon radius while for large
it would tend to the corresponding Einstein solution. It reduces to the degeneracy case for and for it is the pure th
order Lovelock black hole solution with . It is the solution of the pure Lovelock vacuum equation with and we would
henceforth simply call it pure Lovelock black hole dad pure L . The driving consideration for the dimensionally continued black holes was
to have all the couplings given in terms of the unique value of so that the thermodynamical information could be easily extracted. The
pure Lovelock black hole also has only one parameter, and hence is as benign and interesting as the dimensionally continued one. In
addition it has the desired asymptotic limit as it goes over to the corresponding Einstein solution for large .
Besides this, the pure Lovelock black hole has the remarkable characterizing property that its thermodynamical parameters bear an universal relation to the horizon radius in the critical dimensionskpdlett . That is the thermodynamical parameters, temperature and entropy bear the same scaling relation with the horizon radius for odd () and even () dimensions irrespective of the Lovelock order . For instance, the entropy would always scale as and respectively for odd and even dimensions. The thermodynamics is therefore entirely insensitive or neutral to the Lovelock order. Not only this, the converse is also true. The universality of thermodynamics uniquely characterizes the pure Lovelock black hole. This is in line with the first universal feature of higher dimensional gravity discovered for the uniform density fluid sphere dmk . The Schwarzschild interior solution always describes the uniform density sphere for the Einstein as well as the EinsteinLovelock gravity.
Note that are the critical dimensions for the th order Lovelock gravity in the same sense as are for the Einstein gravity. The vacuum solution in dimension is trivially flat and it becomes nontrivial in dimension. This is true in general for in general for the th order analogue of the Riemann curvature (i.e. the th order vacuum is trivial in dimension). This is what has recently been shown odd and it has been motivated by the universality of thermodynamics of the pure Lovelock black holes. Thus are the critical dimensions having the similar behavior for the th order Lovelock gravity.
Further we would also show that all vacuum solutions with have the universal asymptotic behavior tending to the corresponding Einstein limit irrespective of whether it is pure Lovelock or EinsteinLovelock (summing over all Lovelock ) solution. In the latter case if there are repeated roots of the master polynomial (which means all ’s are not independent), there won’t exist the asymptotic Einstein limit for that root. The nondegenerate character of the polynomial is necessary for the asymptotic Einstein limit. In the case of pure Lovelock, the polynomial is required to be nondegenerate ensuring the proper Einstein limit.
The paper is organised as follows: Sec.II summarizes the LL theory which is followed in Sec.III by the discussion of the
static Lovelock black holes. Next we consider the thermodynamical universality in Sec.IV followed by the asymptotic limit of the
solutions in Sec.V. We conclude with a discussion.
Ii LanczosLovelock gravity
Consider the dimensional spacetime to be equipped with form vielbeins (such that the metric ) and a local form spin connection . The torsion and curvature are defined as:
(1a)  
(1b) 
where represent the Lorentz indices in a local orthonormal frame. In the first order formalism, the general LLgravity Lagrangian density can be simply written as:
(2) 
where are the couplings for the various terms, and denotes the order polynomial composed of dimensionally continued Euler densities:
(3) 
The maximum order polynomial that contributes in dimensions to the field equations is given by:
(4) 
Any term of order greater than is either zero or at best a topological invariant and hence does not contribute to the classical field equations. In what follows the limits on the sum are omitted, and assumed from to unless otherwise stated.
Written with only spacetime indices, we have the Lagragian ; , the EinsteinHilbert and the next quadratic GaussBonnet, . Restricting to the case of vanishing torsion , the variation of the action with respect to identically vanishes and variation with yields the field equations of the form:
(5) 
Written in terms of spacetime indices, we have the familiar terms
(6a)  
(6b)  
(6c) 
The exact solutions describing black holes have been found for the EinsteinGaussBonnet (EGB) as well as for the general Lovelock equations
(see boul deser whitt ). In what follows we shall consider the case of static spherically symmetric solutions in vacuum.
Iii Static Lovelock black holes
Let us consider the general static spherically symmetric metric to find the vacuum solution in LLgravity. All vacuum spacetimes satisfy the null energy condition, which means and that in turn implies . Thus the metric takes the form wheeler ; whitt :
(7) 
with
(8) 
The vacuum equation then reduces to solving the master algebraic equation,
(9) 
for . Here is the mass parameter of the solution. When , Eq.(9) gives the vacua of the theory as the zeros
(say ) of the master polynomial . Thus in general there will be as many vacua as there are zeros of the function .
These vacua can be antideSitter (AdS), Minkowski or deSitter (dS) depending on whether the particular is negative, zero or positive
respectively.
The vacua for which is not a simple zero behave differently. In this case, the field equations do not determine the gravitational potential, which remains completely arbitrary and free. Hence these were termed geometrically free solutions by Wheeler wheeler . Such solutions correspond to the metric:
(10) 
where is an undetermined arbitrary function and . Geometrically free vacua also arise in other situations in higher order
gravity theories dmk ; BH soln .
The LanczosLovelock Lagrangian Eq.(2) has a large number of arbitrary dimensionful parameters given by the independent ratios
of coupling constants . To reduce this arbitrariness in the theory, it becomes necessary to prescribe a relation among all the
LLcoefficients. One way is to restrict to the lowest order and the th order;i.e. pure Lovelock gravity with
dad pure L . The other is the case of the prescription of a unique giving the dimensionally continued black holes CTZ ; BTZ .
The general prescription for the Lovelock couplings to have a unique was given by Crisóstomo, Troncoso and Zanelli CTZ . Their choice of LLcouplings is given by:
(11) 
where is a length scale. In dimensions, this describes a family of gravity theories, labeled by the integer which represents the highest
power of curvature that appears in the Lagrangian. Another prescription due to Bañados, Teitelboim and Zanelli BTZ (hereafter referred to
as BTZcontinuation) is a special case of the above with , taking its maximum possible value. The
black hole solutions and their thermodynamic properties in these theories were discussed in CTZ ; BTZ .
All these cases could be knit together in the degeneracy character of the master equation, Eq.(9). The dimensionally continued black holes follow from the degeneracy of the master equation;viz
(12) 
while for the pure Lovelock black holes, it is
(13) 
where is the maximum order of the Lovelock term which contributes in dimensions, given by Eq.(4). Note that is up to a numerical factor the comological contant and we have set to unity the coefficient of the th Lovelock term. Hence the solutions of Eq. (9) will depend on the two parameters, and . Black hole solutions in such theories have been studied in dad pure L ; pure LL . Both these cases are synthesized in the generalization we consider which is the derivative degeneracy of the equation, and so we write
(14) 
The solution then takes the form
(15) 
Now gives the dimensionally continued black hole CTZ ; BTZ while the pure Lovelock black hole dad pure L ; pure LL .
The latter therefore requires the
derivative degeneracy which in general combines the two. It is clear that the former cannot go to the Einstein solution asymptotically because there is
nothing to expand around while the latter has the correct Einstein limit because it can be expanded around for large . Then we readily get which is the dimensional Schwarzschild black hole in dS/AdS spacetime. The presence of which acts
as the cosmological constant is therefore essential for existence of the correct Einstein limit asymptotically. It is the characterizing property of the
derivative degeneracy. At the high energy end for , it goes over to the dimensionally continued black hole solution while at the low energy
end for large to the corresponding Einstein solution. This is what is expected of the higher order terms in the Lagrangian that they should be
significant at the ultraviolet end while their effect should wean out at the infrared end.
Thus the derivative degeneracy generalization makes the black hole to have the right behavior at both high and low energy ends. For the pure Lovelock black hole with in Eq. (15), let us explicitly write the solution in odd and even dimensions as follows:
(16a)  
(16b) 
It is interesting to note that under the radical sign the potential due to the mass parameter is the same as that for the Einstein gravity in
and dimensions. This indicates that the potential is essentially th root of the Einstein potential in odd and even dimensions and it is this that makes the thermodynamics universal for the Lovelock black holes. That is what we take up in the next section.
There however remains the question of the unique fixing of the vacuum; i.e. the sign of . As has been argued in the appendix,
for the odd dimension while its sign for the even dimension is fixed by requiring the solution to go over to the corresponding Einstein solution
with positive mass. We would in the Sec.V consider the asymptotic limit of the pure Lovelock black holes which would require mass,
as given below in Eq.(27) to be positive. That fixes for the even dimension uniquely (for ). Thus there remains no more ambiguity about the unique realization of the vacuum.
Iv Thermodynamics and the Characterization of the pure Lovelock black holes
The causal structure and thermodynamical properties of general spherically symmetric solutions in LLgravity have been extensively studied (whitt ; caitherm ). The temperature and entropy can be easily computed using the methods given in jacmy ; caitherm and so we have
(17a)  
(17b) 
where is the radius of the horizon. The entropy is in general a polynomial series given in caitherm and does not simplify.
In the case of BTZcontinuation (), the temperature and entropy can be written as:
(18a)  
(18b) 
Note that in the above the temperature and entropy in the odd dimension have the universal relation to the horizon radius but not so for the
even dimension. In the odd dimension, gravitational potential due to mass is constant while that due to the cosmological constant has the
universal dependence and that is what defines the black hole in this case. Thus it is no surprise that thermodynamics has the universal
character in odd dimension. In contrast the BTZ continuation black holes in even dimension where mass also plays active role with potential
being dependent, they do not have universal thermodynamical behavior as is clear from the above expressions for temperature and entropy. We
shall now show the universality for the pure Lovelock black holes.
In the case of pureLovelock where , the thermodynamical quantities are given by
(19a)  
(19b) 
This demonstrates the universal thermodynamical behavior in terms of the event horizon radius. That is temperature and entropy always bear the same relation to the horizon radius. This is the remarkable property of the pure Lovelock black holes dad pure L . ( In odd dimension has to be negative for positive . So was the case for the famous BTZ black hole in dimension BTZ0 where which clearly indicated negative . Thus in odd dimension the cosmological constant has always to be negative;i.e. an AdS. In even dimension, the condition for the temperature to be positive is guaranteed by Eq. (30). ) Not only that the converse is also true, the universality uniquely characterizes the Lovelock black holes. That is what we show next.
Using the general formulae obtained in caitherm , it can be easily shown that pureLovelock theory is the unique one which has such universal thermodynamical behaviour. Explicity these are given by a series in terms of powers of the horizon radius as (
(20a)  
(20b) 
For universality of thermodynamics we would now demand that the temperature and entropy are always given in terms of the horizon radius as for the Einstein gravity in and dimensions. That is their horizon radius dependence is entirely free of the spacetime dimension and the Lovelock order. This means in the series for the black hole entropy , for we must have
(21) 
Thus and so the only terms that
contribute are and ; i.e. and the maximal order Lovelock. This is what characterizes the pure Lovelock black hole. This
proves the sufficient condition that the universality uniquely singles out the pure Lovelock gravity.
It should however be noted that this universality is exhibited when we express thermodynamics in terms of mass and the radius , which are the
natural black hole parameters. Instead had we written it all in terms of and , it would have been missed. It is therefore important to tag
on the right black hole parameters to study its thermodynamics.
Further like the BTZcontinuation black holes, the EinsteinGaussBonnet black hole also does not have this universal behavior. The temperature and entropy for the EinsteinGaussBonnet theory, , read as follows:
(22a)  
(22b) 
where is the GaussBonnet coupling parameter. Clearly thermodynamics in terms of the horizon radius is not universal. Here we have considered the GaussBonnet case but the same would be the case for any order which means the EinsteinLovelock black holes will not in general respect the thermodynamical universality.
Thus universality is the unique characterizing property of the pure Lovelock black holes. It is both necessary and sufficient condition.
V Asymptotic behavior
When the spacetime contains a mass source , near spatial infinity the spacetime tends to one of its vacua (from Eq.(9)). If has no real zero, then the spacetime does not contain any spatial infinity. The behaviour far away from the mass source, or equivalently, near spatial infinity, is given by the Taylor expansion of around the particular vacuum , (see whitt ):
(23) 
where .
Note that this is the asymptotic limit only when , i.e. a simple zero of . It represents a dimensional Einstein
black hole of mass in the deSitter spacetime with . Consequently, near spatial
infinity corresponding to a simple zero, every static spherically symmetric LLspacetime with the cosmological constant behaves like the familiar
Einstein one.
If is a degenrate zero of order, the expansion has to be carried down to this order,
(24) 
and the asymptotic limit would then be given by
(25) 
Clearly, this is not the Einstein limit unless . Thus, general LanczosLovelock spacetimes will not have asymptotic Einstein limit when the polynomial has degenerate zeros.
There could however be the case of both a simple zero and degenerate zero, for example
(26) 
In this case, is a simple zero, but is doublydegenerate. The vacuum solution will then have two
spatial infinities corresponding to each and . Expansion near the infinity corresponding to will show an Einsteinlike
behavior (as in Eq.(23)), but that near will not.
The special case of derivative degenerate theory given by Eq.(14) will give vacua at . These will
correspond to a simple zero iff and hence will always have Einsteinlike limit. The case corresponds to BTZcontinuation where there is a unique spatial infinity . This spatial infinity always
corresponds to a degenerate zero unless (viz. Einstein gravity in ). Thus degeneracy of the master polynomial (12) defines a unique spatial infinity but these solutions will not have the Einsteinlike asymptotic behavior unless
which is anyway the Einstein gravity.
For pureLovelock gravity, Eq.(13) gives and . Thus, has a single simple zero if and only if , and all + pureLovelock theories will have an asymptotic limit similar to the Einstein theory with the effective values:
(27a)  
(27b) 
Thus for all orders in the Lovelock polynomial, the asymptotic limit of the pureLovelock solution, is the same as the Einstein solution with
effective cosmological constant and mass given by Eq.(27). This generalizes the analysis and bears out the
expectation of Dadhich dad pure L that in presence of a cosmological constant the spherically symmetric vacuum solution in any theory
whether strictly pureLovelock, + only th order, or EinsteinLovelock, , always goes over to the
corresponding Einstein solution asymptotically. This is a universal asymptotic behavior of the LL vacuum solutions.
This general analysis shows why the cosmological constant is needed for this to work. If , then any order pureLovelock gravity will have a degenerate zero at . The solution (from Eq.(25)), has nonEinsteinlike asymptotic behavior:
(28) 
Thus the presence of a nonzero cosmological constant in a pureLovelock theory, prevents the occurrence of degenerate zeros and hence all
pureLovelock theories have asymptotic behavior like the Einstein theory.
Conclusion
There had been very extensive and rich literature on the Lovelock black holes and hence it is pertinent to ask what is it that this paper does which is
new and nontrivial? First of all it synthesizes the dimensionally continued and the pure Lovelock black holes into the derivative degeneracy property
of the master equation (9). The characterizing feature of this property is that it renders the black hole in contrast to the BTZ continuation have the right
limits at both ends, and large . This is what it should be from physical point of view because the higher order curvature effects should
wean out at low energy. For the latter limit, the presence of in Eq. (14) is essential which marks the derivative degeneracy so that the
potential could be expanded around it. The realization that at the root of these two family of solutions is simply the degeneracy character of the
master equation is new and rather novel. The pure Lovelock black holes or the black holes having proper Einstein limit asymptotically could be viewed
as physical realization of the the derivative degeneracy condition.
The most remarkable feature of the pure Lovelock black holes is their thermodynamical universality which means the temperature and entropy bear
the same relation to the event horizon radius as the Einstein black hole in odd and even dimension irrespective of the Lovelock order
. And the universality of thermodynamics uniquely identifies the pure Lovelock black holes kpdlett . Thus the necessary and sufficient
condition for the thermodynamical universality of a black hole is that it be a pure Lovelock black hole. Further it is interesting to note that
the potential inside the radical in Eq. (16) for odd/even dimension has the same form as the Einstein potential for dimension. This is
what makes the thermodynamics universal and more importantly it signifies that the gravity in dimensions for the
th order Lovelock has a kind of universal character. This is a remarkable realization which has been established firmly and explicitly in odd .
The universality of the asymptotic behavior imitating the Einstein gravity as it was envisaged in dad pure L has been established in general for the pure Lovelock as well as for the EinsteinLovelock for any order. For the latter it was verified earlier only upto while our analysis extends this to all orders. It is remarkable that the derivative degeneracy beautifully imbibes the asymptotic Einstein limit and also brings out the essential role of the nonzero .
Further very recently the universality of the Lovelock vacuum in the odd critical dimension has been established odd . As vacuum is trivial in dimension for the Einstein gravity, similarly the same is also true for higher order pure Lovelock gravity for the corresponding higher order Riemann analogue. That is, the static vacuum solution in dimension has vanishing th order analogue of Riemann. However it is not Riemann flat.
We conclude by reiterating that the derivative degeneracy leads to an interesting synthesis of the BTZ continuation and pure Lovelock black holes and it renders the black hole to have proper right limits at both the ends, small and large . The realization of the thermodynamical universality of pure Lovelock black holes is very enlightening and insightful for understanding gravitational dynamics in higher dimensions. It would be interesting to see whether the universality bears out even when entropy is computed by quantum calculations. We believe that it should hold true because it is computed by integrating the First Law of black hole thermodynamics which should transcend to the quantum computations as well.
Appendix
Here we study some features of eq. (16) for even () and odd (), with .
v.1 , even
For the background to have physical sense, must be positive at the maximum an upper limit for the mass of the BH, , which implies for given
(29) 
The horizon of the BH is located at a value for the radial coordinate, for which . In terms of the BH parameters, the allowed values are
(30) 
From onwards, decreases until it reaches again the zero value ar , which is a coordinate singularity similar to the one present in the de Sitter background in static coordinates (see for instance BD ).
v.2 , odd
To makes sense of the metric, we need . There is only the coordinate singularity at 19) that precisely is required for a positive temperature. We conclude that this case must be excluded. but no BH horizon. Notice from equation (
v.3 , even
We write . For small, , is large and negative. It increases with increasing and its vanishing defines the BH horizon. From odd to keep real. Thus is . onwards we will need
v.4 , odd
For small the quantity is positive. If there will be no BH horizon. If there is a BH horizon at odd to keep real. This means that . onwards we will need . From
Note from the analysis above that for one always needs to be odd, in agreement with previous findings in section V (see (27)).
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