Black-Hole Solutions with Scalar Hair in Einstein-Scalar-Gauss-Bonnet Theories
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1 Black-Hole Solutions with Scalar Hair in Einstein-Scalar-Gauss-Bonnet Theories Athanasios Bakopoulos Physics Department University of Ioannina In collaboration with: George Antoniou and Panagiota Kanti arxiv: to appear in Phys. Rev. D. arxiv: to appear in Phys. Rev. Lett. March Athanasios Bakopoulos arxiv: March / 20
2 Outline 1 Black Holes in General Relativity 2 No-Scalar Hair Theorem 3 The Einstein Scalar Gauss-Bonnet Theory 4 Numerical Solutions Athanasios Bakopoulos arxiv: March / 20
3 Black Holes in General Relativity Black Holes in General Relativity Black holes in General Relativity may be described only by three physical quantities: Mass, E/M charge and Angular Momentum. Black holes are very special objects: Two stars with the same mass are, in general, very different, but two black holes with the same characteristics (M, Q and J) will be identical. No hair theorems: Uniqueness theorems which state that in General Relativity only four possible solutions for black holes may exist. Athanasios Bakopoulos arxiv: March / 20
4 No-Scalar Hair Theorem No-Scalar Hair Theorem Adding new matter/energy forms in the theory could lead to new Black holes solutions. The simplest matter form is a Scalar field coupled to the gravitational field: S = d 4 x [ g R 1 ] 2 µφ µ φ V (φ). Assumptions: Asymptotically flatness, The scalar field has the same symmetries with the spacetime, Minimal coupling. Under these assumptions black holes with scalar hair could not exist 1. 1 J. D. Bekenstein, Phys. Rev. Lett. 28 (1972) 452 J. D. Bekenstein, Phys. Rev. D 51 (1995) no.12 R6608 Athanasios Bakopoulos arxiv: March / 20
5 The Einstein Scalar Gauss-Bonnet Theory The Horndenski Theory The more general scalar-tensor theory in four dimensions with second order equations of motion: S = d 4 x g [L 2 + L 3 + L 4 + L 5] with: L 2 =G 2(φ, X), L 3 =G 3(φ, X) 2 φ, [ L 4 =G 4(φ, X)R + G 4,X ( 2 φ) 2 ( µ ν φ) 2] L 5 =G 5(φ, X)G µν µ ν φ 1 [ 6 G5,X ( 2 φ) 3 3( 2 φ)( µ ν φ) 2 + 2( µ ν φ) 3] X = 1 2 µφ µ φ, and G i,x = Gi X. Athanasios Bakopoulos arxiv: March / 20
6 The Einstein Scalar Gauss-Bonnet Theory The Einstein Scalar Gauss-Bonnet Theory If we make the choices: G 2 = X + 8f (4) X 2 (3 ln X), G 3 = 4f (3) X(7 3 ln X), G 4 =1 + 4 fx(7 ln X), G 5 = 4f ln X, we get the Einstein Scalar Gauss-Bonnet Theory: S = d 4 x [ g R 1 ] 2 µφ µ φ + f(φ)r 2 GB, where: R 2 GB = R µνρσr µνρσ 4 R µνr µν + R 2. Athanasios Bakopoulos arxiv: March / 20
7 The Einstein Scalar Gauss-Bonnet Theory The equations of motion are: R µν 1 gµνr = Tµν, 2 where: 2 φ + d f(φ) dφ R2 GB = 0, T µν = 1 4 gµν ρφ ρ φ µφ νφ (gρµg λν + g λµ g ρν) η κλαβ Rργαβ γ κf(φ), R µν κλ = ηµνρσ R ρσκλ, η µνρσ = gε µνρσ. Athanasios Bakopoulos arxiv: March / 20
8 The Einstein Scalar Gauss-Bonnet Theory We assume a spherically symmetric form for the line-element: ds 2 = e A(r) dt 2 + e B(r) dr 2 + r 2 ( dθ 2 + dϕ 2 sin 2 (θ) ), The (rr) equation can be solved for e B : with e B = β ± β 2 4γ, and B = γ + β e B 2 2e 2B + βe, B β = r2 φ 2 4 (2 fφ + r)a 1, γ = 6 fφ A. If we replace the above relations in the field equations, the (t, t) and (θ, θ) equations reduce to: A = g(r; φ, A, f, f) φ = h(r; φ, A, f, f) Athanasios Bakopoulos arxiv: March / 20
9 The Einstein Scalar Gauss-Bonnet Theory Asymptotic Solutions at the horizon Assuming that e B at the horizon, we find that the solutions take the form: e A(r) = a 0(r r h ) +..., e B(r) = b 0(r r h ) +..., φ (r) = φ h + φ h(r r h ) +..., the regularity condition for the scalar field demands that: φ h = r h 4f 1 ± 1 96 f h 2 h r 4 h The above solutions will serve as boundary conditions for our numerical integration. From the condition for the derivative we get a constraint for the radius and thus the mass of the black hole: r 4 h > 96 f 2 h Athanasios Bakopoulos arxiv: March / 20
10 The Einstein Scalar Gauss-Bonnet Theory Asymptotic Solutions at Infinity In the limit r we demand that the space-time is flat and the scalar field assumes a constant value. We find: e A(r) =1 2M r + MD2 12r 3 + O ( 1/r 4), e B(r) = 1 + 2M r + 16M 2 D 2 4r 2 + O ( 1/r 3), φ (r) = φ + D r + MD r 2 + O ( 1/r 3). The exact form of the coupling function does not enter in the expansions earlier than the order O(1/r 4 ). The No-Scalar Hair theorem may be evaded and the two asymptotic solutions may be smoothly connected assuming that near the horizon 2 : f h φ h < 0, and r( fφ ) rh > 0. 2 P. Kanti talk. G. Antoniou, A. B. and P. Kanti, arxiv: [hep-th], to appear in Phys. Rev. Lett. Athanasios Bakopoulos arxiv: March / 20
11 Numerical Solutions Numerical Solutions The system of field equations can not be solved analytically. In order to get exact solutions, valid over the entire radial domain, we use numerical integration methods. Our integration starts at a distance very close to the horizon of the black hole r r h + O(10 5 ) For simplicity we set r h = 1. We use as boundary conditions the asymptotic solutions near the horizon together with the condition for the φ h The integration proceeds towards large values of the radial coordinate until the form of the derived solution matches the asymptotic solutions at infinity. Known Numerical Solutions: P. Kanti et al., Phys. Rev. D 54 (1996) 5049 (for exponential coupling) T.P. Sotiriou and S.Y. Zhou, Phys. Rev. D 90, (2014) (for linear coupling) Athanasios Bakopoulos arxiv: March / 20
12 Numerical Solutions Solutions for f(φ) = αe φ We reproduce the old results by P.Kanti et al. (1996) 6 5 α=0.01 g tt α=0.1, a 1=1 ϕ h= g rr ϕ(r) ϕ h=1 ϕ h= r r α=0.01 R 2 GB R 2 GB α=0.1, a 1=1, ϕ h=0.5 T r r T t t T θ θ r r The behavior of the solutions for the scalar field, near the horizon, is dictated by the constraint: f h φ h < 0. Athanasios Bakopoulos arxiv: March / 20
13 Numerical Solutions Solutions and Physical Characteristics for f(φ) = α/φ ϕ(r) α=0.01, a1=1 ϕh=+1.00 ϕh=+1.05 ϕh=+1.10 D/α 1/ <α<0.919 φh=3, rh=1 ϕh= r M/α 1/ A h =4πr 2 h, S h = A h 4 + 4π f(φ h) <α<0.919 φh=3, rh=1 Ah/Asch Sh/Ssch M/α 1/2 Athanasios Bakopoulos arxiv: March / 20
14 Numerical Solutions Entropy ratios for more coupling functions Sh/Ssch 0.96 α ϕ, ϕ h=0.2 α ϕ 2, ϕ h=0.2 α ϕ 3, ϕ h=0.2 α ϕ 4, ϕ h=0.4 Sh/Ssch α lnϕ, ϕ h=2 α e ϕ, ϕ h=0.25 α ϕ -2, ϕ h=2 α ϕ -3, ϕ h= M/α 1/2 M/α 1/2 For large values of the φ h there is always a mass range in which the entropy ratio is above unity. We expect the solutions with higher entropy than the Schwarzschild solution to be thermodynamically stable. for more informations about the quadratic and the exponential coupling function: H.O.Silva et al. arxiv: [gr-qc]. D. D. Doneva and S. S. Yazadjiev, arxiv: [gr-qc]. Athanasios Bakopoulos arxiv: March / 20
15 Conclusions Conclusions Regular, asymptotically flat black hole solutions were found for a large number of choices for the coupling function. The profile for the scalar field, in all of the cases considered, was found to be regular over the entire radial domain. The scalar charge D was determined in each case, and its dependence on M was studied In all cases the scalar charge is an M-dependent quantity, and therefore our solutions have a non-trivial scalar field but with a secondary hair. In the large-mass limit, the horizon area and the entropy of all black-hole solutions approached the Schwarzschild value. The entropy ratio S h /S Sch may provide hints for the stability of our solutions. Athanasios Bakopoulos arxiv: March / 20
16 Thank You! Athanasios Bakopoulos arxiv: March / 20
17 Athanasios Bakopoulos arxiv: March / 20
18 1.00 Sh/Ssch f=α ϕ ϕ h =0.1 ϕ h =0.5 ϕ h =1 ϕ h = M/α 1/2 Athanasios Bakopoulos arxiv: March / 20
19 Athanasios Bakopoulos arxiv: March / 20
20 In the cosmological Friedmann- Robertson-Walker (FRW) background the propagation speed of the gravitational waves in the Horndeski theory is: G c 2 4 X ( ) φ G5,X + G 5 gw = G 4 2XG 4,X X ( H φg ) 5,X G 5 Using the above relation for the Scalar Gauss bonnet theory we find: c gw c, This result is valid for cosmological solutions with time-dependent scalar field. The GW and GRB A signals Athanasios Bakopoulos arxiv: March / 20
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