BLACK HOLES. BH in GR and in QG BH...

• BLACK HOLES. BH in GR and in QG

• BH formation• Trapped surfaces

• WORMHOLES• TIME MACHINES• 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.20082-nd lecture


I.Aref’eva BH/WH at LHC, Dubna, Sept.2008

• (1965) Penrose introduces the idea of trapped surfaces to complete his singularity proofs.

• (1972) Hawking introduces the notion of event horizons, to capture the idea of a black hole.

History


I.Aref’eva BH/WH at LHC, Dubna, Sept.2008

• 1. Ric(N;N) >= 0 for all null vectors N;• 2. There is a non-compact Cauchy hypersurface H in M• 3. There is a closed trapped surface S in M.

Th.(singularity th. or incompleteness th.) A spacetime (M; g) cannot be future null geodesically complete if:

Th. (Hawking-Penrose) A spacetime (M; g) with a complete future null infnity which contains a closed trapped surface must contain a future event horizon (the interior of which contains the trapped surface)

Trapped surfaces

• A trapped surface is a two dimensional spacelike surface whose two null normals have negative expansion (=Neighbouring light rays, normal to the surface, must move towards one another)


The cross-sectional area enclosing a congruence of geodesics.


Expansion Rotation Shear

A

A

Expansion

0, 0,

is the tangent vect

( )

or to the null geo desic

a b aa a a a

aa

D

d

D

xd

x x x x

xt

º Ñ= =

=

Null geodesics

ab [a b

a

]

b

can be reduced to three types:

the , =D - antisymmetric part

D ab bab

ab a

an

gh

d

h xq x xw xs

= +=

Transverse deformations of a bundle of null geodesics are characterized by

expansionrotation

shear

a bD ξ

ab (a b)=D - symmetric partx


Expansion

The quantity ,in other words measures the focusing of light by gravity.

If our sign conve

0 ( )ntion the light

measures how light rays exp

rays are being focused tog

and or conv

ether i

er

ns

g

tead of s

e

qq

q<pread apart by the spacetime geometry

When for both "incoming" and "outgoing" light rays, it means that the light , that the escape velocity from that gravitational field has become greater than the speed of light.

0has been trapped

q<

When for null geodesics orthogonal to a smooth spacelike surface, that surface

both incoming and is calle

0a tra

outgopped s

inurd ce

gfa

q£

Any closed trapped surface must lie inside a black hole. I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture

Raychaudhuri equation

The Raychaudhuri equation for a null geodesics (focusing equation)

2 ab ab a ba ab ab ab

1 D , 2

aab ab a b

d R h gdqx q q s s w w x x x xt

= =- - + - = +

No rotation, the matter and energy density is positive

21 02

ddq qt

+ £0

1 1 1 2

tq q

³ +


then in a finite distance along the light ray, nearby light rays will be focused to a point, such that they cross each other with zero transverse area A

0 If 0q <

Apparent horizon.

.

• The trapped region is the region containing trapped surfaces.

• A marginally trapped surface is a closed spacelike D-2-surface, the outer null normals of which have zero expansion (convergence).

[ A trapped surface is a two dimensional spacelike surface whose two null normals have negative expansion]

The boundary of (a connected component of) the trapped region is an apparent horizon

• In stationary geometries the apparent horizon is the same as the intersection of the event horizon with the

chosen spacelike hypersurface. • For nonstationary geometries one can show that the apparent

horizon lies beyond the event horizon (Gibbons, 1972)


Expansion and the second fundamental form (extrinsic curvature)

μ

μμν μν μ ν μ

Σ a hypersurface with normal vector nProjection tenzor (the first fundamental )

P = g - σn n , σ = n n

form

)12

The second fundamental form (the extrinsic curvature of

nK L P

μμν μ ν μ ν μ= n - σ n , = n

One can prove

K a a n

Expansion of null geodesics;

ˆ aaK


Black Hole Formation

Two BHs

REFs: Brill and Lindquist (1963) Bishop (1982)

The metric of a time-symmetric slice of space-time representing two BHs

2 4 2 2 2 2( )ds d d dz

The vacuum eq. reduces to

(3) 20 0R Solution 1 2

1 2

2 2 2 21 2

12 2

, ( )

m mr r

r z r z a

1-st Example


3+1 decomposition3+1 decomposition

ADM 3+1 decomposition Arnowitt, Deser, Misner (1962)

3-metric ijglapse shift iN

lapse, shift Gauge

Einstein equations 6 Evolution equations

4 Constraints

2 2( ) ( )( )i i k kADM ikds Ndt g dx N dt dx N dt

N

0 (3) 20 2

1 1 [( ) ]2 2

i j ii i jG R K K K

N

Time-symmetric metric =inv. (t->-t)

Lemma (Gibbons). If on Riemannian space V there is an isometry which leaves fixed the points of a submanifold W then W is a totally geodesicsubmanifold (extremal surface).

0K 0 (3)0

12

G R

Vacuum (3) 0R

Black Hole Formation. Example: two BHs

A cylindrically symmetric surface ( ), ( ), ( )z z

2 2

2

2 2 1/2

: ( , ,0), (0,0, )

: ( , ,0),( )

Tangent vectors t z t

Normal vector n zz

The induced metric

2 4 2 2 2 4 2 2( )i jijd dx dx z d d

, ,( , ),

The extrinsic curvature ofb a b

ij i j j j aK t n n t n



Zero covergence Zero trace of extrinsic curvature

0ijijK


0ijijK


0ijijK

Theorem: Zero covergence Surface is minimal

Area:0

2 4 2 4 2 1/2

0

2 [ ]A dz d

2 2

, ,

2 2 4, ,

( ) 2 0,

( ) 2 0,z

z

Qz Q z Q z

Q Q z Q z Q

δA=0



2 2, ,

2 2 4, ,

( ) 2 0,

( ) 2 0,z

z

Qz Q z Q z

Q Q z Q z Q

δA=0

The first integral

2 2 2( )z Q

2 .,

2 .,

( ) ,

( )zQ Q z Q

Q Q Q

BC.+I.C.:1 10 0 0 0

0 0

| | 0, | | 0,

|

z z

z z


CTS for 2 Black Holes

From Bishop (1982)

a) returnes to = 0 with z = 0 - an extremal surface;b) goes to infinity;c) goes to the singularity


Advantage of CTS (Closed Trapped Surface) Approach

• The existence and location of BH can be found by a global analysis

• TS can be found by a local analysis (within one Cauchy surface)


2-nd Example: BH Formation in Ultra-relativistic Particle Collisions

Particle

4222 )(,)()( D

iii cxduuxdxdudvds

Penrose, D’Eath, Eardley, Giddings

Shock waves

tvu

x

Z


4-dim Aichelburg-Sexl Shock Wave

2 2 2( ) ( ) , ( ) 4 ln | |i i ids dudv dx x u du x p x

12 2 2 2 2

21 ( ) 1 ( )S Sr rds dt dr r d

r r

4-dim Schwarzschild

,u t y v t y

Aichelburg-Sexl, 1970

2

2 2 4 2 2 222

2

(1 ) 1 ( )(1 )

(1 / 4 ) , / 4s s

Ads dt A d dA

r r A r

1-st step



4-dim Schwarzschild

2 2,

1 1

t Vy y Vtt yV V

2 1, lim

1 V

mE EV

2

11 , lim

Vm p V E p

,u t y v t y

2

2 2 4 2 2 22

(1 ) 1 ( )(1 )

Ads dt A dx dy dzA

2-nd step

4-dim Aichelburg-Sexl Shock Wave4-dim Schwarzschild

2 2,

1 1

t Vy y Vtt yV V

21m p V

2

2 2 4 2 2 22

(1 ) 1 ( )(1 )

Ads dt A dx dy dzA

2-nd step(details)

1

22 2 2 21

lim 0;

lim , , .2 ( ) (1 )( ) 2

V

V

A

p p y t y t divy Vt V x z y t

2 22 2 2 2 2 2 4

2 2

2 24

22 2 2 2

(1 ) ( )1 ( ) (1 ) ,(1 ) 1

(1 ) (1 ), (1 ) 4 ...(1 )2 ( ) (1 )( )

A dt Vdyds A dt dx dy dz AA V

p V AA A AAy Vt V x z



4-dim Schwarzschild

Aichelburg-Sexl, 1970

2 2 2 2 2

2

2 2 2 2 2 2

( )

1 1 ( ) ,( ) (1 )( ) ( ) (1 )

ds dt dx dy dz

dt Vdyy Vt V x z y Vt V


Black Hole Formation (Particle = Shock waves)

4222 )(,)()( D

iii cxduuxdxdudvds

vtu

x

Z

,)()()()( 2222 dvvxduuxdxdudvds iii


,)()()()( 2222 dvvxduuxdxdudvds iii

Two Aichelburg-Sexl shock waves

XxVvUu ,

)()(4

)()(4

,))()((41)()(

,))()((41)()(

2

2

XVVXUUXx

XVVVXVv

XUUUXUu

iiii

,][ )2()2()1()1(2 jiijjkikjkik dXdXHHHHdUdVds

)(2

),(21),(

21

)2,1()2,1(2

)2()2(

)1()1(

X

VVHUUH jiijikjiijik


VU

X

Z

Trapped surface in two Aichelburg-Sexl shock waves

,][ )2()2()1()1(2 jiijjkikjkik dXdXHHHHdUdVds

Trapped marginal surface

1 2 1 1

2 2

: 0, ( ) 0

: 0, ( ) 0

S S S S U V X

S V U X

1,2

2 ( 2)1,2 (1,2

1,2

1 2

)

0 ,

( ,

,

)

0

0D

the outer null normals have zero convergence

insi

no function in conve

ode C n C

X X i

onrge

nsi

ncC

e

de C

Ref.:Eardley, Giddings;


The shape of the apparent horizon C on (X1, X2)-plane in the collision plane U = V = 0 for D = 4, 5. Incoming particles are located on the horizontal line X2 = 0.As the distance b between two particles increases, the radius of C decreases.Figure shows the relation between b and rmin for each D. The value of bmax/r0 ranges between 0.8 and 1.3 and becomes large as D increases.

Yoshino, Nambu gr-qc/0209003


3rd Example: Colliding Plane Gravitational Waves I.A, Viswanathan, I.Volovich, 1995

Plane coordinates; Kruskal coordinates

Regions II and III contain the approaching plane waves. In the region IV the metric (4) is isomorphic to the Schwarzschild metric.

2 2

-1 2

2 2 2

ds = 4m [1 + sin(u (u)) + v (v)]dudv

- [1 - sin(u (u)) + v (v)][1 + sin(u (u)) + v (v)] dx

-[1 + sin(u (u)) + v (v)] cos (u (u)) - v (v))dy ,

q qq q q qq q q q

where u < /2, v < /2, v + u < /2p p p

D-dim analog of the Chandrasekhar-Ferrari-Xanthopoulos duality?


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