E. Rakhmetov, S. Keyzerov

SINP MSU, Moscow

Linear approximation in Möller gravity

QFTHEP 2011, 24-30 September, Luchezarny, Russia

MotivationsBrief introduction in Möller (vielbien) gravityEquations for the small variations over an

arbitrary backgroundMinkovsky space as a backgroundSchwarzschild solution as a backgroundSelf-consistent solutions for Kaluza-Klein

theoriesConclusions

Outline

Searching for such generalization of General Relativity, which might be useful in understanding what “Dark Matter” and “Dark Energy” might be.

Obtaining some important exact solutions in Möller Gravity, and making a comparison with GR.

Trying to use linear approximation for interpretation of some results for this solutions.Make an attempt to find Schwarzschild solution in Möller

Gravity with an arbitrary constants in LagrangianChecking whether self-consistent solution appeared in

this theory in 7 dimensional space-time can be treated as spontaneous compactification of Freund-Rubin type or not.

Finally, we demonstrate, that the small variations over backgraund, can be considered as an antisymmetric second rank tensor field.

Motivations

Möller (vielbein ) Gravity IntrodutionWas suggested by C. Möller in 1978. Metric theory with metric tensor

Constructed from four vielbiene (frame) vectors Which are orthonormal : Here indexes in brackets are frame indexes,

from 1 to 4, and summation is considered over repeated indexes

Stress tensor for this vector fields isCoordinate indexes can be turned into frame

indexes:

3

0

( ) ( ) ( ) ( ) ( )g g g g g

( ) ( ) ( )g g

( )g ( ) 1,1,1,1diag

( ) ( )[ ]

f g

( ) ( )C g C

Vielbein (Möller) Gravity IntrodutionCurvature tensor can be obtaned from stress

tensor :

We can produce from stress tensor 3 different scalars:

Then the simplest action

1 1 1 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2 2

1 1( ) ( ) ( ) ( )2 21 1 1( ) ( ) ( ) ( ) ( ) ( )2 2 4

R f f f f

f f f f

f f f f f f

2L f f 3L f f 1L f f

0 1 1 2 2 3 3S k k L k L k L gdxX

Vielbein (Möller) Gravity IntrodutionFinally action:Motion equations: Symmetric part:

Asymmetric part:

If k’1= k’2 = 0, asymmetric part vanish, and symmetric part gives us General Relativity

0 3 1 1 2 2S k k R k L k L gdx

X

( ) ( ) [ ]14 2 2 23 1 2 1 2 [ ]2( )2 2 2

1 2 2 1 2

X k R g R k k f k k f f

k k f f k f f g k f f k f f

1( ) 0( )SX

g g

[ ] [ ] [ ]2 2 4 2 31 2 2 1 2

X k k f k f k k f f

One can divide variation on 2 parts: -metric variation, - some remnant,

then . , - antisymmetric tensor. After integration over obtain for variation

- is a pure rotations, because of - generators of the

rotation Thus, in Moller Gravity, we can consider frame

rotations as dynamical variables

Linear approximation 1( ) ( ) ( )2

g g g g ( )

1 ( ) ( )[ ]2

g g 1( ) ( ) ( ) ( ) ( )2g g g g

( ) 1( ) ( ) ( )

2g e g g g

( ) ( )( ) 2i T

e e

( ) ( ) ( ) ( ) ( )T i

If and are close solutions

of motion equations, then is background and

are small variations over this background. If we consider

pure rotations , then from asymmetric

part of motion equations we obtain equation for small

deviations:

Where

Equations for the small variations over an arbitrary background

( ) ( ) ( )g g g ( )g ( )g ( )g

( ) ( )g g

[ ]121 2 3 2 32[ ] [ | | ]2 2

1 1[ ] [ ]

2 3 2 3[ [ ]

2 3[ ] [ ]

2 3

k k k k k

k C k C

k k C k k C

k k C C C

k k C C C

0

( ) ( )C g

Because of for Minkovsky space we can take frames with

If 2k1+k2+k3=0, we have the eq. for massless

antisymmetric second rank tensor field with spin 1:

that have, as as usual, two different transverse polarizations. In common case we have longitudinal polarizations also, but later we will consider only last case, because of spherically symmetric Shwarzshield-like solutions appears, only if 2k1+k2+k3=0

Minkovsky space[ ]12 0

1 2 3 2 32k k k k k

0Ñ

[ ] 01k

Schwarzschild solution in Möller GravityMetric:Metric tensor:

Ansatz for the vielbein:

Where

2 ( ) 2 ( )2 2 2 2 2 2 2sinr rds e dt e dr r d d

2 0

20

eg

e gpq

(0)0

g e0(0)g e

( ) 00

g a ( ) ( )g a e g aq q

( ) ( )q qg a e g a ( ) 0g a q

cos

(1) sin

0

g rq

sin cos

(2) cos cos

sin sin

g rqr

sin sin

(3) cos sin

sin cos

g rqr

Finally obtain 3 equations for two parameters

WhereSystem has solution only if K=0 or (this is the

same) 2k1+k2+k3=0 and this is pure GR Schwarzschild solution:

where

Schwarzschild solution in Möller Gravity

2 1 2 2 2 12 4 2 2 2 4, , , , , , , , ,2r e rrr r r r rr r r r r

2 1 1 2 2 2 12 2 2 2 4, , , , , , , ,2r r e rr r r r r r r r

11 1 2 2 2 12 2 2, , , , , , , , , ,2r r e rrr r r rr r rr r r r r

21 223

k k

k

12 2 2 2 2 2 201 sin1 /

0ds dt d d d

2 2 1 10 0 0 0

r r

40 0 0

A r

For Shwarzschield reference frame we have two non trivial expressions for

Thus the equation for small variations is where

As we can see, this eq. describe the waves, which have some additional interaction with background and some effective mass, that depends from radial component.

Equations for the small variations over Schwarzschild background

1 0[ ] [ ]1

A B

2,001

C e r ,1

C g rab ab

2 241 02 2( )0

e rA

r r r

2 1 2 2240 22 2 2 2 2 20 0 0 0

B e rr r r r r r r r r r r

Self-consistent Solutions for Kaluza-Klein theories

Here large latin letters are changed from 0 to 7, except 4, small latin letters -from 5 to 7, small greece letters - from 0 to 3

Manifold is M4XSn . Ansatz for the vielbein

Then nontrivial equations:

If constants are

we have a searching solution.

( ) ( ) ( ) ( )Ag x h x ( ) 0g a ( ) 0g q ( ) ( ) ( 4) ( )A lg a x rg a xq q

1 12[ ]( ) [ ] 6 ( ) [ ]( )2 4

3R h R h r Y h

k

12 2 2 222 ( ) [ ] 6 ( ) 4 ( ) 6 ( )2

3

kr pq R h r pq r pq r pq

k

12 33k k 1

1 36k k 12

6r

Schwarzschild solution appears not only in case, when the constants of the Möller theory are small, as was shown by Möller, but in case of arbitrary constants too, when some relations is valid. There are wide area of theory parameters, in wich we have not find spherically symmetric Shwarzshield-like solutions. Linear approximation does not help us whether such solutions exist or not.

Small variations over backgraund, can be considered as an antisymmetric second rank tensor field with spin 1, which “feels” reference frame structure, and in the arbitrary case have an effective mass, depending from coordinates.

With large spectra of theory parameters, 4-dimantional dynamics allows solutions like plane Minkovsky space, and the other dimensions are spontaneously compactifed in to n-dimentional sphere. This is very similar on compactification of Freund-Rubin type, but our antisymmetric second rank tensor field can not play role of Freund-Rubin field.

Conclusions

Thanks for your attention