Gerard ’t Hooft Spinoza Institute Utrecht University CMI, Chennai, 20 November 2009 arXiv:

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

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

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

imploding matter matter in out out implosion decay

imploding matter matter Hawkingradiation

imploding Hawkingradiation Penrose diagram ?

Black hole complementarity

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.

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

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

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.

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

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

The outside, macroscopic world also has the scale factor: What are the equations for ? 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)?

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

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

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:

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

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:

flat in out in Schwarzschild space time

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 scale ω (x ) cannot be observed locally, but it must be identified by “global” observers! 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..

arXiv:

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

Stephen Hawking’s great discovery: the radiating black hole