1. BLACK HOLES. BH in GR and in QG BH formation Trapped surfaces WORMHOLES TIME MACHINES Cross-sections and signatures of BH/WH production at the LHC I-st lecture. 2-nd lecture. 3-rd lecture. BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

2. BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 (1965) Penrose introduces the idea of trapped surfaces to complete his singularity proofs. (1972) Hawking introduces the notion of event horizons, to capture the idea of a black hole. History

3. BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 1. Ric(N;N) >= 0 for all null vectors N; 2. There is a non-compact Cauchy hypersurface H in M 3. There is a closed trapped surface S in M. Th. (singularity th. or incompleteness th.) A spacetime (M; g) cannot be future null geodesically complete if: Th. (Hawking-Penrose) A spacetime (M; g) with a complete future null infnity which contains a closed trapped surface must contain a future event horizon (the interior of which contains the trapped surface)

4. Trapped surfaces A trapped surface is a two dimensional spacelike surface whose two null normals have negative expansion (= Neighbouring light rays, normal to the surface, must move towards one another ) I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

5. The cross-sectional area enclosing a congruence of geodesics. I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture Expansion Rotation Shear A A

6. Expansion I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

7. Expansion Any closed trapped surface must lie inside a black hole. I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

8. Raychaudhuri equation The Raychaudhuri equation for a null geodesics (focusing equation) No rotation, the matter and energy density is positive I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture then in a finite distance along the light ray, nearby light rays will be focused to a point, such that they cross each other with zero transverse area A

9. Apparent horizon.. The trapped region is the region containing trapped surfaces. A marginally trapped surface is a closed spacelike D-2-surface, the outer null normals of which have zero expansion (convergence). [ A trapped surface is a two dimensional spacelike surface whose two null normals have negative expansion] The boundary of (a connected component of) the trapped region is an apparent horizon In stationary geometries the apparent horizon is the same as the intersection of the event horizon with the chosen spacelike hypersurface. For nonstationary geometries one can show that the apparent horizon lies beyond the event horizon (Gibbons, 1972) I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

10. Expansion and the second fundamental form (extrinsic curvature) Expansion of null geodesics I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

11. Black Hole Formation Two BHs REFs: Brill and Lindquist (1963) Bishop (1982) The metric of a time-symmetric slice of space-time representing two BHs The vacuum eq. reduces to Solution 1-st Example I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

12. 3+1 decomposition ADM 3+1 decomposition Arnowitt, Deser, Misner (1962) 3-metric lapse shift lapse, shift Gauge Einstein equations 6 Evolution equations 4 Constraints Time-symmetric metric =inv. (t->-t) Lemma (Gibbons). If on Riemannian space V there is an isometry which leaves fixed the points of a submanifold W then W is a totally geodesic submanifold (extremal surface). Vacuum

13. Black Hole Formation. Example: two BHs A cylindrically symmetric surface The induced metric I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

14. Black Hole Formation. Example: two BHs Theorem: Area: I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

15. Black Hole Formation. Example: two BHs The first integral BC.+I.C.: I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

16. CTS for 2 Black Holes From Bishop (1982) I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

17. Advantage of CTS (Closed Trapped Surface) Approach The existence and location of BH can be found by a global analysis TS can be found by a local analysis (within one Cauchy surface) I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

18. 2-nd Example: BH Formation in Ultra-relativistic Particle Collisions Particle Penrose, D’Eath, Eardley, Giddings Shock waves t v u x Z I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

19. 4-dim Aichelburg-Sexl Shock Wave4-dim Schwarzschild Aichelburg-Sexl, 1970 1-st step

20. 4-dim Aichelburg-Sexl Shock Wave4-dim Schwarzschild 2-nd step

21. 4-dim Aichelburg-Sexl Shock Wave4-dim Schwarzschild 2-nd step(details)

22. 4-dim Aichelburg-Sexl Shock Wave4-dim Schwarzschild Aichelburg-Sexl, 1970 I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

23. Black Hole Formation (Particle = Shock waves) v t u x Z I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

24. Two Aichelburg-Sexl shock waves I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

25. V U X Z Trapped surface in two Aichelburg-Sexl shock waves Trapped marginal surface Ref.:Eardley, Giddings; I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

26. The shape of the apparent horizon C on (X 1, X 2 )-plane in the collision plane U = V = 0 for D = 4, 5. Incoming particles are located on the horizontal line X 2 = 0. As the distance b between two particles increases, the radius of C decreases. Figure shows the relation between b and r min for each D. The value of b max /r 0 ranges between 0.8 and 1.3 and becomes large as D increases. Yoshino, Nambu gr-qc/0209003 I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture

27. 3 rd Example: Colliding Plane Gravitational Waves I.A, Viswanathan, I.Volovich, 1995 Plane coordinates; Kruskal coordinates Regions II and III contain the approaching plane waves. In the region IV the metric (4) is isomorphic to the Schwarzschild metric. D-dim analog of the Chandrasekhar-Ferrari-Xanthopoulos duality? I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture