BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC I.Ya.Aref’eva Steklov Mathematical Institute,...

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

,

/ 1P

N

E M

or 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)


BLACK HOLES in GR. Schwarzschild solution to vacuum Einstein eq.


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.


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


0R 1

2 2 2 2 221 1S Sr r

ds dt dr r dr r

2 2

6

12G MR R

r

22

221

3232 )(1)(1

D

DSDS drdrr

rdt

r

rds

D-dimensional Schwarzschild Solution

Schwarzschild radiusSr is the

0, , 0,1... 1R D

BH

D

BH

DBHS M

MM

Dr )(1

)(3

1

DBH

)3/(1

2

)2/1(81)(

D

BH D

DD

2

1D D

D

GM

Meyers,…

2

1

PlNewton M

G



• Geodesics

( ) : 0,

( ),

x x n D n

dxn n n

d


12 2 2 2 2

21 1S Sr rds dt dr r d

r r

2

2

( ) ( ) ( )0,

d x dx dx

d d d

12 2 20 1 1S Sr r

ds dt drr r

Null Geodesics



1

1 Sdt r

dr r

Null Geodesics

Regge-Wheeler coordinates

Sr



Eddington-Finkelstein coordinates

The determinant of the metric is is regular at r=2GM

u~

r=0r=2GM

constu ~

r



Kruskal coordinates

Kruskal diagram

Kruskal diagram represents the entire spacetime corresponding to the Schwarzschild metric



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



Each point on the diagram is a two-sphere

Penrose diagram for Schwarzschild



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.


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


AQ . Analogy with Solitons

2a particle with mass m

22 2

2

1( ) (cos 1)2 2

mL

22

24

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

BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC I.Ya.Aref’eva Steklov Mathematical Institute,...

Documents

Transcript of BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC I.Ya.Aref’eva Steklov Mathematical Institute,...