Transcript Geen diatitel

Utrecht University
and
Gerard ’t Hooft
Spinoza Institute, Utrecht University
The 4 Force Laws:
1. Maxwell:
Q1  Q2
Force  C 
R2
nst
Force
2. Weak:
T12  MW R
Force  C  2 e
R
nst
3. Strong:
Force  C
nst
R
4. Gravitation:
M1  M 2
Force  G 
R2
Distance
M1  M 2
Force  G 
2
R
However, mass is energy ...
E
h/c
M 2 
c
Wavelength
2
Gh
1
Force  2  4
c
R
Gravity becomes more important
at extremely tiny distance scales !
Quantum
Gravity
The
highway
across
the
desert
Today’s
Limit …
LHC
Planck length : 10 35 m
10 33 m
10 30 m
10 27 m
GUTs
10 24 m
10 21 m
10
18
10
m
15
m
h / 2    1.05461034 kg m2 sec-1
GN  6.672 1011 m3 kg-1 sec-2
c  2.99792458  108 m / sec
Planck Units
LPlanck 
GN

3
c
M Planck 
c
GN
TPlanck 
1.616  10 33 cm

21.8  g
GN

5
c
5.39  10 44 sec
Electromagnetism: like charges repel,
opposite charges attract → charges
tend to neutralize
Gravity: like masses attract
→ masses tend to accumulate
The Black Hole
The Schwarzschild Solution to Einstein’s equations
2
d
r
2
2
2
2
ds 2 = - (1 - 2rM )dt 2 +
+
r
(
d
q
+
si
n
q
d
j
)
2
M
1- r
Karl Schwarzschild
1916
“Über das Gravitationsfeld
eines Massenpunktes nach
der Einsteinschen Theorie”
dr
d ;
r  2M
   2 r  2M
 2  r  2M
The Schwarzschild Solution to Einstein’s equations
2
d
r
2
2
2
2
ds 2 = - (1 - 2rM )dt 2 +
+
r
(
d
q
+
si
n
q
d
j
)
2
M
1- r
Karl Schwarzschild
1916
“Über das Gravitationsfeld
eines Massenpunktes nach
der Einsteinschen Theorie”
Black Hole
Universe I
“Time”
stands still
at the
horizon
or wormhole?
Universe II
So, one cannot travel from
one universe to the other
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
kT H
hc 3
=
8pG M BH
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 ...
In a black hole:
compare Hawking’s particle emission process
with the absorption process:
9
12
6
3
Black hole plus matter
9
12
6
3
¨
Heavier black hole
| Amplitude |2
Probability =
× (Volume of Phase Space)
The black hole as an information processing machine
One bit of
information
on every
0. 724  10 - 65 cm2
The constant of
integration: a few
“bits” on the side ...
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”
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 !
❖
Horizon
Region II
Region I
time
space
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 ...
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
Black Holes require new axioms for the
quantization of gravity
How do we reconcile these with LOCALITY?
paradox
Unitarity,
Causality,
...
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 ...
Here, gravitational interactions
become strong !!
horizon
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 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!
length   ( x, t )  length
An exact local symmetry transformation,
which does not leave the vacuum invariant,
unless:
 ( x)  ( x 1a ) 2 ; x  ( x, t )
(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? ???????????????
By taking back reaction into account, one can
obtain a unitary scattering matrix
b
❖
Gravitational effect from ingoing objects
in
particles
out
The coordinate shift can be calculated
to be :
 x   4G p  log x
which obeys :
2
  x   8 G p  2 ( x )

The non-commucativity between x  ( ) and p ( )
lleads to a Horizon Algebra :
[x

in
out
( ), p
[x

in
( ')]  i  (   ')  ;
2

in
out
( ), p

out

( ')]  0 ;
 p  in ( ), p  out ( ')   i  2 2 (   ')  


Also for electro-magnetism:
 in ( ), out ( ')   i  2 2 (   ')
The string world-sheet
Black Hole Formation & Evaporation by Closed Strings
The Difference between
BLACK HOLE
WHITE HOLE
A black hole is a quantum superposition of
white holes and vice versa !!
Black holes and extra dimensions
4-d world on
“D -brane”
y
y
x
Horizon of
“Big Hole”
“Little Hole”
These would have a thermal distribution with equal probabilities
for all particle species, corresponding to Hawking’s temperature