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Web End = Eur. Phys. J. C (2016) 76:178DOI 10.1140/epjc/s10052-016-4007-y

Regular Article - Theoretical Physics

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Web End = Higher-derivative gravity with non-minimally coupled Maxwell eld

Xing-Hui Fenga, H. LbDepartment of Physics, Center for Advanced Quantum Studies, Beijing Normal University, Beijing 100875, China

Received: 14 January 2016 / Accepted: 8 March 2016 / Published online: 30 March 2016 The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract We construct higher-derivative gravities with a non-minimally coupled Maxwell eld. The Lagrangian consists of polynomial invariants built from the Riemann tensor and the Maxwell eld strength in such a way that the equations of motion are second order for both the metric and the Maxwell potential. We also generalize the construction to involve a generic non-minimally coupled p-form eld strength. We then focus on one low-lying example in four dimensions and construct the exact magnetically charged black holes. We also construct exact electrically charged z = 2 Lifshitz black holes. We obtain approximate dyonic

black holes for the small coupling constant or small charges. We nd that the thermodynamics based on the Wald formalism disagrees with that derived from the Euclidean action procedure, suggesting this may be a general situation in higher-derivative gravities with non-minimally coupled form elds. As an application in the AdS/CFT correspondence, we study the entropy/viscosity ratio for the AdS or Lifshitz planar black holes, and nd that the exact ratio can be obtained without having to know the details of the solutions, even for this higher-derivative theory.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . 1

2 Non-minimally coupled Maxwell eld . . . . . . . 32.1 The general construction . . . . . . . . . . . . 32.2 A low-lying example . . . . . . . . . . . . . . 4

3 Non-minimally coupled p-form eld strength . . . . 4

4 Electric and magnetic black holes . . . . . . . . . . 54.1 Static ansatz and reduced equations of motion . 54.2 General properties . . . . . . . . . . . . . . . . 54.3 Exact general magnetic black holes . . . . . . . 64.4 Exact electric z = 2 Lifshitz black holes . . . . 7

a e-mail: mailto:[email protected]

Web End [email protected]

b e-mail: mailto:[email protected]

Web End [email protected]

4.5 Dyonic black holes . . . . . . . . . . . . . . . 75 AdS/CFT application: viscosity/entropy ratio . . . . 86 Conclusions . . . . . . . . . . . . . . . . . . . . . 10References . . . . . . . . . . . . . . . . . . . . . . . . 10

1 Introduction

The spacetime metric g, the nonlinear generalization of the massless spin-2 eld, is the fundamental eld in the Einstein formulation of gravity. Electric-magnetic interactions of the U(1) Maxwell eld A underly almost all the phenomena in condensed matter physics. With the development of the AdS/CFT correspondence [13], the EinsteinMaxwell theory with a negative cosmological constant has become one of the most important playgrounds in relating classical gravity to certain strongly coupled condensed matter theories (CMT) at the quantum level, from superconductivity [4] to non-Fermi liquids [5,6].

While there has been great progress in studying condensed matter physics via gravity, the successes are mainly of a qualitative nature. To match a condensed matter phenomenon quantitatively as well, it is likely that one needs to generalize the EinsteinMaxwell theory, by introducing additional elds and/or couplings. One generalization, without breaking the general coordinate invariance, is to consider higher-derivative extensions. Higher-derivative gravity arises naturally in string or M-theory, where the AdS/CFT correspondence has the most solid foundation. The low-energy effective theories of string or M-theory are supergravities as the leading-order expansions, with some specic but innite sequences of higher-derivative corrections. EinsteinMaxwell gravities in both four and ve dimensions can be supersymmetrized and embedded in M-theory [7,8] or the type IIB string [9, 10]. (A specic extra F F A term is necessary in D =

5 for supersymmetrization.) It is thus natural to con-

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sider higher-derivative extensions for the EinsteinMaxwell

theory.

When a linear theory involves higher derivative terms, there are inevitable ghost excitations. This problem can easily be circumvented via nonlinear construction for scalar, vector, and anti-symmetric tensor elds. This is because for these elds, the rst derivative is also a tensor or can be made a tensor, without breaking the gauge symmetries. One can then construct a higher-derivative theory by adding higher-order polynomial invariants of these elds and/or their tensorial rst derivatives. Although the theory may involve high-order total derivatives through nonlinearity, each eld has at most two derivatives acting upon directly in the equations of motion. Consequently, the linearized theory in any background is of the second order. The situation is rather different for the metric. The rst derivative of the metric cannot be a non-vanishing tensor and only two derivatives of the metric may yield a tensor, namely the Riemann tensor. It follows that a typical higher-order polynomial invariant of the Riemann tensor tends to give rise to linear ghost excitations in a generic background.

There are different approaches concerning the ghost issue in higher-derivative gravities. In supersymmetric theories, ghosts may not be fatal [11]. In fact in four dimensions, gravity extended with quadratic curvature invariants was shown to be renormalizabe [12,13]. Recently a new static black hole over and above the usual Schwarzschild black hole was obtained in the four-dimensional theory [14,15]. When there is a cosmological constant, higher-derivative gravities in AdS backgrounds can have a critical point in the parameter space for which the ghost modes become log modes and may be truncated out by some strong boundary conditions. However, this process was more successful in three dimensions [16,17] than in four or higher dimensions [1820].

In perturbative string theory, the coupling constants of higher-order terms are regarded as small. One may use the eld redenition of the metric

g g + Rg + R (1.1) to simplify the theory order by order. In this approach, the propagators are not modied and hence the ghost issue does not arise, even though the theory would have ghosts when treated own its own. The shortcoming is that the contributions from the higher-order terms can only be regarded as small.This is too restrictive in the applications of the AdS/CFT correspondence, since in the discussion of gravity/CMT the purpose of introducing higher-order terms is not simply to add a small perturbation.

It turns out that there are combinations of polynomial invariants that are ghost free. The most famous example is the GaussBonnet term. Einstein gravity extended with the GaussBonnet term has a total of four derivatives via nonlinearity, but it is ghost free since the theory involves only

two derivatives at the linear level. Consequently the coupling constant of the GaussBonnet term does not have to be small. (Causality consideration may provide further restrictions on the coupling constant [2124].) The GaussBonnet term is one of a class of Euler integrands that give rise to general Lovelock gravities [26]. These theories make sense only in the context of string theory. First of all, GaussBonnet gravity violates causality on general grounds and the only way to avoid this problem is by adding an innite tower of massive higher-spin particles [25]. Second, D = 10, N = 1 super-

gravity with the string worldsheet correction indeed has a Riemann-squared [27]

R R

correction. Using the eld redenition (1.1), one can generate the GaussBonnet term at the quadratic order of the curvature polynomials. In other words, the GaussBonnet term or higher-order Euler integrands arise naturally in string theory. One may then appeal to the enormity of the string landscape and argue that in some string vacua, the GaussBonnet term dominates and hence the EinsteinGaussBonnet gravity can be treated on its own.

In this paper, we generalize this line of approach to include the Maxwell eld, or more general p-form eld strengths as well. We construct general higher-derivative gravities coupled to the Maxwell eld with the Lagrangian built from polynomial invariants of the Riemann tensor and the Maxwell eld strength. We require that in all the equations of motion both the metric g and A have at most two derivatives acting directly so that the theory may be ghost free. Since the eld strength couples to the curvature tensor directly, the Maxwell eld is non-minimally coupled, and also the gauge symmetry is preserved. Such couplings arise naturally in string theory and we expect that through a eld redenition analogous to (1.1), ghost-free combinations can also emerge, as in the case of EinsteinGaussBonnet gravity or more general Lovelock gravities.

We now give the outline of the paper. In Sect. 2, we construct higher-derivative gravities whose Lagrangian consists of the polynomial invariants of Riemann tensor and the eld strength F = A A. Analogous to the Euler inte

grands in Lovelock gravities, the combination of the polynomials is such that the equations of motion are second order. In Sect. 3, we generalize the construction to involve a generic non-minimally coupled p-form eld strength. In Sect. 4, we consider a low-lying example in which the EinsteinMaxwell theory with a cosmological constant is augmented with the polynomial of the Riemann tensor with a bilinear of F

so that the theory has at most four total derivatives. The equations of motion nevertheless remain second order. We construct static charged black holes in four dimensions with isometries of 2-sphere, 2-torus, and hyperbolic 2-space. In Sect. 5, we study an application of the AdS/CFT correspon-

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dence and derive the boundary viscosity/entropy ratio for the AdS and Lifshitz planar black holes. We conclude the paper in Sect. 6.

2 Non-minimally coupled Maxwell eld

2.1 The general construction

Our construction is analogous to Lovelock gravities, whose basic ingredients are Euler integrands, dened by

E(k) =

2k c1d1ckdka1b1akbk Ra1b1c1d1 Rakbkckdk , (2.1)

where Rabcd denotes the Riemann tensor Rabcd and

1s1s = s!1[1 ss]. (2.2) The Euler integrands can also be expressed as

E(k) =

(2k)!

2k R[a1b1a1b1 Rakbk]akbk. (2.3) The low-lying examples are

E(0) = 1, E(1) = R,E(2) = R2 4R R + R R , etc. (2.4)

The term gE(k) in the Lagrangian contributes

E(k) =

2k+1

the Z tensor satises the property

[[ Z]] = [ F[]F] + 2F[ R][ F].

(2.10)

In other words, although each term involves a total of four derivatives, both A and g have at most two derivatives.

This property is crucial in our construction.

With these preliminaries, we consider polynomial invariants of the tensor Rabcd and Zabcd analogous to the Euler integrands, namely

L(m,n) =

2m+n

c1d1cmdm c1 d1cn dna

1b1ambm a1 b1an bn

Ra1b1c1d1 Rambmcmdm Z a1 b1c1d1

Z am bmcmdm

(2(m + n))!

2m+n

R[a1b1a1b1 Rambmambm Z a1 b1a1b1

(2.11)

It is clear that when n = 0, the above gives rise to the Euler

integrands, i.e.

L(k,0) = E(k). (2.12)

It is easy to perform the variation of both the metric and A:

gL(m,n) = g L(m,n)()g + L(m,n) A

+total derivatives. (2.13)

We nd

L(m,n) =

2 g L(m,n) +

Z am bm]ambm

a1b1akbk c1d1ckdk Ra1b1c1d1 Rakbkckdk (2.5) to the Einstein equation of motion. A striking property is that no Riemann-tensor factor acquires any derivative in the equations of motion, such that the theory remains second order in derivatives. This is a consequence of the fact that the variation of the Riemann tensor, namely

R = , (2.6) yields a total derivative in the Lagrangian for the polynomial combinations of the Euler integrands. This is largely due to the Bianchi identity of the Riemann tensor, namely

[ R] = 0 = [ R]. (2.7) In order to include the Maxwell eld A in an analogous construction, we introduce a bilinear tensor of the eld strength F = dA

Zabcd = FabFcd. (2.8)

This tensor shares some similar properties of the Riemann tensor, but the properties (2.7) and Ra[bcd] = 0 of the Rie

mann tensor do not extend to the Z tensor. Nevertheless, owing to the Bianchi identity of the Maxwell eld, namely

[ F] = 0 = [ F], (2.9)

2n 2m+n

c1d1cmdm c1 d1cn dna

1b1ambm a1an bn

Ra1b1c1d1

Rambmcmdm Z a1c1 d1

Z an bncndn

m 2m+n

c1d1cmdm c1 d1cn dna

1ambm a1 b1an bn

Ra1c1d1

Rambmcmdm Z a1 b1c1d1

Z an bncndn

2m 2m+n

gc1c1d1cmdm c1 d1cn dnb

1 ambm a1 b1an bn

Ra2b2c2d2

Rambmcmdm b1d1(Z a1 b1c1d1

Z an bncndn

L(m,n) =

F =

4n 2m+n

c1d1cmdmcn dna

1b1ambm a1 b1an bn

Ra1b1c1d1

. (2.14)

It follows from (2.7) and (2.10) that neither the metric nor A has more than two derivatives in all terms in L(m,n)() and

L(m,n) .

The Lagrangian for the general theory is then given by

L = g

k=0

m+n=k

Rambmcmdm F a1 b1 Z a2 b2c2d2

Z an bncndn

mn L(m,n), (2.15)

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where mn are coupling constants. The full set of equations of motion are

hence the theory is of the second order. In Sect. 4, we shall construct charged black holes of this theory.

3 Non-minimally coupled p-form eld strength

The construction in the previous section can be easily generalized to general (p1)-form potential A(p1) whose p-form

eld strength is given by

F(p) = d A(p1), Fa

mn L(m,n)() = 0,

m+n=k

mn L(m,n) = 0.

(2.16)

Again, in all these equations, the metric and A have at most two derivatives acting on directly, with the total higher derivatives achieved through nonlinearity. The theories are thus of the second order.

We note that the non-minimally coupled Maxwell eld can also have the following structure:

g R[a1b1a1b1 Ranbnanbn Fc1c1 Fck]ck . (2.17) When k = 1, this term is a total derivative. When k = 2n +1

with n 1, this term vanishes. For k = 2n, this term

is proportional to the gL(m,n) owing to the identity

R[abc]d = 0. Thus we shall not consider the terms (2.17).

It is also worth pointing out again that any polynomial structures involving purely the Maxwell eld strength without the Riemann tensor are allowed and hence we shall not list them all.

2.2 A low-lying example

Having constructed general higher-derivative gravities with non-minimally coupled Maxwell eld, we shall study a lowlying example in detail. The Lagrangian is

L = g R 2 0

14 F2 + L(1,1) , (2.18)

where

L(1,1) =

]. (3.1)

For simplicity of notation, we construct the corresponding Z tensors

Za1apb1bp =

Fa1ap Fb1b

= p [a

1 Aa2a

p . (3.2)

The generalizing polynomial of the p-form to L(m,n) of the

2-form eld strength is then given by

L(m,n),p =

(2m + pn)!

2m(p!)n

R[a1b1a1b1 Rambmambm Za

1 1

a p1 a11a p1

(3.3)

Owing to the Bianchi identity,

[ap+1 Fa1ap] = 0 = [b

p+1 Fb1b

1 4cd c dabab

Rabcd Z a bcd =

RF2 4Rab Fac Fbc

+ Rabcd Fab Fcd. (2.19) In other words, the theory is the EinsteinMaxwell theory with a cosmological constant, together with an additional L(1,1) term. The Einstein equations of motion are

G + 0g

2(F2

], (3.4) it is straightforward to verify that in the equations of motion associated with the Lagrangian

gL(m,n),p

neither the metric nor A(p1) has more than two derivatives,

even though the theory involves higher-order derivatives through nonlinearity. When p is odd, we have L(m,n),p = 0

for n 2, since the wedge product of an odd form with

itself vanishes. Note that for p = 1, we must have n = 0, 1.

The series L(m,1),1 H(m) was rst constructed by Horn-

deski [28]. The p = 2 series was constructed in the previous

section.

It should be pointed out that the non-minimal coupling terms L(m,n),p are not the only possible structures that one can build for ghost-free combinations. For example, when p = 3, we can also have terms like(2m + 3n)!

2m6n R[a1b1a1b1 Rambmambm Y a

1 1

a31 a11a31

14 g F2) + L(1,1)() = 0, (2.20)

where

L(1,1) =

2 g L(1,1) +

2cd c daba Rabcd Fc d F a

Y a

1 a3n]

n , (3.5)

1 4cd c daab

Racd Z a bcd +

2 gccd c dbab bd

(Z a bcd

(2.21)

with

Y a1a2a3b1b2b3 = Fa1a2b

F =0, with

F F cdababRabcd F a b. (2.22)

Owing to the Bianchi identity of the Riemann tensor, the differential operator can only land on F, but not R, and

1 Fa3b2b3 . (3.6)

It is fairly straightforward to verify that the equations of motion are second order. The most dangerous term that can arise in the equations of motion is

[a1[b

1 Y a

The Maxwell equation is

2a3a4]

b2b3b4]

. (3.7)

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It is useful to note that

Fabc = 2[a Ab]c + c Aab. (3.8)

It then becomes obvious that (3.7) does not involve three or more derivatives.

As p increases, more and more possible ghost-free polynomial structures can be built. We shall not in this paper classify all such terms for general p-forms. It is also worth pointing out that in the construction, we can replace the p-form eld strength with the p-form potentials, whose kinetic term needs to be further introduced. The corresponding theory may also be ghost free. In particular the Einstein-vector theory was constructed in [29].

4 Electric and magnetic black holes

4.1 Static ansatz and reduced equations of motion

In this section, we focus on the low-lying four-derivative theory (2.18) in four dimensions, where the GaussBonnet term is a total derivative and hence irrelevant. We construct static black holes that carry electric and magnetic charges.The ansatz is given by

ds2 = hdt2 +

dr2f + r2d 22, ,

A = dt + p (1), d(1) = (2), (4.1) where p is a constant. The metric functions (h, f ) and the electrostatic potential are functions of r. The metric d 2n2, of the level surfaces is

d 22, =

dx21 x2 +

(1 x2) dy2. (4.2)

The topology parameter takes values of 1, 0, 1, for the

unit S2, the 2-torus or the unit hyperbolic 2-space. The 1-form (1) is simply (1) = xdy and (2) = dx dy is the

volume 2-form for the metric (4.2). With these conventions,

we see instantly that the ansatz carries the magnetic charge

Qm =

116 p (2) =

fh = 0. (4.5)

The rst integral can easily be obtained as a quadrature,

q8 ( f ) + r2

8 ( f ) + r2

hf , (4.6)

where q is an integration constant. This determines the electric charge, given by

Qe =

1 16

F01 =

16 q =

14q, (4.7)

where

F is dened in (2.22).

The Einstein equations (2.20) can now be reduced to one rst-order nonlinear differential equation and one quadrature:

f =

14(r4 2 p2)

4r3( f + 0r2)

q2r3

+ 8 ( f ) + r2 +

p2(48 f + r2)

h = u f

u u =

4r4 2 p2

3p2r

q2r3

8 ( f ) + r2

(4.8)

4.2 General properties

Much information can be extracted without solving Eqs. (4.8). The general solution is expected to be parametrized by three quantities, namely the mass and electric and magnetic charges, (14q, 14 p). The near-horizon geometry is then specied by the horizon radius r0, for which f (r0) = 0, and

(q, p). It follows from (4.8) that

h (r0) = u(r0) f (r0),

f (r0) =

r0 4(r402 p2)

4r20( 0r20) p2

(4.9)

The temperature of the black hole can then easily be determined by the standard technique:

T = h (r

0) f (r0) 4 =

q21 8 r20

14 p. (4.3)

Throughout this paper, we set, without loss of generality, the volume 2, of level surfaces to be independent of the topology, namely

2, = 4, for = 1, 0, 1. (4.4) For = 1, it is the true volume of the S2. For = 0 the

extensive quantities such as mass and charges are then density quantities per 4 area.

The ansatz (4.1) is the most general one for the static conguration with isometries of either S2, T 2 or H2. The

Maxwell equation (2.22) becomes

16 p =

f (r0)u(r0)

4 . (4.10)

The entropy can be obtained using the Wald entropy formula [30,31],

S =

1 8

dn2x h

(L/g)

Rabcd ab cd, (4.11)

which yields

S = r20

1 +

2 p2 r40

= r20

1 +

32 Q2m

r40

. (4.12)

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It is worth commenting that the Wald entropy formula is not always valid. It was shown to be invalid in EinsteinHorndeski gravity, owing to the unusual behavior of the scalar in black hole horizon [32,33]. In our charged black holes, however, the Maxwell eld behaves in a similar fashion to the ReissnerNordstrm (RN) black hole on the horizon and hence we expect that the Wald entropy formula holds in our black hole solutions.

The asymptotic region is less universal. For generic parameters, the large r expansions for f and u are

f =

13 0r2 +

and

f = u

(3p2 + 48 r2 16 0r4)u

48r2

(3 32 0)p216r2 2 F1

1 4,

1 4;

5 4;

2 p2 r4

. (4.19)

The solution becomes the usual magnetic RN black hole when = 0. For < 0, the curvature singularity is located

at r = 0 and hence there must be a horizon r = r0 > 0,

where r0 is the largest root of f . If > 0, there is an additional curvature singularity located at r = (2 p2)

14 , and we

2 p2r4 2 F1

1 4,

7 4;

2 p2 r4

112(32 0 3)p2 +

3q2

4(8 0 3)

r2 + ,

uu0 = 1

3 r4

p2 3q2

(8 0 3)2

+ (4.13)

This expansion becomes singular when

8 0 = 3, and q = 0. (4.14) As we shall see presently that the solution describes the z =

2 charged Lifshitz black hole for these special parameters (4.14).

It is worth commenting that as was shown in [34] for purely electric AdS planar black holes (p = 0 and = 0),

there is a global scaling symmetry whose conserved Noether charge is given by

QN =

1 4

must require that r0 > r. This implies

> (3 32 0)p

3 2

14 p (34)2

. (4.20)

(Analogous bound can be found in [33,35].) Once the event horizon r0 exists, the temperature and entropy are given by

T =

32 2

1 4

p2 + 4 r20 4 0r20 16r20(r40 2 p2)

1 4

, S = r20

1 +

2 p2 r40

2rh+r2h (r2+8 f ) +4 r f 2 .(4.15)

It is easy to verify that evaluating both on the horizon and asymptotic (A)dS innity yields

+ =

T S, S = r20,

(4.21)

The solution becomes extremal with T = 0 ifp2 = p2 4( 0)r20. (4.22)

For a general non-extremal black hole, we must have p p

so that the temperature is non-negative. It follows that the condition (4.20) can be satised provided that it is satised for the extremal solution for a given horizon radius r0. The mass and magnetic charge of the black hole are given by

M =

2, Qm =

32 M e Qe. (4.16)

The conservation of the Noether charge implies the following generalized Smarr relation:

M =

3(T S + e Qe). (4.17)

4.3 Exact general magnetic black holes

When q = 0, the ansatz (4.1) carries only magnetic charges.

In this case, the equations can be solved completely, given by

u = 1

2 p2 r4

20q =

14 p. (4.23) We do not have an independent way of determining the thermodynamical potential m for the magnetic charge, and we determine it by completing the rst law of the black hole thermodynamics,

dM = T dS + mdQm. (4.24) We nd a complicated expression:

m =

7r40 + 2 (p2 32 r20 + 32 0r40)

16r40(r40 2 p2)

1 4

2 p r30

2 F1

7 4;

2 p2 r40

. (4.25)

Although we determine the m using the rst law (4.24), the result is nontrivial since the rst law (4.24) involves two

3p16r0 (3 32 0)

2 F1

1 4;

5 4;

2 p2 r40

3 2

(4.18)

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Eur. Phys. J. C (2016) 76 :178 Page 7 of 11 178

independent parameters and hence a nontrivial integrability condition. To be specic, it is nontrivial in our case that (dM T dS) does not involve terms proportional to dr0,

which would make the rst law invalid.

4.4 Exact electric z = 2 Lifshitz black holes

When p = 0, the ansatz carries only the electric charge

Qe = 14q. We have not found the general exact solutions for

generic parameters. However, when

8 0 = 3, (4.26) we nd an exact solution for general q:

f = g2r2 +

general parameters. We nd two approximate solutions, one for small and the other for small charges (p, q).

4.5.1 Small- black holes

We rst present the small solutions. When = 0, the

solutions are the dyonic RN black holes. At the linear order of , we nd

f = f(1 + f) + O( 2), f= g2r2 +

r +

q2 + p2 4r2 ,

(q2 3p2)r4 + O( 2),

= 0

r0 +

u = 1 +

(7p2q2 + 3q3 20qr + 80g2qr4

10r5

+ O( 2), (4.32)

where

f =

7p2 q2

r2 + 42r , h = (r2 + 4) f,

= g(r2 r20), (4.27)

where the constant g is dened by 0 = 3g2 and r0 is the

location of the event horizon dened by f (r0) = 0. When

= 0 = q, the solution describes a Lifshitz vacuum of

z = 2, namely

ds2 = r2(g2r2 + )dt2 +

dr2g2r2 + +

2r4 +

c14r

(p2 + q2)(3p2 q2 20 r2)

40r6

3g2(3p2 q2) 2r2

r2 22, . (4.28)

(To be precise, the Lifshitz metric is given by = 0, in which

case the spacetime is homogeneous. For non-vanishing , the metric has a curvature singularity at r = 0.) The large-r

expansion of f is given by

f = g2r2 +

2 gq

f . (4.33)

For the small- approximation to be valid for all regions on and out of the horizon, f must be well-dened for r r0

where f(r0) = 0. This condition restricts the parameter c1,

namely

c1 =

(p2 + q2)(3p2 q2 20 r20)

10r50

6g2(3p2 q2) r0 .

gqr2 + . (4.29) It follows from [36] that the mass can be read off as M =

2 gq. The rst law of thermodynamics

dM = T dS + edQe (4.30) can easily be veried where the thermodynamics quantities are

T =

f (r0)h (r0)4 , S = r20,

M =

(4.34)

Now the solution describes a dyonic black hole for sufciently small . The asymptotic large-r expansion of the function f is given by

f = g2r2 +

r +

14q, e = g(r20 + 2). (4.31)

For = 0, one has generalized Smarr relation M = 12(T S +

e Qe). Note that this is different from the generalized Smarr relation for the AdS planar black holes (4.17).

4.5 Dyonic black holes

In four dimensions, the Maxwell eld in a black hole can carry both electric and magnetic charges, giving rise to dyonic solutions. We do not have exact solutions for such

2 gq, Qe =

p2 + q2 + 8g2 (4p2 q2)

4r2 + ,(4.35)

where = 14c1 . Thus the mass and electric and mag

netic charges are

M =

2, Qe =

14q, Qm =

14 p. (4.36)

The other thermodynamic quantities, up to the linear order, are given by

T =

4r20(3g2r20 ) p2 q2 16r20

(12g2r40 p2 + q2)(p2 + q2) + 4 (p2 3q2) 32r70

123

178 Page 8 of 11 Eur. Phys. J. C (2016) 76 :178

S = r20

1 +

2 p2 r40

, e =

For the expansion to be valid, the horizon r = r0 with f (r0) = 0 should not be altered. This implies that

c1 =

14 Q(r0)

q(7p2 + 3q2 20

r0 + 80g2r40)

10r60

p2(2 (7

8 r0 16g2r30) r30)

4r30

2 p

5(3g2r20 )r20 + p2 + 2q2 5r50

. (4.37)

It is then straightforward to verify that the rst law of black hole thermodynamics,

dM = T dS + edQe + mdQm, (4.38) is valid up to and including the linear order of .

The purely electric small- solution (p = 0) was obtained

in [37], where thermodynamical properties were analyzed using Euclidean action approach based on the quantum statistic relation (QSR) [38]. Our results disagree with this approach. Such a phenomenon also occurred in Einstein Horndeski gravity and it was suggested that the culprit is that the theory may not have a Hamiltonian formalism [32,33]. We expect that the same situation occurs here. Our example serves the further lesson that the QSR becomes problematic in theories with non-minimally coupled derivative matter elds.

4.5.2 Small charge black holes

An alternative approximation is to consider small charges. The leading-order solution is then the Schwarzschild black hole with

f = g2r2 +

m =

pr0 +

(4.42)

The thermodynamical quantities can now be easily calculated, given by

M =

2 =

+ c1), T =

f (r0)u(r0)

4 ,

S = r20

1 +

2 p2 r40

Qe =

14q, e =

Q(r0)

q ,

Qm =

14 p, m =

p((6 g2 + 1)r20 2 ) r30

. (4.43)

It is now straightforward to verify that the rst law (4.38) is indeed satised up to and including the quadratic order of the electric and magnetic charges.

5 AdS/CFT application: viscosity/entropy ratio

Having constructed theory and obtained many charged black hole solutions, we are in the position to discuss applications in the AdS/CFT correspondence. One such an application is that the AdS planar or Lifshitz black holes are dual to some ideal uid and the linear response of a graviton in the SO(2)-rotational invariant directions can be used to calculate the shear viscosity of the uid [39,40]. In two-derivative gravities, various arguments were given that the viscosity/entropy ratio is xed, given by

S =

r . (4.39) We nd that up to and including the quadratic order of electric and magnetic charges, the solutions are

f = f0(1 + f), u = 1 + , =

14 . (5.1) This value is no longer held in higher-derivative gravities [41]. There is no universal answer; it depends on the details of theories such as coupling constants, as well as the integration constants of the solution such as the mass and charges.

For higher-derivative gravities, there is typically the shortcoming in the literature that the results are applicable only for small coupling constants of the higher-derivative terms [37,4245]. This may be a consequence of two obstacles. One is that the higher-derivative theory is only dened for the small couplings, as in the case of perturbative string theory. The theory would have a ghost issue when treated on its own. This issue is resolved by our construction so that the theory can be ghost free. Another obstacle is that exact solutions may be lacking for higher-derivative gravities for general parameters. This is indeed the case for our theory. Although we have found many exact examples of special

f =

1 r f

c1 +

14 Q(r) +

p2(2 (7

8 r 16g

2r3) r

4r3

q2r2

u = 1 +

Q(r) 6

3 p2r4 +

((8g2 + 1)r

3 8

) ,

Q(r)q , (4.40)

where

Q(r) =

q2r

(8g2 + 1)r 3 8

= 123(

(8 g2 + 1)2)

arctan

1 + ( 8g

13 r

+33 log (1 + 8 g2)

)

3 log (8 g2 + 1)r3 8

. (4.41)

123

Eur. Phys. J. C (2016) 76 :178 Page 9 of 11 178

solutions, we do not have the exact solutions of the most general dyonic black holes for the generic parameters.

Recently a new technique was developed where the viscosity can be calculated without knowing the exact solutions [34]. This technique was developed mainly for two-derivative gravities. The key point of this technique is that AdS planar black holes or Lifshitz black holes have a scaling symmetry that gives rise to a Noether charge which relates the quantity on the horizon to that on the asymptotic innity. The consequence is a generalized Smarr relation, which can be viewed as the bulk dual to the boundary viscosity/entropy relation.Since the existence of the Noether charge associated with the scaling symmetry is independent of the number of derivatives of the theory, we nd that this technique can be adopted for our higher-derivative gravities as well. Thus although we do not have the general solutions for equations (4.8), the equations themselves are enough for us to determine the viscosity/entropy ratio.

To proceed, we set 0 = 3g2. It is important to note

that we are now dealing with the case = 0. It follows from

Eq. (4.8) that we have

f (r0) =

r0(3g2r40 p2 q2) 4(r40 2 p2)

Qe Qm = 0. (5.7) It turns out that the wave Eq. (5.5) can be analyzed without imposing the condition (5.7), and hence we shall thus proceed. Making a Fourier transformation in time,

(r, t) = eit(r), (5.8) we nd, near the horizon, that satises

(r r0)2 + (r r0) +

2162T 2 = 0. (5.9)

This equation can be solved exactly, implying

= 0e

i4T log(rr0) e

i4T log(

g2r2 )

0 . (5.10)

In other words, we select only the ingoing modes. To extend the horizon solution to asymptotic innity, we make the following ansatz:

= 0e

i4T log(rr0) e

i4T log(

g2r2 )

. (5.2)

1 i U(r) + O(2) , (5.11) where U should be regular on the horizon and vanish at the asymptotic innity. At the linear order of , we nd that the function U is a quadrature, given by

U =

r2(h + 2 f

The temperature is therefore

T =

r0(3g2r40 p2 q2)u(r0) 16(r40 2 p2)

hf ,

. (5.3)

To derive the shear viscosity, we consider the traceless and transverse perturbation on the metric,

d 22, =0 = dx21 + dx22 dx21 + dx22 + 2 (r, t) dx1dx2.(5.4)

The graviton mode (r, t) satises

h f

V = V0

r4 + 2 (8r2 f + q2) + 642 f 2

16T (r4 2 p

2)(r2 + 8 f )

2r4

2(r2 + 32 f ) u.(5.12)

In order for U to be regular on the horizon, we must have V (r0) = 0, which implies

V0 =

+(r2 + 8 f ) 12r

4(g2r2 f ) p

r(h f ) 4h f + 2 f (4h f ) r(h f 5 f h )

2r(h 2 f 2)

= 0,

(5.5)

(r40 2 q2)(12g2r40 p2 q2)u(r0)

16T r0(r40 2 p2)

= r20

1 +

2 q2 r40

together with the constraint

h 2

= 0. (5.6)

The constraint arises in the linearized Einstein equations in the diagonal (x1, x1) and (x2, x2) directions, while the wave

Eq. (5.5) arises in the off-diagonal (x1, x2) direction. The constraint (5.6) is automatically satised for general dyonic black holes in the EinsteinMaxwell theory, corresponding to = 0. For non-vanishing , the constraint is satised

only for either a purely electric solution or a purely magnetic solution, but not for the general dyonic solution, i.e. we need to impose

. (5.13)

To extract the information of the shear viscosity of the boundary eld theory, we consider the effective Lagrangian for , given by

116 r2

fh (h + 2 f 2) 2 + . (5.14) Thus the action can be evaluated,

1 16

=0(2) r2

fh (h + 2 f 2)

= i

r(h + 2 f 2)

16h f 4r f U +

r f 2 f T

+ O(2),

123

178 Page 10 of 11 Eur. Phys. J. C (2016) 76 :178

= i

14 V0 + O(2). (5.15)

The shear viscosity can then be read off:

14 V0 =

holes in four dimensions with isometries of S2, T 2, and H2. Although we do not have the most general exact solutions, we obtained many exact special ones, including the magnetic black holes and also electrically charged Lifshitz black holes with critical exponent z = 2. We then constructed

analytic approximate dyonic solutions with small charges or with small parameter . We studied the thermodynamics of the black holes and obtained the general rst law. An important lesson is that the rst law based on the Wald formalism disagrees with that from the Euclidean action procedure based on QSR. Such a phenomenon was rst observed in EinsteinHorndeski gravity and it was suspected that EinsteinHorndeski gravity may not admit a Hamiltonian formalism [32]. Our results suggest this may be a widespread situation for theories involving non-minimally coupled form elds.

We then studied an application of the theory in the AdS/CFT correspondence by deriving the boundary viscosity/entropy ratio for AdS or Lifshitz planar black holes. The purpose of our work is that higher-derivative terms in our theory do not have to be small and the theory can stand on its own right. The lack of the exact general solution appears to produce an obstacle to get general results for all allowed parameters. We nd that the viscosity/entropy ratio can be fully determined without the need to know the black hole solutions; the equations of motion sufce. We thus obtain the viscosity/entropy ratio for all parameters, including the coupling constant and electric and magnetic charges, none of which has to be small.

Form elds arise naturally in string and M-theory. They typically couple to gravity non-minimally in higher-order expansions of the low-energy effective theories of the perturbative strings. Our ghost-free construction makes it possible to treat the theories in nite order and study the theories on their own right. The explicit results of black holes and their certain AdS/CFT application in the low-lying example shows rich structures that deserve further investigation.

Acknowledgments The work is supported in part by NSFC Grants No. 11475024 and No. 11235003.

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

Web End =http://creativecomm http://creativecommons.org/licenses/by/4.0/

Web End =ons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP3.

14r20 1 +

32 Q2e

r40

. (5.16)

It follows from the denition of the entropy (4.12) that the viscosity/entropy ratio is then given by

S =

1 4

r40 + 32 Q2e

r40 + 32 Q2m

. (5.17)

Thus we obtain the ratio without using any exact solution. Owing to the constraint (5.7), the result is applicable for all purely electric or purely magnetic black holes, for all ranges of where a black hole exists.

The viscosity (5.16) was obtained also in [37] for the small parameter for which the approximate solution was found. Our general result conrms this. However, our viscosity/entropy ratio (5.17) disagrees with [37] even for vanishing Qm and small . This is because the entropy in [37]

was obtained using the Euclidean action procedure, which we believe is invalid in this theory. It should also be emphasized that as we mentioned in the introduction, higher-derivative gravities in general give a bound on the coupling constants such as due to the causality consideration. This leads to a further constraint on the allowed value for the viscosity/entropy ratio. (See, for examples, [22,23,46,47].) We expect the analogous causality bound also exists in our case.

6 Conclusions

In this paper we constructed higher-derivative gravities with a non-minimally coupled Maxwell eld A(1) = Adx. The

general Lagrangian consists of invariant polynomials built from the Riemann tensor and the eld strength F(2) = dA(1).

These polynomials are analogous to the Euler integrands in Lovelock gravities in that the eld equations of motion remain of second order for both the metric and A. The total higher derivatives are achieved through nonlinearity. The linearized equations of motion in any background involve only two derivatives and hence the theories can be ghost free. We also generalize the construction to involve a generic non-minimally coupled p-form eld strength. We noted that as p increases, more and more invariant polynomials could be constructed to give rise to ghost-free theories. However, we did not classify all possible structures.

As an application in black hole physics, we focused on a low-lying example in which the EinsteinMaxwell gravity with a cosmological constant was augmented by a polynomial built from the Riemann tensor and bilinear F(2), with a coupling constant . We constructed charged static black

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(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

We construct higher-derivative gravities with a non-minimally coupled Maxwell field. The Lagrangian consists of polynomial invariants built from the Riemann tensor and the Maxwell field strength in such a way that the equations of motion are second order for both the metric and the Maxwell potential. We also generalize the construction to involve a generic non-minimally coupled p-form field strength. We then focus on one low-lying example in four dimensions and construct the exact magnetically charged black holes. We also construct exact electrically charged ...... Lifshitz black holes. We obtain approximate dyonic black holes for the small coupling constant or small charges. We find that the thermodynamics based on the Wald formalism disagrees with that derived from the Euclidean action procedure, suggesting this may be a general situation in higher-derivative gravities with non-minimally coupled form fields. As an application in the AdS/CFT correspondence, we study the entropy/viscosity ratio for the AdS or Lifshitz planar black holes, and find that the exact ratio can be obtained without having to know the details of the solutions, even for this higher-derivative theory.

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Title

Higher-derivative gravity with non-minimally coupled Maxwell field

Author

Feng, Xing-hui; Lü, H

Pages

1-11

Publication year

2016

Publication date

Mar 2016

Publisher

Springer Nature B.V.

ISSN

14346044

e-ISSN

14346052

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1140/epjc/s10052-016-4007-y

ProQuest document ID

1776784334

SIF and Springer-Verlag Berlin Heidelberg 2016

Higher-derivative gravity with non-minimally coupled Maxwell field

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