Black hole solutions in the N>4 gravity models with higher order curvature corrections and possibilities for their experimental search S.Alexeyev *, N.Popov, S ternberg Astronomical Institute, Moscow, Russia) A.Barrau, J.Grain, … Fourth Meeting on Constrained Dynamics and Quantum Gravity, September 12-16, 2005

Main publications S.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 (M D – D dimensional fundamental Planck mass, M Pl – 4D Planck mass) M D = [M Pl 2 / V D-4 ] 1/(D-2)

Planck Energy shift Planck energy in 4D representation ↓ 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: ds 2 = - R(r) dt 2 + R(r) -1 dr 2 + r 2 dΩ n+2 2 Metric function: R(r) = 1 – [ r s / r ] n+1

The Schwarzschild radius r s is related to the mass M BH r s = π -½ M * -1 γ(n) [ M BH / 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 hole Hawking temperature and entropy T H = (n+1) [ 4 π r s ] -1 S = [ (n+1) / (n+2) ] M BH / T H 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 ∫ d D x (-g) ½ [ R + λ ( R μναβ R μναβ – 4R αβ R αβ + R 2 ) + … ] Gauss-Bonnet term

(4+n)D Schwarzschild- Gauss-Bonnet (SGB) black hole Metric representation: ds 2 = - e 2ν dt 2 + e 2α dr 2 + r 2 h ij dx i dx j Metric functions:

Corresponding (4+n)D SGB black hole parameters Mass Temperature

Hawking Temperature M/M Pl T with GB /T without 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, ρ 2 = r 2 + a 2 cos 2 θ N.Deruelle, Y.Morisawa, Class.Quant.Grav.22: ,2005, S.Alexeyev, N.Popov, A.Barrau, J.Grain, in preparation ds 2 = - (du + dr) 2 + dr 2 + ρ 2 dθ 2 + (r 2 + a 2 ) sin 2 θdφ a sin 2 θ dr dφ + β(r,θ) (du – a sin 2 θ dφ) 2 + r 2 cos 2 θ (dx sin 2 x 5 (dx sin 2 x 6 (…dx N 2 )…)

(UR) equation for β(r,θ) For 6D case h 1 = 24 α r 3 h 0 = r ρ 2 (r 2 + ρ 2 ) g 2 = 4 α (3r r 2 a 2 cos 2 θ – a 4 cos 4 θ) / ρ 2 g 1 = (r 2 + ρ 2 ) (2r 2 + ρ 2 ) g 0 = Λ r 2 ρ 4 [ h 1 (r,β) β + h 0 (r, β) ] (dβ/dr) + [ g 2 (r, β) β 2 + g 1 (r, β) β + g 0 (r, β) ] = 0

When Λ = 0 (Analogously to Myers-Perry solution) β(r,θ)  μ / [r N-5 (r 2 + a 2 cos 2 θ)] + … When Λ ≠ 0 β(r,θ)  C(N) Λ r 4 / [r 2 + a 2 cos 2 θ] + … Behavior at the infinity

Behavior at the horizon β(r,θ) = 1 + b 1 (θ) (r - r h ) + b 2 (θ) (r – r h ) 2 + … For 6D case b 1 = [ 4 α (3 r h r h 2 a 2 cos 2 θ – a 4 cos 4 θ) (r h 2 + a 2 cos 2 θ) -1 + (2 r h 2 + a 2 cos 2 θ) (3 r h 2 + a 2 cos 2 θ) + Λ r h 2 (r h 2 + a 2 cos 2 θ) 2 ] / [ 24 α r h 3 + r h (2 r h 2 + a 2 cos 2 θ) ]

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

Conclusions In 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!