Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qfthep.sinp.msu.ru/talks/Pozdeeva.pdf Äàòà èçìåíåíèÿ: Tue Sep 14 20:44:31 2010 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:35:52 2012 Êîäèðîâêà: IBM-866

Sho ck wave in the FriedmanníRobertson-Walker space-time E.O. Pozdeeva

Moscow Aviation Institute

QFTHEP 2010
based on work by I. Ya. Aref 'eva, E.O. Pozdeeva and A.A. Bagrov

1


§ According to 't Hooft 1 shock waves in the Minkowski space-time can be used to describe ultra relativistic particles collisions. § Shock waves in AdS 2 and in dS 3 can be used to describe ultra relativistic particles collisions too.

1

G. 't Hooft, Phys. Lett. B. 198, 61, 1987. S. S. Gubser, S. S. Pufu, A. Yarom, Phys.Rev.D, 78, 2008, 066014 P.C. Aichelburg and R.U. Sexl, Gen. Relat. and Grav., V.24, 1971, 303.

2

3

I.Ya. Aref 'eva, A.A. Bagrov and E.A. Guseva, JHEP, 0912,009, 2009.
2


1) P.C. Aicelburg and R.U. Sexl, Geral Relativity and Gravitation, V.24, 1971, 303. 2)G. 't Hooft, Phys. Lett. B. 198, 61, 1987. M. Hotta, M. Tanaka, Clas. Quan. Grav., 10 (1993) 307í314. 3) J. Podolsky, M. Ortaggia, Clas. Quan. Grav., 18 (2001) 2689í2706. 4) G. Esposito, R. Pettorino and P. Scudellaro, Int. J. Geom. Meth. Mod. Phys., 4 (2007), 361. 5) I.Ya. Aref 'eva, K.S. Viswanwthan, I.V. Volovich, Nucl. Phys. B 452 (1995) 346í368. 6) J. ChoquetíBruhat, A. Fisher and A. Marsden, Proc. Enrico Fermi Summer School of the Italian Physical Society, Varenna, ed. Y. Ehlers (1978). 7) 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 ultrarelativistic particles in the Friedmann-Robertson-Walker spacetime is presented.

4


§ The shock gravitational waves are ultrarelativistic limits to the solutions of classical gravitation theory 4

P.C. Aicelburg and R.U. Sexl, Geral Relativity and Gravitation, V.24, 1971, 303.
4

G. Esposito, R. Pettorio and P. Scudellaro, Int.J.Geom.Meth.Mod.Phys. 4,361,2007, arXiv:0606126[gr-qc] I.Ya. Aref 'eva, A.A. Bagrov and L.V. Joukovskaya, Algebra and analysis 22(3), 3, 2010.
5


§ McVittie metric 5 in cosmological coordinates is dS 2 d2
2 m 1- 4 m 2a(t) dt2 + a(t)2 1 + (2d2 + d2), =- 2 2a(t) m 1+ 2a(t) = sin2 d2 + d2,

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[hep-th].
6


Some interesting cases of function a(t) corresponds to the following types of universes expansion: § for a(t) = 1, the Hubble constant H = 0, reduces McVittie metric to the Schwarzschild black hole of mass m, § for a(t) = eH t, the Hubble constant H = const, reduces McVittie metric to de Sitter-Schwarzschild black hole of mass m, n, the Hubble constant H = a = n . § for a(t) = k2t at
7


Sho ck wave in Minkowski space-time The Schwarzschild black hole metric in Minkowski space-time: (1 - A2) 2 ds2 = - dt + (1 + A)4(dx2 + dy 2 + dz 2), 4 1 + A2 m A = , r2 = x2 + y 2 + z 2. 2r The first order small mass approximation ds2 = ds2M + 4A(ds2M + 2dt2), 1 4 4 ds4M = ds4|A=0. (1)

8


Sho ck wave in Minkowski space-time

The Lorenz transformation is ï ï t = (t - v x), x = (t - v x), ï ï ïï In terms of t, x the function A p(1 - v 2) A= ï 2 (x - v t)2 + (1 - v ï and 2 = (d dt is

=

1 1 - v2

.

2)(y 2 + z 2) ï ï

, where p = m

ï ï t - v dx)2 . 2 1-v

Shock wave in Minkowski space-time 1 2 = ds2 + 4p ï ïï - 2 ln(y 2 + z 2)1/2 (t2 - x2) (d(t - x))2, ï ï ï ds 4M ïï |t - x| is obtained by the ultra relativistic limit .
9


Sho ck wave in dS space-time The Schwarzschild black hole metric in dS space-time: dS 2 = - 2m R2 1- -2 R b dt2 + dR2 2m R 2 1- -2 R b +

+ R2(d2 + sin2 d2). The first order small mass approximation of Schwarzschild black hole metric in dS 2m 2 2m dR2 ds2 = ds2 + dt + , ds2 = dS 2|m=0 dS dS R R 22 R 1- 2 b
10


Sho ck wave in dS space-time

§ In the plane coordinates representation the metric is:
2 ds2 = ds2M + ds2 , where ds2M = - dZ0 + p 5 5 4 i=1 2 - Z4 )3/2 4)dZ 2 - 4 0 2 2 4 22 2 + (b2(Z4 + Z0 ) + Z0 - Z0 Z4 )dZ4 ). 2 dZi ,

ds2 = p

2mb2
2 2 Z0 )2(b2 + Z0 22 ) + Z0 Z4 - Z 2 - Z4 )dZ0dZ4

2 (Z4 - 2 2 ((b2(Z4 + Z0 2 -2(2b2 + Z0

½

§ The 4D hyperboloid condition to the coordinates in dS:
2 -Z0 + 4 i=1
11

2 Zi = b 2 .


Sho ck wave in dS space-time

§ The Lorenz transformation along Z1 coordinate: Z0 = (Y0 + v Y1), Z1 = (v Y0 + Y1).

(2)

is applied to first order small mass approximation of Schwarzschild black hole in dS with mass rescaling m = p/ . § Shock wave in Minkowski space-time is ds2 = -d Y02 +
4 i=1 Y4

d Yi2 + (Y0 + Y1)(d(Y0 + Y4))2.

b + Y4 + 4p -2 + ln b b - Y4

12


Sho ck wave in Friedmann-Rob ertson-Walker space-time Co ordinates relations § For description ultrarelativistic particles movement by boost in plane coordinates representation we need in relation 5D Minkowski space-time coordinates with 4D FRW coordinates. § Connection between four-dimensional spatially flat cosmology and five-dimensional Minkowski space-time has been proposed by M.N. Smolyakov at 2008.

13


Sho ck wave in Friedmann-Rob ertson-Walker space-time Co ordinates relations

§ Consider the 5D Minkowski metric and 4D FRW metric:
2 2 2 2 2 2 dS5M = -dZ0 + dZ1 + dZ2 + dZ3 + dZ4 , M5, D=5,

ds2 RW = -dt2 + a2(t)(dx2 + dy 2 + dz 2), FRW, D=4. F § If a(t) is arbitrary function of t, then the hyperboloid condition becomes non-stationary:
2 2 2 2 2 -Z0 + Z1 + Z2 + Z3 + Z4 = b2(t)

14


Sho ck wave in Friedmann-Rob ertson-Walker space-time Co ordinates relations

Figure 1:

Hyperboloid for different t.

15


Sho ck wave in Friedmann-Rob ertson-Walker space-time Co ordinates relations

§ The surface is 1 Z0 = 2 1 Z4 = 2 Z1 = a

defined by: 1 b2(t) 1a(t) - + 2 1a(t) 1 b2(t) 1a(t) + - 2 1a(t) (t)x, Z2 = a(t)y ,

1 a(t)(x2 + y 2 + z 2) , 2 1 1 a(t)(x2 + y 2 + z 2) , 2 1 Z3 = 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) db(t) b(t) da(t) b(t) 2 +2 + 1 = 0. - dt a(t) dt dt a(t) § In the case a(t) = 2tn, we get b(t) = ‘ t . n(n-2)
16


Sho ck wave in Friedmann-Rob ertson-Walker space-time

McVittie metric in small mass approximation § McVittie metric (1 - ²)2 2 dt + a2(t) (1 + ²)4 (dx2 + dy 2 + dz 2), ds2 = - (1 + ²)2 m ²= . 2a(t) § First order approximation (m2 0), 1 - 4², (1 + ²)4 1 + 4², (1 + ²)2 to McVittie's metric is ds2 = ds2 RW + 4²(ds2 RW + 2dt2). 1 F F
17

(1 - ²)2


Sho ck wave in Friedmann-Rob ertson-Walker space-time McVittie metric in small mass approximation

§ For a(t) = k2tn the metric can be written in plane coordinates: ds2 = ds2M + 5 2m
2 Zi

2d(Z0 + Z4)2 ds2M + 2 2 2 5 n 12(n(n - 2)b2(t))n-1

,

2 2 2 where b2(t) = -Z0 + Zi + Z4 , i = 1, 3.

18


Sho ck wave in Friedmann-Rob ertson-Walker space-time

Lorentz transformation § Boost in the 5-dimensional Minkowski space-time: Z0 = (Z0 + v Z1), Z1 = (Z1 + v Z0), = 1 1 - v2 .

§ We apply the Lorentz transformation to the McVittie metric in the first order small mass approximation: 2m ~ ds2 = ds2M + 5
~ ~ ~2 2 + 2 d( (Z0+v Z1)+Z4) ds5M p222(p(p-2)b2(t))p-1 12

~ ~ ~2 ~2 2(v Z0 + Z1)2 + Z2 + Z3
19

, m = m ~


Sho ck wave in Friedmann-Rob ertson-Walker space-time Lorentz transformation

or 2m ~ ds2 = ds2M + 5

~ 2 + 2 d( (Z0+v ds5M p222 12

~ ~ Z1)+Z4)2 t2(p-1)

~ ~ ~2 ~2 2(v Z0 + Z1)2 + Z2 + Z3

.

§ For , it is evidently that: ds2 | 1 ds2M + 5 4m ~ ~ ~ ~2 ~2 2(Z0 + Z1)2 + Z2 + Z3 ~ ~ d(Z0 + Z1)2 p222t2(p-1) 12

20


Sho ck wave in Friedmann-Rob ertson-Walker space-time

Limiting pro cess § Limiting process in generalized function meaning:
2 4 f (U )dU = f (0) ln 2 + 2U 2 + X 2 X -

-

1 f (U ) dU |U | reg

where

- 1

1 f (U ) dU |U | reg f (U ) - f (0) dU + |U |
-1


-1

-
21

1 f (U )dU + |U |



1

1 f (U )dU. |U |


Sho ck wave in Friedmann-Rob ertson-Walker space-time Limiting pro cess

The result can be presented by the Dirac-delta function


lim

X2 + - (U ) ln 2 = - (U ) ln 4 2U 2 + X 2

1 . |U | reg

22


Sho ck wave in Friedmann-Rob ertson-Walker space-time

Lorentz transformations in the ultrarelativistic limit the McVittie metric § After the regularization we have the gravitational waves metric ds2 = ds2M + 5 p 2 2 1 where t= Z0 + Z4 1/n , k1k2
2 2 3 2 2 t2 = n(n - 2)(-Z0 + Z1 + Z2 + Z3 + Z4 )

(U )d(U )2, m = m ln 2, U = Z0+Z1, ï ~ 2(t)2(p-1) 2

4m ï

23


Sho ck wave in Friedmann-Rob ertson-Walker space-time Lorentz transformations in the ultrarelativistic limit the McVittie metric

§ The obtained metric can be presented with cosmological coordinates: 4m ï 2 = ds2 ds (U )d(U )2, + F RW n222(t)2(n-1) 12 1 1 t2 1 k2tn(x2 + y 2 + z 2) U = k1k2tn - + + k2tnx 2 2 n(n - 2)k1k2tn 2 k1 The most interesting case U = 0. Shock wave profile F (U = 0) is 1 proportional to : 2/n (V /2 + Z4) 1 F (U )|u0 . (V /2 + Z4)2/n
24


Sho ck wave in Friedmann-Rob ertson-Walker space-time

Conclusion § It is proposed to use the boosted McVittie metric such as model of ultrarelativistic particle in the Friedmann-Robertson-Walker space-time with a(t) = k tn.

§ The shock wave corresponding ultrarelativistic particle in the Friedmann-Robertson-Walker space-time is constructed.

25


Sho ck wave in the FriedmanníRob ertson-Walker space-time E.O. Pozdeeva

Thank you for attention!

26