Thermodynamics of Einstein-Proca AdS black holes

Abstract

We study static spherically-symmetric solutions of the Einstein-Proca equations in the presence of a negative cosmological constant. We show that the theory admits solutions describing both black holes and also solitons in an asymptotically AdS background. Interesting subtleties can arise in the computation of the mass of the solutions and also in the derivation of the first law of thermodynamics. We make use of holographic renormalisation in order to calculate the mass, even in cases where the solutions have a rather slow approach to the asymptotic AdS geometry. By using the procedure developed by Wald, we derive the first law of thermodynamics for the black hole and soliton solutions. This includes a non-trivial contribution associated with the Proca "charge". The solutions cannot be found analytically, and so we make use of numerical integration techniques to demonstrate their existence.

Published for SISSA by Springer

Received: March 6, 2014

Revised: April 23, 2014 Accepted: May 16, 2014 Published: June 18, 2014

Thermodynamics of Einstein-Proca AdS black holes

Hai-Shan Liu,a H. Lb and C.N. Popec,d

aInstitute for Advanced Physics & Mathematics, Zhejiang University of Technology, Hangzhou 310023, China

bDepartment of Physics, Beijing Normal University,

Beijing 100875, China

cGeorge P. & Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX 77843, U.S.A.

dDAMTP, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge CB3 OWA, U.K.

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We study static spherically-symmetric solutions of the Einstein-Proca equations in the presence of a negative cosmological constant. We show that the theory admits solutions describing both black holes and also solitons in an asymptotically AdS background. Interesting subtleties can arise in the computation of the mass of the solutions and also in the derivation of the rst law of thermodynamics. We make use of holographic renormalisation in order to calculate the mass, even in cases where the solutions have a rather slow approach to the asymptotic AdS geometry. By using the procedure developed by Wald, we derive the rst law of thermodynamics for the black hole and soliton solutions. This includes a non-trivial contribution associated with the Proca charge. The solutions cannot be found analytically, and so we make use of numerical integration techniques to demonstrate their existence.

Keywords: Black Holes, AdS-CFT Correspondence, Classical Theories of Gravity

ArXiv ePrint: 1402.5153

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP06(2014)109

Web End =10.1007/JHEP06(2014)109

JHEP06(2014)109

Contents

1 Introduction 1

2 Einstein-Proca AdS black holes 5

3 Holographic energy 8

4 Thermodynamics from the Wald formalism 11

5 Solutions outside 0 < 1 14

6 Numerical results 196.1 Solitonic solutions 206.2 Black hole solutions 23

7 Conclusions 26

A Proca solutions in AdSn, and the Breitenlohner-Freedman bound 27

1 Introduction

There has been a resurgence of interest recently in constructing black holes in a variety of theories that, in one way or another, are more general than those typically considered heretofore. One particular aspect that is now receiving considerable attention is the study of black holes in theories admitting ant-de Sitter (AdS) rather than Minkowski backgrounds, since asymptotically AdS solutions play a central role in the AdS/CFT correspondence [13].

Higher-derivative theories of gravity provide a fertile ground for constructing asymptotically AdS solutions, but they typically su er from the drawback that the concomitant massive spin-2 modes have the wrong sign for their kinetic terms in a linearised analysis around an AdS background, and thus the theories are, in general, intrinsically plagued by ghosts. Nevertheless, as a framework for purely classical investigations, they can provide interesting starting points for the study of black-hole solutions and their dynamics. For example, in some recent work on the existence of black hole solutions in Einstein-Weyl gravity with a cosmological constant, it was found through numerical studies that asymptotically-AdS black holes can arise whenever the mass-squared m22 of the massive spin-2 mode is negative [6]. In general one may expect a negative mass-squared to exhibit another undesirable feature, namely the occurrence of tachyonic run-away instabilities that grow exponentially as a function of time. However, in anti-de Sitter backgrounds there is

1

JHEP06(2014)109

a window of negative mass-squared values m22BF m22 < 0, where the negative mass-

squared m22BF is the limiting value of the so-called Breitenlohner-Freedman bound [4, 5], above which the linearised modes in the AdS background still have oscillatory rather than real exponential time dependence, and thus the run-away behaviour is avoided.

In this paper, we shall study a di erent type of theory where again certain modes of non-tachyonic negative mass-squared play a central role, but now in the framework of a more conventional two-derivative theory that has no accompanying ghost problems. Specifically, we shall focus on the n-dimensional Einstein-Proca theory of a massive spin-1 eld coupled to Einstein gravity, in the presence also of a (negative) cosmological constant.1 This theory exhibits many features that are similar to those of a higher-derivative theory of gravity, with the massive Proca eld now playing the rle of the massive spin-2 mode. The solutions we study approach anti-de Sitter spacetime at large distance, with R

(n 1)[lscript]2 g as the radius tends to innity. We nd two distinct kinds of spherically-

symmetric static solutions, namely asymptotically AdS black holes, and smooth asymptotically AdS solitons. The metrics for both classes of solution take the form

ds2 = h(r) dt2 +

dr2f(r) + r2 d 2n2 . (1.1)

It is useful rst to consider the situation where the mass is small and the Proca eld is weak, so that its back-reaction on the spacetime geometry can be neglected. In this limit, the large-distance behaviour of the Proca potential A = (r) dt approaches that of a massive vector in AdS, which takes the form

(r) !

1 r(n3)/2

JHEP06(2014)109

Xp=0bpr2p +1 r(n3+)/2

1

Xp=0~bpr2p , (1.2)

1

where

p4 ~m2 [lscript]2 + (n 3)2 , (1.3)

and ~m is the mass of the Proca eld. The Breitenlohner-Freedman window of non-tachyonic negative mass-squared values lies in the range m2BF ~m2 < 0, with

m2BF =

=

14[lscript]2 (n 3)2, (1.4)

thus ensuring that remains real. The sums in (1.2) can actually be expressed in closed forms as hypergeometric functions (see appendix A).

It follows from the equations of motion that the back-reaction of the Proca eld on the geometry will rst appear in the metric functions h and f at order 1/rn3, together with associated higher inverse powers of r. The back-reacted geometry will in turn modify

1Theories with a massive vector eld coupled to gravity been considered extensively in discussions of holographic descriptions of non-relativistic boundary theories, arising from gravity duals where the space-time approaches a Lifshitz or Schrdinger geometry. See, for example, [79]. In these cases the mass-squared of the massive vector eld is taken to be positive.

2

the Proca solution, with the onset beginning at order 1/r(3n53)/2. Thus at large r the

Proca and metric functions will include terms of the general form

= q1

r(n3)/2 +

q2

r(n3+)/2 +

a1 q31 r(3n53)/2 + [notdef] [notdef] [notdef] ,

h = r2 [lscript]2 + 1 + m2 rn3 +

a2 q21 rn3 [notdef] [notdef] [notdef] ,

a3 q21rn3 [notdef] [notdef] [notdef] . (1.5)

The terms with coe cients m2 and n2 in h and f are associated with the mass of the black hole or soliton.

The ellipses denote all the remaining terms in the large-r expansions. These will include the direct descendants of the q1/r(n3)/2 and q2/r(n3+)/2 terms, at orders 1/r(n3)/2+2p and 1/r(n3+)/2+2p for all integers p 1, as in (1.2), and also higher-

order back-reaction terms and descendants of these. Depending on the value of the index , dened by (1.3), some of these remaining terms may intermingle with orders already displayed in (1.5), or they may all be at higher orders than the displayed terms. The discussion of the asymptotic forms of the solutions can therefore become quite involved in general. It will be convenient for some of our calculations to focus the cases where is su ciently small that the displayed terms in (1.5) are in fact the leading order ones, and all the terms represented by the ellipses are of higher order than the displayed ones. This will certainly be the case, in all dimensions n 4, if we choose

1 . (1.6)

This allows us to investigate the asymptotic structure of the solutions systematically for a non-trivial range of Proca masses corresponding to 0 1, near to the Breitenlohner-

Freedman bound.

In fact the special case when = 1 is particularly nice, since then the characterisation of inverse powers of r that arise in the asymptotic expansions of the Proca and metric functions is very simple. In this special case the leading-order term in the expansion of

(r) is 1/r(n4)/2, with each successive term having one extra power of 1/r. The terms in the metric functions occur always at integer powers of 1/r:

(r) = 1

r(n4)/2

1

f = r2 [lscript]2 + 1 + n2 rn3 +

JHEP06(2014)109

Xp=0qp+1 rp ,

h(r) = r2 [lscript]2 + 1 + 1 rn4

1

Xp=0np+1rp . (1.7)

Accordingly, we rst study the asymptotic expansions in this special case with = 1, where it is rather straightforward to see how one can systematically solve, order by order in powers of 1/r, for all the coe cients in terms of q1, q2 and m2. We then look at the leading orders in the expansions for the general case 1, su cient for our subsequent

purposes of computing the physical mass and deriving the rst law of thermodynamics.

3

Xp=0mp+1rp , f(r) = r2 [lscript]2 + 1 +1 rn4

1

We also study some isolated examples with > 1, showing that even though the structure of the inverse powers of r in the asymptotic expansions can then be rather involved, the solutions can still be found in these cases.

We then turn to the problem of calculating the physical mass of the solutions in terms of the free adjustable parameters of the asymptotic expansions, which can be taken to be q1,

q2 and m2. The calculation turns out to be somewhat non-trivial, because of the presence of the terms in the metric functions h and f at order 1/rn3, which represents a slower fall-o than the usual 1/rn3 of a normal mass term in n dimensions. Indeed, we nd that a naive calculation using the prescription of Ashtekar, Magnon and Das (AMD) [10, 11], in which a certain electric component of the Weyl tensor in a conformally-related metric is integrated over the boundary (n 2)-sphere, leads to a divergent result. Presumably

a more careful analysis, taking into account boundary contributions that can normally be neglected, may give rise to a nite and meaningful result. In the present paper we opt instead for calculating the mass using the method of the holographic stress tensor, and thereby obtain a well-dened nite result.

Having obtained an expression for the physical mass in terms of the parameters characterising the asymptotic form of the solution we then study the rst law of thermodynamics, using the methods developed by Wald. We nd that the Proca eld makes a non-trivial contribution in the rst law, which, for the static solutions we consider in this paper, will now take the form

4 q1 dq2 , (1.8) where !n2 is the volume of the unit (n2)-sphere. The way in which the Proca eld con

tributes to the rst law is analogous to a phenomenon that has been encountered recently when studying the thermodynamics of dyonic black holes in certain gauged supergravi-ties, where parameters characterising the asymptotic behaviour of a scalar eld enter in the rst law, thus indicating the presence of a scalar charge or hair [12]. (See [13] for higher-dimensional generalisations.) In the present case, one can think of the asymptotic parameters q2 and q1 as being like a thermodynamic conjugate charge and potential pair characterising Proca hair.

The equations of motion for the Einstein-Proca theory appear not to be exactly solvable even in the static spherically-symmetric situation that we study in this paper. We therefore turn to a numerical analysis in order to establish more explicitly the nature of the solutions to the theory. We do this by rst developing small-distance series expansions for the metric and Proca functions, and then using these to set initial data for a numerical integration out to large distance. We nd two di erent kinds of regular short-distance behaviour. In one type, we integrate out from a black-hole horizon located at some radius r0 > 0 at which the

Proca and metric functions all vanish. In the other type, we start from a smooth coordinate origin at r = 0, where the Proca and metric functions all begin with non-vanishing constant values. In each of these types of solution, the numerical integration indicates that the elds stably approach the expected asymptotic forms we discussed above, thus lending condence to the idea that such well-behaved black hole and soliton solutions do indeed exist. By matching the numerical solutions to the expansions (1.5) in the asymptotic region, we can

4

JHEP06(2014)109

dM = T dS

!n2

relate the mass and Proca charge to black-hole area and surface gravity, and thereby obtain numerical conrmation of the rst law (1.8).

The organisation of the paper is as follows. In section 2 we present the Lagrangian and equations of motion for the Einstein-Proca theory, and the consequent ordinary di erential equations that are satised by the Proca potential and the metric functions in the static spherically-symmetric ansatz. We then examine the asymptotic forms of the solutions that arise when both of the parameters characterising the Proca eld are turned on. In section 3, we show how the holographic renormalisation procedure may be used to calculate the mass of the solutions. In section 4, we use the formalism developed by Wald in order to derive the rst law of thermodynamics for the black hole and soliton solutions. We rst obtain results for values of the parameter, dened in (1.3), lying in the range 0 < 1. In section 5

we extend these calculations to values of outside the 0 < 1 range, showing how some

new features can now arise. For example, we nd that at certain values of , the conuence of generically-distinct inverse powers of r can lead to the occurrence of logarithmic radial coordinate dependence in the solutions, which can then require new kinds of counterterm in order to cancel divergences. In section 6, we carry out some numerical studies, in order to see how the asymptotic forms of the solutions we studied so far match onto the short-distance forms that arise either near the horizon, in the case of black holes, or near the origin, in the case of solitons. The paper ends with conclusions in section 7. Some details of the exact static and spherically-symmetric solutions of the Proca equation in AdS are given in an appendix.

2 Einstein-Proca AdS black holes

We shall study black hole and soliton solutions in the n-dimensional Einstein-Proca theory of a massive vector eld coupled to gravity, together with a cosmological constant. The Lagrangian, viewed as an n-form in n dimensions, is given by

L = R 1l + (n 1)(n 2)[lscript]2 1l 2 F ^ F 2 ~m2 A ^ A . (2.1)

This gives rise to the equations of motion

E R 2 F 2 12(n 2)

F 2 g

[parenrightbigg] 2 ~m2 AA + (n 1)[lscript]2 g = 0 , (2.2)

d F = (1)n ~m2 A . (2.3)

We shall consider spherically-symmetric solutions of the Einstein-Proca system, described by the ansatz

ds2 = h(r)dt2 +

dr2f(r) + r2d 2n2 , A = (r)dt , (2.4)

5

JHEP06(2014)109

where d 2n2 is the metric of the unit (n2)-sphere. The Ricci tensor of the metric in (2.4)

is given by

Rtt = hf

h[prime][prime]2h

h[prime]24h2 +h[prime]f[prime]4hf +(n 2)h[prime] 2rh

,

Rrr =

h[prime][prime]

2h +

h[prime]2

4h2

h[prime]f[prime]

4hf

(n 2)f[prime]

2rf ,

Rij =

(n 3)

r(hf)[prime]2h (n 3)f

ij , (2.5)

whereij is the metric of the unit (n 2)-sphere. The equations of motion following

from (2.2) and (2.3) can then be written as

2 = n2

4 ~m2r (fh[prime]hf[prime]) , (EttErr = 0) ,

r(hf)[prime]

JHEP06(2014)109

2h

2r2f [prime]2

(n2)h

+(n3)(1f)+(n1)[lscript]2 r2 =0 , (Eij = 0) ,

phf rn2

rf

h [prime][parenrightbigg][prime]= ~m2 , (Proca eom) . (2.6)

The remaining equation, which may be taken to be Ett = 0, is in consequence automatically satised.

We shall be interested in studying two kinds of solutions of these equations, namely black holes and solitons. In the black hole solutions the functions h(r) and f(r) will both vanish on the horizon. The Proca equation implies that the potential will vanish on the horizon also. In the solitonic solutions the radial coordinate runs all the way down to r = 0, which behaves as the origin of spherical polar coordinates. The functions h, f and all approach constants at r = 0.

It is not hard to see, by constructing power-series solutions of the equations near innity, and also in the vicinity of a putative horizon at r = r0, where one assumes that h(r0) = 0 and f(r0) = 0, that black holes could be expected to arise. Of course these series expansions do not settle the question of precisely how the interior and exterior solutions join together. This can be studied by means of a numerical integration of the equations, and we shall discuss this in greater detail in section 6. For now, we just remark that the numerical analysis indeed conrms the existence of black-hole solutions.

Our immediate interest is in studying the asymptotic behaviour of the solutions, with a view to seeing how to calculate the mass of the black holes. This will also be relevant for studying the thermodynamics of the solutions. As we discussed in the introduction, a rather straightforward case arises if the mass ~m of the Proc eld is chosen so that the index , dened by (1.3), is equal to 1. This means that the inverse powers of r in the asymptotic expansions of the Proca and metric functions take the simple form given in (1.7). It is achieved by taking the mass of the Proca eld to be given by

~m2 =

14(n 2)(n 4)[lscript]2. (2.7)

6

rn2

One can now systematically solve for the coe cients in the expansions (1.7) in terms of the free parameters q1, q2 and m2. We nd that the leading coe cients are given by

n1 = 12m1 =

n 4

n 2

q21 , n2 = m2

2(n 4)

n 1

q1q2 . (2.8)

At the next two orders, we nd

q3 = 18(n 2)(n 4)[lscript]2 q1 , n3 = q22 +

14(n 2)(n 4)[lscript]2 q21 ,

m3 = 2(n 2) q22

n +

(n 4)(n2 6n + 10) [lscript]2 q21

2(n 2)

,

JHEP06(2014)109

13(n 1)(n 4)[lscript]2 q1q2 ,

m4 = (n 4)(2n3 8n2 + 7n + 11)[lscript]2 q1q2

3(n2 1)

q4 = 1

24(n 2)(n 4)[lscript]2 q2 , n4 =

. (2.9)

It is straightforward to continue the process of solving for the further coe cients to any desired order. In fact, for the purposes of computing the mass and deriving the rst law of thermodynamics, it turns out to be unnecessary to go beyond the orders give in (2.8).

It should be noted that the specic choice (2.7) gives a rather natural higher-dimensional generalisation of the ordinary massless Einstein-Maxwell system in four dimensions. Setting n = 4 we have ~m = 0 and = 0, and the Maxwell potential A = dt is simply given by

= q1 + q2r . (2.10)

Thus in this case q2 is the ordinary electric charge of the Reissner-Nordstrm AdS black hole in four dimensions, and q1 represents an arbitrary constant shift in the gauge potential.

In the more general case where the Proca mass is not xed to the special value (2.7) that implies = 1, but is instead allowed to lie anywhere in the range that corresponds to 1 (see (1.3)), the leading orders in the asymptotic expansions take the form given

in (1.5). Specically, writing

= q1

r(n3)/2 +

q2

r(n3+)/2 + [notdef] [notdef] [notdef] ,

h = r2 [lscript]2 + 1 + m1 rn3 +

m2rn3 [notdef] [notdef] [notdef] ,

f = r2 [lscript]2 + 1 + n1 rn3 +

n2rn3 + [notdef] [notdef] [notdef] , (2.11)

then substituting into the equations of motion we nd

n1 = n 3 n 2

q21 , m1 = 2(n 3 )

n 1

q21 ,

n2 = m2

2(n 3 )(n 3 + )

(n 1)(n 2)

q1q2 . (2.12)

One can continue solving for higher coe cients to any desired order in 1/r. Unlike the = 1 case we discussed previously, this will in general be a somewhat less neatly ordered process, because of the intermingling of powers of 1/r from di erent sources. However, it

7

turns out for our present purposes, of calculating the mass and deriving the rst law of thermodynamics, that the coe cients given in (2.12) are su cient. Note that they reduce to those given in (2.8) in the case that = 1.

The free parameter m2 can be thought of as the mass parameter of the black hole or soliton; it is associated with the r(n3) fall-o in the metric coe cients h(r) and f(r). However, it is not itself directly proportional to the physical mass of the object. In fact, it can be seen from the expansions (2.11) that there is the potentially troubling feature that, unlike in a normal asymptotically-AdS black hole, here the metric functions have terms with a slower asymptotic fall o than the mass terms, namely the m1/rn3 and n1/rn3 terms. Naively, these might be expected to give rise to an innite result for the physical mass.

A simple and usually reliable way to calculate the mass of an asymptotically-AdS black hole is by means of the AMD procedure devised by Ashtekar, Magnon and Das [10, 11]. This involves integrating a certain electric component of the Weyl tensor over the sphere at innity in an appropriate conformal rescaling of the metric. If we naively apply the AMD procedure to the solutions described above, we indeed obtain an innite result for the physical mass. Rather than pursuing this further here, we shall instead use a di erent approach to calculating the physical mass, using the technique of the holographic stress tensor. This forms the subject of the next section.

3 Holographic energy

One way to calculate the mass of an asymptotically-AdS metric is by constructing the renormalised holographic stress tensor, via the AdS/CFT correspondence [1417]. Thus one adds to the bulk Lagrangian given in (2.1) the standard Gibbons-Hawking surface term and the necessary holographic counterterms. At this stage it is more convenient to write the extra Lagrangian terms as scalar densities rather than as n-forms. Thus in the gravitational sector we have [1618]

Lbulk = 1 16Gpg

R F F 2 ~m2 AA (n 1)[lscript]2 [bracketrightbig]

, (3.1)

JHEP06(2014)109

18Gph K , (3.2)

Lct = 1 16Gph [bracketleftbigg]

2(n 2)

Lsurf =

[lscript] +

[lscript](n 3) R

+ b2 [lscript]3

R R (n 1)4(n 2) R2[parenrightbigg]

b3 [lscript]5[parenleftbigg]

(3n 1)

4(n 2) RR R

(n2 1)16(n 2)2 R3

2R R R

+ (n 3)

2(n 2) R rr R R

R +

1

2(n 2) R

R

[parenrightbigg]

+ [notdef] [notdef] [notdef]

[bracketrightbigg]

,

(3.3)

where K = h K is the trace of the second fundamental form K = r(n ), R

and its contractions denote curvatures in the boundary metric h = g nn , and

b2 = 1

(n 5)(n 3)2

, b3 = 2

(n 7)(n 5)(n 3)3

. (3.4)

8

The ellipses in (3.3) denote terms of higher order in curvature or derivatives, which are only needed in dimensions n > 9. The expressions in (3.3) should only be included when they yield divergent counterterms. This means that the terms with coe cient c2 should be included only in dimensions n > 5, and those with coe cient c3 should be included only in dimensions n > 7.

We should also include counterterms for the Proca eld. There is, furthermore, an option also to add a boundary term for the Proca eld, analogous to the Gibbons-Hawking term Lsurf for the gravitational eld. Thus we can take

LAsurf =

8G ph nF A . (3.5)

In the case of Dirichlet boundary conditions, where the value of the potential is xed on the boundary, the coe cient would be taken to be zero. For the counterterms, it turns out that for most of the solutions we shall be interested in we may simply take

LAct =

e116G[lscript] ph A A . (3.6)

We have left the constants and e1 arbitrary for now. It will turn out that one (dimension-dependent) linear combination of and e1 will be determined by the requirement of removing a divergence in the expression for the energy. The remaining combination then represents an ambiguity in the denition of the energy, corresponding to the freedom to perform a Legendre transformation to a di erent energy variable.

The variation of the surface and counterterms with respect to the boundary metric h

gives the energy-momentum tensor T of the dual theory, with T = (2/ph) I/ h .

In our case, since we just wish to compute the energy, and since our metrics are spherically-symmetric and static, many of the terms that come from the variations of the counterterms turn out to vanish. In particular, when we compute T00 the only surviving contributions from the variations of the quadratic and cubic curvature terms will be those coming from the variation of ph. This greatly simplies the calculations. The upshot is that we

may write

T = 1

8G

K Kh (n 2)[lscript]1 h +[lscript]

n 3

R 1 2Rh [parenrightbigg]

JHEP06(2014)109

1

2b2 [lscript]3[parenleftbigg]R R

4(n2) R2[parenrightbigg]

h + 12b3 [lscript]5[parenleftbigg]

(n1)

(3n1)

4(n2) RR R

(n21) 16(n2)2 R3

2R R R +

(n 3)

2(n 2) R rr R R

R +

1

2(n 2) R

R

h

+ n F A 12e1[lscript]1 A A [parenrightbigg]

h 2 nF( A ) + e1 [lscript]1 A A + [notdef] [notdef] [notdef] [bracketrightbigg]

, (3.7)

where the ellipses denote terms that will not contribute to T00 for our solutions, and terms that are needed in dimensions n > 9. The holographic mass is obtained by integrating T00

over the volume of the (n 2) sphere that forms the spatial boundary of the boundary

metric. The boundary metric for our solutions, and the normal vector to the boundary, are given simply by

h dxdx = h dt2 + r2 d 2n2 , n@ =

pf @

@r . (3.8)

9

We are now ready to insert the asymptotic expansions that we discussed earlier. We can use either (1.7), in the special case = 1, or more generally (2.11) for all the cases with 1. It turns out that the terms displayed in (2.11) are su cient for the purpose,

with the coe cients m1, n1 and n2 given by (2.12).

Substituting into (3.7), we nd that the counterterms subtract out all divergences, provided that we impose the relation

e1 = (n 3 )(1 ) (3.9)

on the two coe cients and e1 associated with the counterterms for the Proca eld. We then nd that the masses of the black holes in dimensions 5 n 9 are given by2

n = 5 : M = 3

8

m2 +

23 +16(2 )(6 + )

JHEP06(2014)109

q1q2 + 14[lscript]2[bracketrightbigg]

,

n = 6 : M = 2

3

m2 +

12 +110(3 )(8 + )

q1q2[bracketrightbigg]

,

n = 7 : M = 52

16

m2 +

25 +115(4 )(10 + )

q1q2 1 8[lscript]4[bracketrightbigg]

,

n = 8 : M = 22

5

m2 +

13 +121(5 )(12 + )

q1q2[bracketrightbigg]

,

q1q2 + 564[lscript]6[bracketrightbigg]

. (3.11)

Note that the term proportional to [lscript]n3 in each odd dimension is the Casimir energy. It would be natural to omit this if one wants to view the mass as simply that of a classical black hole.

Although the general expressions for the counterterms at the quartic or higher order in curvatures are not readily available, we can in fact easily calculate the holographic mass for the static spherically-symmetric solutions in any dimension. Any invariant constructed from p powers of the curvature R of the boundary metric h will necessarily just be

a pure dimensionless number times r2p, and so the contributions T gct00 to T00 coming from the gravitational counterterms to all orders can simply be written as

T gct00 = 1

8G [lscript] h00

1

n = 9 : M = 73

48

m2 +

27 +128(6 )(14 + )

Xp=0cp [lscript]2pr2p . (3.12)

The constants cp are then uniquely determined by the requirement of removing all divergences in the holographic expression for the mass. Together with the contributions from

2Our convention for the denition of mass in dimension n is such that an ordinary AdS-Schwarzschild black hole, whose metric is given by (2.4) with h = f = r2 [lscript]2 + 1 2mr3n, has mass M = (n 2)

m!n2/(8), where

!n2 =

2(n1)/2

[(n 1)/2]

(3.10)

is the volume of the unity (n 2)-sphere.

10

the surface term and the counterterms for the Proca eld, the complete expression for T00

for our metrics is given by

T00 = 1 8G

(n 2)hpf

r

1

pf [prime] + 12e1 [lscript]1 2 + h [lscript]1

Xp=0cp [lscript]2p r2p

[bracketrightbigg]

. (3.13)

From this, we can calculate the mass of the Einstein-Proca black holes in arbitrary dimensions, nding in n dimensions

M = (n 2) !n216

[bracketleftbig]

m2 + (b1 + b2 )q1q2

[bracketrightbig]

+ ECasimirn , (3.14)

JHEP06(2014)109

where e1 = (n 4)(1 ) and

b1 = 2(n 3 )(2n 4 + )

(n 1)(n 2)

, b2 = 2 n 2

. (3.15)

Note that if we write the energy in terms of the metric parameter n2 rather than m2, we obtain the simpler expression

M = (n 2) !n216

[bracketleftbigg]

n2 + 2(n 3 ) + 2

(n 2)

q1q2

[bracketrightbigg]

. (3.16)

The Casimir energies are zero for even n, while for odd n we nd

ECasimirn = (1)(n1)/2 (n3)/2 [lscript]n3 (n 1)(n 2) (n 4)!!

2(n+3)/2 [(n 1)/2]!

2 . (3.17)

4 Thermodynamics from the Wald formalism

Wald has developed a procedure for deriving the rst law of thermodynamics by calculating the variation of a Hamiltonian derived from a conserved Noether current. The general procedure was developed in [19, 20]. Its application in Einstein-Maxwell theory can be found in [21]. Starting from a Lagrangian L, its variation under a general variation of the

elds can be written as

L = e.o.m. + pg rJ, (4.1) where e.o.m. denotes terms proportional to the equations of motion for the elds. For the theory described by (2.1), J is given by

J = gg (r g r g ) 4F A . (4.2)

From this one can dene a 1-form J(1) = Jdx and its Hodge dual

(n1) = (1)n+1 J(1) = grav(n1) + A(n1) , A(n1) = 4(1)n F ^ A . (4.3)

We now specialise to a variation that is induced by an innitesimal di eomorphism x = . One can show that

J(n1) (n1) i L0 = e.o.m. d J(2) , (4.4)

11

where i denotes a contraction of on the rst index of the n-form L0, andJ(2) = d(1) 4(iA)F , (4.5)

where (1) = dx. One can thus dene an (n 2)-form Q(n2) J(2), such that

J(n1) = dQ(n2). Note that we use the subscript notation (p) to denote a p-form.

To make contact with the rst law of black hole thermodynamics, we take to be the time-like Killing vector that is null on the horizon. Wald shows that the variation of the Hamiltonian with respect to the integration constants of a specic solution is given by

H =

1 16

Zc J(n1) [integraldisplay]cd i (n1)

[parenrightbig]

= 1

, (4.6)

where c denotes a Cauchy surface and (n2) is its boundary, which has two components, one at innity and one on the horizon. In particular

QA(n2) i A(n1) = 4iA F + 4(1)ni F ^ A . (4.7)

For the case of our ansatz (2.4), we nd

Qgrav(n2) = h[prime][radicalbigg]

JHEP06(2014)109

16

Z (n2)

Q(n2) i (n1)

[parenrightbig]

f

h rn2 (n2) ,

QA(n2) = 4 [prime][radicalbigg]

f

h rn2 (n2) ,

i grav(n1) = rn2[parenleftBigg]

h[prime]

rf h

[parenrightbigg]

+ n 2

r

sh

f f

! (n2) ,

i A(n1) = 4( ) [prime][radicalbigg]

f

h rn2 (n2) . (4.8)

In the asymptotic region at large r, this gives

Q i = rn2[radicalBigg]

h

f

n 2

r f

4f

h [prime] 2 [prime][parenleftbigg]

f

h

f h h2

[parenrightbigg][parenrightBigg]

(n2) . (4.9)

From the boundary on the horizon, one nds

1 16

Zr=r0( Q i ) = T S . (4.10)

(This result is a generalisation of that for Einstein-Maxwell theory obtained [21]. Analagous results were obtained in [13] for Einstein gravities coupled to a conformally massless scalar.)

Substituting the asymptotic expansions (2.11) into (4.9), we nd from (4.6) that the contribution to H from the boundary at innity is given by sending r to innity in the

expression

H1 !

!n2 16

r

[bracketleftbig]

(n 2) n1 + (n 3 ) q21 [bracketrightbig]

(n 2) n1 + 2(n 3 )q2 q1 + 2(n 3 + )q1 q2

[bracerightbig]

. (4.11)

12

The ostensibly divergent r term in fact vanishes, by virtue of the relation between n1 and q1 given in (2.12).

If we now use our expression (3.16) for the holographic mass, the variation H1 can

be rewritten as H1 = M +

8 (q1q2) , (4.12)

and so, together with (4.10), we obtain the rst law of thermodynamics in the form

dM = T dS

!n2

4 q1 q2

!n2

8 d(q1q2) . (4.13)

It should be recalled that the parameter can be chosen freely, with di erent choices corresponding to making Legendre transformations which redene the mass, or energy, of the black hole by the addition of some constant multiple of q1 q2 (see (3.14)). A simple choice is to take = 0, in which case we have

dM = T dS

!n2

4 q1dq2 +

!n2

4 q1 dq2 . (4.14)

As mentioned previously, taking = 0 corresponds to the case where the potential at innity is held xed in the variational problem.

It is worth remarking that if we specialise to = 1 in four dimensions, as we observed earlier, then ~m2 = 0 and our Einstein-Proca model reduces to the ordinary Einstein-Maxwell system, then the q1 dq2 term in (4.14) reduces simply to the standard dQ contribution in the rst law for charged black holes. Namely, q2 is then electric charge Q, and the potential di erence = h 1 is equal to q1, since in our general calculation the

potential vanishes on the horizon and here, in four dimensions, the potential at innity is equal to q1 (see eqn (2.10)). More generally, choosing ~m = 0 in any dimension n gives = n 3. For this Einstein-Maxwell system we have A = q1 + q2 r3n, and again the rst

law (4.14) becomes the standard one for a charged black hole, derived in the gauge where the potential vanishes on the horizon.

Another choice for the constant that might be considered natural is to choose it so that the q1 dq2 and q2 dq1 terms in the rst law (4.13) occur in the ratio such that they vanish if one imposes a dimensionless relation between q1 and q2. Since these quantities have length dimensions [q1] = L(n3)/2 and [q2] = L(n3+)/2, such a relation must take the form

qn3+1 = c qn3, (4.15)

where c is a dimensionless constant. Thus if we choose

= n 3 +

n 3

, (4.16)

JHEP06(2014)109

!n2

then the rst law (4.13) becomes

dM = T dS

(n 3 ) q1 dq2 (n 3 + ) q2 dq1[bracketrightbig]

, (4.17)

reducing simply to dM = T dS if (4.15) is imposed.

13

!n2 8(n 3)

Finally in this section, we remark that the Wald type derivation of the rst law that we discussed above can be applied also to the case of solitonic solutions to the Einstein-Proca system. In these solutions there is no inner boundary, and instead r runs outwards from r = 0 which is simply like an origin in spherical polar coordinates. The behaviour of the metric and Proca functions at large r takes the same general form as in (2.11). Thus when we apply the procedures described earlier in this section, we can derive a rst law of thermodynamics that is just like the one for black holes, except that the T dS term that came from the integral (4.10) over the boundary on the horizon. If we make the simple choice dening the energy of the system, the rst law (4.13) for the black hole case will simply be replaced by

4 q1 dq2 . (4.18)

5 Solutions outside 0 < 1Until now, our discussion of the solutions to the Einstein-Proca system has concentrated on the cases where the Proca mass is such that the index , dened in (1.3), satises 1. This was done in order to allow a relatively straightforward and uniform analysis

of the asymptotic structure of the solutions. However, it should be emphasised that black hole and soliton solutions of the Einstein-Proca equations exist also if the index lies in a wider range. Looking at the form of the expansions in (2.11), we see that the e ects of the back-reaction of the Proca eld on the metric components sets in at a leading order of 1/rn3. Clearly, if this were to be of order r2 or higher, then the back-reaction would be overwhelming the r2[lscript]2 terms in h and f that establish the asymptotically-AdS nature of the solutions. Thus we can expect that in order to obtain asymptotically-AdS black holes or solitons, we should have

< n 1 , (5.1) which, from (1.3), implies that the Proca mass must satisfy

~m2 < m2

(n 3)2

JHEP06(2014)109

dM =

!n2

n 2

[lscript]2 . (5.2)

Thus the full Proca mass range where we may expect to nd stable black hole and soliton solutions is

[lscript]2 < ~m2 <

n 2

[lscript]2 . (5.3) Below the Breitenlohner-Freedman bound which forms the lower limit, we expect the solutions to be unstable against time-dependent perturbations, on account of the tachyonic nature of the Proca eld.

In the next section, where we carry out a numerical study of the various solutions, we nd that indeed the upper bound in (5.3) represents the upper limit of where we appear to obtain well-behaved black hole and soliton solutions. At the lower end, the numerical integrations appear to be stable not only for the entire Breitenlohner-Freedman window of negative ~m2 in (5.3), but also for arbitrarily negative ~m2 below this, where the Proca eld has become tachyonic. Presumably if we were to extend our numerical analysis to include

14

the possibility of time-dependent behaviour we would nd exponentially-growing timelike instabilities below the limit in (5.3), but these cannot be seen in the numerical integration of the static equations that we study here.

We may thus divide the range of possible values for ~m2, the square of the Proca mass, as follows:

(1) 0 < ~m2 < m2 ; (n 3) < < (n 1)

(2) 14(n 2)(n 4)[lscript]2 ~m2 < 0 ; 1 < (n 3)(3) m2BF < ~m2 < 14(n 2)(n 4)[lscript]2 ; 0 < < 1(4) ~m2 = m2BF ; = 0

(5) ~m2 < m2BF ; imaginary

When the Proca mass lies in the range (1), the leading term in the fall-o in the metric functions h and f due to back reaction from the Proca eld occurs at a positive power of r, lying between 0 and 2. In the range (2), the leading powers of r in the metric functions due to back reaction are negative, and so the rate of approach to AdS is more conventional, but there can still be a rather complicated sequence of back-reaction terms at more dominant orders than the mass term m2/rn3 in the metric functions. The range (3) corresponds to the cases we have already discussed in general, where the leading-order terms in the asymptotic expansions take the form (1.5). A special case arises in (4), where the Proca mass-squared precisely equals the negative-most limit of the Breitenlohner-Freedman bound. Finally, in the range (5), the Proca mass-squared is more negative than the Breitenlohner-Freedman bound, and there is tachyonic behaviour.

It is instructive to examine some examples of the asymptotic behaviour of solutions where the Proca mass lies in the various ranges outside the cases in (3) that we have already studied. As a rst example, we shall consider

Dimension n = 5 with = 32 . Note that in this example, the mass-squared ~m2 of the Proca eld is still negative, with ~m2 = 7/(4[lscript]2), and it lies within the range (2) described

above. We nd that the asymptotic expansions for the Proca and metric functions take the form

= q1

r1/4 +

JHEP06(2014)109

q2

r7/4 +

7[lscript]2 q1

16r9/4 +

7[lscript]2 q31 120r11/4 +

[lscript]2 q2

16r15/4 +

7(16[lscript]2 m2 q1+22[lscript]2 q21 q29[lscript]4 q1

2560r17/4 + [notdef] [notdef] [notdef] ,

h = r2 [lscript]2 + 1 + 2q21

m2r2 +

49[lscript]2 q21

60r5/2 +

7[lscript]2 q41

5r1/2 +

50r3 + [notdef] [notdef] [notdef] ,

720r3 + [notdef] [notdef] [notdef] . (5.4)

This example illustrates how when > 1 we can get an intermingling of fall-o powers, with the r9/4 descendant of the leading q1 r1/4 term in the expansion for appearing prior to the rst back-reaction term in (2.11), which is at order r11/4. Nevertheless, we nd that the holographic mass and the Wald formula for the rst law continue to give nite results which agree with the general expressions (3.14) and (4.13).

15

f = r2 [lscript]2 + 1 + q21

35[lscript]2 q21

48r5/2 +

91[lscript]2 q41

6r1/2 +

24m2 7q1 q224r2 +

As a second example we consider

Dimension n = 5 with = 52 . In this case the Proca mass-squared is positive, with ~m2 = +9/(16[lscript]2). It is still less than the upper limit ~m2 = m2 = 3/[lscript]2 in ve dimensions

that we described above, and so it lies within the range (1). The asymptotic forms of the Proca and metric functions are

= q1 r1/4

[lscript]2 q31 8r5/4 +

9[lscript]2 q1

16r7/4 +

q2

r9/4 +

21[lscript]4 q51 128r11/4 + [notdef] [notdef] [notdef] ,

h = r2[lscript]2

2

3q21 r1/2 + 1

[lscript]2 q41

6r +

[lscript]2 q21 4r3/2 +

m2r2 +

55[lscript]4 q61 144r5/2 + [notdef] [notdef] [notdef] ,

229[lscript]4 q61

576r5/2 + [notdef] [notdef] [notdef] . (5.5)

In this case there is even more intermingling of the orders in the expansions, with the rst back-reaction term in , at order r5/4, preceding a descendant of the leading order q1 r1/4 term and also preceding the second independent solution that begins with q2 r9/4. In the

metric functions the rst back-reaction term is at order r1/2, which is prior even to the usual constant term of the pure AdS metric functions h = f = r2[lscript]2 + 1.

A new feature that arises in this example is that we must now add further counterterms in the calculation of the holographic mass, in order to obtain a nite result. Specically, we now need to include terms

LA,extract =

JHEP06(2014)109

16q21 r1/2 + 1

11[lscript]2 q41

f = r2[lscript]2

48r +

9[lscript]2 q21 16r3/2 +

8m2 + 3q1 q2

8r2 +

8Gph (AA)2 +

e2 [lscript]

e3 [lscript]

8Gph R AA . (5.6)

The divergences in the holographic mass are then removed if we take

e1 = 12( 1) , e2 =

118(3 2 ) , e3 =

13(1 ) . (5.7)

The resulting expression for the holographic mass then agrees with the general formula (3.14), after specialising to n = 5 and = 52. Using the Wald procedure described in section 4, we obtain the specialisation of (4.13) to n = 5 and = 52.

For another example, we consider

Dimension n = 6 with = 2. This lies within the range of category (2) above, and has ~m2 = 5/(4[lscript]2). Another new feature arises here, namely that we nd also log r behaviour

in the asymptotic expansions of the metric and Proca functions. To the rst few orders at large r, we nd

h = [lscript]2r2 + 1 + 2q213r

5[lscript]2q21 log r

4r3 +

m2r3 [notdef] [notdef] [notdef] ,

f = [lscript]2r2 + 1 + q21

4r

15[lscript]2q21 log r16r3 +

n2r3 + [notdef] [notdef] [notdef] ,

= r

1

2

q1 5[lscript]2q1 log r8r2 +q2r2 + [notdef] [notdef] [notdef]

,

748[lscript]2q21 . (5.8)

16

m2 = n2 + 12q1q2

The Wald formula still gives a convergent result, with

16

!4 H1 = 4 n2 + 2q2 q1 + 10q1 q2 +

52[lscript]2q1 q1 . (5.9)

Now, however, using just the counterterms we have discussed so far we nd that there is still an order log r divergence remaining in the holographic mass. It can be removed if we make a specic choice for the coe cient in (3.5), namely = 1. Thus in this case we are e ectively obliged to view the surface term (3.5) as being instead a counterterm, which serves the purpose of removing the logarithmic divergence. We then nd that all the divergences in the holographic mass are removed if we take

e1 = 0 , = 1 , (5.10)

leading to

M = 2

3

n2 +32 q1q2 +116 (48e3 + 5) [lscript]2 q21[bracketrightbigg]

. (5.11)

Note that although the ph R AA counterterm in (5.6), with its coe cient e3, is not

required for the purpose of subtracting out any divergence, it is now making a contribution to the holographic mass, representing an ambiguity in its denition.

If we use (5.11) to eliminate n2 from (5.9), this leads to the rst law

dM = T dS

2

3

JHEP06(2014)109

3q1 dq2 + q2 dq1 + 6e3 q1dq1[bracketrightbig]

. (5.12)

A natural choice would be to take the free parameter e3 to vanish.

For the next example, we consider case (4) above, where the Proca mass satises exactly the Breitenlohner-Freedman bound:

Dimension n with = 0. In n dimensions, we therefore have ~m2 = (n 3)2/(4[lscript]2). The

large-distance expansions take the form

h = [lscript]2r2 + 1 + m1(log r)2 + ~m1 log r + m2

rn3 + [notdef] [notdef] [notdef] ,

f = [lscript]2r2 + 1 + n1(log r)2 +[notdef]1 log r + n2

rn3 + [notdef] [notdef] [notdef] ,

= q1 log r + q2

r(n3)/2 + [notdef] [notdef] [notdef] , m1 = 2n1 =

2(n 3)q21 n 1

,

~m1 = 4q1 (n 1)(n 3)q2 2q1 (n 1)2

,[notdef]1 = 2q1

(n 3)q2 q1 n

2

,

m2 = n2 + (n 3)2 2q21 + 2(n 1)q1q2 + (n 1)2q22

(n 1)3(n 2)

. (5.13)

We see that here also, there is logarithmic dependence on the r coordinate in the asymptotic expansions. Nevertheless, the Wald formula turns out to be convergent, and we nd

16 !n2

H1 = (n 2) n2 4q2 q1 + 2(n 3)q2 q2 . (5.14)

17

Logarithmic divergences proportional to log r and (log r)2 arise in these = 0 examples, and as in the earlier case of = 2 in n = 6 dimensions, it is necessary to make a specic choice for the coe cient in the surface term (3.5) in order to remove them, namely by setting

= 4

n 2

. (5.15)

Again it turns out that the original Proca counterterm (3.6) is not required in this case, and so we take e1 = 0 here. The holographic mass is then given by

M = (n 2) !n216

[bracketleftbigg]

and hence from (5.14) we arrive at the rst law

dM = T dS

is real. We nd that the asymptotic expansions take the form

h = [lscript]2r2 + 1 + m1 cos(~ log r) + ~m1 sin(~ log r) + m2

rn3 + [notdef] [notdef] [notdef] ,

f = [lscript]2r2 + 1 + n1 cos(~ log r) +[notdef]1 sin(~ log r) + n2

rn3 + [notdef] [notdef] [notdef] ,

= q1 cos

log r

r(n3)/2 + [notdef] [notdef] [notdef] ,

m1 = (q21 q22)~2 4q1q2~

+ (n 1)(n 3)(q21 q22) ~

2 + (n 1)2

~m1 = 2(q21 q22)~ + 2q1q2 (n 1)(n 3) + ~

~ 2 + (n 1)2

18

n2 + 1n 2

(n 3)q22 2q1q2

, (5.16)

[parenrightbig]

q2

[bracketrightbigg]

!n2

8 (q1 dq2 q2 dq1) . (5.17)

The reason why q1 and q2 enter in a rather symmetrical way here may be related to the fact that in this = 0 case the dimensions of the two quantities q1 and q2 are the same, as can be seen from the expansion for (r) in (5.13). Also, the di erent way in which the logarithmic singularities in the holographic mass are handled in these = 0 examples could be related to the fact the that the original solution for itself, prior to taking back reactions into account, already has the logarithmic dependence associated with the coe cient q1. Again, this arises because q1 and q2 have the same dimensions when = 0. By contrast, in an example such as n = 6, = 2 discussed previously, the logarithmic dependences arose only via the back reactions, as a result of a conuence of powers of r in these sub-leading terms.

Finally, we consider the cases (5) where the Proca mass-squared is more negative than the Breitenlohner-Freedman bound. We make take

Dimension n with = i ~

. Here we have

~

=

r14 ~m2[lscript]2 (n 3)2 = i , (5.18)

with ~m2 < m2BF and so ~

JHEP06(2014)109

1

2 ~

log r

[parenrightbig]

+ q2 sin 12 ~

,

2

,

n1 = (n 3)(q21 q22) 2q1q2~

,

2(n 2)

[notdef]1 = (q21 q22)~ + 2(n 3)q1q2

2(n 2)

m2 = n2 + (q21 + q22) (n 3)2 + ~

2(n 1)(n 2)

. (5.19)

The resulting Wald formula is given by

16 !n2

(5.21)

~ 16 (q2dq1 q1dq2) . (5.22)

6 Numerical results

It does not appear to be possible to solve the Einstein-Proca equations of motion for static spherically-symmetric geometries analytically, and so we now resort to numerical integration in order to gain more insight into the solutions. Two distinct kinds of regular solutions can arise; rstly black holes, and secondly what we shall refer to as solitons.

The black hole solutions can be found numerically by rst assuming that there exists an horizon at some radius r = r0, at which the metric functions h(r) and f(r) vanish, then performing Taylor expansions of the metric and Proca eld functions h(r), f(r) and

(r) around the point r0, and then using these expansions to set initial conditions just outside the horizon for a numerical integration out to innity. The criterion for obtaining a regular black hole solution is that the functions should smoothly and stably approach the asymptotic forms (1.7) that we assumed in our discussion in section 2. Since, as we have seen, the general asymptotic solutions, with all three independent parameters q1, q2 and m2 nonvanishing, are well-behaved at innity (provided the Proc mass ~m satises ~m2 < m2 , where m is dened in eqn (5.2), there is no reason why a solution that is well-behaved

on the horizon will not integrate out smoothly to a well-behaved solution at innity, and indeed, that is what we nd in the numerical analysis.3

3Note that the situation would be very di erent in the absence of a cosmological constant. The asymptotic form of the Proca solutions is then given by (A.6), and so one of the two solutions diverges exponentially at innity, assuming ~m2 > 0. The analogous evolution from a well-behaved starting-point on the horizon would then inevitably pick up the diverging solution at innity, leading to a singular behaviour. The exponential divergence could be avoided if ~m2 were negative, but in a Minkowski background this would always be tachyonic, and so there would be instabilities because of exponential run-away behaviour as a function of time.

19

,

2

1

2(n 3) (q21 + q22) . (5.20)

Again, as in the previous example, the original Proca counterterm (3.6) is not required to regularize the holographic mass, and so we may take e1 = 0. Terms proportional to sin(~ log r) and cos(~ log r) can be removed by taking the coe cient of the Proca surface term (3.5) to be again given by (5.15). This yields the nite result

M = (n 2) !n2

16

n2 +(n 3)2(n 2)(q21 + q22) [bracketrightbigg]

H1 = (n 2) n2 + ~

(q2 q1 q1 q2) +

for the holographic mass, and hence, from (5.20), we arrive at the rst law

dM = T dS

JHEP06(2014)109

!n2

The solitonic solutions have a very di erent kind of interior behaviour, in which the functions h(r), f(r) and (r) all approach constant values at the origin at r = 0. To study these numerically we start by obtaining small-r expansions for the functions, using these to set initial conditions just outside the origin, and then integrating out to large r. The asymptotic forms of the metric and Proca functions will again be of the general form given in (1.7) in the case of smooth solitonic solutions. Again, since the generic asymptotic solutions with Proca mass satisfying ~m2 < m2 are well-behaved, the smooth solutions

near the origin will necessarily evolve to solutions that are well-behaved at innity. Since the solitonic solutions are somewhat simpler than the black holes, we shall begin rst by investigating the solitons. For the rest of this section, we shall, without loss of generality, set the AdS scale size by taking

[lscript] = 1 . (6.1)

6.1 Solitonic solutions

The soliton solutions we are seeking have no boundary at small r; rather, r = 0 will be like the origin of spherical polar coordinates. We begin by making Taylor expansions for the metric and Proca functions, taking the form

h = (1 + b2r2 + b4r4 + [notdef] [notdef] [notdef] ) , f = 1 + c2r2 + c4r4 + [notdef] [notdef] [notdef] ,

= p (a0 + a2r2 + a4r4 + [notdef] [notdef] [notdef] ) . (6.2) Substituting into the equations of motion (2.6), one can systematically solve for the coe -cients (a2, a4, . . .), (b2, b4, . . .) and (c2, c4, . . .) in terms of the coe cient a0. Thus we nd

a2 =

(n 2)(n 4)a0

8(n 1)

JHEP06(2014)109

, b2 = 2(n 1)[lscript]2 (n 2)(n 4)a20

2(n 1)

,

c2 = 2(n 1)[lscript]2 + a20(n 4)

2(n 1)

, (6.3)

with progressively more complicated expressions for the higher coe cients that we shall not present explicitly here. The coe cient represents the freedom to rescale the time coordinate. In an actual numerical calculation, when we integrate out to large r, we may, without loss of generality, start out by choosing = 1, and then, by taking the limit

lim

r!1

h

r2 = , (6.4)

determine the appropriate scaling factor that allows us the to redene our by setting = 1/ . Thus we see that the soliton solution has only the one free parameter, a0.

To illustrate the numerical integration process, let us consider the example of the soliton in dimension n = 5. We determined the coe cients in the expansions (6.2) up to the r8 order, and used these to set initial conditions for integrating the equations (2.6) out to large r. We found that indeed the solitons, parameterised by the constant a0, have a well-behaved and stable asymptotic behaviour, in which the functions h, f and approach the forms given in (1.7) with n = 5. In particular, we have

h ! r2 + 1 +

m1r +

m2r2 + [notdef] [notdef] [notdef] , !

q1

r1/2 +

q2

r3/2 + [notdef] [notdef] [notdef] . (6.5)

(Recall that we are setting the AdS scale [lscript] = 1 in this section.)

20

25

[Minus]0.2

20

[Minus]0.4

15

[Minus]0.6

10

[Minus]0.8

5

[Minus]1.0

[Minus]1.2

1 2 3 4 5 r

2 4 6 8 10 12 14 r

Figure 1. Smooth soliton in n = 5 dimensions, with a0 = 1 and = 1.6285. The left-hand plot

shows the metric functions h(r) and f(r). The upper line is h(r), starting from h(0) = , and the lower line is f(r), starting from f(0) = 1. To leading order, they coalesce at large r. The right-hand plot shows the potential function (lower line), and compares it with a best t of the function ~ = q1/r 12 + q2/r 32 (upper line) that represents the leading-order terms in the large-r expansion in (6.2), achieved by taking q1 = 2.0980 and q2 = 1.6018. The numerically-obtained potential

function converges at small r to (0) = p .

In principle, for a given solitonic solution determined by the choice of the free parameter a0, we can match the numerically-determined asymptotic form of the solution to the expansions (6.5), and hence read o the values of the coe cients q1, q2, m1 and m2. It is quite delicate to do this, especially to pick up the coe cient m2 which occurs at four inverse powers of r down from the leading-order behaviour of h(r). A useful guide to the accuracy of the integration routine is to match the numerical results for h(r) at large r to an assumed form

h = 0 r2 + 1 + m1r +

m2r2 . (6.6)

Ideally, one should nd 0 = 1 and 1 = 1. There will in fact, of course, be errors. We rst rescale the numerically-determined h(r) and (r) by the factors 1/ 0 and 1/p 0

respectively. A test of a reliable solution is then that to high accuracy we should nd (see (2.8))

1 1 0 , m1

2

3q21 0 . (6.7)

We now present an explicit example of a smooth soliton solution, for which we shall make the choice with a0 = 1 in (6.2). (We choose a negative value of a0 so that q2 is

positive, and in fact q2 > 0 > q1.) For this choice of a0, we nd that the free scaling parameter should be chosen to be = 1.6285 in order to ensure that = 1.0000. The behaviour of the metric functions (h, f) and the potential function is then displayed in gure 1. In the left-hand plot, can be seen that the metric functions are indeed running smoothly from constant values at the origin, and at large r they approach their expected AdS forms. By a careful matching of the asymptotic forms of the function h(r) to the expansion (6.2), we can read o a value for the mass parameter m2, nding m2 =

6.2630.

21

JHEP06(2014)109

q

M

0.5 1.0 1.5 2.0 2.5

q

8

[Minus]0.5

[Minus]1.0

6

[Minus]1.5

4

[Minus]2.0

[Minus]2.5

2

[Minus]3.0

0.5 1.0 1.5 2.0 2.5

q

Figure 2. The left-hand plot shows q1 as a function of q2, whilst the right-hand plot shows the mass M as a function of q2, for 0 < q2 < 3. Correspondingly, a0 runs from a0 = 0+ to a0 = 1.3.

Repeating this calculation for a range of values for a0, we obtain reasonably trustworthy numerical expressions for

m2 = m2(a0) , q1 = q1(a0) , q2 = q2(a0) . (6.8)

We can then choose to use q2 to parameterise the solutions, rather than a0, so that we can write

m2 = m2(q2) , q1 = q1(q2) . (6.9)

We are now in a position to attempt a numerical verication of the rst law of thermodynamics for the solitonic solutions. Choosing the parameter in section 4 to be zero for simplicity, the energy of the ve-dimensional soliton is given, from (3.11), by

M = 3

8

m2 +76q1 q2

. (6.10)

Viewing M and q1 as functions of q2, we can now check how accurately the rst law (4.18), which in n = 5 dimensions reads

dM =

1

2 q1 dq2 , (6.11)

is satised. In gure 2, we display plots of q1(q2) and M(q2) for a range of q2 values with 0 < q2 < 3.

After tting the data for these curves, we obtain the approximate relations

q1 = 1.486q2 + 0.1094q22 , M =

1

2(0.7449q22 0.03764q32) . (6.12)

Thus we nd

JHEP06(2014)109

@M

2

@q2 = 1.490q2 + 0.1129q22 , (6.13) which should, according to (6.11), be equal to q1. It is indeed in reasonable agreement with the approximate form for q1 given in (6.12).

It is worth pointing out that as we increase the negative value of a0, the mass becomes divergent, with the solution becoming singular around a0 = 1.8.

22

6.2 Black hole solutions

The static black hole solutions that we are seeking are characterised by the fact that the Killing vector @/@t will become null on the horizon at r = r0. Thus the metric function h(r) in (2.4) will have a zero at r = r0. It follows from the equations of motion (2.6) that the functions f(r) and (r) will vanish at r = r0 also. We are therefore led to consider near-horizon series expansions of the form

h = b1

(r r0) + b2(r r0)2 + [notdef] [notdef] [notdef]

[bracketrightbig]

, f = c1(r r0) + c2(r r0)2 + [notdef] [notdef] [notdef] ,

=

pb1

a1(r r0) + a2(r r0)2 + [notdef] [notdef] [notdef]

[bracketrightbig]

. (6.14)

The constant b1 parameterises the freedom to rescale the time coordinate by a constant factor. It can be used in order to rescale the solution, after numerical integration out to large distances, so that the time coordinate is canonically normalised.

Substituting the expansions (6.14) into the equations of motion (2.6), we can systematically solve for the coe cients (a2, a3, . . .), (b2, b3, . . .) and (c1, c2, . . .) in terms of r0 and a1. In dimension n the solution for c1 is

c1 = (n 2)(n 3) + (n 1)r20

(2r0a21 + n 2)r0

a1(4r40a41 + 12r30a21 + 153r20 + 72) 48r0(2r20 + 1)

2r40a41 93r30a21 32r0a21 + 24r20 + 36

24r0(2r20 + 1)

. (6.16)

In our actual numerical calculations, we have expanded up to and including the (r r0)4 order. These expansions are then used in order to set initial conditions just outside the horizon. We then integrate out to large r. The criterion for a good black hole solution is that the metric and Proca functions should approach the asymptotic forms given in (1.7). We nd that indeed such solutions arise, and they are stable as the parameters r0 and a1 are adjusted.

As we did in the case of the solitonic solutions, here too we can attempt a numerical conrmation that these black hole solutions obey the rst law of thermodynamics that we derived in section 4. Taking the simple choice = 0 again, the rst law is given by (4.14). We shall present an example calculation in n = 5 dimensions, for which the rst law becomes

JHEP06(2014)109

. (6.15)

(Recall, again, that we have set the AdS scale [lscript] = 1 in this section.) The expressions for the higher coe cients are all quite complicated in general dimensions. Here, as an example, we just present (a2, b2, c2) in the special case of n = 5:

a2 =

,

b2 =

,

c2 = 6r40a41 + 105r30a21 + 32r0a21 24r20 36

4r20(2r20a21 + 3)

1

2 q1 dq2 . (6.17)

23

dM = T dS

5

[Minus]0.4

4

[Minus]0.6

3

[Minus]0.8

2

[Minus]1.0

1

1.2 1.4 1.6 1.8 2.0 r

2 4 6 8 10 12 14 r

Figure 3. Black hole in n = 5 dimensions, with a1 = 10 and b1 = 2.163. The left-hand plot shows

the metric functions h(r) and f(r). The upper line is h(r), starting from h(r0) = 0, and the lower line is f(r), starting from f(r0) = 0. To leading order, they coalesce at large r, which we did not present in this gure. The right-hand plot shows the potential function (lower line), and compares it with a best t of the function ~ = q1/r 12 + q2/r 32 (upper line) that represents the leading-order terms in the large-r expansion in (6.2), achieved by taking q1 = 3.222 and q2 = 3.742. The actual

numerically-obtained potential function vanishes at small r = r0, although ~(r0) [negationslash]= 0.

The easiest case to consider is when we x the entropy. This corresponds to holding r0 xed, and the rst law becomes

dM = T dS

1

2q1dq2 ! dM =

JHEP06(2014)109

1

2q1dq2 . (6.18)

As a concrete example, let us set r0 = 1. The solution then has one non-trivial adjustable parameter, a1, remaining. The parameter b1 should be xed so that we have

h r2

|r!1 = 1 at large r. To see in more detail how the metric function and behave, let

us take as an example a1 = 10, implying c1 = 18/203. We nd that we should then take

b1 = 2.163. Thus the black hole has a temperature

T = b1c14 = 0.01526 . (6.19)

The plots for the metric functions (h, f) and the Proca potential are given in gure 3.

From the numerical solution, we can read o m2 = 23.93 and (q1, q2) = (3.222, 3.742),

and hence nd that the mass is given by M = 11.63.

We can now solve numerically for a range of values for the parameter a1 and hence read o (q1, q2, M) and the temperature, all depending on the chosen values of a1. In particular, when a1 = 0, the solution becomes the usual Schwarzschild AdS-black hole, with r0 = 1. Here we present the results for a1 in the range 34 a1 0. We can then express the

quantities (M, q1, T ) as functions of q2. The results are given in gures 4 and 5.The data tting for small q2 implies that

M = 12[parenleftbigg]

32 + 0.420q22[parenrightbigg]

, q1 = 0.843q1 . (6.20)

Note that

2

@M

@q2 = 0.840q2 , (6.21)

24

q

M

0.5 1.0 1.5

q

4

[Minus]0.5

3

2

[Minus]1.0

1

[Minus]1.5

0.5 1.0 1.5

q

JHEP06(2014)109

Figure 4. Black hole for r0 = 1 and 0 < q2 < 2. The solution is Schwarzschild-AdS when q2 = 0. For small q2, the function q1(q2) is linear and M(q2) is parabolic.

0.48

T

0.46

0.44

0.42

0.40

0.38

0.5 1.0 1.5

q

Figure 5. Temperature of the black hole for r0 = 1 and 0 < q2 < 2.

q

0.355

T

[Minus]3.23550

0.350

[Minus]3.23555

0.345

[Minus]3.23560

0.340

0.335

[Minus]3.23565

0.330

[Minus]3.23570

3.75006 3.75008 3.75010 3.75012 3.75014

q

3.75006 3.75008 3.75010 3.75012 3.75014

q

Figure 6. Black hole for r0 = 1 and larger q2. Note that for a given q2, there can be two values for q1 or T respectively, suggesting that a phase transition can occur.

If we decrease a1 from 0 to negative values, eventually the solution becomes singular when a1 reaches about a 1 20.12. We obtained data from a0 = 18 to a1 = 20, and

the results for T and q1 are plotted in gure 6.

What is curious is that as a1 approaches a 1, the parameters (q1, q2, m2) remain nite, approaching certain xed values. Yet the solution becomes singular once a1 passes over a 1.

25

7 Conclusions

In this paper, we have investigated the static spherically-symmetric solutions of the theory of a massive Proca eld coupled to gravity, in the presence of a negative cosmological constant. The fact that the solutions are asymptotic to anti-de Sitter, rather than Minkowski, spacetime has a profound e ect on their geometry and stability. In the absence of a cosmological constant, a generic static spherically-symmetric solution of the Proca equation will take the form = A0 e ~mr/rn3 + e ~mr/rn3, and so without a ne-tuning to

set = 0, the solution will diverge exponentially at innity. There will in turn be back-reaction on the metric that leads to analogous singular behaviour. This implies that even if one nds a solution that is well behaved on the horizon of a black hole, its evolution out to large r will inevitably pick up some component of the diverging asymptotic solution, thus implying that it will be singular. The asymptotic solutions would instead be decaying and oscillatory in r if ~m2 were negative, but then, the Proca eld would be tachyonic and so the solutions would exhibit runaway behaviour with real exponential time dependence.

By contrast, with the cosmological constant turned on, two factors come into play that radically change the picture. First of all, the asymptotic behaviour of the Proca solutions in the AdS background involve power-law rather than exponential dependence on r. Secondly, there is now a window of negative mass-squared values for the Proca mass, extending in the range m2BF ~m2 < 0, where m2BF is the Breitenlohner-Freedman bound

given in (1.4), within which the Proca eld is still non-tachyonic, and thus is not subject to exponential time-dependent runaway behaviour. Between them, these two factors imply that perfectly well-behaved black hole solutions exist, provided that the Proca mass-squared lies in an appropriate range. There also exist solitonic solutions, that extend smoothly to an asymptotically AdS region from an origin of the radial coordinate at r = 0.

We performed some numerical integrations to demonstrate the existence of such well-behaved black hole and solitonic solutions, but in fact, one can see on general grounds that they must exist. Namely, by making a general expansion of the Proca and metric functions in the vicinity of the horizon, in the black hole case, or of the origin, in the solitonic case, one can rst establish that well-behaved short-distance solutions exist in each case. Although one does not know precisely how these join on to the solutions at large distance, which are known only asymptotically, the fact that the general asymptotic solutions are well-behaved means that the evolution from small to large r will necessarily be a smooth one. In other words, there is no issue in this asymptotically-AdS situation of needing a ne-tuning to avoid an evolution to a singular solution at innity, since all large-r solutions are non-singular.

The calculation of the mass of the static spherically-symmetric solutions can be somewhat delicate, because the way in which they approach AdS at innity may involve fall-o s that are slower than in a typical AdS black hole such as Schwarzschild-AdS. In the case of Schwarzschild-AdS, the metric functions h and f in (2.4) take the form h = f = r2 [lscript]2 + 1 m/rn3 in n dimensions, and one might think that any approach

to pure AdS that was slower than the 1/rn3 rate would lead to a diverging mass. We performed our calculations of the mass using the renormalised holographic stress tensor,

26

JHEP06(2014)109

and it turns out that when the contribution from the Proca eld is properly taken into account, the result in general is perfectly nite and well dened. We also carried out a derivation of the rst law of thermodynamics using the techniques developed by Wald, and we found that this gives consistent and meaningful results.

An important feature in the rst law is that there is a contribution from the Proca eld, giving a result typically of the form dM = T dS + (const) q1dq2, where q1 and q2 are the two arbitrary coe cients in the asymptotic form of the Proca solution. One might think of q2 as being like a charge for the Proca eld and q1 as a conjugate potential.

However, since q2 is not associated with a conserved quantity it is not necessarily clear whether it should really be thought of as a charge, or whether instead it should be viewed as being a parameter characterising a Proca hair. In any case, the inclusion of the q1dq2 term is necessary in order to obtain an integrable rst law. The situation is in fact analogous to one that was encountered for the gauged dyonic black hole in [12], where a careful examination of the contribution of a scalar eld in the rst law showed that it gave an added contribution, with the leading coe cients in the asymptotic expansion of the scalar eld being the extra thermodynamic variables.

Acknowledgments

We are grateful to Sijie Gao and Yi Pang for useful discussions. H-S.L. is supported in part by NSFC grant 11305140 and SFZJED grant Y201329687. The research of H.L. is supported in part by NSFC grants 11175269 and 11235003. The work of C.N.P. is supported in part by DOE grant DE-FG02-13ER42020.

A Proca solutions in AdSn, and the Breitenlohner-Freedman bound

Here we record some results for spherically-symmetric solutions of the Proca equation in a pure AdSn background metric. These allow us to study the asymptotic form of the Proca eld in our black hole and soliton solutions, and also to give a simple derivation of the Breitenlohner-Freedman bound for massive vector modes in n-dimensional anti-de Sitter spacetime.

From the last equation in (2.6), we see that in a pure AdS background, which has h = f = 1+r2 [lscript]2, the Proca potential for a static spherically-symmetric eld A = (r)dt satises the equation

1rn2 (1 + r2 [lscript]2) rn2 [prime]

[prime] = ~m2 . (A.1)

This can be solved straightforwardly in terms of hypergeometric functions, giving

= q1

r(n3)/2 F [parenleftbigg]

[parenrightbigg]

p4 ~m2 [lscript]2 + (n 3)2 , (A.3)

27

JHEP06(2014)109

3 n

4 ,

4 ,

2 ,

n 3

2

[lscript]2 r2

+ q2

r(n3+)/2 F [parenleftbigg]

=

3 n +

n 3 +

2 +

4 ,

4 ,

2 ,

[lscript]2 r2

, (A.2)

where

and ~m is the mass of the Proca eld. (Since our focus is principally on the large-r asymptotic behaviour of the solutions, we have presented them in the form where the hyper-geometric functions are analytic functions of 1/r2.) Note that the leading-order terms q1 r(3+n)/2 and q2 r(3n)/2 associated with the two independent solutions then have descendants falling o with the additional factors of integer powers of 1/r2.

The Breitenlohner-Freedman bound for the vector modes is determined by the requirement that the parameter appearing in (A.2) should be real. Thus from (A.3) we see that the bound is given by

~m2 m2BF

(n 3)2

4[lscript]2 . (A.4)

A case of particular interest in this paper is when the Proca mass ~m is chosen to be given by (2.7), in order to ensure that the sequence of terms in the power-series expansion of (r) at large r should involve inverse powers of r that increase in steps of 1/r. Thus we see from (A.4) that our mass parameter ~m lies within the Breitenlohner-Freedman bound, with

~m2 = m2BF + 1

4[lscript]2 > m2BF . (A.5)

We then study a more extended range of values for ~m.Note that in the limit where the cosmological constant goes to zero (i.e. [lscript] ! 1), the

Proca solution (A.2) becomes

= e ~mrr +

JHEP06(2014)109

e ~mr

r . (A.6)

Unlike the AdS case, where both solutions of the Proca equation can be well-behaved at innity, in an asymptotically-Minkowski background one of the solutions always diverges exponentially.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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