MANY-BODY PROBLEM in KALUZA-KLEIN MODELS with TOROIDAL COMPACTIFICATION

Astronomical Observatory and Department of Theoretical Physics , Odessa National University ,Odessa, Ukraine

andNorth Carolina Central University, CREST and NASA Research

Centers, Durham, USA

Alexander Zhuk, Alexey Chopovsky and Maxim Eingorn

1. For a massive body, any gravitational theory should possess solutions which generalize the Schwarzschild solution ofGeneral Relativity.

Evident statements:

2. These solutions must satisfy the gravitational experiments (the perihelion shift, the deflection of light, the time delay of radar echoes) at the same level of accuracy as General Relativity.

What about multidimensional Kaluza-Klein models

?

2

Multidimensional KK models with toroidal compactification:3

4DM M T

Our external space-time(asymptotically flat)

Compact internal space(mathematical tori)

Class. Quant. Grav. 27 (2010) 205014, Phys. Rev. D83 (2011) 044005, Phys. Rev. D84 (2011) 024031, Phys. Lett. B 713 (2012) 154 :

To satisfy the gravitational tests, a gravitating mass should have tension (negative pressure) in the internal space. E.g. black strings/branes have EoS . In this case, the variations of the internal space volume is absent. If ,such variations result in fifth forth contradiction with experiments.

1/ 2 0

Can we construct a viable theory for a many-body system ?

3

Gravitational field of the many-body system

,2== 00

2000

2

dxdxgdxdxgdxgdxdxgds ki

ik

Metrics:

No matter sources -> Minkowski spacetime:,1== 0000 g ,0== 00 g ==g

Weak-field perturbations in the presence of gravitating masses:

,,,1 000000000 hgfhgfhg

2(1/ )O c

4(1/ )O c3(1/ )O c

4

Energy-momentum tensor of the system of N gravitating masses:

,,30,=,,~= 2 kiuucT kiik

,,4,=,3;0,=,~= 2 DiuucT ii

DuucgpT ,4,=,,~= 23)(

Gravitating bodies are pressureless in the external space : 0, 1,2,3.p

)(1)(~00

1/2

1=p

ml

lmpD

N

p

xxdx

dx

dx

dxgmg

dsdxu ii /=(D+1)-velocity

They have arbitrary EoS in the internal spaces:2

( 3) ( 3)= , 4, , ,p c D

where

5

Multidimensional Einstein equation:

,1

1~

2=

4

TgD

Tc

GSR ikik

Dik

Solution:

||||

2)(2)(21

4

2

4

2

200qp

q

pqp

p

p

N

rr

m

rr

m

c

G

c

r

c

rg

,||2

2

4p

pp

p

N

rr

vm

c

G

D

D

,)(||2||22

23330 ppp

p

p

p

Np

p

p

p

N vnnrr

m

c

Gv

rr

m

c

G

D

Dg

,)(2

2

11

2

c

r

Dg

,

)(2

2

11)(1

2

3)(

c

r

D

Dg

00h

h

3D radius-vector

3D velocity

r

v

(*)

6

( 3)=4

D

Non-relativistic gravitational potential:

,||

=)(p

pN

p rr

mGr

Newton gravitational constant:

.~

1)(

)2(2=4

3)(GaD

DSG DN

periods of the tori

7

Gauge conditions and smearing

To get the solutions (*), we used the standard gauge condition:

,,0,1,=,,0=2

1Dkihh k

ill

kik

This condition is satisfied:0i up to ; 3(1/ )O c i identically;

i .0=2

2

1)(=

2

12

3)(

cD

Dhh kll

kk

0.1 3)( -- is not of interest. 0=.2

.203)(

The gravitating masses should be uniformly smeared over the extra dimensions. Excited KK modes are absent !!!

8

Lagrange function for a many-body system

The Lagrange function of a particle with the mass in the gravitational field created by the other bodies is given by:

aam

2/1

20002 2==

arr

aaaa

aaa c

vvg

c

vggcm

dt

dscmL

(*)

||82=

2

422

p

pa

pN

aaaaaa rr

mmG

c

vmvmcmL

||||

1

||||2

1 22

22

qpp

qpa

pqpN

qp

qpa

qpN rrrr

mmmG

crrrr

mmmG

c

22

2 2

1),(2),(

||2

1ap

p

pa

pN vDvD

rr

mmG

cba ))(())(,( apppap vnvnvvD

c

9

,2

),(

D

DDa

.2

1),(,

2

23),(

D

DD

DD bc

Two-body system

3)(4=

D

The Lagrange function for the particle “1”:

10

||||

1

||2

1

||)(f=

212

2212

222

2212

22

21211 rrrr

mmG

crr

mmG

crr

mmGvL NNN

2

122

2

212 2

1),(2),(

||2

1vDvD

rr

mmG

cN ba ))(())(,( 122221 vnvnvvD

c

)/(8/2=)(f 2411

211

21 cvmvmv

L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, § 106:

The total Lagrange function of the two-body system should be constructed so that it leads to the correct values of the forces acting on each of the bodies for given motion of the others .

To achieve it, we, first, will differentiate with respect to , setting after that. Then, we should integrate this expression with respect to .

arra rL

=/

1L r

1= rr

1r

11

The two-body Lagrange function from the Lagrange function for the particle "2":

(2)2L 2L

The two-body Lagrange function from the Lagrange function for the particle “1":

(2)1L 1L

212

22121

2

12

2122

21

(2)1 2

)(),(f

~=

rc

mmmmG

r

mmGvvL NN

21

22

122

21 1),(2),(2

vDvDrc

mmGN ba ,))(())(,( 21211221 vnvnvvD

c

1(2)1

1=1 /=/ rLrL

rr

212

22121

2

12

2122

21

(2)2 2

)(),(f

~=

rc

mmmmG

r

mmGvvL NN

22

21

122

21 1),(2),(2

vDvDrc

mmGN ba .))(())(,( 21211221 vnvnvvD

c

12

(2) (2)1 2L L GR 3, 0.D if

(2)1L

(2)2L and should be symmetric with respect to permutations

of particles 1 and 2 and should coincide with each other

1),(2=),( DD ba is satisfied identically for any values of

13

We construct the two-body Lagrange function for any value of the parameters of the equation of state in the extra dimensions.

,D

Gravitational tests

It can be easily seen that the components of the metrics coefficients in the external/our space as well as the two-body Lagrange functions exactly coincide with General Relativity for

( 3)= = ( 3) / 2D The latent soliton value.

E.g. black strings/braneswith ( 3) 1/ 2

How big can a deviation be from this value?

3=

2

D ?

14

15

1. PPN parameters

00

1=1, =

2

h

h D

Eqs. (*):

as in GR !

41

1D

Shapiro time-delay experiment (Cassini spacecraft):

5 511= (2.1 2.3) 10 | | 10 .

2

D

2. Perihelion shift of the Mercury16

For a test body orbiting around the gravitating mass , the perihelion shift for one period is

2 2

61 1= 2 2 2 23 (1 ) 3

NGR

G m

c a e

m

1 8= 13 2 3( 1)GR GR

D

D D

33( 1)| | 10

8

D

In GR, a predicted relativistic advance agrees with the observations to about 0.1%

3. Periastron shift of the relativistic binary pulsar PSR B1913+16

1 22 2

6 ( )1 8= 13 2 (1 ) 3( 1)

NGR

G m mD

D c a e D

Two-body Lagrange function:

For the pulsar PSR B1913+16 the shift is degree per year4.226598 0.000005

Much bigger than for the Mercuryand with extremely high accuracy!

Unfortunayely, masses are calculated from GR!1 2,m m

In future, independent measurements of these masses will allow us to obtain a high accuracy restriction on parameter .

17

Conclusion:1. We constructed the Lagrange function of a many-body system for any value of in the case of Kaluza-Klein models with toroidal compactification of the internal spaces.

18

( 3)=

2. For , the external metric coefficients and the Lagrange function coincide exactly with GR expressions.

= ( 3) / 2D

3. The gravitational tests (PPN parameters, perihelion and periastron advances) require negligible deviation from the value .

4. The presence of pressure/tension in the internal space results necessarily in the smearing of the gravitating masses over the internal space and in the absence of the KK modes. This looks very unnatural from the point of quantum physics!!!

= ( 3) / 2D

A big disadvantage of the Kaluza-Klein models with the toroidal compactification.