1. BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC I.Ya.Aref’eva Steklov Mathematical Institute, Moscow

2. BH/WH production in Trans-Planckian Collisions BLACK HOLES. BH in GR and in QG BH formation Trapped surfaces WORMHOLES TIME MACHINES WHY on LHC? 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

3. I-st lecture. Outlook: BLACK HOLES in GR Historical Remarks SCHWARZSCHILD BH Event Horizon Trapped Surfaces BH Formation BLACK HOLES in QG BLACK HOLES in Semi-classical approximation to QG I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 Refs.: L.D.Landau, E.M.Lifshitz, The Classical Theory of Fields, II v. Hawking S., Ellis J. The large scale structure of space-time. R.Wald, General Relativity, 1984 S. Carroll, Spacetime and Geometry. An introduction to general relativity, 2004

4. BLACK HOLES in GR. Historical Remarks The Schwarzschild solution has been found in 1916. The Schwarzschild solution(SS) is a solution of the vacuum Einstein equations, which is spherically symmetric and depends on a positive parameter M, the mass. In the coordinate system in which it was originally discovered, (t,r,theta,phi), had a singularity at r=2M In 1923 Birkoff proved a theorem that the Schwarzschild solution is the only spherically symmetric solution of the vacuum E.Eqs. In 1924 Eddington, made a coordinate change which transformed the Schwarzschild metric into a form which is not singular at r=2M In 1933 Lemaitre realized that the singularity at r=2M is not a true singularity In 1958 Finkelstein rediscovered Eddington's transformation and realized that the hypersurface r=2M is an event horizon, the boundary of the region of spacetime which is causally connected to infinity. I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008

5. BLACK HOLES in GR. Historical Remarks In 1950 Synge constructed a systems of coordinates that covers the complete analytic extension of the SS In 1960 Kruskal and Szekeres (independently) discovered a single most convenient system that covers the complete analytic extension of SS In 1964 Penrose introduced the concept of null infinity, which made possible the precise general definition of a future event horizon as the boundary of the causal past of future null infinity. In 1965 Penrose introduced the concept of a closed trapped surface and proved the first singularity theorem (incompleteness theorem). Hawking-Penrose theorem: a spacetime with a complete future null infnity which contains a closed trapped surface must contain a future event horizon (H-E-books, 9-2-1) I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008

6. BLACK HOLES in GR. Schwarzschild solution to vacuum Einstein eq. I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008 A general metric in a spherically symmetric spacetime (Ch.13 of Weinberg) Christoffel symbols, Riemann tensor, Ricci tensor 1) 2) 3) 5) 2) 1) 4)

7. BLACK HOLES in GR. Schwarzschild solution to vacuum Einstein eq. I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008 A general metric in a spherically symmetric spacetime (Ch.13 of Weinberg) Christoffel symbols, Riemann tensor, Ricci tensor

8. BLACK HOLES in GR. Schwarzschild solution Asymptoticaly flat Birkhoff's theorem: Schwarzschild solution is the unique spherically symmetric vacuum solution Singularity I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008

9. D-dimensional Schwarzschild Solution Meyers,… I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008

10. BLACK HOLES in GR. Schwarzschild solution Geodesics I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008 Null Geodesics

11. BLACK HOLES in GR. Schwarzschild solution I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008 Null Geodesics Regge-Wheeler coordinates

12. BLACK HOLES in GR. Schwarzschild solution I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008 Eddington-Finkelstein coordinates The determinant of the metric is is regular at r=2GM r=0 r=2GM r

13. BLACK HOLES in GR. Schwarzschild solution I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008 Kruskal coordinates Kruskal diagram

14. Kruskal diagram represents the entire spacetime corresponding to the Schwarzschild metric BLACK HOLES in GR. Schwarzschild solution I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008 Each point on the diagram is a two-sphere WE –region I; Future-directed null rays reach region II, Past-directed null rays reach region III. Spacelike geodesics reach region IV. Regions III is the time-reverse of region II. ``White hole.'' Boundary of region II is the future event horizon. Boundary of region III is the past event horizon.

15. Penrose (or Carter-Penrose, or conformal) diagram BLACK HOLES in GR. Schwarzschild solution I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008 Each point on the diagram is a two-sphere

16. Penrose diagram for Schwarzschild BLACK HOLES in GR. Schwarzschild solution I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008 Penrose diagram for Minkowski t=const

17. Event Horizons The event horizon is always a null hypersurface. The event horizon does not depend on a choice of foliation. The event horizon always evolves continuously. The event horizon captures the intuitive idea of the boundary of what can reach observers at infinity. I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008

18. BH production in QFT Q - can QFT (in flat space-time) produce BH I.Aref’eva BH-QCD,CERN,Sept.2008 A Q - QFT(in flat space-time) cannot produce BH/WH, etc. Answer : QFT + "SQG" (semicl.aprox. to QG) can Question We can see nonperturbatively just precursor (предвестник) of BHs

19. BH in Quantum Gravity Background formalism in QFTIA, L.D.Faddeev, A.A.Slavnov, 1975 I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008

20. AQ. Analogy with Solitons I.Aref’eva BH-QCD,CERN,Sept.2008 perturbative S-matrix for phi-particles - no indications of solitons Exact nonperturbative S-matrix for phi-particles – extra poles - indications of a soliton-antisoliton state How to see these solitons in QFT? IA, V.Korepin, 1974