BLACK HOLES. BH in GR and

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BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC

• 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.

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History
BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC

• (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.

I.Arefeva
BH/WH at
LHC, Dubna, Sept.2008
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Th.(singularity th. or incompleteness th.) A
spacetime (M g) cannot be future null
geodesically complete if
BLACK HOLES and WORMHOLES PRODUCTION AT THE LHC

• 1. Ric(NN) gt 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. (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)
I.Arefeva
BH/WH at
LHC, Dubna, Sept.2008
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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)

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The cross-sectional area enclosing a congruence
of geodesics.
Expansion Rotation Shear


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Expansion
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Expansion
Any closed trapped surface must lie inside a
black hole.
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Raychaudhuri equation
The Raychaudhuri equation for a null geodesics
(focusing equation)
No rotation, the matter and energy density is
positive
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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)

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Expansion and the second fundamental form
(extrinsic curvature)
Expansion of null geodesics
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Black Hole Formation
1-st Example
REFs Brill and Lindquist (1963)
Bishop (1982)
Two BHs
The metric of a time-symmetric slice of
space-time representing two BHs
The vacuum eq. reduces to
Solution
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31 decomposition

• ADM 31 decomposition

Arnowitt, Deser, Misner (1962)
3-metric
Time-symmetric metric inv. (t-gt-t)
lapse
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
geodesic submanifold (extremal surface).
shift

• lapse, shift Gauge

• Einstein equations

6 Evolution equations
4 Constraints
Vacuum
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Black Hole Formation. Example two BHs
A cylindrically symmetric surface
The induced metric
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Black Hole Formation. Example two BHs
Theorem
Area
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Black Hole Formation. Example two BHs
The first integral
BC.I.C.
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CTS for 2 Black Holes
From Bishop (1982)
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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)

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2-nd Example BH Formation in Ultra-relativistic
Particle Collisions
Particle
Shock waves

Penrose, DEath, Eardley, Giddings
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4-dim Aichelburg-Sexl Shock Wave
4-dim Schwarzschild
Aichelburg-Sexl, 1970
1-st step
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4-dim Aichelburg-Sexl Shock Wave
4-dim Schwarzschild
2-nd step
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4-dim Aichelburg-Sexl Shock Wave
4-dim Schwarzschild
2-nd step(details)
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4-dim Aichelburg-Sexl Shock Wave
4-dim Schwarzschild
Aichelburg-Sexl, 1970
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Black Hole Formation (Particle Shock
waves)
t
v
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Two Aichelburg-Sexl shock waves
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Trapped surface in two Aichelburg-Sexl shock waves
Ref.Eardley, Giddings
V
Trapped marginal surface
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Yoshino, Nambu gr-qc/0209003
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.
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3rd Example Colliding Plane Gravitational Waves
I.A, Viswanathan, I.Volovich, 1995
D-dim analog of the Chandrasekhar-Ferrari-Xanthopo
ulos duality?