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Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

Black holes production in transplanckian collisionsin (anti)-de Sitter spacetime

Andrey Bagrov

Steklov Mathematical Institute of RAS

(based on joint works withI. Aref’eva, E. Guseva, and L. Joukovskaya,

hep-th: 0905.1087, 0909.1294)

Kolomna, 09 June 2010

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

1 Introduction

2 Structure of spacetime

3 Trapped surface

4 Cross sections of BH production

5 Summary and outlook

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

Problem definition

We consider non-expanding gravitational neutral and chargedshock waves in the D-dimensional AdS and dS spacetime

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

Problem definition

We consider non-expanding gravitational neutral and chargedshock waves in the D-dimensional AdS and dS spacetime

Structure of null-geodesics beams in such perturbed spacetimeis being analyzed

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

I d i S f i d f C i f BH d i S d l k

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

Problem definition

We consider non-expanding gravitational neutral and chargedshock waves in the D-dimensional AdS and dS spacetime

Structure of null-geodesics beams in such perturbed spacetimeis being analyzed

In the case of two head-on colliding shock waves we deriveand study the equation on formed trapped surface

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

I t d ti St t f ti T d f C ti f BH d ti S d tl k

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

Problem definition

We consider non-expanding gravitational neutral and chargedshock waves in the D-dimensional AdS and dS spacetime

Structure of null-geodesics beams in such perturbed spacetimeis being analyzed

In the case of two head-on colliding shock waves we deriveand study the equation on formed trapped surface

The volume of the trapped surface is being interpreted as thecross section of black holes production

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introd ction Str ct re of spacetime Trapped s rface Cross sections of BH prod ction S mmary and o tlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

Motivation

Shock-waves seem like plausible approximation forgravitational fields of ultrarelativistic particles

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

Motivation

Shock-waves seem like plausible approximation forgravitational fields of ultrarelativistic particlesIt has been shown by several authors (Banks, Fishler - 1999;Aref’eva - 1999) that in collisions of particles at TeV energiesblack holes can be formed

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

http://slidepdf.com/reader/full/andrey-bagrov-black-holes-production-in-transplanckian-collisions-in-anti-de 9/40

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

Motivation

Shock-waves seem like plausible approximation forgravitational fields of ultrarelativistic particlesIt has been shown by several authors (Banks, Fishler - 1999;Aref’eva - 1999) that in collisions of particles at TeV energiesblack holes can be formed

Within the framework of AdS /QCD holography the model of shock wave is often applied to description of heavy ions(Gubser, Kovchegov etc.)

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

http://slidepdf.com/reader/full/andrey-bagrov-black-holes-production-in-transplanckian-collisions-in-anti-de 10/40

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

Motivation

Shock-waves seem like plausible approximation forgravitational fields of ultrarelativistic particlesIt has been shown by several authors (Banks, Fishler - 1999;Aref’eva - 1999) that in collisions of particles at TeV energiesblack holes can be formed

Within the framework of AdS /QCD holography the model of shock wave is often applied to description of heavy ions(Gubser, Kovchegov etc.)Beyond the AdS /CFT -duality there is also dS /CFT -duality(Strominger - 2001). Therefore there is a chance to extendthe holographic approaches to QCD on the de Sitter case

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

Motivation

Shock-waves seem like plausible approximation forgravitational fields of ultrarelativistic particlesIt has been shown by several authors (Banks, Fishler - 1999;Aref’eva - 1999) that in collisions of particles at TeV energiesblack holes can be formed

Within the framework of AdS /QCD holography the model of shock wave is often applied to description of heavy ions(Gubser, Kovchegov etc.)Beyond the AdS /CFT -duality there is also dS /CFT -duality(Strominger - 2001). Therefore there is a chance to extendthe holographic approaches to QCD on the de Sitter casedS background can be considered as the simplest model of spacetime with dark energy. In this context we analyzepossible influence of dark energy on the process of black holes

production in collisionsAndrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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p pp p y

Metric of spacetime

It is convenient to immerse D -dimensional (A)dS spacetimewith shock wave as submanifold to (D + 1)-dimensionalMinkowski space. In these terms the metric is following:

ds 2 =

−dUdV + d Z 2 + dZ D

2+ F (Z D )δ(U )dU 2,

Z = Z 2, ...,Z D −1, −UV + Z 2 ± Z D 2

= ±a2,

U = Z 0 + Z 1, V = −Z 0 + Z 1

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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y

Metric of spacetime

It is convenient to immerse D -dimensional (A)dS spacetimewith shock wave as submanifold to (D + 1)-dimensionalMinkowski space. In these terms the metric is following:

ds 2 =

−dUdV + d Z 2 + dZ D

2+ F (Z D )δ(U )dU 2,

Z = Z 2, ...,Z D −1, −UV + Z 2 ± Z D 2

= ±a2,

U = Z 0 + Z 1, V = −Z 0 + Z 1

The shape of shock-wave could be obtained by boosting of

the (A)dS -Schwarzschild solution (Hotta, Tanaka - 1992):

limv →1

1√1− v 2

f

(Z 0 + vZ 1)2

1− v 2;λ

= δ(Z 0+Z 1)

∞ −∞

f (x 2;λ)dx

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

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Shape function properties

Explicit integral representation:

F (Z ) = 2Ma2 · p .v .∞

−∞

(a2(±Z 2 + x 2) + Z 2(x 2 Z 2))

(Z 2 x 2)2(±a2 + x 2 Z 2)D −1

2

dx

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

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Shape function properties

Explicit integral representation:

F (Z ) = 2Ma2 · p .v .∞

−∞

(a2(±Z 2 + x 2) + Z 2(x 2 Z 2))

(Z 2 x 2)2(±a2 + x 2 Z 2)D −1

2

dx

For example:

F 4,dS (Z 4) = 4MG 4

−2 +

Z 4

aln

1 + Z 4

a

1

−Z 4

a

F 5,AdS (Z 5) =

3πMG 52a

2Z 5

2

a2 − 1

Z 5

2

a2 − 1− 2

Z 5

a

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

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Shape function properties

In the charged case:

F D (Q ,Z ) = F D (Z )−Q 2

2M F 2D −3(Z ),

where rescaled energy and charge are

M =8πG D m

(D − 2)ΩD −2, Q 2 =

8πG D q 2

(D − 2)(D − 3)

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

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Definition of trapped surface

A trapped surface is a (D − 2)-dimensional spacelike surfacewhose each two null normals have zero convergence(Neighboring light rays, normal to the surface, must move

towards one another)

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

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Definition of trapped surface

A trapped surface is a (D − 2)-dimensional spacelike surfacewhose each two null normals have zero convergence(Neighboring light rays, normal to the surface, must move

towards one another)Th. (Hawking-Penrose) A spacetime (M; g) with a completefuture null infinity which contains a closed trapped surfacemust contain a future event horizon, the interior of whichcontains the trapped surface

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

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Definition of trapped surface

A trapped surface is a (D − 2)-dimensional spacelike surfacewhose each two null normals have zero convergence(Neighboring light rays, normal to the surface, must move

towards one another)Th. (Hawking-Penrose) A spacetime (M; g) with a completefuture null infinity which contains a closed trapped surfacemust contain a future event horizon, the interior of whichcontains the trapped surface

This has been proved for asymptotically flat spacetimes, butthere is a common opinion that it is valid for dS /AdS too

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Geometrical structure

Parts of trapped surface:

S = S 1 ∪ S 2;

S 1 :

U = 0,

V = −Ψ1(ρ),

S 2 : V = 0, U = −Ψ2(ρ),

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

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Geometrical structure

Parts of trapped surface:

S = S 1 ∪ S 2;

S 1

: U = 0, V =−

Ψ1

(ρ),

S 2 : V = 0, U = −Ψ2(ρ),

On the surface of gluing:

Ψ1 = Ψ2d Ψ

d ρ

2

= 4

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

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Basic mathematical objects

Trapped surface is a surface of null convergence:θ = hMN N ξM ; θ = 0

Here:

ξM is a vector tangent to geodesic flow,M is a covariant derivation in the sense of D -dimensionalmetric

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

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Basic mathematical objects

Trapped surface is a surface of null convergence:θ = hMN N ξM ; θ = 0

Here:

ξM is a vector tangent to geodesic flow,M is a covariant derivation in the sense of D -dimensionalmetric

And (here M ,N = 0..D − 1; α, β = 0..D − 3):

hMN = K M

α

g αβ K N

β

,

K M α = (0,−∂ αΨ, δM

α ),

g αβ =∂ Z M

∂ s α∂ Z N

∂ s β g MN

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

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Dirichlet problem

Equation on the trapped surface:

SD −2/HD −2

±D − 2

a2 Ψ− H

1± ρ2

2a2

= 0, H =1

2

(1

±ρ2

2a2

)F

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

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Dirichlet problem

Equation on the trapped surface:

SD −2/HD −2

±D − 2

a2 Ψ− H

1± ρ2

2a2

= 0, H =1

2

(1

±ρ2

2a2

)F

Boundary conditions in arbitrary dimension give us thefollowing equation on the radius of the trapped surface:

14

(1± ρ20

2a2 )F (ρ0)± ρ02a2 ρ2

0

F (ρ0) + 2 = 0

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

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Criticality in de Sitter space

In de Sitter space we deal with equation1

4(1 +

ρ20

2a2)F (ρ0) +

ρ0

2a2 − ρ20

F (ρ0) + 2 = 0

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

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Criticality in de Sitter space

In de Sitter space we deal with equation1

4(1 +

ρ20

2a2)F (ρ0) +

ρ0

2a2 − ρ20

F (ρ0) + 2 = 0

There is the explicit general formula:

(2 + x 2)D −

2

x D −3(2− x 2)= C D a

D −

3

p ; x = ρ/a

C 3 = 2/π, C 4 =√

2,C 5 = 32/3π, C 6 = 6√

2

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Criticality in de Sitter space

In de Sitter space we deal with equation1

4(1 +

ρ20

2a2)F (ρ0) +

ρ0

2a2 − ρ20

F (ρ0) + 2 = 0

There is the explicit general formula:

(2 + x 2)D −

2

x D −3(2− x 2)= C D a

D −

3

p ; x = ρ/a

C 3 = 2/π, C 4 =√

2,C 5 = 32/3π, C 6 = 6√

2

These functions have positive minima at the points

x min,D =

2(−1 + D − 2√D − 2)

√D − 3

, D > 3;

x min,3 = 0, D = 3

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

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When does the trapped surface exist in dS?

Figure: Critical effect of trapped surface non-formation.

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

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Charged shock waves in dS

Figure: Influence of charge on the cross section of BH production in dS

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Charged shock waves in AdS

Figure: Influence of charge on the cross section of BH production in AdS

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Volume of the trapped surface

Volume of the trapped surface can be evaluated in generalway and does not depend explicitly on the form of Ψ. Weconsider particular cases of dS 4 and AdS 5 both in the limit of low energy M aD −3:

Atrap = 2

ρ<ρ0

det g αβ dV

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Volume of the trapped surface

Volume of the trapped surface can be evaluated in generalway and does not depend explicitly on the form of Ψ. Weconsider particular cases of dS 4 and AdS 5 both in the limit of low energy M aD −3:

Atrap = 2

ρ<ρ0

det g αβ dV

In particular:

AdS 4 ≈ 4πρ20(1−ρ2

02a2 ) ≈ 32πM 2(1−4M 2

a2 −3π

64

Q 2

M 2 ))

AAdS 5 ≈ 4πa3(3π

2

M

a2)2/3

1− 1

24(1 +

5Q 2

Ma2) (

4√

2

3π

a2

M )2/3

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Conclusions

Equation on the trapped surface forming in transplanckiancollisions in (A)dS has been derived and analyzed

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Conclusions

Equation on the trapped surface forming in transplanckiancollisions in (A)dS has been derived and analyzed

For dS spacetime it was shown that if energy density of

neutral shock waves is very large or if the spacetime isstrongly curved then black hole could not be formed even inhead-on collision. For D = 3 the critical value of energydensity does not depend on the cosmological radius

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Conclusions

Equation on the trapped surface forming in transplanckiancollisions in (A)dS has been derived and analyzed

For dS spacetime it was shown that if energy density of

neutral shock waves is very large or if the spacetime isstrongly curved then black hole could not be formed even inhead-on collision. For D = 3 the critical value of energydensity does not depend on the cosmological radius

Influence of charge on the cross section of black holesproduction has been calculated

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Outlook

Analysis of shock waves with more complicated innerstructure (spin, scalar phantom charge etc.)

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Outlook

Analysis of shock waves with more complicated innerstructure (spin, scalar phantom charge etc.)

Search for possible catalysts for black hole production. Howcan we get big black holes at LHC? (Arefeva - 2009)

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Outlook

Analysis of shock waves with more complicated innerstructure (spin, scalar phantom charge etc.)

Search for possible catalysts for black hole production. Howcan we get big black holes at LHC? (Arefeva - 2009)

Analysis of different possibilities to produce more exoticobjects (like wormholes)

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)

Introduction Structure of spacetime Trapped surface Cross sections of BH production Summary and outlook

8/3/2019 Andrey Bagrov- Black holes production in transplanckian collisions in (anti)-de Sitter spacetime

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Outlook

Analysis of shock waves with more complicated innerstructure (spin, scalar phantom charge etc.)

Search for possible catalysts for black hole production. Howcan we get big black holes at LHC? (Arefeva - 2009)

Analysis of different possibilities to produce more exoticobjects (like wormholes)

Holographic description. Application of shock wave technics

to solution of QCD problems (Gubser, Yarom, Pufu - 2008)

Andrey Bagrov Black holes production in transplanckian collisions in A(dS)