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Cold Black Holes in Einstein-Maxwell-Scalar Theory

Tales Gomes

Programa de P´os-Gradua¸ao em F´ısica Universidade Federal do Esp´ırito Santo

V Jos´e Pl´ınio Baptista School of Cosmology, September 30, 2021

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Table of Contents

1 Einstein-Maxwell-Scalar Theroy

2 Field’s equations

3 Cold Black Holes

4 Conclusion

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Table of Contents

1 Einstein-Maxwell-Scalar Theroy

2 Field’s equations

3 Cold Black Holes

4 Conclusion

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Starting from the action of Einstein-Maxwell-Scalar Tensor Theory with a minimally coupled massless scalar field, in Einstein’s frame,

S = Z

d4x√

−g (R − ∇αΦ∇αΦ + FαβFαβ), (1) The constant  can assume the values +1, which will give us a positive kinetic energy for the scalar field, and −1, that will represent the ”phantom” scalar field with negative kinetic energy.

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Using the variational principle, one obtain, from the action (1), Rµν = ∇µΦ∇νΦ −



2FµλFνλ−1

2gµνFαβFαβ



, (2)

αFαβ = 0, (3)

ααΦ = 0. (4)

To solve these equations, we consider a static, spherically symmetric metric

ds2= edt2+ edr2+ ed Ω2 (5)

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Table of Contents

1 Einstein-Maxwell-Scalar Theroy

2 Field’s equations

3 Cold Black Holes

4 Conclusion

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Maxwell’s equation

The general form of Maxwell’s tensor for a point charge in a static spherically symmetric space-time can be written as,

Fαβ = E (u)(−δµ0δν1+ δµ1δν0), (6) in this sense, the Maxwell’s equation (3) gives us

uE (u) − (γ0+ α0− 2β0)E (u) = 0 (7) which can be integrated to obtain the solution

E (u) = Qeα+γ−2β; (8)

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Klein-Gordon equation

On the other hand, since we are working with a static, spherically symmetric space-time,Φ ≡ Φ(u), thus

Φ00− (α0− γ0− 2β00 = 0, (9) where the solution is

Φ0= Ceα−γ−2β (10)

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Einstein’s filed equations

A suitable choice of coordinate is the harmonic coordinates, where α(u) = γ(u) + 2β(u), with this choice the equations become

γ00= Q2e, (11)

− 2β00+ 2(β0)2+ 4β0γ0 = C2, (12)

γ00+ β00 = e2γ+2β; (13)

where the scalar and eletric field are writen

E (u) = Qe Φ(u) = Cu + Φ0 (14)

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The solutions for Einstein’s equations are

e−γ−β = s(k, u) =





k−1sinh(ku), k > 0,

u, k = 0,

k−1sin(ku), k < 0;

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e−γ = h(u) = Qs(λ, u + u0) =





Q

λ sinh[λ(u + u0)], λ > 0,

u, λ = 0,

Q

λ sin[λ(u + u0)], λ < 0;

; (16) where λ and k are constants of integration,

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thus, we can write the metric as ds2 = s2(λ, u0)dt2

s2(λ, u + u0)− s2(λ, u + u0) s2(λ, u0)s2(k, u)

 du2

s2(k, u)+ d Ω2

 (17) where all constants are related by

m2− Q2 = λ2sign λ = k2sign k −C2

2 ; (18)

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Table of Contents

1 Einstein-Maxwell-Scalar Theroy

2 Field’s equations

3 Cold Black Holes

4 Conclusion

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All BH are found in the Phantom sector, where  = −1. The first BH solution is for the case λ > k = 0. The metric in this case is written as,

ds2 = sinh2(λu0)dt2

sinh2[λ(u + u0)] −sinh2[λ(u + u0)]

sinh2(λu0)u2

 du2

u2 + d Ω2

 (19) Where we have the relation

m2− Q2= λ2 = C2

2 (20)

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The appropriate coordinate transformation for this solution is x = 1

u (21)

In consequence the metric become ds2= h(x )dt2− dx2

h(x )− x2

h(x )d Ω2 (22) where

h(x ) = 4λ2e−2λx

[(m + λ) − (m − λ)e−2λx]2

, (23)

and the geometric mass is

m = λ coth (λu0) (24)

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Figure:Carter-Penrose Diagram for the case λ > k = 0, with one event horizon.

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The other solution, for λ > k > 0, has the metric in the form

ds2= sinh2(λu0)dt2

sinh2[λ(u + u0)]− k2sinh2[λ(u + u0)]

sinh2(λu0) sinh2(ku)

 k2du2

sinh2(ku)+ d Ω2

 (25) with constants relation

m2− Q2 = λ2 = k2+C2

2 (26)

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the transformation for the another solution is e−2ku = 1 −2k

ρ = P(ρ) (27)

so the metric writes

ds2 = f (ρ)dt2− d ρ2

f (ρ)−P(ρ)

f (ρ)ρ2d Ω2; (28) where

f (ρ) = 4λ2P(ρ)a

[(m + λ) − (m − λ)P(ρ)a]2 (29) and a = λ/k.

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Figure:Carter-Penrose Diagram for an even integer a in the case where λ > k > 0.

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Figure: Carter-Penrose Diagram for λ > k > 0 and a an odd integer.

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Table of Contents

1 Einstein-Maxwell-Scalar Theroy

2 Field’s equations

3 Cold Black Holes

4 Conclusion

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Conclusions

We study the Cold Black Holes in Einstein-Maxwell-Scalar theory;

This BHs have interesting characteristics, such as infinity surface area and zero Hawking temperature;

We will study the geodesics and stability of these solutions, We look forwards to obtain a rotating solution from the static ones we studied;

Investigate the methods of rotate solutions, such as JNA.

Study the thermodynamics of this BHs.

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Obrigado!

Cytaty

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