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 3-rd lecture

2. Wormholes Lorentzian Wormhole is a region in spacetime in which 3-dim space-like sections have non-trivial topology. By non-trivial topology we mean that these sections are not simply connected In the simplest case a WH has two mouths which join different regions of the space-time. We can also imagine that there is a thin handle, or a throat connected these mouths. Sometimes people refer to this topology as a 'shortcut' through out spacetime I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture

3. Wormholes The term WH was introduced by J. Wheeler in 1957 Already in 1921 by H. Weyl ( mass in terms of EM) The name WH comes from the following obvious picture. The worm could take a shortcut to the opposite side of the apple's skin by burrowing through its center, instead of traveling the entire distance around.

4. The traveler just as a worm could take a shortcut to the opposite side of the universe through a topologically nontrivial tunnel.

5. Wormholes The first WH solution was found by Einstein and Rosen in 1935 (so-called E-R bridge) There are many wormhole solutions in GR. A great variety of them! With static throat, dynamic throat, spinning, not spinning, etc Schwarzschild WHs (E-R bridges) The Morris-Thorne WH The Visser WH Higher-dimensional WH Brane WH I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture

6. Traversable Wormholes Morris, Thorne, Yurtsever, Visser,.. The embedding condition together with the requirement of finiteness of the redshift function lead to the NEC violation on the WH throat I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture

7. Time Machine. Definition Spacetime: (M,g), M – manifold, g – metric. Einstein equations for g. Time machine is a region of space- time (M,g) that has a closed timelike curve (CTC). CTC suggests the possibility of time travel with its well known paradoxes Example: time is circle. I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture

8. Time Machine TM is impossible in special relativity. Indeed, to make a loop, a curve must somewhere leave the null cone as shown in this picture. A particle with such a world line would exceed the speed of light that is impossible in SR. I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture

9. Time Machine In general relativity the situation is much less trivial. According to GR, our spacetime must be a smooth Lorentzian manifold small regions is “approximately Minkowskian”, at large scale could be any geometry and topology ( holes, handles, almost whatever one wants). I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture

10. Solutions of Einstein eqs. with Closed Timelike Curves (CTC) / Time Machine. Godel's solution [1949] van Stockum-Tipler cylinder [1937, 1974]; Kerr solutions; 2 axially symmetric, stationary Kerrs Gott's time machine; Wheeler wormholes; Morris-Thorne-Yurtsever's TM Ori's dust asymptotically-flat space-time Violation of normal chronology is such an objectionable occurrence that any of such solutions could be rejected as nonphysical. I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture

11. Summation over topologies Theorem (Geroch, Tipler): Topology-changing spacetimes must have CTC (closed timelike curve) Theorem (Gammon) : If asymptotically flat spacetimes has a Cauchy surface with a nontrivial topology, then spacetime is geodesically incomplete ( under assumption of NEC ) I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture

12. Mathematical solution of Grandfather paradox Recent overcoming of the grandfather paradox: There are spacetimes having CTC for which smooth, unique solutions to the scalar wave eq. exist for all data on generalized Cauchy surface I.A., I. Volovich, T. Ishiwatari I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture

13. Time Machine Surgery in the Minkowski spacetime Make two cuts and glue the left edge of left cut to the right edge of the right cut and vice verse, This space contains timelike loops x t I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture

14. Cauchy problem on not globally hyperbolic spacetimes x t Cauchy problem:

15. I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture

16. Example: 2 dim scalar wave equation Theorem: Under assumption of minimal singularity the Cauchy problem for t<b has a unique solution The Cauchy problem for t>b is not well posed I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture

17. A possibility of production in ultra-relativistic particle collisions of some objects related to a non-trivial space-time structure is one of long-standing theoretical questions In 1978 collision of two classical ultra relativistic particles was considered by D'Eath and Payne and the mass of the assumed final BH also has been estimated In 1987 Amati, Ciafaloni, Veneziano and 't Hooft conjectured that in string theory and in QG at energies much higher than the Planck mass BH emerges. Aichelburg-Sexl shock waves to describe particles, Shock Waves ------ > BH Colliding plane gravitation waves to describe particles Plane Gr Waves ----- > BH I.A., Viswanathan, I.Volovich, 1995 BH in Collisions I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture

18. BLACK HOLE PRODUCTION Collision of two fast point particles of energy E. BH forms if the impact parameter b is comparable to the Schwarzschild radius r s of a BH of mass E. The Thorn's hoop conjecture gives a rough estimate for classical geometrical cross-section I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture

19. BLACK HOLE PRODUCTION To deal with BH creation in particles collisions we have to deal with trans-Planckian scales. Trans-Planckian collisions in standard QG have inaccessible energy scale and cannot be realized in usual conditions. TeV Gravity to produce BH at Labs (1999) Banks, Fischler, hep-th/9906038 I.A., hep-th/9910269, Giuduce, Rattazzi, Wells, hep-ph/0112161 Giddings, hep-ph/0106219 Dimopolos, Landsberg, hep-ph/0106295 I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture

20. Conclusion TeV Gravity opens new channels – BH, WH, TM Wheeler foam at TeV scale. WH/TM production at LHC is of the same order of magnitude as BH production (under assumption of geometrical crossection) The important question on possible experimental signatures of spacetime nontrivial objects deserves further explorations. I.Aref’eva BH/WH at LHC, Dubna, Sept.2008 3-rd lecture