Gerard ’t HooftSpinoza Institute, Utrecht University

Utrecht University

and

The 4 Force Laws:

Distance

Forcenst 1 22

CQ Q

ForceR

1. Maxwell:

1 22

M MForce G

R

4. Gravitation:

nst 122

C WM RTForce e

R

2. Weak:

nstCForce 3. Strong:

R

Gravity becomes more importantat extremely tiny distance scales !

2

2

2 4

1

/

Wavelength

G

E h cM

hForce

c R

c

However, mass is energy ...

1 22

M MForce G

R

1510 m

1810 m

2110 m

2410 m

2710 m

3010 m

3310 m

The highwa

y across the

desert

Today’sLimit …

GUTs

3510 mPlanck length :Quantum Gravity

LHC

Planck Units

-12 34 sec m kg 100546.12/ h

11 3 1 2NG 6.672 10 m kg sec- -

33Planck 3

Planck

44Planck 5

1.616 10 cm

21.8 g

5.39 10 sec

N

N

N

GL

c

cM

G

GT

c

82.99792458 10 m / secc

The Black Hole

Electromagnetism: like charges repel, opposite charges attract → chargestend to neutralize

Gravity: like masses attract → masses tend to accumulate

The Schwarzschild Solution to Einstein’s equations

( )2

2 2 222

2 2 2d sid

d 1 d ( )d1

nMr M

r

rs t r q q j= - - + + +

-

Karl Schwarzschild1916

“Über das Gravitationsfeldeines Massenpunktes nachder Einsteinschen Theorie”

2

dd ;

2

2 2

2

r

r M

r M

r M

The Schwarzschild Solution to Einstein’s equations

( )2

2 2 222

2 2 2d sid

d 1 d ( )d1

nMr M

r

rs t r q q j= - - + + +

-

Karl Schwarzschild1916

“Über das Gravitationsfeldeines Massenpunktes nachder Einsteinschen Theorie”

Universe I

Universe II“Time” stands still at the horizon

So, one cannot travel from

one universe to the other

Black Hole

or wormhole?

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

3

HBH8

ckT

G Mp=

h

While emitting particles, the black hole loosesenergy, 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

639

12

639

¬Black hole plus matter ® Heavier black hole

compare Hawking’s particle emission process with the absorption process:

In a black hole:

of the final states

time reversal

symmetry (PCT):

forwards and

backwards in time:

the same

Probability =| Amplitude |2 × (Volume of Phase Space)

65 2

One bit of

information

on every

cm0 724 10 -.

The black hole as an information processing machine

The constant of integration: a few“bits” on the side ...

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)

Dogma: We should be able to derive all propertiesof these states simply by applying General Relativityto 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 evolveinto a pure final state?

The calculation of the Hawking effect suggests thatpure states evolve into mixed states !

❖

Region IRegion II

Horizon

The quantum states in regions I and II are coherent.

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 ...

space

time

Alternative theories:

1. 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)

3. Information is in the Hawking radiation

How do we reconcile these with LOCALITY?

paradox

Black Holes require new axioms for thequantization of gravity

Unitarity,Causality, ...

paradox

Black Hole Quantum Coherence is realized in String/Membrane Theories !

-- at the expense of locality? -- How does Nature process information ?

❖

The physical description of the horizon problem ...

horizon

Here, gravitational interactions become

strong !!

brick wall

interaction

horizon

2-d surface

Particles and horizons, the hybrid picture

Black hole complementarity principle

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

length ( , ) lengtht x

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

21

( )( ) ; ( , )

x ax x t

x

(the conformal transformation)

This local scale invariance is a local 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?

????????

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

b

By taking back reaction into account, one can obtain a unitary scattering matrix

❖

Gravitational effect from ingoing objects

particlesout

in

2 2

The coordinate shift can be calculated

to be :

which obeys :

4 log

8 p ( )

x G p x

x G x

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

( )x ( )p

2 2

in out( ), ( ') ( ')p p i

2 2

in out( ), ( ') ( ')i

2in in

out out

in out

[ ( ), ( ')] ( ') ;

[ ( ), ( ')] 0 ;

x p i

x p

Also for electro-magnetism:

The string world-sheet

Black Hole Formation & Evaporation by Closed Strings

BLACK HOLE WHITE HOLE

A black hole is a quantum superposition ofwhite holes and vice versa !!

The Difference between

y

Black holes and extra dimensions

xy

4-d world on “D -brane”

Horizon of “Big Hole” “Little

Hole”

These would have a thermal distribution with equal probabilities for all particle species, corresponding to Hawking’s temperature