black holes. bh in gr and in qg

• 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.20083-rd lecture

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


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 itscenter, instead of traveling the entire distance around.

The traveler just as a worm could take a shortcut to the opposite side of the universe through a topologically nontrivial tunnel.

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


Traversable Wormholes

Morris, Thorne, Yurtsever, Visser,..

)sin()(

1

2222

2

22)(22 ddr

rrb

drdteds r

The embedding condition together with the requirement of finiteness of theredshift function lead to the NEC violation on the WH throat


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.


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.


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).


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.


Summation over topologies

'|;"|

,]},[exp{','|","

'" hghg

dggSi

hh

topologiesoversum

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)


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


Time MachineSurgery 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,

This space contains timelike loops

x

t


Cauchy problem on not globally hyperbolic spacetimes

x

t

2 2

0 0 0 1

( ) ( , ) 0, 0 ,

( , ) | ( ), ( , ) | ( )

t x

t t t

u t x t x

u t x u x u t x u x

1 2 1 2

1 2 1 2

2 1 2 1 1 1

2 1 2 1

( , ) | ( , ) | , ( , ) | ( , ) | , ,

( , ) | ( , ) | , ( , ) | ( , ) | ,

x a x a x x a x x a

x a x a x x a x x a

u t x u t b b x u t x u t b b x b t b l

u t x u t b b x u t x u t b b x

Cauchy problem:

1 1( , )a b

2 2( , )a b

1 2 1 1 2 1 2 1 2 2 1 2

3 3 2 1 2 1 4 4 1 2 1 2

( , ) ( ) ( ), ( , )

, 0; , 0;

0, ; 0, ;

0, 0, 5,6,7.

f gi i i

f g f g

f g f g

f gi i

t x f x t g t x t x D

a a b b a a b b

a a b b a a b b

i

0 0

0 1 0 1

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

2 2

x x

x x

f x u x u s ds g x u x u s ds


Example: 2 dim scalar wave equation

Theorem: Under assumption of minimal singularity the Cauchy problem for t<b has a unique solution

The Cauchy problem for t>b is not well posed


• 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 ------ > BH

• Colliding plane gravitation waves to describe particles

Plane Gr Waves ----- > BH I.A., Viswanathan, I.Volovich, 1995

BH in Collisions


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

r~) BH1(1 2S

2BH

SD

Mr

M


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


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.


black holes. bh in gr and in qg

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