Utrecht University Gerard ’t Hooft CERN Black Hole Institute.
Gerard t Hooft Spinoza Institute, Utrecht University Utrecht University and.
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Transcript of Gerard t Hooft Spinoza Institute, Utrecht University Utrecht University and.
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