Classical tests of multidimensionalgravity: negative result

Maxim Eingorn and Alexander Zhuk

Astronomical Observatory andDepartment of Theoretical Physics

Odessa National University

Odessa, Ukraine

arXiv: 1003.5690 (gr-qc)

Setup

DM M= ×ℝ

Classical gravitational tests:

1. frequency shift2. perihelion shift3. deflection of light4. PPN parameters

GR – good agreement with experiments !

Kaluza-Klein models – ?⇒

33

DM T −×

Metrics:

( )22 0 000 02i k

ikds g dx dx g dx g dx dx g dx dxα α βα αβ= = + +

c dt⋅

our space(asymptotically flat)

internal space

:

Topology:

2

1

( )21; ;ik ik ik ikg h h O

cη⇒ ≈ + =

2002 ;h

cϕ≡ 0, , ?h hα αβϕ −

III. Energy-momentum tensor of N moving point masses:

( ) ( )1

2

1

1 ,i kN

Dikpp

p

dx dx cdtT m g r r

dt dt dsδ

−

=

= − − ∑

1 2( , , , )Dr x x x= …

pdxc

dt

α

≪

rest mass density: ( )1

N

p pp

m r rρ δ=

≡ −∑

I. No matter source

00 00 0 01, 0,g g gα α αβ αβ αβη η η δ= = = = = = −

⇒ Minkowski spacetime:

Three assumptions :3

II. Weak field limit:1. gravitational field is weak;2. velocities of test bodies are small.

? ? ?h hα αβϕ − − − ⇐ Einstein equation:

4

2 1

1D D

ik ik ik

S GR T g T

c D = − −

ɶ

( )/22 / / 2DDS Dπ= Γ – square of (D-1)-dimensional sphere of a unit radius

DGɶ – gravitational constant in ( 1)D D= + – dimensional spacetime

4

0

21/ c correction terms

4

2 1

1D D

ik ik ik

S GR T g T

c D = − −

ɶ

( )21/O c ( )21/O c

10

2k l ki l ik

h hx

δ∂ − = ∂ – gaugecondi tions

00 00

1,

2R h≈ ∆ 0 0

1,

2R hα α≈ ∆ 1

2R hαβ αβ≈ ∆

2 / x xαβ α βδ ∂ ∂ ∂

2 200 0, 0, 0T c T T T cα αβρ ρ≈ ≈ ≈ ⇒ ≈

⇓

00 02 2

2 21, 0,

2D DD DS G S G

h h hc D cα αβ αβρ ρδ∆ = ∆ = ∆ =

−

D DS Gϕ ρ⇒ ∆ =200

2hc

ϕ≡

0 00 2

1 1 20,

2 2h h h

D D cα αβ αβ αβϕδ δ= = =

− −

( )( )

2 2

1 D

DG

D

−−

ɶ

– D-dimensional Poisson equation

5

31/ c 41/ cand correction terms

( )2200 02i k

ikds g dx dx g cdt g cdtdx g dx dxα α βα αβ= = + +

4(1/ )O ccorrection terms: 3(1/ )O c 2(1/ )O c

?

Energy-momentum tensor

( )

( ) ( ) ( ) ( )

00 004 21

2 2 44

1

2 1

1

3 41/ 1/

2 2 2

ND DD D

p pp

ND D

p p p pp

S G S GT g T m r r

c D c

S G D Dm v r r O c O c

c D Dϕ δ

=

=

− ≈ − + −

−+ + − = + − −

∑

∑

ɶ

00h 00f/p pv dx dtα α≡

-component:00

( )pr rϕ = – ?

6

( ) ( )4

1

1/N

p p p pp

T m v v r r O cαβ α βδ=

≈ − ⇒∑

correction terms in hαβ

is not of interest

( ) ( )30 04 3

1

2 1 11/

1 2

ND DD D

p p pp

S G S GDT g T m v r r O c

c D D cα α αδ=

− − ≈ − − = − − ∑

ɶ

0f α – ?

⇓

ik ik ik ikg h fη≈ + +

( )400 1/f O c= ( )3

0 1/f O cα =

? ⇐

0α -components:

αβ -components:

Einstein equations

7

Ricci tensor:

( )2

00 002 4 4

1 1 1 2 2,

2 2R f

c D c cϕ ϕ ϕ ϕ≈ ∆ + ∆ + ∆ − ∇

−2

0 0 3

1 1

2 2R f

c t xα α αϕ∂≈ ∆ +

∂ ∂

Einstein equations:

( )

( ) ( )

2

00 4 4

24

1

1 4 4

2

2 3 4,

2 2 2

ND D

p p p pp

fD c c

S G D Dm v r r

c D D

ϕ ϕ ϕ

ϕ δ=

∆ + ∆ − ∇ =−

−= + − − − ∑

( ) ( )2

0 3 31

2 11

2

ND D

p p pp

D S Gf m v r r

c t x D cα ααϕ δ

=

−∂∆ + = − −∂ ∂ − ∑

8

metric coefficients:⇒

( ) ( )

( ) ( )

200 2 4

24 4

1 1

2 21

2 1,

2

N N

p p p pp p

g r rc c

Dr r v r r

c D c

ϕ ϕ

ϕ ϕ ϕ= =

≈ + + +

′ ′ ′+ − + −−∑ ∑

( ) ( )2

0 3 31

2 1 1 1

2

N

p pp

D fg v r r

D c c t xα α αϕ=

− ∂′≈ − − −− ∂ ∂∑

pϕ′pr

( )pr rϕ′ − r ( )D p pDS G m r rϕ δ′∆ = −

( ) ( )1

N

pp

r r rϕ ϕ=

′= −∑

( )f rϕ∆ =

3D =

⇓Eqs. (106.13) and (106.14) in: L. Landau and E. Lifshitz,Theory of fields

9

– potential in a point produced by all particles, except of the p-th.

– potential in a point produced by p-th particle:

One particle at rest

11 0, , ( )p f F tϕ ϕ′= ⇒ = ≠

the main conclusions will not change if particlehas nonzero componentsof velocity in extra dimensions

⇓

( ) ( )200 02 4

2 21 , 0,g r r g

c c αϕ ϕ≈ + + ≈

( )2

1 21

2g r

D cαβ αβϕ δ ≈ − − −

( )2

D DS G m rx x

αβα β

ϕϕ δ δ∂∆ = =∂ ∂

where is a solution of -dimensional Poisson equation:D( )rϕ

10

1

1/22

3 1 313

2

11

1 3 3

( , , , ) exp 2

22cos cos

2

d

dN i

dk k i i

gd Nd

d

G m kr r

r a

r ck G mk

a a r r

ϕ ξ ξ π

ππ ξ ξ

+∞ +∞

=−∞ =−∞ =

= − × − ×

× ≈ − = −

∑ ∑ ∑… ⋯

⋯

3 dDM T= ×ℝFor spatial topology solution is ( M.E.&A.Z., CQG, 2010):

3 3| |r r= in solar system

3 1 2, , , dr a a a≫ …

22 /g Nr G m c=

“Smeared” extra dimensions: ( )( )3 1 31

/d

m a r rαα

ρ δ=

= −

∏

⇒ ( ) ( )2

33 32

gNr cG m

r rr r

ϕ ϕ= = − = −

Newton gravitation constant:1

4 /d

N D DG S G aααπ

== ∏

periods of tori

11

Metrics in isotropic 3-D spherical coordinates:

( )2 2 2 2 2 2 2 2 23 3 3 sindx dy dz dr r d r dθ θ ψ+ + = + +

⇓

( )

( ) ( ) ( )( )

22 2 2

23 3

2 2 2 2 2 23 3 3

3

2 2 24 5

3

12

11 sin

2

11

2

g g

g

g D

r rds c dt

r r

rdr r d r d

D r

rdx dx dx

D r

θ θ ψ

≈ − + −

− + + + − −

− + + + + −

…

( )∗

Asymptotic form of the metric coefficients for delta-shaped matter source.Non-relativistic gravitational potential has correct Newtonian limit .

⇑

3

( ) NG mr

rϕ ≈ −

12

Three classical gravitational tests

I. Frequency shift

1 2.

( ) ( )1/2 1/2

1 00 2 001 2

g gω ω =

⇓ page 12, Eq.

( )21/ :O c1 2

2 1 21

c

ϕ ϕω ω − ≈ + ⇒

( )∗

1, 1ω ϕ 2, 2ω ϕ

no deviation from GR !

13

II. Perihelion shift

m m′3rR 3r R≫ 86.96 10R ≈ ×

⊙m

( ) ( ) 103 4.6 6.9 10Mercuryr ≈ ÷ × m

Hamilton-Jacobi equation:

2 2 0iki k

S Sg m c

x x

∂ ∂ ′− =∂ ∂

2 222

2 2 23 3 3 3 3 3

2 22 2

43

1 1 1 11 1 1

2 2 2

11 0

2

g g g g

g

D

r r r rS S S

c r r t D r r r D r

r S Sm c

D r x x

ψ ∂ ∂ ∂ + + − − − − − ∂ − ∂ − ∂

∂ ∂ ′− − + + − ≈ − ∂ ∂ …

/ 2θ π= ⇓ ( )∗

separation ofvariables: ( ) ( ) ( )

3

43 4

Dr DS E t M S r S x S xψ′= − + + + + +…

2E E m c′ ′≈ + requires corrections in( )41/O c 00g

14

up to : ( )21/O c ( )

( )( )

3

2 22 24 2

3

2 2 22 2 2

23 3

2

2 11 1.

2 2 2

rD

gg g

dS Em E p p

dr c

Dm c rDm c r m Er M

r D r D

′≈ − + + + +

′ − ′ ′+ + − − − −

…

( )

( )( )

3

22 24 2

1/22 2 2

2 2 232

3 3

2

2 11 1.

2 2 2

r D

gg g

ES m E p p

c

Dm c rDm c r m Er M dr

r D r D

′≈ − + + + +

′ − ′ ′+ + − − − −

∫ …

-relativisticcorrection

ε

⇒

( ) ( )( )

( )

( )( )

3 3

3 3 3 3

0 02 2 20 02

2 4 2r rg

r r r r

S SDm c rS S M S S

M D M Mε ε

′∂ ∂= − ≈ − = −

∂ − ∂

0ε =

15

Angular momentum is the integral of motion:M

3rSS

constM M

ψ∂∂ = + =

∂ ∂– trajectory of the test body

– the change of the angle during one revolution

⇓3r

SM

ψ ∂∆ = − ∆( )3

3 3

02 2 20

4 2rg

r r

SDm c rS S

D M M

′ ∂∆∆ ≈ ∆ −

− ∂

2π−

⇓

( )2 2 2

22

2 2gD m c r

D M

πψ π

′∆ ≈ +

−Perihelion shift

Mercury:

observed value: arcsec per century43.11 0.21±

3 42.94

4 28.63

9 18.40

D

D

D

′′= ⇒

′′= ⇒

′′= ⇒contradict observations !

General Relativity !⇐

⇐

∂

16

( )

( )

III. Deflection of light

m

πδψρ

Hamilton-Jacobi equation

Eikonal equation: 0iki k

gx x

∂Ψ ∂Ψ =∂ ∂

2 222

2 2 23 3 3 3 3 3

2 2

43

1 1 1 11 1 1

2 2 2

11 0

2

g g g g

g

D

r r r r

c r r t D r r r D r

r

D r x x

ψ ∂Ψ ∂Ψ ∂Ψ + + − − − − − ∂ − ∂ − ∂

∂Ψ ∂Ψ − − + + ≈ − ∂ ∂ …

/ 2θ π= ⇓( )∗

⇓

17

separation ofvariables: ( ) ( ) ( )

3

400 3 4

Dr Dt r x x

c

ρωω ψΨ = − + + Ψ + Ψ + + Ψ…

0 /M k cρ ρω≡ = – “angular momentum”of the light beam

Light propagates in our 3-D space / 0, 4, ,p d dx Dαα α α⇒ = Ψ ≡ = …

3

2 1 22 2 20 02 2 2

3 3 3 3 3

2 202 2

3 3

11 1

2 2

11

2

r g g g

g

d r r r

dr c D r r r c r

rD

c D r r

ω ρ ω

ω ρ

− Ψ

⇒ ≈ − + + − −

−≈ + − − up to ( )41/O c

It is sufficient to keep in and correction terms 00g gαβ2

00 , (1/ )h h O cαβ ∼

are defined by non-relativistic potential ϕ

Finite size of the gravitating mass results in exponentially small correction terms (M.E.&A.Z., CQG, 2010)

⇓!

( )3r R aα≥ ≫

18

3

1/220

323 3

11

2g

r

rDdr

c D r r

ω ρ −⇒ Ψ ≈ + − ≈ −

∫

small relativistic correction

( )

( ) ( ) ( )

( )3 3

1/20 0 0 02 2 33 3

1 1arccosh

2 2 2 2g g

r r

r r rD Dr dr

D c D c

ω ωρ

ρ−− −≈ Ψ + − ≈ Ψ +

− −∫

3

1/21/2 22 2(0) 0 0

3 32 23 3

1r

Mdr dr

c r c r

ω ωρ Ψ = − = −

∫ ∫

0gr ≡

nonrelativistictrajectory: ( )3

(0)(0)(0) (0) (0)

3 3arccos / 0 cosr r rM M

ψ ψ ρ ρ ψ∂Ψ∂Ψ = + = − = ⇒ =

∂ ∂

mρ(0)ψ

3r

(0)ψ π∆ =

3

(0) (0) /r Mψ π⇒ ∆ = −∂∆Ψ ∂ =

19

R Rρ ⇒

3 3

0(0) 1arccosh

2g

r r

rD R

D c

ωρ

−∆Ψ = ∆Ψ +−

0/Mc ω

M is integralof motion ⇒ 3r const

M Mψ

∂Ψ∂Ψ = + =∂ ∂

( )3 3

(0)1/22 21 1

2 2r r g gr R rD D

RM M D D

ψ ρ πρ ρ

−∂∆Ψ ∂∆Ψ − −∆ = − ≈ − + − ⇒ +∂ ∂ − −

R → +∞

deflection of light δψ

3 1.75

4 1.31

9 1.00

D

D

D

′′= ⇒

′′= ⇒

′′= ⇒

Sun:

observed value: arcsec1.75δψ ≈

General Relativity !

contradict observations !

⇐

⇐

m

20

PPN parameters and :γ β

( )2 3 22 2 22

13 3 3

1 12

g g g i

i

r r rds c dt dx

r r rβ γ

=

= − + − +

∑

( )∗Eq. , page 12: ( )1/ 2Dγ = −

Observed value: (Cassini spacecraft)( ) 51 2.1 2.3 10γ −− = ± ×Bertotti et all, Nature (2003),C. Will, gr-qc/0504086,Jain&Khoury, 1004.3294

3 1 0

4 1 1/ 2

9 1 6 / 7

D

D

D

γγγ

= ⇒ − == ⇒ − = −= ⇒ − = −

General Relativity !

contradict observations !

⇐

⇐

γ β

GR:1

1

γβ

==

21

IV. Parameterized post-Newtonian (PPN) parameters and

The classical test formulas via PPN parameters (C.Will, 1993)

1. The perihelion shift :

( )( ) ( ) ( )22

3 12 2

3 2 11

g gr D r

D a ea e

π πδψ γ β= + − =

− −−our formula on page 16

2. The deflection of light :

( )1/ 2 , 1Dγ β= − =

( ) 11

2g gr rD

Dδψ γ

ρ ρ−= + =−

our formula on page 20

22

3. The time delay of radar echoes :

time difference of propagation of electromagnetic sign als between two points (or for a round trip ) in the curved and flat spaces

Sun

EarthPlanet or satellite

SunREarthrplanetr

C.Will :

( ) 2 2

4 411 ln ln

2g Earth planet g Earth planet

Sun Sun

r r r r r rDt

c R D c Rδ γ

−≈ + = −

This formula is used to estimatethe upper limit of γ

( )1/ 2Dγ = −

The time delay explicitly depends on total number of spatial dimensions !

23

Solitons and black strings

5-D static metric ansatz in isotropic coordinates:

( ) ( )( ) ( )2 2 2 2 2 2 23 3 3ds A r c dt B r dx dy dz C r dξ= + + + +

2 2 23r x y z= + +

metric coefficients do not depend on the extra dimension

non-relativistic gravitational potential does not depend on the extra dimension

“smeared” extra dimensions – gravitating mass is uniformly smeared over the extra dimension

⇓

⇓CQG (2010)

ξ

3r

1a

non-relativistic picture ( M.E.&A.Z., CQG, 2010) :

( ) ( )1 3/m a rρ δ=

( ) ( )33

NG mr r

rϕ ϕ= = −

– rest mass density

– gravitational potential

24

Soliton metrics

Kramer (1970),Gross&Perry (1983),Davidson&Owen (1985)

5-D solution of the vacuum Einstein equation 0ikR =

⇓

M.E.&A.Z., arXiv: 1003.5690 : there is only one particular case

when the metric coefficients have the asymptotic form in page 12 !

( )

( )2 2 2 1 2

2 2 2 2 2 2 23 3 33 3 22 2

3 3 3 3

1 1 111 ,

1 1 1

k kar ar ar

ds c dt dr r d dar a r ar ar

ε ε ε

ξ−

− + += − − + Ω − + − −

( )2 2 1 1k kε − + =

( )2, 1/ 3 , 8 / 3 gk a rε= = =

( )∗

delta-shaped matter source

( )∗∗

25

( ) ( )

00 003 3

3 300 00 2 2

1 1 , 1 1 , 1, 2,3, 42

4 21 1,

2 2

g g

N N

r rg h g h

r r

G m r G m rR h R h

c c

αα αα

αα αα

α

π δ π δ

≈ − = + ≈ − − = − + =

≈ ∆ = ≈ ∆ = ( )21/O c ⇒

(I)

(II)

a) (I) provides the correct asymptotic form for metric coefficients in the case of the delta-shaped matter source !

b) (II) + gravitating body at rest is the only non-zero component of the energy-momentum tensor.

c) (II) + Einstein equations

d) (I) results in contradiction with observations !

e) All other soliton metrics (different from ) do not correspond to the asymptotic metrics for the delta-shaped source. Besides non-zero

they have other non-zero components with unclear physical origin.

00T⇒

( ) 2 200 3

1

1T m r c c

aδ ρ⇒ ≈ =

( )∗∗( )∗

00T

26

0, , 1k kε ε→ → +∞ →Black string:

( )2 4

2 2 2 2 2 2 23 33 3 2

3 3

1 1

1

ar ards c dt dr r d d

ar arξ

− +⇒ = − + Ω − +

4-D Schwarzschild metric for 4 / ga r=

( ) ( ) 2

3 33 1 / 4gSchr r r r = + Schwartzschildian radial coordinate:

( ) ( )200 44

3 3

1/ 1 ; 1 , 1, 2,3 ; 1g gr rO c g g g

r rαα α⇒ ≈ − ≈ − − = = −

00 11 22 33 443

, 0grh h h h hr

= = = = − =⇓

( )300 11 22 33 442

4; 0NG m r

R R R R Rc

π δ= = = ≈ =

⇓

27

Gravitating mass at rest + the form of + Einstein eqs.ikR

⇓

( ) ( )2 200 3 44 3

1 1

4 2,

3 3

m mT r c T r c

a aδ δ≈ ≈ −

Black string tensionThe only non-zero components

a) BS metrics does not contradict observations !

b) Metric coefficients do not have the asymptotic form which corresponds to the delta-shaped source.

c) BS tension has unclear physical origin !

( )∗

28

Conclusions:

1. The metric coefficients in a weak field limit were obtained for KK multidimensional models in the case of delta-shaped matter source.

2. It was found the formulas for the perihelion shift, the deflection of light and PPN parameters.

3. These formulas demonstrate good agreement with experimental dataonly in the case of ordinary 3-D space.

4. This result does not depend on the size of extra dimensions.

5. It was obtained the exact 5-D soliton solution with correct non-relativistic Newtonian limit. The energy-momentum tensor for this solution has clear physical interpretation (delta-shaped source). However, the classical tests for this metric do not satisfy the experimental data.

6. Black string solution does not contradict the classical gravitational tests.However, this solution does not have asymptotic metric coefficients corresponding to a delta-shaped source. Matter source has non-zero tension. It is hard to imagine that astrophysical objects, e.g. the Sun, can have such energy-momentum tensor.

⇒ KK models face severe problems !

29

Supplement

Static spherically symmetric perfect fluid matter source

Perfect fluid of radius is uniformly smeared over extra dimension

R

ξ

3r

R

1a ( )3

3 31 3 1

;4 / 3

m m

a V a R

ρρ ρπ

= ≡ =

Energy-momentum tensor:

( )4 3

3

diag , , , , ; ,

0 ; ,i

k

p p p p r RT

r R

ε ξξ

− − − − ≤ ∀= > ∀

We consider astrophysical objects where 4,p pε ≫

Rest mass density:

30

Static configuration î non-diagonal metric coefficients are absent:

0 0 , 1,2,3,4g α α= =“Smeared” extra dimensions î metric coefficients depend only on : 3r

( )3ik ikg g r= spherical symmetry

Weak field limit:

( )200 00 , 00 00 2

21 1 ; , 1/ ;g h g h h h O c h

cαα αα ααϕ≈ + ≈ − + ≡∼

⇓

00 00

1 1,

2 2R h R hαα αα≈ ∆ ≈ ∆

?

3D

31

Einstein equations:

0000 00 00 00 00 00 00

1 1 2

3 3 3R k T Tg k T T g g k T

′′ ′′ ′′= − ≈ − =

( ) ( )4 44 5 1 12 / 3 / 4 8 / 3 / 4N Nk S G c a G c a kπ′′ = = × ≡ ×ɶ

00 0 ,T T Tα αβ≫

0000 00

1 1 1

3 3 3R k T Tg k T g g k Tαα αα αα αα

′′ ′′ ′′= − ≈ − ≈

00 00 2

1 1

2 2R R h h

cαα ααϕ= ⇒ = =

32

Inner region3 :r R≤ ( ) 2 2

00 3 1/inT c c aε ρ ρ≈ ≈ =

( ) ( )2 300 3 32

1

84 14

3in inN

N

Gh k c G

a c

π ρρ ϕ π ρ′′⇒ ∆ = = ⇒ ∆ =

Outer region3 :r R> ( )

00 0outT =

( ) ( )00 0 0out outh ϕ⇒ ∆ = ⇒ ∆ =

should match at 3r R=

( )3/out

NG m rϕ = −⇓

( ) ( )00

3 3

, , 1,2,3,42

out g out gr rh h

r rαα α⇒ = − = − =

Spherically symmetricperfect fluid matter sourcewith

results in our asymptoticmetric coefficients!

00 0 ,T T Tα αβ≫

33

Conclusion:

Neither multidimensional delta-shaped matter source nor multidimensional compact perfect fluid with condition

can produce in KK models the gravitational field whichcorresponds to classical gravitational tests.

00 0 , ; , 1, ,T T T Dα αβ α β =≫ …

Possible solutions of the problem:

1. To localize matter source on the brane î bran e world models !

2. To violate the condition00 0 ,T T Tα αβ≫ î realistic KK models ?

34

Thank you for attention!