gerard ’t hooft spinoza institute
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Utrecht University. QUANTUM GRAVITY WITHOUT SPACE-TIME SINGULARITIES OR HORIZONS. arXiv:0909.3426. CMI, Chennai, 20 November 2009. Gerard ’t Hooft Spinoza Institute. Entropy = ln ( # states ) = ¼ (area of horizon). Are black holes just “elementary particles”?. - PowerPoint PPT PresentationTRANSCRIPT
Gerard ’t HooftSpinoza Institute
Utrecht University
CMI, Chennai, 20 November 2009arXiv: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 nEI II
n
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
matterimploding imploding mattermatter
horizonhorizon
singulsingul-arity-arity
inin
outout
inin
outout
outoutoutoutCauchy 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 complementarity
Ingoing particlesvisible; Horizon to future,Hawking particlesinvisible
space
time
Outgoing particlesvisible; Horizon to past,Ingoing particlesinvisiblespace
time
Extreme 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 observersThis means that the light cones must be the same
Light cone:2 ( ) 0ds g x dx dx
1/4ˆ ˆ; det( ) 1
( det( ))g
g gg
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
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
outThe 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