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On the global geometr y of Brane Universe models
A. Smirnov V. Berezin

Depar tment of Theoretical Physics Institute for Nuclear Research

14-th Lomonosov Conference on Elementary Par tcile Physics, 2009


Aim Investigate possible global geometries of "brane universe" scenario. Our Universe is a thin shell/membrane/brane embedded into the space-time of larger number of dimensions ("bulk"). Strategy Simplify everything as much as possible and construct some exactly solvable model - the source of our physical intuition. Mo d e l (N+1)-dimensional space-time contains N-dimensional brane (= thin shell). Brane is time-like and have have the "cosmological symmetry" (homogeneity + isotropy).


Simplification: step I Outside the shell - bulk geometry posseses some symmetry: bulk does not depend on the brane position everywhere in the normal Gaussian coordinate system
2 ds2 = -dn2 + B 2 (n, t )dt 2 - A2 (n, t )dlN -1

,

where

is k k k

th = = =

dr 2 + r 2 d 2 -2 -1 N 1 - kr 2 e Rober tson-Walker line element. As usual +1 - (N-1)-dimensional sphere -1 - rotational hyperboloid 0 - flat space. Energy-momentum tensor
2 dlN

=

T

µ

= Sµ (n) + [Tµ ] (n) + T

-µ

Sµ - surface energy-momentum tensor of the brane (n = 0). Notations: [X ] = X+ - X- , n > 0 ("+"), n < 0 ("-")


Local approach (connected to the brane) Decomposition of Einstein equations -([Kik ] - ik [K ]) = 8 GSik ,
(N )

G

ik

= 8 GT
-i l

-i k

+ 8 GT

i nd -i k

,

1 j l K + ik (K-j K-l +K )). 2 Cosmological time and scale factor on the brane T
i nd -i k

= -(K

-i k , n

-ik K

-, n

+ 2K

K

l -k

-K

-i k

d = B (n, t )dt ,
(N )

a( ) = A(n, t )

G G

0 0 2 2

= =

(N )

(N - 1)(N - 2) a2 + k , 2 a2 a2 + k N - 2 2a + (N - 3) 2 2 a a
(N ) 2 G2 = (N ) 3 G3 = ... = (N ) N GN .

,

Note, by symmetry


Global approach Cosmological symmetry allows us to deals with the invariants of bulk geometry only. Namely (d+2)-decomposition (d+2 = N+1)
2 ds2 = AB (x )dx A dx B - R 2 (x )dld ,

A, B = 0, 1,

2 dld - Rober tson-Walker, curvature d (d - 1)k , k = ±1, 0, R scale factor = radius. Two invariants

R (t , q ),

=

AB

R,A R

,B

Einstein equations + integrability condition R
d -1

( + k )
AC

,A

R

| | CB

16 G d B R T R,A - TA R d 8 G A RTB =- d =

,B


R,T-regions , R allow us to classify regions of curved space-time.


R-regions: < 0 R =const, R can be chosen as a spatial coordinate. Sign R,q cannot be changed. R+ -region if R- -region if R,q > 0 R,q < 0



T-regions: > 0 R =const, R can be chosen as a time coordinate. Sign R,t cannot be changed. T+ -region if R,t > 0 - inevitable expansion T- -region if R,t < 0 - inevitable contraction



Apparent horizons = 0 Global geometry = set of R,T-regions and apparent horizons


Simplification: step II n = 0 - vacuum = -k + 2 Gm 2 R 2 . + N -2 N (N - 1) R

Different k - different bulk different global structure Two-dimensional metric in R-regions,
2 ds2 = (-) dt 2 - d R 2

,

dR = ±

dR , ||

Two-dimensional metric in T-regions,
2 ds2 = d R 2

- dq

2

.

Single brane: absence of singularity at R = 0 m = 0 = R
,n

1 R 2 (n, t ),t - R 2 (n, t ),n = f 2 (t ) - R,2 , n B (n, t )
2

= ± f 2 (t ) - =

f 2 (t ) - ,

= sign of R-region


Simplification: step III
0 2 0 S0 = S2 S0 =const - vacuum brane. Set of equations

R,n (±) = - R,n R S
0 0

±

f 2 (t ) + k -

2 R2 , N (N - 1)

=

8 G 0 1 (R,n (-) - R,n (+)) = S, R N -1 0 N -2 N N (4 G)2 S 2(N - 1)
2

= const , R (0, t ) = a( ) , = t , = +
0 0

(N - 1)(N - 2) a2 + k 2 a2

.

R,+ = -R,+ = - = -+ Z2 -symmetry n n 0 signS0 = Inner evolution on the brane does not depend on but the bulk does


Solutions > 0 R=R
0

f 2 (t ) + k sin

n + (t ) , R0

R0 =

N (N - 1) 2

On the brane cot = R0 a2 + k a2 = 4 G 0 S, N -1 0 2 + N (N - 1) 4 G N -1 cosh t e a0 , sinh
2

S

0 0

2

=

1 R sin2
2 0

.

R = R0 sin a

n + 0 R0

t a0

,

for for , for

k = +1 k =0 k = -1

t a0

0

= R0 sin 0

Different different matching


Bulk: Carter-Penrose diagrams > 0, k = +1

T R

+

+

R T
-

-


0 > 0 , k = +1 , S 0 > 0

R=

T R=R

+

T

+ 0

0

R=R

R=0

R

+

R

-

R=0

R=R T
-

0

R=R T
-

0

R=


0 > 0 , k = +1 , S 0 < 0

R=

R=

T R=0 R

+

T R
-

+

+

R

+

R

-

R=0

T

-

T

-

R=

R=

Non-traversable wormholes on both sides


Bulk: Carter-Penrose diagrams > 0, k = 0

R = , R = 0 T
+

R=0 R =
-

T

R=0 R = - R = , R = 0


0 > 0, k = 0, S0 > 0

R=

R=0

R=0


0 > 0, k = 0, S0 < 0

R=

R=

R=0 R=0

R=0


Bulk: Carter-Penrose diagrams > 0, k = -1 T± -regions everywhere R = ± arctan R R , = R = ±R0 tan , R0 R0
0

R = , R = R 2 T
+

R = 0, R = 0 T
- 0

R = 0, R = 0

R = , R = - R 2


0 0 > 0, k = 0, S0 > 0, S0 < 0

R = , R = R 2

0

R = 0, R = 0

R = 0, R = 0
0 S0 > 0

R = , R = - R 2
0 S0 < 0

0


Solutions < 0, f 2 (t ) + k > 0 R,n = f 2 (t ) + k - R=R On the brane n coth (t ) = R0 a2 + k f 2 (t ) + k = 2 2 a a = 4 G 0 S, N -1 0 2 + N (N - 1) 4 G N -1
2

2 R2 = N (N - 1)

f 2 (t ) + k +

R2 , 2 R0

0

f 2 (t ) + k sinh

n + (t ) . R0

S

02 0

=

1 R sinh2 (t )
2 0

.

"Heavy brane/shell"
0 If S0 > for > 0 (N -1) || 2 G N

evolution inside the brane is the same as


Bulk: Carter-Penrose diagrams < 0, k = +1, f 2 (t ) + k > 0

R=0 R = 0

R= R
+

R=
0

R=0
0

R = R
2

R = - R
2

R

-

R = 0

Unfolded AdS


0 < 0 , k = +1 , S 0 > 0

R=0

R=0


0 < 0 , k = +1 , S 0 < 0

R

+

R

-

R

+

R

-


Bulk: Carter-Penrose diagrams < 0, k = 0

R=0 R = -


R=0 R = R= R = 0 R= R = 0 R
-

R

+


0 < 0, k = 0, S0 > 0

R=

R=0

R=0

R

+

R

-

R=0

R=0


0 < 0, k = 0, S0 < 0

R=

R=

R

-

R

+

R=

R=

R=0


Solutions < 0, k = -1, f 2 (t ) + k > 0, f 2 (t ) + k < 0 "Heavy shell" S R = R0 sinh
0 0

>

(N -1)|| 2 G N

t R0 sinh 0
0 0

sinh <

n + 0 , R0

"Light shell" S

(N -1)|| 2 G N

R = R0 sin

t n + 0 , cosh R0 cosh 0 R0 R R = ±R0 tanh , , 0 R R0 , R0 R , R0 R < . R = R0 coth R0


Bulk: Carter-Penrose diagrams < 0, k = -1 R = 0, R = 0 T

-

R= R = 0 R
-

R = - R = R = - R R = T
+

R= R = 0

+

R = 0, R = 0


0 "Heavy brane" < 0, k = -1, S0 > 0

R=0 T R= R
- -

R=

R=0 T
-

R

+

R

-

R T
+

+

R=

T

+

R=0

R=0


0 "Heavy brane" < 0, k = -1, S0 < 0

R=

R=

R

-

R

+

R=R R=

0

R=R T
+

0

T

+

R=

R=0


0 "Light brane" < 0, k = -1, S0 > 0

R=0

T

-

T

-

R=

R

-

R

+

R

-

R

+

R=

T

+

T R=0

+


0 "Light brane" < 0, k = -1, S0 < 0

R=0 T
-

T

-

R=

R

-

R

+

R=

T

+

T R=0

+


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