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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Wormholes

• Lorentzian Wormhole is a region in spacetime in
  which 3-dim space-like sections have non-trivial
  topology.
• By non-trivial topology we mean that these
  sections are not simply connected
• In the simplest case a WH has two mouths which
  join different regions of the space-time.
• We can also imagine that there is a thin handle,
  or a throat connected these mouths.
• Sometimes people refer to this topology as a
  'shortcut' through out spacetime

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Wormholes

• The term WH was introduced by J. Wheeler in
  1957
• Already in 1921 by H. Weyl (mass in terms of EM)
• The name WH comes from the following obvious
  picture.

The worm could take a shortcut to the opposite
side of the apple's skin by burrowing through
its center, instead of traveling the entire
distance around.
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The traveler just as a worm could take a
shortcut to the opposite side of the universe
through a topologically nontrivial tunnel.
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Wormholes

• The first WH solution was found by Einstein and
  Rosen in 1935 (so-called E-R bridge)
• There are many wormhole solutions in GR.
• A great variety of them! With static throat,
  dynamic throat, spinning, not spinning, etc
• Schwarzschild WHs (E-R bridges)
• The Morris-Thorne WH
• The Visser WH
• Higher-dimensional WH
• Brane WH

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Traversable Wormholes
Morris, Thorne, Yurtsever, Visser,..
The embedding condition together with the
requirement of finiteness of the redshift
function lead to the NEC violation on the WH
throat
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Time Machine. Definition

• Spacetime (M,g), M manifold, g metric.
• Einstein equations for g.
• Time machine is a region of space-time (M,g)
  that has a closed timelike curve (CTC).
• CTC suggests the possibility of time travel with
  its well known paradoxes
• Example time is circle.

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Time Machine

• TM is impossible in special relativity.
• Indeed, to make a loop, a curve must somewhere
  leave the null cone as shown in this picture.
• A particle with such a world line would exceed
  the speed of light that is impossible in SR.

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Time Machine

• In general relativity the situation is much less
  trivial.
• According to GR, our spacetime must be a smooth
  Lorentzian manifold small regions is
  approximately Minkowskian, at large scale could
  be any geometry and topology (holes, handles,
  almost whatever one wants).

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Solutions of Einstein eqs. with Closed Timelike
Curves (CTC) / Time Machine.

• Godel's solution 1949
• van Stockum-Tipler cylinder 1937, 1974
• Kerr solutions 2 axially symmetric, stationary
  Kerrs
• Gott's time machine
• Wheeler wormholes
• Morris-Thorne-Yurtsever's TM
• Ori's dust asymptotically-flat space-time

Violation of normal chronology is such an
objectionable occurrence that any of such
solutions could be rejected as nonphysical.
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Summation over topologies
Theorem (Geroch, Tipler) Topology-changing
spacetimes must have CTC (closed timelike curve)

Theorem (Gammon) If asymptotically flat
spacetimes has a Cauchy surface with a nontrivial
topology, then spacetime is geodesically
incomplete (under assumption of NEC)
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Mathematical solution of Grandfather paradox

Recent overcoming of the grandfather paradox
There are spacetimes having CTC for which
smooth, unique solutions to the scalar wave eq.
exist for all data on generalized Cauchy surface
I.A., I. Volovich, T. Ishiwatari
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Time Machine
Surgery in the Minkowski spacetime
Make two cuts and glue the left edge of left cut
to the right edge of the right cut and vice
verse,
t
x
This space contains timelike loops
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Cauchy problem on not globally hyperbolic
spacetimes
t
x
Cauchy problem
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Example 2 dim scalar wave equation
Theorem Under assumption of minimal singularity
the Cauchy problem for tltb has a unique solution
The Cauchy problem for tgtb is not well posed
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BH in Collisions

• A possibility of production in ultra-relativistic
  particle collisions of some objects related to a
  non-trivial space-time structure is one of
  long-standing theoretical questions
• In 1978 collision of two classical ultra
  relativistic particles was considered by D'Eath
  and Payne and the mass of the assumed final BH
  also has been estimated
• In 1987 Amati, Ciafaloni, Veneziano and 't Hooft
  conjectured that in string theory and in QG at
  energies much higher than the Planck mass BH
  emerges.
• Aichelburg-Sexl shock waves to describe
  particles,
• Shock Waves ------ gt
  BH
• Colliding plane gravitation waves to describe
  particles
• Plane Gr Waves ----- gt BH I.A.,
  Viswanathan, I.Volovich, 1995

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BLACK HOLE PRODUCTION

• Collision of two fast point particles of energy
  E.
• BH forms if the impact parameter b is comparable
  to the Schwarzschild radius rs of a BH of mass E.
• The Thorn's hoop conjecture gives a rough
• estimate for classical geometrical
  cross-section

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BLACK HOLE PRODUCTION

• To deal with BH creation in particles collisions
  we have to deal with trans-Planckian scales.
• Trans-Planckian collisions in standard QG have
  inaccessible energy scale and cannot be realized
  in usual conditions.
• TeV Gravity to produce BH at Labs (1999)
• 
  Banks, Fischler, hep-th/9906038
• 
  I.A., hep-th/9910269,
• 
  Giuduce, Rattazzi, Wells, hep-ph/0112161
• 
  Giddings, hep-ph/0106219
• 
  Dimopolos, Landsberg, hep-ph/0106295
• 
  

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Conclusion

• TeV Gravity opens new channels BH, WH, TM
• Wheeler foam at
  TeV scale.
• WH/TM production at LHC is of the same order
  of magnitude as BH production (under assumption
  of geometrical crossection)
• The important question on possible experimental
  signatures of spacetime nontrivial objects
  deserves further explorations.