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BLACK HOLES

and

Quantum Physics

Gerard ’t Hooft

Spinoza Institute, Utrecht University

Gravity becomes more importantat extremely tiny distance scales !

However, mass is energy ...

The Black Hole

Electromagnetism: like charges repel,

oppositecharges attract → charges

tend to neutralize

Gravity: like masses attract

→ masses tend to accumulate

The Schwarzschild Solution to Einstein’s equations

Karl Schwarzschild

1916

“Über das Gravitationsfeld

eines Massenpunktes nach

der Einsteinschen Theorie”

The Schwarzschild Solution to Einstein’s equations

Karl Schwarzschild

1916

“Über das Gravitationsfeld

eines Massenpunktes nach

der Einsteinschen Theorie”

“Time” stands still at the horizon

So, one cannot travel from

one universe to the other

Black Hole

or wormhole?

Universe I

Universe II

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

Stephen Hawking’s great discovery:

the radiating black hole

While emitting particles, the black hole looses

energy, hence mass ... it becomes smaller.

Lighter (smaller) black holes emit more intense

radiation than heavier (larger) ones

The emission becomes more and more intense,

and ends with ...

12

3

3

9

9

6

6

In a black hole:

compare Hawking’s particle emission process

with the absorption process:

Black hole plus matter

Heavier black hole

| Amplitude |2 × (Volume of Phase Space)

of the

final states

time reversal symmetry (PCT):

forwards and backwards in time: the same

The black hole as an information processing machine

The constant of

integration: a few

“bits” on the side ...

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”

Dogma: We should be able to derive all properties

of these states simply by applying General Relativity

to the black hole horizon ... [ isn’t it ? ]

That does NOT seem to be the case !!

For starters: every initial state that forms a black

hole generates the same thermal final state

But should a pure quantum initial state not evolve

into a pure final state?

The calculation of the Hawking effect suggests that

pure states evolve into mixed states !

❖

space

Horizon

The quantum states in regions I and II are coherent.

Region II

Region I

This means that quantum interference experiments in region I cannot be carried out without considering the states in region II

But this implies that the state in region I is not a “pure quantum state”; it is a probabilistic mixture of different possible states ...

- Alternative theories:
- No scattering, but indeed loss of quantum coherence
- (problem: energy conservation)

2. After explosion by radiation:

black hole remnant

(problem: infinite degeneracy of the

remnants)

- Information is in the Hawking radiation

Black Holes require new axioms for the

quantization of gravity

How do we reconcile these with LOCALITY?

paradox

Black Hole Quantum Coherence is

realized in String/Membrane Theories !

-- at the expense of locality? --

How does Nature process information ?

paradox

Unitarity,

Causality, ...

❖

horizon

Black hole complementarity principle

An observer going into a black hole can detect all other material that went in, but not the Hawking radiation

An observer outside the black hole can detect the Hawking particles, but not all objects that have passed the horizon.

Yet both observers describe the same “reality”

Elaborating on this complementarity principle:

An observer going into a black hole treats ingoing matter as a source of gravity, but Hawking radiation has no gravitational field.

An observer outside the black detects the gravitational field due to the Hawking particles, but not the gravitational fields of the particles behind the horizon.

Yet both observers describe the same “space-time”

Space-time as seen by ingoing observer

Space-time as seen by late observer outside

This may be a conformal transformation of the interior region:

Light-cones remain where they are, but distances and time intervals change!

An exact local symmetry transformation, which does not leave the vacuum invariant, unless:

(the conformal transformation)

This local scale invariance is a local interior region:U (1) symmetry: electromagnetism as originally viewed by H. Weyl.

Fields may behave as a representation of this U (1) symmetry.

????????

Is this a way to unify EM with gravity?

The cosmological constant (“Dark energy”) couples directly to scales

Is this a way to handle the cosmological constant problem?

???????????????

By taking back reaction into account, one can interior region:

obtain a unitary scattering matrix

b

❖

The non-commucativity between and lleads to a Horizon Algebra :

Also for electro-magnetism:

The string world-sheet lleads to a

WHITE HOLE lleads to a

BLACK HOLE

The Difference between

A black hole is a quantum superposition of

white holes and vice versa !!

y lleads to a

y

x

Black holes and extra dimensions4-d world on

“D -brane”

Horizon of “Big Hole”

“Little Hole”

These would have a lleads to a thermal distribution with equal probabilities

for all particle species, corresponding to Hawking’s temperature

THE END lleads to a

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