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BLACK HOLES. BH in GR and in QG BH formation Trapped surfaces

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.

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BLACK HOLES. BH in GR and in QG BH formation Trapped surfaces

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  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-stlecture. 2-nd lecture. 3-rd lecture. I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC

  2. BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC History • (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. I.Aref’eva BH/WH at LHC, Dubna, Sept.2008

  3. BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC Th.(singularity th. orincompleteness th.) A spacetime (M; g) cannot be future null geodesically complete if: • 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. (Hawking-Penrose) A spacetime (M; g)with a complete future null infnity which containsa closed trapped surface must contain a future event horizon (the interior of which containsthe trapped surface) I.Aref’eva BH/WH at LHC, Dubna, Sept.2008

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

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

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

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

  8. 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 I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture Raychaudhuri equation The Raychaudhuri equation for a null geodesics (focusing equation) No rotation, the matter and energy density is positive

  9. I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture Apparent horizon. . • The trapped region is the region containing trapped surfaces. • A marginally trapped surfaceis 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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  26. I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 2-nd lecture Yoshino, Nambu gr-qc/0209003 The shape of the apparent horizon C on (X1, X2)-plane in the collision plane U = V = 0for D = 4, 5. Incoming particles are located on the horizontal line X2 = 0. As the distance b betweentwo particles increases, the radius of C decreases. Figure shows the relation between b and rmin for each D. The value of bmax/r0 rangesbetween 0.8 and 1.3 and becomes large as D increases.

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

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