Lifshitz black holes in Brans-Dicke theory

0.28 MB·English

by  Hideki Maeda

January 20, 2013
Lifshitz black holes in Brans-Dicke theory
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Hideki Maedaa and Gaston Giribetb
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q a Centro de Estudios Cient´ıficos (CECs), Casilla 1469, Valdivia, Chile
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r b Instituto de F´ısica de Buenos Aires, CONICET, Buenos Aires, Argentina.
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[ hideki-at-cecs.cl, gaston-at-df.uba.ar

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Abstract
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We present an exact asymptotically Lifshitz black hole solution in Brans-Dicke
1
: theory of gravity in arbitrary n( 3) dimensions in presence of a power-law
v ≥
potential. In this solution, the dynamical exponent z is determined in terms of
i
X
the Brans-Dicke parameter ω and n. Asymptotic Lifshitz condition at infinity
r
a requires z > 1, which corresponds to (n 1)/(n 2) ω < n/(n 1). On
− − − ≤ − −
the other hand, the no-ghost condition for the scalar field in the Einstein frame
requires 0 < z 2(n 2)/(n 3). We compute the Hawking temperature of
≤ − −
the black hole solution and discuss the problems encountered and the proposals
in defining its thermodynamic properties. A generalized solution charged under
the Maxwell field is also presented.
Contents
1 Introduction 2
2 Preliminaries 5
2.1 Brans-Dicke theory with a Maxwell field . . . . . . . . . . . . . . . . . . . 5
2.2 Ansatze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Decomposed field equations . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Exact asymptotically Lifshitz vacuum black hole solutions 9
3.1 Causal structure of the Lifshitz spacetime . . . . . . . . . . . . . . . . . . 9
3.2 Lifshitz solution without potential . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Asymptotically Lifshitz solution with power-law potential . . . . . . . . . . 13
3.4 Thermodynamical properties of the Lifshitz black hole . . . . . . . . . . . 15
4 Lifshitz black holes with Maxwell field 18
4.1 Solution for the Maxwell equation . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Asymptotically Lifshitz solutions with Maxwell field . . . . . . . . . . . . . 19
5 Summary 21
A Generalized Brans solution contains (asymptotically) Lifshitz solutions 23
A.1 Generalized Brans solution without potential . . . . . . . . . . . . . . . . . 23
A.2 Generalized Brans solution with potential . . . . . . . . . . . . . . . . . . 25
B No Lifshitz solution in the Einstein frame with a real scalar field 26
1
C Solution with z = 1 28
1 Introduction
One of the most interesting ideas that have been proposed in the last years is that of ex-
tending the AdS/CFT holographic duality [1, 2, 3] to the area of condensed matter physics.
This idea has attracted considerable attention, and much effort has been made to construct
holographic gravity duals for such systems [4, 5, 6, 7]. The reason why this area has be-
came one of the most active ones in the field is that it raises the hope to explicitly work out
the details of the non-perturbative dynamics of interesting scale invariant non-relativistic
models. Among the class of models one could try to study with this holography inspired
methodology, there are systems of strongly correlated electrons, which have received a re-
newed interest recently and permit to address questions that are accessible experimentally.
In particular, for the case of systems exhibiting anisotropic scale invariance, but not
Galileaninvariance, there exists a concise proposal for their holographiccounterparts. Such
models, known as Lifshitz fixed points, present symmetry under
t L2zt , xi L2xi. (1.1)
→ →
where L is the scale parameter and xi are the spatial coordinates i = 1,2,...d; here we will
use the convention d = n 2. The time and the spatial coordinates suffer from different
−
scaling transformations, and this is characterized by the value of the dynamical exponent
z.
Among the physical systems that exhibit symmetry under (1.1) one finds several strongly
correlated models in condensed matter, like the Rokhsar-Kivelson dimer model in d = 2
dimensions. Critical points of this kind were also conjectured to exist in d = 1; such is
the case of the transition from a Luttinger liquid to a ferromagnetic state. Being strongly
coupled models, having a tool to compute observables, such as correlation functions, is
clearly of importance. For instance, the question as to whether the previously reported
ultralocal behavior of correlation functions in z = 2 models manifest itself or not could be
addressed if a holographic calculation of these observables was at hand.
In[4],thespacetimecandidatestorepresentgravitydualsforthistypeof(n 1)-dimensional
−
scale invariant fixed points were proposed. The metrics of these n-dimensional spacetimes
naturally depend on the dynamical exponent z of the model they want to be in correspon-
dence with. The metric takes a simple form
r2z l2
ds2 = dt2 + dr2 +r2dxidx , (1.2)
−l2z r2 i
2
where xi are coordinates of the d = n 2 flat directions, with i = 1,2,...n 2, and where
− −
l and z are real parameters. Metric (1.2) generalizes that of n-dimensional Anti-de Sitter
spacetime (AdS ), which corresponds to the special case z = 1.
n
Then, (1.2) happens to realize the symmetry transformation (1.1) geometrically, provided a
similar transformation on the bulk coordinate r is performed as well; namely r L 2r. In
−
→
the holographic realization, the dual scale invariant theory would be formulated on a n 1-
−
dimensional space that can be regarded as being located at infinite r, say the boundary,
whose physical coordinates are the time t and the n 2 spatial directions xi. This is totally
−
analogtoAdS/CFT, andithappenstocoincidewithitwhenz = 1. Infact, theholographic
recipe follows the lines of the standard AdS/CFT prescription of [2, 3]. In particular, one
can compute the scaling dimensions of the dual fields in the boundary theory by studying
the asymptotic dynamics of fields in the bulk. For example, if one considers a scalar field
ϕ of mass m in the bulk, and one demands suitable boundary conditions obeying a specific
falling-off
ϕ(t,x,r) ϕ (t,x) r ∆ + (r ∆ 1), (1.3)
0 − − −
∼ O
it is easy to derive the scaling dimension ∆ of the field O dual to ϕ, which turns out to be
ϕ
1
∆ = ((z +n 2) (z +n 2)2 +4m2l2 ). (1.4)
± 2 − ± −
p
In the original paper [4], the two-point function of operators O was calculated in the case
ϕ
of d = 2 Lifshitz point with z = 2. This function was shown to have the expected functional
dependence
O (x ,t ) O (x ,t ) x x 8. (1.5)
ϕ 1 1 ϕ 2 1 1 2 −
h i ∼ | − |
Notice that for n = 4 and z = 2 (1.4) actually gives ∆ = 4.
+
In [12], on the other hand, the system at finite temperature T was studied as well, and it
was shown there that, in the regime of large separation, the expected exponential damping
induced by the finite temperature effects arises; namely
O (x ,t ) O (x ,t ) e √4πT x1 x2 x x 3/2; (1.6)
ϕ 1 1 ϕ 2 1 − | − | 1 2 −
h i ∼ | − |
while in the short distance regime the limit (1.5) is consistently recovered, and no evidence
of ultralocality was found.
Also resembling the standard AdS/CFT correspondence, adding finite temperature to the
problem amounts to replace the Lifshitz metric (1.2) by a black hole solution that asymp-
totestoit. Then, theHawkingtemperatureT ofsuchaLifshitzblackholewouldcorrespond
to the temperature of the boundary scale invariant dual model. Nevertheless, an immediate
obstruction appears here: Because of the form of the metric (1.2) and the needs of holo-
graphic realization, such an asymptotically Lifshitz black hole would be a static spherically
symmetric solution with a z-dependent asymptotic, and, usually, Birkhoff-like theorems
3
and no-hair theorems prevent such spaces from appearing in natural models. In fact, the
Lifshitz black holes found so far use to exist in models whose matter contents correspond
to contrived stress-energy tensors. A manifestation of this is how specific the fine tuning of
the coupling constants in the models analyzed in the literature has to be for Lifshitz black
holes to be admitted as exact solutions.
Lifshitz black holes were extensively discussed in the literature in the last two years, and
both analytic and numerical solutions were found [8, 9, 10, 11, 12]. Models involving
Einstein-Hilbert action coupled to matter were analyzed, together with models consisting
of higher-curvature modifications [13, 14, 15, 16, 17, 18, 19, 20, 21, 13, 22, 23, 24]. Charged
Lifshitz black holes were also reported, and their thermodynamics and consistency studied
[25, 26, 27, 28, 29, 30, 31].
In this paper, we will be concerned with Lifshitz black holes in Brans-Dicke theory. Brans-
Dicke theory is the prototype of all the scalar-tensor theories of gravity and parametrized
by the so-called Brans-Dicke parameter ω [32]. This theory was originally formulated in
four dimensions and the theory with sufficiently large ω passes all the observational and
experimental tests for gravitation theories [33]. (See [34] for the relation between Brans-
Dicke theory and general relativity.) This theory is also interesting theoretically because
it contains the dilatonic theory (ω = 1) or f(R) gravity (ω = 0 in the presence of a
−
potential) [35]. In the asymptotically flat case in vacuum, Hawking showed a “no-scalar-
hair theorem” claiming that a black hole cannot support a non-trivial Brans-Dicke scalar
field if the scalar field is regular on and outside the event horizon [36]. Therefore, in order
to have a hairy black hole configuration in Brans-Dicke theory, one has to consider (I) non-
asymptotically flat spacetime, (II) non-regular scalar field on the horizon, (III) potential
for the scalar field, or (IV) other matter fields [37, 38].
We will show that asymptotically Lifshitz solutions exist in this theory provided the scalar
field potential to have a specific, though simple, functional form. The motivation is to show
that analytic Lifshitz black hole solutions with a continuous and finite range of z, and in
arbitrary number of dimensions n do exist in a familiar field theory, where the value of z
is determined by ω and n. Here, we report black holes with or without a Maxwell field,
whose metrics can be regarded as generalizations of solutions previously reported in the
literature.
The paper is organized as follows. In section II, we present our assumptions and derive the
basic equations. In section III, we derive an exact asymptotically Lifshitz vacuum solution
and discuss its properties. In section IV, we generalize the solution in the presence of
a Maxwell field. Our conclusions and future prospects are summarized in section IV. In
appendix A, we present a vacuum solution which is a generalization of the Brans solution in
the four-dimensional spherically symmetric spacetime and show that it certainly contains
the (asymptotically) Lifshitz vacuum solution. In appendix B, we show that there is no
Lifshitz solution in the Einstein frame with a real scalar field. In appendix C, the solution
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for the AdS case (z = 1) is presented. Our basic notations follow [39]. The conventions of
curvature tensors are [ , ]Vµ = Rµ Vν and R = Rρ . The Minkowski metric is
ρ σ νρσ µν µρν
∇ ∇
taken to be the mostly plus sign, and Roman indices run over all spacetime indices.
2 Preliminaries
2.1 Brans-Dicke theory with a Maxwell field
In this paper, we consider Brans-Dicke theory with a Maxwell field in arbitrary n( 3)
≥
dimensions. We include a potential V(φ) for a Brans-Dicke scalar field φ in the action,
which is given by
1 ( φ)2 1
I = dnx√ g φR ω ∇ V(φ) F Fµν , (2.1)
− 16π − φ − −ζ2 µν
Z (cid:20) (cid:18) (cid:19) (cid:21)
where( φ)2 := φ µφ,ω istheBrans-Dickeparameter, andζ isarealcouplingconstant.
µ
∇ ∇ ∇
F := ∂ A ∂ A is the Faraday tensor defined by the gauge potential Aµ, The field
µν µ ν ν µ
−
equations given from this action are
8π
G = (T(φ) +T(em)), (2.2)
µν φ µν µν
Fµν =0, (2.3)
ν
∇
1 dV 16π(n 4)
φ = (n 2)φ nV − F Fρσ , (2.4)
2(n 2)ω +2(n 1) − dφ − − ζ2 ρσ
− − (cid:18) (cid:19)
where φ := µφ, G := R (1/2)g R is the Einstein tensor, and
µ µν µν µν
∇ ∇ −
1 ω 1 1
T(φ) := φ φ g ( φ)2 Vg +( φ g φ) , (2.5)
µν 8π φ ∇µ ∇ν − 2 µν ∇ −2 µν ∇µ∇ν − µν
(cid:20) (cid:18) (cid:19) (cid:21)
4 1
T(em) := F ρF g F Fρσ . (2.6)
µν ζ2 µ νρ − 4 µν ρσ
(cid:18) (cid:19)
The trace of the gravitational equation (2.2) gives
( φ)2 n 2(1 n) φ 16π(4 n)
R =ω ∇ V + − + − F Fµν. (2.7)
φ2 − (2 n)φ 2 n φ (2 n)ζ2φ µν
− − −
As seen in the action (2.1), the inverse of φ is the effective gravitational constant and
therefore we assume that φ is positive. The action with ω = 1 gives dilatonic gravity
−
5
shown as follows. By defining Φ := lnφ, we obtain the gravitational part of the action
−
(2.1) with ω = 1 in the form of
−
1
I = dnx√ ge Φ R+( Φ)2 W(Φ) , (2.8)
−
16π − ∇ −
Z (cid:18) (cid:19)
W(Φ) :=eΦV(φ). (2.9)
This action with W(Φ) = 0 and n = 10 appears as the dilaton-graviton sector in the
low-energy action of string theories [40].
The action (2.1) for (g ,φ), which is called the Jordan frame, can be conformally trans-
µν
formed into the Einstein frame for (gˆ ,ψ), where ψ is a minimally coupled scalar field. By
µν
the conformal transformation g = Ω(ψ)2gˆ with
µν µν
Ω(ψ) =(G φ)1/(2 n), (2.10)
n −
8πG (n 2)
n
φ =exp − (ψ ψ ) , (2.11)
0
±s(n 1)+(n 2)ω −
(cid:18) − − (cid:19)
where G and ψ are constants, the gravitational action in the Jordan frame (2.1) is trans-
n 0
formed into the Einstein frame as
1 1
Iˆ= dnx gˆ Rˆ (ˆψ)2 Vˆ(ψ) , (2.12)
− 16πG − 2 ∇ −
Z (cid:20) n (cid:21)
p
where ˆ is covariant derivative in the Einstein frame and the potential in the Einstein
∇
frame is given by
1
Vˆ(ψ) = ΩnV(φ),
16π
1
= (G φ)n/(2 n)V(φ). (2.13)
n −
16π
It is seen that G is the n-dimensional gravitational constant in the Einstein frame. If the
n
Brans-Dicke scalar field is real, ψ is real only if
n 1
ω − . (2.14)
≥ −n 2
−
ψ becomes a ghost scalar field for ω < (n 1)/(n 2). In the present paper, we impose
− − −
the condition (2.14) for the physical theory.
2.2 Ansatze
In this paper we consider an n-dimensional spacetime ( n,g ) which is a warped product
µν
M
of an (n 2)-dimensional maximally symmetric space (Kn 2,γ ) and a two-dimensional
− ij
−
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orbit spacetime (M2,g ) under the isometries of (Kn 2,γ ). The line element in this
ab − ij
spacetime may be written as
g dxµdxν = g (y)dyadyb+r2(y)γ (z)dzidzj, (2.15)
µν ab ij
where a,b = 0,1 while i,j = 2,...,n 1. Here r is a scalar on (M2,g ) and γ is the metric
ab ij
−
on (Kn 2,γ ) with its sectional curvature k = 1,0.
− ij
±
The (n 2)-dimensional maximally symmetric space satisfies
−
(n 2)
R− = k(γ γ γ γ ). (2.16)
ijkl ik jl il jk
−
The superscript (n 2) means that the geometrical quantity are defined on (Kn 2,γ ).
− ij
−
The non-zero components of the Einstein tensor are given by
D D r (D2r)g (n 2)(n 3) k (Dr)2
a b ab
G = (n 2) − − − g − , (2.17)
ab − − r − 2 ab r2
(cid:18) (cid:19)
1 k (Dr)2 (D2r) (2)
Gi = (n 3)(n 4) − 2(n 3) + R δi , (2.18)
j − 2 − − r2 − − r j
(cid:20) (cid:18) (cid:19) (cid:21)
Here D is a metric compatible linear connection on (M2,g ), (Dr)2 := gab(D r)(D r),
a ab a b
and D2r := DaD r. The contraction was taken over on the two-dimensional orbit space
a
and (2)R is the Ricci scalar on (M2,g ).
ab
Inthis symmetric spacetime, φis a scalar on(M2,g ). We obtainthenon-zero components
ab
of µ φ as
ν
∇ ∇
D r
a φ = DaD φ, i φ = a Daφδi (2.19)
∇ ∇b b ∇ ∇j r j
and then
D r
φ = D2φ+(n 2) a Daφ. (2.20)
− r
The non-zero components of T(φ)µ are given by
ν
1 ω 1 1 D r
T(φ)a = DaφD φ δa (Dφ)2 Vδa +DaD φ δa D2φ+(n 2) d Ddφ ,
b 8π φ b − 2 b −2 b b − b − r
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
(2.21)
1 ω 1 D r
T(φ)i = (Dφ)2 V D2φ (n 3) d Ddφ δi . (2.22)
j 8π −2φ − 2 − − − r j
(cid:20) (cid:21)
We assume that the electromagnetic field has the form
A dxµ = A (y)dya+A (z)dzi, (2.23)
µ a i
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which in turn implies that the Faraday tensor reads
F dxµ dxν = F (y)dya dyb+F (z)dzi dzj. (2.24)
µν ab ij
∧ ∧ ∧
Here F (y) and F (z) are identified with the electric and magnetic components, respec-
ab ij
tively. For the magnetic component, the compatibility with the field equations requires the
following form;
γklF F = Q2 γ , (2.25)
ik jl m ij
where Q is a real constant [41, 42]. The Maxwell invariant scalar is given as
m
(n 2)Q2
F Fµν = F Fab + − m, (2.26)
µν ab r4
while the non-zero components of T(em)µ are given by
ν
1 (n 2)Q2
T(em)a = F Fcd − m δa , (2.27)
b ζ2 cd − r4 b
(cid:18) (cid:19)
1 (n 6)Q2
T(em)i = F Fcd + − m δi , (2.28)
j − ζ2 cd r4 j
(cid:18) (cid:19)
where we used F Fcd = 2F F01.
cd 01
2.3 Decomposed field equations
Now we are ready to derive the decomposed field equations. The gravitational equation
(2.2) gives
DaD r (D2r)δa (n 2)(n 3) k (Dr)2
(n 2) b − b − − δa −
− − r − 2 b r2
(cid:18) (cid:19)
1 ω 1 1 D r
= DaφD φ δa (Dφ)2 Vδa +DaD φ δa D2φ+(n 2) d Ddφ
φ φ b − 2 b −2 b b − b − r
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
8π (n 2)Q2
+ F Fcd − m δa , (2.29)
ζ2φ cd − r4 b
(cid:18) (cid:19)
1 k (Dr)2 (D2r) (2)
(n 3)(n 4) − 2(n 3) + R
− 2 − − r2 − − r
(cid:20) (cid:18) (cid:19) (cid:21)
1 ω 1 D r 8π (n 6)Q2
= (Dφ)2 V D2φ (n 3) d Ddφ F Fcd+ − m ,
cd
φ −2φ − 2 − − − r −ζ2φ r4
(cid:20) (cid:21) (cid:18) (cid:19)
(2.30)
8
while the scalar field equation (2.4) reduces to
D r 1 dV
D2φ+(n 2) a Daφ = (n 2)φ nV
− r 2(n 2)ω +2(n 1) − dφ −
− − (cid:20)
16π(n 4) (n 2)Q2
− F Fab + − m . (2.31)
− ζ2 ab r4
(cid:18) (cid:19)(cid:21)
The Maxwell equation (2.3) reduces to
D Fab = 0, D¯ Fij = 0, (2.32)
b j
where D¯ is a metric compatible linear connection on (Kn 2,γ ). By eliminating the
j − ij
potential V from Eqs. (2.29) and (2.30), we obtain the following useful equation without
the potential term;
(D2r) k (Dr)2 (2)
(n 4) 2(n 3) − + R
− − r − − r2
(cid:18) (cid:19)
1 ω D r 32π 2Q2
= (Dφ)2 +D2φ 2 d Ddφ + F Fab m . (2.33)
ab
φ φ − r ζ2φ − r4
(cid:20) (cid:21) (cid:18) (cid:19)
The decomposed field equations (2.29)–(2.33) are the basic equations in this study.
3 Exact asymptotically Lifshitz vacuum black hole so-
lutions
(em)
In this section, we consider the vacuum case, namely the case with T 0. Hereafter,
µν
≡
we focus our attention on the case with k = 0 because it allows for Lifshitz solution.
3.1 Causal structure of the Lifshitz spacetime
To begin, let us clarify the basic properties of the background Lifshitz spacetime. The
Lifshitz spacetime is defined by
r2z l2
ds2 = dt2 + dr2 +r2dxidx , (3.1)
−l2z r2 i
where z and l are real constants and l is assumed to be positive. dxidx is the line element
i
of the (n 2)-dimensional flat space. We note that the constant l cannot be set to one
−
by rescaling of the coordinates. The curvature invariants are all constant in this space-
time, however, some of the components of the Riemann tensor in the parallelly propagated
9