1. TeV Gravity at LHC,
Hawking’s Chronology Protection Conjecture
and Renormalizations Group
I. Aref’eva

2. PREDICTIONS
Micro-Black hole production at CERN's Large Hadron Collider (LHC)
Micro-Wormhole/time machine production at LHC
I.A. and I.V.Volovich, Time Machine at the LHC,
arXiv: , Int.J.Geom.Meth.Mod.Phys. (2008)

3. Outlook:
TeV gravity
TIME MACHINES
Chronology Protection Conjecture and Renormalizations

4. Time Machine
In GR a TM is a region of space-time that has a Closed Timelike Curve (CTC)
CTC suggests the possibility of time-travel with its well known paradoxes
CTC
Time Machine

5. 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).
Example. Time is circle:

6. Causality Time machines violate the standard causality condition.
“It seems that there is a Chronology Protection Agency which prevents the appearance of CTCand so makes the universe safe for historians”
Hawking, Phys.Rev. (1992)

7. Causality Cauchy problem. Global hyperbolic: Causality in QFT
Bogoliubov, Shirkov
Local commutativity:
Bogoliubov, Tavkhelidze, Vladimirov,
Whightman,…
Locality in string theory: Gross, Veneziano, Susskind, ‘t Hooft
Nonlocality at the Planck scale: Bronstein,
Wheeler, Blokhintzev, Markov, ‘t Hooft,
p-adic space-time

8. Time Machine in Special Relativity
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.

9. Time Machine in GR
In general relativity the situation is much less trivial.
According to GR, our spacetime must be a smooth Lorentzian manifold
small regions `approximately Minkowskian',
large scale any geometry and topology.
There may be holes, handles, almost whatever one wants.
A direction of null cones may change. t x
A simple example is the Minkowski space
rolled into a cylinder.
Locally everything is fine in this spacetime,
but due to its non-trivial global structure,
an observer can meet his younger self

10. van Stockum-Tipler cylinder [1937, 1974];
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.

11. General Relativity and Chronology
In GR one cannot simply assert that chronology is preserved, and causality respected, without doing considerable additional work.
The essence of the problem lies in the fact that the Einstein eqs of GR are local equations, relating some aspects of the spacetime curvature at a point to the presence of stress-energy at that point.
“In the small” GR respects all of the causality constraints of special relativity, but GR does not provide any natural way of imposing global constraints on the spacetime
Without imposing additional principles along GR is completely infested with time machines

12. Paradoxes generated by the possibility of time travel
There are two broad classes of paradox generated by the possibility of time travel
Grandfather paradoxes: Caused by attempts to “change the past”,
and so modify the conditions that lead to the very existence of the
entity that is trying to “modify the timestream”.
Information paradoxes: bring information to past.

13. Proposals
Make radical alterations to our worldview to incorporate at least some versions of chronology violation and “time travel”.
Permit constrained versions of closed timelike curves
Incorporate quantum physics to intervene and provide a universal mechanism for preventing the occurrence of closed timelike curves.
“Chronology Protection Conjecture”
Hawking

14. There are long debate concerning such principles.
Several people participated in these discussions.
Wheeler, Tipler, Gott, Visser, …
Hawking, Deser, Jackiw, ‘t Hooft, …

15. “Chronology Protection Conjecture“ (Hawking)
But there are not enough convincing arguments for this conjecture
QG effects might smear out the divergences.
Moreover, if QG exists, then chronology protection should be settled within the framework of this theory
CTC does appear in a semi-classical approximation

16. “Chronology Protection Conjecture“
It was suggested that large values of expectation value of the energy-momentum tensor occur when one has CTCs. If one fed this energy-momentum tensor into the Einstein eqs. it could prevent one from creating a TM.
Or divergences in the energy-momentum tensor occur. These divergences may create space-time singularities which prevent one from traveling through to the region of CTC

17. Hawking's "chronology protection conjecture“
Hadamard form

18. Hawking's "chronology protection conjecture“
Theorem (Kay,Radzikowski,Wald). There are points on the
chronology horizon where the two-point functions is not
of Hadamard form

19. Renormalization Group in Curved Spacetime
DeWitt
Problems with definition of Renormalization Group flow:
Scaling of global coordinates or momenta (as in Minkowski space) is not well defined;
No preferred vacuum state for Green functions
Proposal:

20. 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

21. 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, t x
This space contains timelike loops

22. The Cauchy problem on spacetimes that are not globally hyperbolicx
Cauchy problem:

23. t x

25. Example: 2 dim scalar wave equation
Theorem: Under assumption of minimal singularity the Cauchy problem has a unique solution
The Cauchy problem for t>b is not well posed

26. Quantization
IR regularization

27. Energy density No new divergences as compare with Minkowski case
Counter example to Hawking's "chronology protection conjecture“

28. Conclusion TeV Gravity opens new channels – BH, WH, TM
Wheeler foam at TeV scale.
No enough arguments for Hawking's "chronology protection conjecture“
WH/TM production at LHC is of the same order of magnitude as BH production.
The important question on possible experimental signatures of spacetime nontrivial objects deserves further explorations.