1 / 27

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.

halia
Télécharger la présentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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?

More Related