2. Black hole solutions in N>4 Gauss-Bonnet Gravity S.Alexeyev* 1, N.Popov 2, T.Strunina 3 1 S ternberg Astronomical Institute, Moscow, Russia 2 Computer Center of Russian Academy of Sciences, Moscow 3 Ural State University, Ekaterinburg, Russia 4th International Seminar on High Energy Physics QUARKS'2006 Repino, St.Petersburg, Russia, May 19-25, 2006

3. 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, Journal of Physics: Conference Series 33, 343 (2006) S.Alexeyev, N.Popov, T.Strunina, A.Barrau, J.Grain, in preparation

4. 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)

5. Planck Energy shift Planck energy in 4D representation ↓ 10 19 GeV Fundamental Planck energy ↓ ≈ 1 TeV

6. Extended Schwarzschild solution in (4+n)D Tangherlini, ‘1963, Myers & Perry, ‘1986 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

7. (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 + λ S GB + … ] Gauss-Bonnet term S GB = R μναβ R μναβ – 4R αβ R αβ + R 2

8. Einstein-GB equations R.Cai, ‘2003 R µν - ½ g µν R - Λg µν – α ( ½ g µν S GB – 2 RR µν + 4 R µγ R γ ν + 4 R γδ R γ µ δ ν – 2 R µγδλ R ν γδν ) = 0 S GB = R μναβ R μναβ – 4R αβ R αβ + R 2

9. (4+n)D Schwarzschild- Gauss-Bonnet black hole solution (Boulware, Dieser, ‘1986, R.Cai, ‘2003) Metric representation: ds 2 = - e 2ν dt 2 + e 2α dr 2 + r 2 h ij dx i dx j Metric functions:

10. Mass and Temperature Mass Temperature

11. Hawking Temperature M/M Pl T with GB /T without GB

12. “Toy model” (4+n)D Kerr-Gauss-Bonnet solution with one momentum (“degenerated solution”). Necessity: to compare with the usual Kerr one in the complete range of dimensions: N=5,…,11

13. “Degenerated” solution here 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.22:933-938,2005, S.Alexeyev, N.Popov, A.Barrau, J.Grain, Journal of Physics: Conference Series 33, 343 (2006) ds 2 = - (du + dr) 2 + dr 2 + ρ 2 dθ 2 + (r 2 + a 2 ) sin 2 θdφ 2 + 2 a sin 2 θ dr dφ + β(r,θ) (du – a sin 2 θ dφ) 2 + r 2 cos 2 θ (dx 5 2 + sin 2 x 5 (dx 6 2 + sin 2 x 6 (…dx N 2 )…)

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

15. Λ = 0 β(r,θ)  μ / [ r N-5 (r 2 + a 2 cos 2 θ) ] + … Λ ≠ 0 β(r,θ)  C(N) Λ r 4 / [r 2 + a 2 cos 2 θ] + … Behavior at the infinity

16. 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 4 + 6 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) ]

17. Usual form of metric ds 2 = - dt 2 (1 – β 2 ) + dr 2 [(r 4 (1 – β 2 ) + a 2 (r 2 + β 2 a 2 cos 4 θ) / Δ 2 ] + ρ 2 dθ 2 – 2aβ 2 sin 2 θ dtd φ + dφ 2 sin 2 θ [r 2 + a 2 + a 2 β 2 sin 2 θ] + r 2 cos 2 θ (dx 5 + …) Δ = r 2 + a 2 - ρ 2 β 2 ρ 2 = r 2 + a 2 cos 2 θ

18. Mass & angular momentum Mass M = µ (N-2) A N-2 /16πG where A N-1 = 2 π N/2 /Γ(N/2) Angular momentum J y i x i = 2 M a i /N  the same as in pure Kerr case

19. 6D plot of β=β(r,a ∙ cosθ) in asymptotically flat case (Λ=0), λ=1

20. 6D plot of β=β(r,a ∙ cosθ) when Λ ≠ 0, λ=1

21. While considering “degenerated solution” there are no any new types of particular points, so, there is no principal difference from pure Kerr case  all the difference will occur only in temperature and its consequences

22. Real angular momentum tensor

23. Number of angular momentums According to the existence of [N s /2] Casimirs of SO(N) (N s is the number of space dimensions) For N=4 (N s =3) there is 1 moment For N=5 (N s =4) there are 2 moments For N=6 (N s =5) there are 2 moments For N=7 (N s =6) there are 3 moments For N=8 (N s =7) there are 3 moments For N=9 (N s =8) there are 4 moments For N=10 (N s =9) there are 4 moments For N=11 (N s =10) there are 5 moments

24. 5D Kerr metric ( complete version ) ds 2 = dt 2 - dr 2 - (r 2 +a 2 ) sin 2 θ dφ 1 - (r 2 +b 2 ) cos 2 θ dφ 2 – ρ 2 dθ 2 - 2 dr (a sin 2 θ dφ 1 + b cos 2 θ dφ 2 ) - β (dt – dr – a sin 2 θ dφ 1 - b cos 2 θ dφ 2 ) Where ρ 2 = r 2 + a 2 cos 2 θ + b 2 sin 2 θ, β = β(r, θ) is unknown function a, b - moments

25. θθ component A β’’ + B β’ 2 + C β’ + D β + E = 0 Where A = r ρ 2 (4 αβ – ρ 2 ) B = 4 α r ρ 2 C = 2 [ 4 αβ (ρ 2 - r 2 ) – ρ 2 (ρ 2 + r 2 ) ] D = 2 r (2 r 2 – 3 ρ 2 ) E = 2 r ρ 4 Λ

26. Solution manipulations This equation could be divided into 2 parts A(r,ρ)β’’+B(r,ρ)β’+C(r,ρ)β+D(r,ρ,Λ)=Z(r,ρ,β) E(r,ρ)(ββ’)’+F(r,ρ)(ββ’) =Z(r,ρ,β)

27. 5D solution

28. 6D metric ds 2 = dt 2 - dr 2 – sin 2 ψ [(r 2 + a 2 ) sin 2 θ d φ 1 2 + (r 2 + b 2 ) cos 2 θ d φ 2 2 ] - (r 2 + a 2 cos 2 θ + b 2 sin 2 θ) sin 2 ψ dθ 2 - [r 2 + (a 2 sin 2 θ + b 2 cos 2 θ) cos 2 ψ] dψ 2 - 2 dr sin 2 ψ (a sin 2 θ d φ 1 + b cos 2 θ d φ 2 ) + 2 (b 2 - a 2 ) sinθ cosθ sinψ cosψ dθ dψ - β(r,θ,ψ) [dt – dr –sin 2 ψ (a sin 2 ψ d φ 1 + b cos 2 θ d φ 2 )] 2

29. Conclusions Taking into account 5D case one can see that in the general form of Kerr-Gauss-Bonnet solution there are no any new types of particular points, so, there is no principal difference from pure Kerr case  all the difference will occur only in temperature and its consequences

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