Exact hairy black holes asymtotically AdS21

2025-04-30 0 0 678.42KB 17 页 10玖币

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Exact hairy black holes asymptotically AdS2+1

Carlos Desa(1), Weyner Ccuiro(1) and David Choque(1)

(1)Universidad Nacional de San Antonio Abad del Cusco,

Av. La Cultura 733, Cusco, Per´u.

January 2, 2023

Abstract

In the context of Einstein’s minimally coupled scalar ﬁeld theory, we present a family of hairy black

holes which are asymptotically AdS2+1. We investigate the boundary conditions and build the thermal

superpotential. Two methods are used to regularize the free energy in the Euclidean section and the Brown-

Newton tensor in the Lorentzian section. Finally, the relevant thermodynamic quantities are calculated and

the diﬀerent phases are analyzed.

Contents

1 Introduction 2

2 Theory 3

3 Asymptotically AdS solution 3

3.1 On-shellscalarpotential......................................... 4

3.2 Thermalsuperpotential ......................................... 6

3.3 Asymptotic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 On-shell ﬁnite action 7

5 Holographic stress tensor 9

6 Thermodynamics 10

7 Discussion 12

A Integration of the scalar potential 13

B Asymptotically ﬂat solution 13

C Schwarzschild AdS2+1 14

References 15

arXiv:2210.06421v4 [hep-th] 29 Dec 2022

1 Introduction

In accordance with the no-hair theorem [1,2], in three dimensions there are no asymptotically ﬂat black hole

solutions for pure gravity. Indeed, the Riemann tensor is purely a function of the Ricci tensor and metric

Rαβγσ(Rµν , gµν ) on this manifold. Then, no solutions emerge unless the Einstein-Hilbert action is supplemented

with a negative cosmological constant, as considered by Ba˜nados, Teitelboim and Zanelli (BTZ), who ﬁrst found

the BTZ black hole solution [3,4], which have been extensively studied [5–10]. The procedure for overcome the

no-hair theorem consists the coupling the other ﬁelds to Einstein-Hilbert action minimally, non-minimally or

conformally [2,11–16]. A further way to circumvent the no-hair theorem is by considering higher-derivative

gravities [17–19], this has been successfully done in D= 3 + 1 dimensions [20–26].

Here we show the construction of the hairy family black hole solution. We begin by integrating the metric

functions, then the scalar ﬁeld φ, and ﬁnally the scalar potential V(φ). This method is the on-shell procedure,

and it was extensively applied in [27–30]. In summary, we present a family of hairy black hole solutions in

three dimensions with a non-trivial scalar potential minimally coupling. This scalar ﬁeld contributes to the

metric and, therefore, to the thermodynamics of the black hole. In [31] was discovered the existence of the ﬁrst

order Hawking-Page phase transitions and their respective dual interpretations [32], based on the conjecture

AdS/CF T [33–35], this led to a great deal of activity in research on black hole phase transitions [36–40]. It

is known that the scalar hair can have relevant eﬀects on phase diagrams. Here [36,41,42] they have shown a

window of parameters in which the asymptotically ﬂat, hairy black holes are stable. These last results encour-

age us to study the phase diagrams of our family of hairy black holes. In general, the most interesting phase

diagrams can be found in the context of V dP formalism [41,43–45], but we focused on phase diagrams with a

ﬁxed cosmological constant. [36,37,40,42,46]

Recently, hairy solutions in D= 3 + 1 was embed in omega-deformed gauged N= 8 supergravity [47], and

the explicit construction of the hairy solution from N= 2 supergravity was demonstrated [48,49]. These last

results open the possibility of ﬁnding the supergravity origin of our solution. Many solutions with diﬀerent

conformal masses can be found in the literature [50–52], in our case, we have an interesting example with a

conformal mass m2=−1/L2that saturates the Breitenlohner-Freedman bound m2

BF =−1/L2.

The well-posed variational principle ensures the construction of the counterterms for the scalar ﬁeld, allow-

ing us to obtain the ﬁnite on-shell action[53]. In the context of supergravity theories, the theorem of positive

energy is ensured by the existence of a superpotential[54–56], curiously, in [57] they proposed a superpotential

as a counterterm; the problem is that in general, we cannot construct this superpotential exactly. The goal of

[58] was to develop a thermal superpotential that considered the black hole horizon’s existence. This thermal

superpotential is easier to construct, and similarly to [49,59] we use it to regularize the on-shell action and the

Brown-York quasilocal stress tensor [60]

We structure this work in the following way. Section 2describes the theory’s action, as well as the poten-

tial V(φ) and its stability conditions, which allow us to avoid the non-hair theorem. We show an exact hairy

solution in the section 3; we also consider the horizon existence conditions, the thermal superpotential, and

the asymptotic AdS boundary symmetry for the scalar ﬁeld. In the middle part, section 4, we get the ﬁnite

on-shell action, and the goal is the construction of the counterterm for the scalar ﬁeld. In the section 5we build

the quasilocal stress tensor properly regularized by two diﬀerent methods. In the last two sections, 6and 7we

study the thermodynamics, show the conclusions, and discuss future directions.

2 Theory

In D= 2 + 1 dimensions, we are interested in modiﬁed Einstein-Hilbert actions with a non-minimal coupling

scalar ﬁeld.

I[gµν , φ] = 1

2κZM

d3x√−gR−(∂φ)2

2−V(φ)+1

κZ∂M

d2xK√−h+Iφ(1)

where V(φ) is the scalar potential, κ= 8πGN, and the last term is the Gibbons-Hawking boundary term. The

boundary metric is hab, and the trace of the extrinsic curvature is K. Then the equations of motion for the

metric are

Eαβ =Gαβ −Tαβ = 0 (2)

Tαβ =1

2∂αφ∂βφ−gαβ

2(∂φ)2

2+V(φ)(3)

where the Einstein’s tensor is Gαβ := Rαβ −Rgαβ /2. And the Klein-Gordon equation for the scalar ﬁeld is

√−g∂α√−ggαβ ∂βφ−∂V

∂φ = 0 (4)

3 Asymptotically AdS solution

We consider the D= 2+1 version of the metric which is static and radially symmetric [61]. The non-dimensional

radial coordinate is xand we will show that we have two branches, x∈(0,1) and x∈(1,∞). Each has a distinct

singularity at x= 0 and x=∞, but they share a boundary at x= 1

ds2= Ω(x)−f(x)dt2+η2dx2

f(x)+dϕ2(5)

Einstein’s equations are

t−Ex

x= 0 ⇒(φ0)2=3Ω02−2ΩΩ00

2Ω2(6)

t−Eϕ

ϕ= 0 ⇒(Ω1/2f0)0= 0

t+Eϕ

ϕ= 0 ⇒V=−1

4η2Ω(x)32Ω00fΩ+2f00Ω2−fΩ02+ 3ΩΩ0f0

The conformal factor Ω(x) was constructed in [27–30], and from it we can integrate f(x) and φ(x). This

conformal factor blowing up at the boundary x= 1, and ηis a constant integration related to the hairy black

hole’s mass

Ω(x) = ν2xν−1

η2(xν−1)2(7)

Ω(x) is the input function, and from (Ω1/2f0)0= 0 we can integrate f(x), with new a constant α.1. There is

no horizon when the theory parameter αis null, as shown in (8); that is, solutions with α= 0 have a naked

singularity.2. Another property is that f(ν) = f(−ν) and Ω(ν) = Ω(−ν), so we can work with ν≥1 without

losing generality

f(x) = 1

L2+α

2ν2ν

ν2−9−1

(3 + ν)x3+ν

2+1

(3 −ν)x3−ν

2(8)

The scalar ﬁeld φ(x) has the simple form (9), where `is the dilaton length. Clearly, if ν= 1, the non-hair limit

is obtained, and the scalar ﬁeld is φ(x) = 0. The scalar ﬁeld is null φ= 0 at the boundary x= 1, and it is

blowing up near the singularity x= 0 or x=∞

φ(x) = `−1ln x, `−1=√2

2pν2−1 (9)

1Actually, αis not a constant integration. Considering f(x) and Ω(x) in Et

t+Eϕ

ϕ= 0 We obtain a scalar potential that is

dependent on α, and we can identify αas a constant of the theory. For details, see the next subsection 3.1

2We obtain the asymptotically ﬂat solution at the limit L!∞; see the appendix B

The existence of the horizon f(xh, α, ν) = 0. is ﬁxed by the correct choice of the theory’s parameters (α, ν),

which are related to the convexity condition on V(φ),3i.e. the non-hair theorem; see Figure (2). We are going

to study the existing conditions on the horizon for each branch; see Figure (1):

Negative branch

The scalar ﬁeld is deﬁnite negative φ≤0 in the region 0 ≤x < 1, with the singularity at x= 0 and the

boundary at x= 1

•For: ν > 3 with x!0 the existence of the horizon f(xh) = 0 is ensured by f(x)<0

lim

x!0f(x)∼ − αL2

2ν(ν−3)x−(ν−3)

2<0⇒αL2>0 (10)

•For: 1 ≤ν < 3 with x!0 the existence of the horizon f(xh) = 0 is ensured by f(x)<0

lim

x!0f(x)∼1

L2−α

9−ν2<0⇒0<9−ν2< αL2(11)

Positive branch

The scalar ﬁeld is deﬁnite positive, φ≥0, in the region 1 ≤x < ∞, where the singularity is at x=infty and

the boundary is at x= 1

•The case 1 ≤ν < 3 and ν > 3 with x!∞the horizon existence f(xh) = 0 is ensured by f(x)<0

lim

x!∞f(x)∼ − α

2ν(3 + ν)x3+ν

2<0⇒αL2>0 (12)

3.1 On-shell scalar potential

In this section, we build our theory’s on-shell scalar potential, V(φ). We use (7), (8), and simplify to get V(x)

in the third equation (6), for more details, see the appendix A

V(x) = −3

4ν21

L2+α

ν2−9x−√2ν2−2

2(1 + ν)1 + ν

3x√2ν2−2

2+ (1 −ν)1−ν

3x−√2ν2−2

−2(1 −ν2)−α·x√2ν2−2

ν(ν2−9) (1 −ν)x√2ν2−2

4−(1 + ν)x−√2ν2−2

4(13)

The correct form of the scalar potential is V(φ) rather than V(x), and we get x(φ) from (9)

φ(x) = `−1ln x⇒x=e`φ (14)

Finally, we obtain the theory’s scalar potential V(φ) with parameters α, Λ = −2/L2, and ν. Another interesting

construction was present in [47,48,62,63]

V(φ) = −3 exp(−φ`)

4ν21

L2+α

ν2−9(1 + ν)1 + ν

3exp (φ`ν) + (1 −ν)1−ν

3exp (−φ`ν)

−2(1 −ν2)−αexp(φ`/2)

ν(ν2−9) (1 −ν) exp φ`ν

2−(1 + ν) exp −φ`ν

2 (15)

3The on-shell scalar potential V(φ) will be constructed in the following subsection 3.1

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摘要：

ExacthairyblackholesasymptoticallyAdS2+1CarlosDesa(1),WeynerCcuiro(1)andDavidChoque(1)(1)UniversidadNacionaldeSanAntonioAbaddelCusco,Av.LaCultura733,Cusco,Peru.January2,2023AbstractInthecontextofEinstein'sminimallycoupledscalareldtheory,wepresentafamilyofhairyblackholeswhichareasymptoticallyAdS2+1...

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Exact hairy black holes asymtotically AdS21

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