Gerard ’t Hooft Spinoza Institute

1. Utrecht University QUANTUM GRAVITY WITHOUT SPACE-TIME SINGULARITIES OR HORIZONS arXiv:0909.3426 CMI, Chennai, 20 November 2009 Gerard ’t Hooft Spinoza Institute

2. Entropy = ln ( # states ) = ¼ (area of horizon) Are black holes just “elementary particles”? Are elementary particles just “black holes”? Imploding matter Hawking particles Black hole “particle”

3. Small region near black hole horizon: Rindler space A quantum field in space splits into two parts, and . The vacuum in space corresponds to an entangled state: time II I space

4. Cauchy surface out horizon out out out singul -arity in in imploding matter imploding matter

5. out decay out in imploding matter implosion

6. Hawking radiation imploding matter

7. Hawking radiation imploding matter Penrose diagram ?

8. Black hole complementarity

9. An observer going in, experiences the original vacuum, Hence sees no Hawking particles, but does observe objects behind horizon An observer staying outside sees no objects behind horizon, but does observe the Hawking particles. They both look at the same “reality”, so there should exist a mapping from one picture to the other and back. Black hole complementarity

10. Extreme version of complementarity Ingoing particles visible; Horizon to future, Hawking particles invisible time space

11. Extreme version of complementarity Outgoing particles visible; Horizon to past, Ingoing particles invisible time space

12. But now, the region in between is described in two different ways. Is there a mapping from one to the other? The two descriptions are complementary.

13. Starting principle: causality is the same for all observers This means that the light cones must be the same Light cone: The two descriptions may therefore differ in their conformal factor. The only unique quantity is

14. Invariance under scale transformations May serve as an essential new ingredient to quantize gravity describes light cones describes scales

15. The outside, macroscopic world also has the scale factor: Einstein equs for massless ingoing or outgoing particles generate singularities and horizons. Question: can one adjust such that all singularities move to infinity, while horizons disappear (such that we have a flat boundary for space-time at infinity)? What are the equations for ?

16. The transformations that keep the equation unchanged are the conformal transformations.

17. out The transform-ation from the ingoing matter description “in ” to the outgoing matter description “out ” is a conformal transform- ation in

18. Why is the world around us not scale invariant ? Empty space-time has , but that does not fix the scale, or the conformal transformations. These are defined by the boundary at infinity. Thus, the “desired” is determined non-locally. How? At the Planck scale, the particles that are familiar to us are all massless. Therefore, the trace of the energy-momentum tensor vanishes:

19. is a constraint to impose on Together with the boundary condition, this fixes . However, , therefore different observers see different amounts of light-like material:

20. This is also why, in one conformal frame, an observer sees Hawking radiation, and in an other (s)he does not. For the black hole, the transformation “in” ⇔ “out” is no longer a conformal one when we include in- and out going matter. Therefore, one can then describe all of space-time in one coordinate frame. To describe , we can impose , but we don’t have to. Then we can describe the metric as follows:

21. out out flat Schwarzschild Schwarzschild time in in space

22. The scale ω(x ) cannot be observed locally, but it must be identified by “global” observers! Space-time is not just “emergent”, but can be, and should be, the essential backbone of a theory. Space-time is topologically trivial perhaps, conceivably, on a cosmological scales Scale invariance is an exact symmetry, not an approximate one! The vacuum state, and the scale of the metric, both play a central role in this theory Note that the Cosmological Constant problem also involves a hierarchy problem, which cannot be addressed this way ..

23. THE END arXiv:0909.3426

24. As seen by distant observer As experienced by astro- naut himself Time stands still at the horizon Continues his way through They experience time differently. Mathematics tells us that, consequently, they experience particles differently as well

26. Stephen Hawking’s great discovery: the radiating black hole