Gerard ’t HooftSpinoza Institute

Utrecht University

CMI, Chennai, 20 November 2009

arXiv:0909.3426arXiv:0909.3426

Are black holes just“elementary particles”?

Black hole“particle”

Implodingmatter

Hawking particles

Are elementary particles just “black holes”?

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

( , )x t

( , )I ( , )II

vac nE

I IIn

C n n e

x

x

Small region near black hole horizon:Rindler space

time

III

22( ) ; 2 1/ nEW n C e kT

space

imploding

imploding

matter

matter

imploding imploding mattermatter

horizonhorizon

singulsingul-arity-arity

inin

outout

inin

outout

outoutoutout

Cauchy surfaceCauchy surface

imploding imploding mattermatter

inin

outout

outout

implosion

decay

imploding

imploding

matter

matter

HawkingHawkingradiationradiation

imploding

imploding

matter

matter

HawkingHawkingradiationradiation

Penrose diagramPenrose diagram

??

Black hole complementarity

An observer going in, experiences the An observer going in, experiences the original vacuumoriginal vacuum,,Hence sees Hence sees no Hawking particlesno Hawking particles, but does observe, but does observeobjects behind horizonobjects behind horizon

An observer staying outside sees An observer staying outside sees no objects behind horizon, no objects behind horizon, but does observe thebut does observe the Hawking particles Hawking particles..

They both look at the same “reality”, so thereThey both look at the same “reality”, so thereshould exist a should exist a mappingmapping from one picture to the from one picture to theother and back.other and back.

Extreme version of complementarityExtreme version of complementarity

Ingoing particlesvisible; Horizon to future,Hawking particlesinvisible

space

time

Outgoing particlesvisible; Horizon to past,Ingoing particlesinvisiblespace

time

Extreme version of complementarityExtreme version of complementarity

But now, the region in between is described intwo 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:

2 ( ) 0ds g x dx dx

1/4ˆ ˆ; det( ) 1

( det( ))

gg g

g

The two descriptions may therefore differ in their conformal factor. The only unique quantity is

2

1/8

ˆ ˆ, det( ) 1 ,

( det( ))

g g g

g

Invariance under scale transformationsMay serve as an essential new ingredient

to quantize gravity

Invariance under scale transformationsMay serve as an essential new ingredient

to quantize gravity

g describes light cones

describes scales

The outside, macroscopic world also has the scale factor:

2 ˆ( ) ; ( ) ( ) ( )x g x x g x

| | : 1x

What are the equations for ?ˆ ( ) , ( )g x x

Einstein equs for massless ingoing or outgoing particles generate singularities and horizons.Question: can one adjust such that allsingularities move to infinity, while horizonsdisappear (such that we have a flat boundaryfor space-time at infinity)?

( )x

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

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.

g

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:

0T

2 1ˆ ˆ6 0R R g D

0R is a constraint to impose onTogether with the boundary condition, this fixes .

However, , therefore different observers see different amounts of light-like material:

T D

0 , 0 .T T

16

ˆg D R

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:

0R

0R 0R

flat

in

outout

in

SchwarzschildSchwarzschild

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, The vacuum state, and the scale of the metric, both play a central role in this theoryboth play a central role in this theory

Note that the Cosmological Constant problem also Note that the Cosmological Constant problem also involves a hierarchy problem, which cannot be involves a hierarchy problem, which cannot be addressed this way ..addressed this way ..

arXiv:0909.3426arXiv:0909.3426

As seen by distantobserver

As

experienced by astro-

naut himself

They experience time differently. Mathematics tells usthat, consequently, they experience particles differently

as well

Time stands stillat the horizon

Continueshis waythrough

Stephen Hawking’s great discovery:the radiating black hole