BLACK HOLES and WORMHOLES
PRODUCTION AT THE LHC I.Ya.Aref’eva Steklov Mathematical Institute, Moscow
BH/WH production in Trans-Planckian Collisions
• BLACK HOLES. BH in GR and in QG
• BH formation• Trapped surfaces
• WORMHOLES• TIME MACHINES• WHY on LHC? Cross-sections and signatures
of BH/WH production at the LHC
• I-st lecture.
• 2-nd lecture.
• 3-rd lecture.
BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC
I.Aref’eva BH/WH at LHC, Dubna, Sept.2008
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P
N
E Mor G s
Trans-Planckian Energy
I-st lecture. Outlook:
• BLACK HOLES in GR• Historical Remarks• SCHWARZSCHILD BH• Event Horizon • Trapped Surfaces• BH Formation
• BLACK HOLES in QG• BLACK HOLES in Semi-classical approximation to QG
I.Aref’eva BH/WH at LHC, Dubna, Sept.2008
Refs.: L.D.Landau, E.M.Lifshitz, The Classical Theory of Fields, II v. Hawking S., Ellis J. The large scale structure of space-time.R.Wald, General Relativity, 1984 S. Carroll, Spacetime and Geometry.An introduction to general relativity, 2004
BLACK HOLES in GR. Historical Remarks• The Schwarzschild solution has been found in 1916. The Schwarzschild solution(SS) is a solution of the vacuum Einstein
equations, which is spherically symmetric and depends on a positive parameter M, the mass.
In the coordinate system in which it was originally discovered, (t,r,theta,phi), had a singularity at r=2M
• In 1923 Birkoff proved a theorem that the Schwarzschild solution is the only spherically symmetric solution of the vacuum E.Eqs.• In 1924 Eddington, made a coordinate change which transformed the
Schwarzschild metric into a form which is not singular at r=2M• In 1933 Lemaitre realized that the singularity at r=2M is not a true
singularity • In 1958 Finkelstein rediscovered Eddington's transformation and
realized that the hypersurface r=2M is an event horizon, the boundary of the region of spacetime which is causally connected to infinity.
I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008
BLACK HOLES in GR. Historical Remarks• In 1950 Synge constructed a systems of coordinates that
covers the complete analytic extension of the SS• In 1960 Kruskal and Szekeres (independently) discovered
a single most convenient system that covers the complete analytic extension of SS
• In 1964 Penrose introduced the concept of null infinity, which made possible the precise general definition of a future event horizon as the boundary of the causal past of future null infinity.
• In 1965 Penrose introduced the concept of a closed trapped surface and proved the first singularity theorem (incompleteness theorem).
• Hawking-Penrose theorem: a spacetime with a complete future null infnity which contains a closed trapped surface must contain a future event horizon (H-E-books, 9-2-1)
I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008
BLACK HOLES in GR. Schwarzschild solution to vacuum Einstein eq.
I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008
0R A general metric in a spherically symmetric spacetime (Ch.13 of Weinberg)
Christoffel symbols, Riemann tensor, Ricci tensor
1) 2) 3)
5)
2)
1)
4)
BLACK HOLES in GR. Schwarzschild solution to vacuum Einstein eq.
I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008
0R A general metric in a spherically symmetric spacetime (Ch.13 of Weinberg)
Christoffel symbols, Riemann tensor, Ricci tensor
2 SG M r
BLACK HOLES in GR. Schwarzschild solution
• Asymptoticaly flat• Birkhoff's theorem: Schwarzschild solution is the unique spherically symmetric vacuum solution• Singularity
I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008
0R 1
2 2 2 2 221 1S Sr rds dt dr r d
r r
2 2
6
12G MR Rr
22
221
3232 )(1)(1
D
DSDS drdrrrdt
rrds
D-dimensional Schwarzschild Solution
Schwarzschild radiusSr is the
0, , 0,1... 1R D
BH
D
BH
DBHS M
MM
Dr )(1)(31
DBH
)3/(1
2)2/1(81)(
D
BH DDD
2
1D D
D
GM
Meyers,…
21
PlNewton M
G
I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008
BLACK HOLES in GR. Schwarzschild solution
• Geodesics
( ) : 0,
( ) ,
x x n D n
dxn n nd
I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008
12 2 2 2 2
21 1S Sr rds dt dr r dr r
2
2
( ) ( ) ( ) 0,d x dx dxd d d
12 2 20 1 1S Sr rds dt dr
r r
Null Geodesics
BLACK HOLES in GR. Schwarzschild solution
I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008
1
1 Sdt rdr r
Null Geodesics
Regge-Wheeler coordinates
Sr
BLACK HOLES in GR. Schwarzschild solution
I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008
Eddington-Finkelstein coordinates
The determinant of the metric is is regular at r=2GM
u~
r=0 r=2GM
constu ~
r
BLACK HOLES in GR. Schwarzschild solution
I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008
Kruskal coordinates
Kruskal diagram
Kruskal diagram represents the entire spacetime corresponding to the Schwarzschild metric
BLACK HOLES in GR. Schwarzschild solution
I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008
Each point on the diagram is a two-sphere
• WE –region I; • Future-directed null rays reach region II,
• Past-directed null rays reach region III.
• Spacelike geodesics reach region IV.
• Regions III is the time-reverse of region II. ``White hole.''
• Boundary of region II is the future event horizon.
• Boundary of region III is the past event horizon.
Penrose (or Carter-Penrose, or conformal) diagram
BLACK HOLES in GR. Schwarzschild solution
I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008Each point on the diagram is a two-sphere
Penrose diagram for Schwarzschild
BLACK HOLES in GR. Schwarzschild solution
I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008
Penrose diagram for Minkowski
t=const
Event Horizons
The event horizon is always a null hypersurface.The event horizon does not depend on a choice of foliation.The event horizon always evolves continuously.The event horizon captures the intuitive idea of the boundary of what can reach observers at infinity.
I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008
BH production in QFT
•Q - can QFT (in flat space-time) produce BH
I.Aref’eva BH-QCD,CERN,Sept.2008
• A Q - QFT(in flat space-time) cannot produce BH/WH, etc.
Answer:
QFT + "SQG" (semicl.aprox. to QG) can
Question
We can see nonperturbatively just precursor (предвестник) of BHs
BH in Quantum Gravity
" " ' '
", ", " | ', ', ' exp{ [ , ]} ,
": ", " ; ' : ', ',
| ", | "; | ', | 'ij ij
ih h S g dg d
h h
g h g h
Background formalism in QFT IA, L.D.Faddeev, A.A.Slavnov, 1975
' ''1 1 ' '',... | ,... exp{ [ , ]} ,
( ')... ( '')..'' '" : ( '') (
.
; ' : ')
as as
ik k S
k kas a
gk
s
dk
2 |particles BH
I.Aref’eva I-st lecture BH/WH at LHC, Dubna, Sept.2008
AQ . Analogy with Solitons
2a particle with mass m
22 2
2
1 ( ) (cos 1)2 2
mL
2
224
mSolitons with mass M
I.Aref’eva BH-QCD,CERN,Sept.2008
• perturbative S-matrix for phi-particles - no indications of solitons
• Exact nonperturbative S-matrix for phi-particles – extra poles - indications of a soliton-antisoliton state
How to see these solitons in QFT?
IA, V.Korepin, 19742222
84 )γ(mmM
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