Utrecht University ar Xiv 0909 3426 CMI Chennai

  • Slides: 26
Download presentation
Utrecht University ar. Xiv: 0909. 3426 CMI, Chennai, 20 November 2009 Gerard ’t Hooft

Utrecht University ar. Xiv: 0909. 3426 CMI, Chennai, 20 November 2009 Gerard ’t Hooft Spinoza Institute

Entropy = ln ( # states ) = ¼ (area of horizon) Are black

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

Small region near black hole horizon: Rindler space time II I space A quantum

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

horizon out Cauchy surface out out singul -arity in in im m plod at

horizon out Cauchy surface out out singul -arity in in im m plod at in te g r imploding matter

out decay out in imploding matter implosion

out decay out in imploding matter implosion

Hawking radiation im m plod at in te g r

Hawking radiation im m plod at in te g r

Penrose diagram Hawking radiation im m plod at in te g r ?

Penrose diagram Hawking radiation im m plod at in te g r ?

Black hole complementarity

Black hole complementarity

An observer going in, experiences the original vacuum, Hence sees no Hawking particles, but

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

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

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

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

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

But now, the region in between is described in two different ways. Is there

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

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

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.

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

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

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

Why is the world around us not scale invariant ? Empty space-time has ,

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,

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

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 inand 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:

out flat Schwarzschild time in in space

out flat Schwarzschild time in in space

The scale ω (x ) cannot be observed locally, but it must be identified

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 cosmological scales perhaps, conceivably, on a 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. .

ar. Xiv: 0909. 3426

ar. Xiv: 0909. 3426

As seen by distant observer Time stands still at the horizon As experienced by

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

Stephen Hawking’s great discovery: the radiating black hole

Stephen Hawking’s great discovery: the radiating black hole