Post on 25-Feb-2016
description
• 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
BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC
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
BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC
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)
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
The cross-sectional area enclosing a congruence of geodesics.
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
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
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
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
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 +
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
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)
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
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
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
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
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
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
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
Black Hole Formation. Example: two BHs
Zero covergence Zero trace of extrinsic curvature
0ijijK
Zero covergence Zero trace of extrinsic curvature
0ijijK
Zero covergence Zero trace of extrinsic curvature
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
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
Black Hole Formation. Example: two BHs
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
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
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
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
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)
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
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
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
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 Aichelburg-Sexl Shock Wave
2 2 2( ) ( ) , ( ) 4 ln | |i i ids dudv dx x u du x p x
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 Aichelburg-Sexl Shock Wave
2 2 2( ) ( ) , ( ) 4 ln | |i i ids dudv dx x u du x p x
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
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
Black Hole Formation (Particle = Shock waves)
4222 )(,)()( D
iii cxduuxdxdudvds
vtu
x
Z
,)()()()( 2222 dvvxduuxdxdudvds iii
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
,)()()()( 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
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
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;
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
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
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture
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?
I.Aref’eva BH/WH at LHC, Dubna, Sept.20082-nd lecture