Black hole solutions in the N>4 gravity models with higher order curvature corrections and possibilities for their experimental search

S.Alexeyev*, N.Popov, Sternberg Astronomical Institute, Moscow, Russia)

A.Barrau, J.Grain, …

Fourth Meeting on Constrained Dynamics and Quantum Gravity, September 12-16, 2005

Main publicationsS.Alexeyev and M.Pomazanov, Phys.Rev. D55, 2110 (1997)

S.Alexeyev, A.Barrau, G.Boudoul, M.Sazhin, O.Khovanskaya, Astronomy Letters 28, 489 (2002)

S.Alexeyev, A.Barrau, G.Boudoul, O.Khovanskaya, M.Sazhin, Class.Quant.Grav. 19, 4431 (2002)

A.Barrau, J.Grain, S.Alexeyev, Phys.Lett. B584, 114 (2004)

S.Alexeyev, N.Popov, A.Barrau, J.Grain, Proceedings of XXII Texas Symposium on Relativistic Astrophysics, Stanford, USA, December 13-17, 2004

S.Alexeyev, N.Popov, A.Barrau, J.Grain, in preparation

String/M Theory (11d)

↓

General Relativity (4d)

Fundamental Planck scale shift

Large extra dimensions scenario (MD – D dimensional fundamental Planck mass, MPl – 4D Planck mass)

MD = [MPl2 / VD-4]1/(D-2)

Planck Energy shift

Planck energy in 4D representation

↓

1019 GeV

Fundamental Planck energy

↓

≈ 1 TeV

Extended Schwarzschild solution in (4+n)D

applicable when the horizon size is compatible with the extra dimensions ones (elementary particles approximation)

Metric:

ds2 = - R(r) dt2 + R(r)-1 dr2 + r2dΩn+22

Metric function: R(r) = 1 – [rs / r]n+1

The Schwarzschild radius rs is related to the mass

MBH

rs = π-½ M*-1 γ(n) [MBH / M*]1/(n+1)

Where

γ(n) = [8 Γ((n+3)/2) / (2+n)]1/(n+1)

Thermodynamics properties of (4+n)D Schwarzschild

black holeHawking temperature and entropy

TH = (n+1) [4 π rs]-1

S = [(n+1) / (n+2)] MBH / TH

So, in extra dimensions black hole is “more hot” its Hawking evaporation speed is greater

(4+n)D Low Energy Effective String Gravity

with higher order (second order in our consideration) curvature corrections

S = (16πG)-1 ∫ dDx (-g)½

[R + λ(Rμναβ Rμναβ – 4Rαβ Rαβ + R2) + …] Gauss-Bonnet term

(4+n)D Schwarzschild-Gauss-Bonnet (SGB) black

holeMetric representation:

ds2 = - e2ν dt2 + e2α dr2 + r2 hij dxi dxj

Metric functions:

Corresponding (4+n)D SGB black hole parameters

Mass

Temperature

Hawking Temperature

M/MPl

M/MPl

Twith GB/Twithout GB

Twith GB/Twithout GB

Flux computation

Spectrum of emitted particles

Number of emitted particles

Integrated flux against the total energy of the emitted quanta for an initial black hole mass M =10 TeV

λ=0 TeV-2 λ=0.5 TeV-2

D=6 D=11

For different input values of (D,) emitted spectra are reconstructed taking into account

fragmentation process λ=1 TeV-2 D=10 λ=5 TeV-2 D=8

Kerr-Gauss-Bonnet solution (Kerr-Shild parametrization)

here β =β (r,θ) is the function to be found, ρ2 = r2 + a2 cos2θ

N.Deruelle, Y.Morisawa, Class.Quant.Grav.22:933-938,2005, S.Alexeyev, N.Popov, A.Barrau, J.Grain, in preparation

ds2 = - (du + dr)2 + dr2 + ρ2dθ2 + (r2 + a2) sin2θdφ2

+ 2 a sin2θ dr dφ + β(r,θ) (du – a sin2θ dφ)2

+ r2 cos2θ (dx52 + sin2x5 (dx6

2 + sin2x6 (…dxN2)…)

(UR) equation for β(r,θ)

For 6D case

h1 = 24 α r3

h0 = r ρ2 (r2 + ρ2)g2 = 4 α (3r4 + 6 r2 a2 cos2θ – a4 cos4θ) / ρ2

g1 = (r2 + ρ2) (2r2 + ρ2)g0 = Λ r2 ρ4

[h1(r,β) β + h0(r, β)] (dβ/dr)

+ [g2(r, β) β2 + g1(r, β) β + g0 (r, β)] = 0

When Λ = 0 (Analogously to Myers-Perry solution)

β(r,θ) μ / [rN-5 (r2 + a2 cos2θ)] + …

When Λ ≠ 0

β(r,θ) C(N) Λ r4 / [r2 + a2 cos2θ] + …

Behavior at the infinity

Behavior at the horizon

β(r,θ) = 1 + b1(θ) (r - rh) + b2(θ) (r – rh)2 + …

For 6D case

b1 = [4 α (3 rh4 + 6 rh

2 a2 cos2θ – a4 cos4θ) (rh2 + a2 cos2θ)-1

+ (2 rh2 + a2 cos2θ) (3 rh

2 + a2 cos2θ)

+ Λ rh2 (rh

2 + a2 cos2θ)2] / [24 α rh3 + rh (2 rh

2 + a2 cos2θ)]

6D plot of β(r, θ) againgt r and a*cosθ in asymptotically flat case (string coupling constant λ is set to be equal to 1)

6D plot of β(r, θ) againgt r and a*cosθ when Λ ≠ 0 (string coupling constant λ is set to be equal to 1)

One can see that there are no any new types of particular points, so, there is no principal difference from pure Kerr case (R.C.Myers, M.J.Perry, Ann.Phys.172, 304 (1986)), all the difference will occur only in temperature and its consequences

ConclusionsIn case the Planck scale lies in the TeV range due to extra dimensions, beyond the dimensionality of space, the next generation of colliders should be able to measure the coefficient of a possible Gauss-Bonnet term in the gravitational action

It is also interesting to notice that this would be a nice example of the convergence between astrophysics and particle physics in the final understanding of black holes and gravity in the Planckian region.

Thank you for your kind attention!And for your questions!