Shock wave in the FriedmannRobertson-Walker space-time E.O. - PowerPoint PPT Presentation

Shock wave in the Friedmann–Robertson-Walker space-time E.O. Pozdeeva Moscow Aviation Institute Bogoliubov Readings – 2010 based on work by I. Ya. Aref’eva, E.O. Pozdeeva and A.A. Bagrov 1

• According to ’t Hooft 1 shock waves 2 in the Minkowski space- time can be used to describe ultrarelativistic particles collisions. • The shock gravitational waves are also know in (A)dS back- ground. They are ultrarelativistic limits of Schwarzschild-(A)dS metrics 3 1 G. ’t Hooft, Phys. Lett. B. 198 , 61, 1987. 2 P.C. Aichelburg and R.U. Sexl, Gen. Relat. and Grav. , V.2 4 , 1971, 303. 3 M. Hotta, M. Tanaka, Clas. Quan. Grav. , 10 (1993) 307–314. K. Sfetsos, Nucl. Phys. B 436 721, 1995. G. T. Horowitz and N. Itzhaki, JHEP 02 453, 1999. J. Podolsky and J.B. Griffiths, Phys. lett A 261 , 1999. R. Emparran, Phys. Rev. D 64 024025, 2001. G. Esposito, R. Pettorino and P. Scudellaro, Int.J.Geom.Meth.Mod.Phys. , 4 ,361, 2007. I.Ya. Aref’eva, A.A. Bagrov and L.V. Joukovskaya, Algebra and analysis 22(3) , 3, 2010. 2

• Shock waves in AdS and in dS can be used to describe ultra relativistic particles collisions too 4 4 S. S. Gubser, S. S. Pufu, A. Yarom, Phys.Rev.D , 78 , 2008, 066014 I.Ya. Aref’eva, A.A. Bagrov and E.A. Guseva, JHEP , 0912 ,009, 2009. 3

• In this talk the generalization of this construction for the ultra- relativistic particles in the Friedmann-Robertson-Walker space- time is presented. 4

• McVittie metric 5 in cosmological coordinates is � � 2 m � � 4 1 − 2 a ( t ) ρ m dS 2 = − � 2 dt 2 + a ( t ) 2 ( ρ 2 d Ω 2 + dρ 2 ) , 1 + � 2 a ( t ) ρ m 1 + 2 a ( t ) ρ d Ω 2 = sin 2 θdφ 2 + dθ 2 , where a ( t ) is arbitrary function of t. 5 G. C. McVittie, Mon. Not. R. Astron. Soc. 93, 325 (1933). N. Kalopery, M. Klebanz and D. Martiny, McVittie’s Legacy: Black Holes in an Expanding Universe, arXiv:1003.4777. 5

Some interesting cases of function a ( t ) corresponds to the fol- lowing types of universes expansion: • for a ( t ) = 1 , the Hubble parameter H = 0 , reduces McVittie metric to the Schwarzschild black hole of mass m, • for a ( t ) = e Ht , the Hubble parameter H = const, reduces McVittie metric to de Sitter-Schwarzschild black hole of mass m, • for a ( t ) = k 2 t n , the Hubble parameter H = ˙ a a = n t . 6

Shock wave in Minkowski space-time The Schwarzschild black hole metric in Minkowski space-time: 4 = − (1 − A 2 ) 1 + A 2 dt 2 + (1 + A ) 4 ( dx 2 + dy 2 + dz 2 ) , ds 2 (1) A = m r 2 = x 2 + y 2 + z 2 . 2 r, The first order small mass approximation ds 2 1 = ds 2 4 M + 4 A ( ds 2 4 M + 2 dt 2 ) , ds 4 M = ds 4 | A =0 . 7

Shock wave in Minkowski space-time The Lorenz transformation is 1 t = γ (¯ x = γ (¯ √ t − v ¯ x ) , t − v ¯ x ) , γ = 1 − v 2 . In terms of ¯ t, ¯ x the function A is p (1 − v 2 ) A = � , where p = mγ t ) 2 + (1 − v 2 )(¯ y 2 + ¯ x − v ¯ z 2 ) 2 (¯ and x ) 2 dt 2 = ( d ¯ t − vd ¯ . 1 − v 2 Shock wave in Minkowski space-time � � 1 y 2 + ¯ t 2 − ¯ ds 2 γ = ds 2 z 2 ) 1 / 2 δ (¯ x 2 ) x )) 2 , ( d (¯ 4 M + 4 p x | − 2 ln(¯ t − ¯ | ¯ t − ¯ is obtained by the ultra relativistic limit γ → ∞ . 8

Shock wave in dS space-time The Schwarzschild black hole metric in dS space-time: � � R − R 2 dR 2 1 − 2 m dS 2 = − dt 2 + � + � b 2 R − R 2 1 − 2 m b 2 + R 2 ( dθ 2 + sin 2 θdφ 2 ) . The first order small mass approximation of Schwarzschild black hole metric in dS dR 2 dS + 2 m R dt 2 + 2 m ds 2 = ds 2 ds 2 dS = dS 2 | m =0 � 2 , � R 1 − R 2 b 2 9

Shock wave in dS space-time • In the plane coordinates representation the metric is: 4 � ds 2 = ds 2 5 M + ds 2 p , where ds 2 5 M = − dZ 2 dZ 2 0 + i , i =1 2 mb 2 ds 2 p = 4 ) 3 / 2 × 0 ) 2 ( b 2 + Z 2 ( Z 2 4 − Z 2 0 − Z 2 (( b 2 ( Z 2 4 + Z 2 0 ) + Z 2 0 Z 2 4 − Z 4 4 ) dZ 2 0 − − 2(2 b 2 + Z 2 0 − Z 2 4 ) dZ 0 dZ 4 + ( b 2 ( Z 2 4 + Z 2 0 ) + Z 4 0 − Z 2 0 Z 2 4 ) dZ 2 4 ) . • The 4D hyperboloid condition to the coordinates in dS: 4 � − Z 2 Z 2 i = b 2 . 0 + i =1 10

Shock wave in dS space-time • The Lorenz transformation along Z 1 coordinate: Z 0 = γ ( Y 0 + vY 1 ) , Z 1 = γ ( vY 0 + Y 1 ) . is applied to first order small mass approximation of Schwarzschild black hole in dS with mass rescaling m = p/γ . • Shock wave in Minkowski space-time is 4 � ds 2 γ = − dY 2 dY 2 0 + i + i =1 � � b + Y 4 �� − 2 + Y 4 δ ( Y 0 + Y 1 )( d ( Y 0 + Y 4 )) 2 . + 4 p b ln b − Y 4 11

Shock wave in Friedmann-Robertson-Walker space-time Coordinates relations • For description ultrarelativistic particles movement by boost in plane coordinates representation can use the relation of 5D Minkowski space-time coordinates with 4D FRW coordinates. 12

Shock wave in Friedmann-Robertson-Walker space-time Coordinates relations • Connection between four-dimensional spatially flat cosmology and five-dimensional Minkowski space-time (see, for example, 6 ). ♦ Consider the 5D Minkowski metric and 4D FRW metric: dS 2 5 M = − dZ 2 0 + dZ 2 1 + dZ 2 2 + dZ 2 3 + dZ 2 4 , M 5 , D=5 , FRW = − dt 2 + a 2 ( t )( dx 2 + dy 2 + dz 2 ) , FRW, D=4 . ds 2 ♦ If a ( t ) is arbitrary function of t, then the hyperboloid condi- tion becomes non-stationary: − Z 2 0 + Z 2 1 + Z 2 2 + Z 2 3 + Z 2 4 = b 2 ( t ) 6 M. N. Smolyakov Class.Quant.Grav.25:238003,2008 13

Shock wave in Friedmann-Robertson-Walker space-time Coordinates relations Figure 1: Hyperboloid for different t . 14

Shock wave in Friedmann-Robertson-Walker space-time Coordinates relations • The surface is defined by: a ( t )( x 2 + y 2 + z 2 ) b 2 ( t ) Z 0 = 1 2 κ 1 a ( t ) − 1 κ 1 a ( t ) + 1 , 2 2 κ 1 a ( t )( x 2 + y 2 + z 2 ) b 2 ( t ) Z 4 = 1 2 κ 1 a ( t ) + 1 κ 1 a ( t ) − 1 , 2 2 κ 1 Z 1 = a ( t ) x, Z 2 = a ( t ) y, Z 3 = a ( t ) z. • The metric in 5D Minkowski space-time is equal to metric in 4D FRW, if the following condition relates a ( t ) with b ( t )): � da ( t ) � 2 b ( t ) + 2 da ( t ) db ( t ) b ( t ) − a ( t ) + 1 = 0 . dt a ( t ) dt dt t • In the case a ( t ) = κ 2 t n , we get b ( t ) = ± √ n ( n − 2) . 15

Shock wave in Friedmann-Robertson-Walker space-time McVittie metric in small mass approximation • McVittie metric ds 2 = − (1 − µ ) 2 (1 + µ ) 2 dt 2 + a 2 ( t ) (1 + µ ) 4 ( dx 2 + dy 2 + dz 2 ) , m µ = 2 a ( t ) ρ. • First order approximation ( m 2 ∼ 0), (1 − µ ) 2 (1 + µ ) 4 ≈ 1 + 4 µ, (1 + µ ) 2 ≈ 1 − 4 µ, to McVittie’s metric is ds 2 1 = ds 2 FRW + 4 µ ( ds 2 FRW + 2 dt 2 ) . 16

Shock wave in Friedmann-Robertson-Walker space-time McVittie metric in small mass approximation • For a ( t ) = k 2 t n the metric can be written in plane coordinates: � � 2 d ( Z 0 + Z 4 ) 2 5 M + 2 m ds 2 = ds 2 ds 2 � 5 M + , n 2 κ 2 1 κ 2 2 ( n ( n − 2) b 2 ( t )) n − 1 Z 2 i where b 2 ( t ) = − Z 2 0 + Z 2 i + Z 2 4 , i = 1 , 3. 17

Shock wave in Friedmann-Robertson-Walker space-time Lorentz transformation • Boost in the 5-dimensional Minkowski space-time: 1 Z 0 = γ ( � Z 0 + v � Z 1 = γ ( � Z 1 + v � √ Z 1 ) , Z 0 ) , γ = 1 − v 2 . • We apply the Lorentz transformation to the McVittie metric in the first order small mass approximation: � � d ( γ ( ˜ Z 0 + v ˜ Z 1 )+ ˜ Z 4 ) 2 ds 2 2 ˜ m 5 M + 2 p 2 κ 2 1 κ 2 2 ( p ( p − 2) b 2 ( t )) p − 1 ds 2 γ = ds 2 5 M + � , ˜ m = mγ Z 1 ) 2 + ˜ γ 2 ( v ˜ Z 0 + ˜ Z 2 2 + ˜ Z 2 γ 3 18

Shock wave in Friedmann-Robertson-Walker space-time Lorentz transformation or � � 5 M + 2 d ( γ ( ˜ Z 0 + v ˜ Z 1 )+ ˜ Z 4 ) 2 ds 2 2 ˜ m p 2 κ 2 1 κ 2 2 t 2( p − 1) ds 2 γ = ds 2 5 M + � . Z 1 ) 2 + ˜ γ 2 ( v ˜ Z 0 + ˜ 2 + ˜ Z 2 Z 2 γ 3 • For γ → ∞ , it is evidently that: � � d ( ˜ Z 0 + ˜ Z 1 ) 2 4 ˜ mγ ds 2 | υ → 1 → ds 2 5 M + � p 2 κ 2 1 κ 2 2 t 2( p − 1) Z 1 ) 2 + ˜ γ 2 ( ˜ Z 0 + ˜ 2 + ˜ Z 2 Z 2 3 19

Shock wave in Friedmann-Robertson-Walker space-time Limiting process γ → ∞ • Limiting process γ → ∞ in generalized function meaning: ∞ ∞ � 1 � � � γ 2 U 2 + X 2 f ( U ) dU = f (0) ln 4 γ 2 γ � X 2 + f ( U ) dU | U | reg −∞ −∞ where ∞ � 1 � � f ( U ) dU ≡ | U | reg −∞ 1 − 1 ∞ � � � f ( U ) − f (0) 1 1 ≡ dU + | U | f ( U ) dU + | U | f ( U ) dU. | U | −∞ − 1 1 20

Shock wave in Friedmann-Robertson-Walker space-time Limiting process γ → ∞ The result can be presented by the Dirac-delta function � � � 1 � = − δ ( U ) ln X 2 γ γ 2 U 2 + X 2 − δ ( U ) ln γ 2 lim � 4 + . | U | γ →∞ reg 21

Shock wave in Friedmann-Robertson-Walker space-time Lorentz transformations in the ultrarelativistic limit the McVittie metric • After the regularization we have the gravitational waves metric 4 ¯ m ds 2 γ = ds 2 2 ( t ) 2( p − 1) δ ( U ) d ( U ) 2 , ¯ m ln γ 2 , U = Z 0 + Z 1 , 5 M + m = ˜ p 2 κ 2 1 κ 2 where � Z 0 + Z 4 � 1 /n t 2 = n ( n − 2)( − Z 2 0 + Z 2 1 + Z 3 2 + Z 2 3 + Z 2 t = , 4 ) k 1 k 2 22