The Hubble law beyond

the General Relativity

L.M.TomilchikB.I.Stepanov Institute of Physics of NAS of BelarusN.G.KembrovskayaBelarusian State University, Minsk

Gomel, July 2009

Gedanken experiment in the history of PhysicsThe experimental discovery of the Universe

expansion was happened not in 1929 but in 1910-1911 (before the creation of GR)

The expansion law:

R(t) is the time-dependent distance between the light signal source and detector.

is their relative radial velocity

10

( )( ) ,

dR tR t H const

dt

R

dRV

dt

Creation of the Special Relativity

(A) Einstein: 1. Relativity Principle2. Speed of light independence on the source velocity3. Starting point: location clock synchronization procedure

(gedanken experiment) for derivation of the Lorentz transformation;

(B) Poincare:1. Relativity Principle2. Group of the isometric symmetry transformations of the

four-dimensional pseudo Euclidian space-time (Lorentz group). Consistently rigorous algebraic consideration

Einstein and Poincare response (most likely version)(A) Einstein modified his initial synchronization

procedure for the pair of mutually motionless clocks.

(B) Poincare extended the group of the space-time transformations taking into account the nonisometric (changing the scales) ones: i.e., the group of the special conformal transformations.

Location clock synchronization procedure

0At

At

Bt

A B

ABR

Location clock synchronization procedure I

01 2

1 2

0

0

( ), ( ).

,

1( )

2

AB B A BA A B

A B B A

B

B A A

R c t t R c t t

c c c

t t t t

t

t t t

-Definitions

- Einstein'sconditions :

(I)

(II)

(II)isalinear equationfor ;its solution is

Location clock synchronization procedure II

2 2 2 2

,

( ) ( ) ,

,

AB

A B

A B AB

Rc

t t

SR

S c t r

t t t r R

Therefore we have :

that corresponds to zero fundamental interval

Location clock synchronization procedure in the expanding space

10

0 0 0

0 00

0

200

1

( )( )

( ) ( ) exp ( ) ,

( ) exp ( ) ,

exp ( ) .

( )exp ( ) ,

( )

AB A B A

BA AB A B AB

BA A BB A

AB B A

dR tR t H

dtR t R t H t t

R R t H t t

R R H t t R

R c t tH t t

R c t t

Theexpansion law :

Thesolution :

Then :

so, it is impossible to satisfy both of

Einstein's conditions simultaneously.

Two possibilities

01 2

02 1

, ;

, ;

A B B A

A B B A

c c c t t t t

c c t t t t

There is an alternative :

(I)

(II)

The case (I) is more attractive by physical

and metrological reasons.

Version (I)

0

20 00

0 0

1.

1 1( ) , ( ).

2 2 2

:

, ,

B

B A A A A

AB

BA

AB BA

H t

t

H tt t t t t t

R

R

R r r R r r r

Approximation :

It is quadratic equation for . Its solution :

where

The distances covered by the signal in direct

and reverse directions

0

200

0

, ,2

W

Wr t

cH

ct

where

Comments for abovementionedThe situation looks as existence of the uniform

acceleration of the clock B with respect to clock A directed to the observation point.

0W cH

The outcomes1. It is noninertiality FR effect;

2. It could be treated as an existence of the homogeneous gravitational field (equivalence principle);

3. The Universe expansion leads to the (local) quadratic time inhomogeneity and (possibly) conformal time inhomogeneity;

0

1 1lim 0

lim

(1 ),

(1 ) ,

BA AB BA

ABBA AB

t t H tt

t t t Ht

From follows :

The outcomes (continue)4. The experimental prediction: there exist the

universal uniform blue-shifted Doppler drift in the location-type experiments.

Linear transformation of light cone generatrixs

2

2 2

2 2 2 2 2 2

1

2

, ,0,0 ,

' , ' , .1 1

1

1

' ' 0,

' |x ct

ct x

Vxtx Vt Vcx t

c

x

c t x c t x

t

Lorentz boosts :

Light cone invariance :

Longitudinal Doppler effect

1 1

1

2

2

2

,

11

1

:

1 1

1 1

'em em obs obs

obs

em

N

z

z

zV

c z

t NT c t NT c

– the number of oscillations in the signal.

The red shift :

Speed of light dependence on

Longitudinal Doppler effect (continue)

0

mod 0

1:

1

: 1

If the uniform acceleration – exist :

( )

obs

em

obs em

obs em em

V

c

V

c

W

V V t V Wt

WtH t

c

Under condition

For the frequencies

The blue frequency drift appears

Nonisometric Subgroup of SO(4,2) (besides Poincare group)

1

:

( , ) ( ) ,

( , ) 1 2( ) ( )( ).

1 1 .

x a x x a x x

a x a x a a x x

Special proper conformal transformation

SCT group S

For simplicity : – subspace

Standard interpretation of SCT — transformations from the comoving inertial RF to the uniformly accelerated RF

2

2

2

2

2 2 2 2

2 2

2

, ,0,0 , 0,- ,0,0 .2

1 -2 2 , ,

1 - 1 -2 2 2 2

1, 1, :

- , .2

wx wtcc

wx ct x a

c

wx wtx

tcx twx wt wx wtc c c c

wtx x t t

w

if we have Galilei -Newton limit

is the 3 - dimentional consta .nt acceleration

Conformal deformationof the light cone

2 20

00

0

0

0  :

, .1 2 1 2

,

:

/ .

1

x x

x xxx

ax axx ct x ct

t c w

tt

tt

In the case we have

Under conditions wehave

the light - co

nonlinear transformation of time

(time inhomogen

ne generatrix deformation

where

eity)

The dependence of distance on red shift I

0

2

0

.1

:́

1 , 1obs

em

tt

tt

t t

tt t z

t

The case of a signal traveling in the direction of negativegeneratrix of light cone :

For small time increments and

The time interval between the moments

1/2

0 1/2

( 1) 1( ) .

( 1)

zt z t

z

of emitting and receivingof the signal :

The dependence of distance on red shift II

1/2

01/2 1/2

2

2

( 1) 1 1( ) 1 ,

( 1) ( 1)

( 1) 1.

( 1) 1

( )

)

:

(

.

u u u

zR z R R R ct

R z

V z

V

z z

R ct

z

c

z

z

Location distance by definition :

The dependence of on the red shift :

The known expression for velocity

The Hubble law I

1/1

0

1 10 0

0

2 2

2 1/ 2

01

2 10

( )( ),

( )

lim ( )

( 1) ( 1) 1( ) ,

( 1) 1 ( 1) 1

1

2 2

2

;

2

z

u

u u

u

z

z zf z

z zV z

cR f z

w cH

VH

R

R ct c

R z

f z

c H c

w

R R H

Compare with the Hubble law

from we obtain

The Hubble law II

The function φ(z): The zero position of φ´(z):

0 0,475.z

0

( )( )

( )V z

zH R z

1 0 2 0( ) ( )f z f z

2 1/ 20

1 22 1/ 2

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

( 1) 1 ( 1)V z z H R z z

f z f zc z c z

z0 1 ,3 1 4 6

f1 zf2 z

0 .5 1 .0 1 .5 2 .0 z

0 .2

0 .4

0 .6

0 .8

f z

Accelerating Universe(Riess et al., 1998; Perlmutter et al. 1999)

The observed phenomenon Deflection from linearity of the conventional

Hubble law V=H0R with maximum at zexp=0.46±0.13

The interpretation zexp is the point of transition from the Universe

expansion deceleration to acceleration. Effect follows from the existence of the Dark Energy, amount of which is ~70 % of the full gravitating energy in the Metagalaxy.

Location-type experiment

0

00

00

0 0

0

, ,11

, .

1( ).

2

A BB A

B A A BB A A B t tt t

tt

A B B A A B B A

A A

t t t tt t t t

t t t t t t t t

t t t

r c t

when then

Timeinterval :

Distance :

Mapping of the world lines

0A B B At t t t

2

0

cr

w 02x r ct

t A0

tB

t A0

x 0 2 r0 x 2 x

t A

2 r0

c tS

0A B B At t t t

tB'

t A'0

t A'

x 0 r0x

c tS

Observable consequencesUnder condition we have time distance

the forward:

the backward:

“Acceleration”

Frequency

,t t

2 20

20

0

2

0

( )

(

( )

( )2

)

2W t

c t

W tc t

tt

t

tt

t

0 1t t

10 0 02 .W ct H c

00

21 .obs

tt

Pioneer Anomaly (PA)(1998)

The observable phenomenon:

Unexpected uniform blue-shifted drift of the frequency obs:

The interpretation:

An anomalous sunward directed accelerationap= (8,741,33)10-10 m/s2

9d(5,99 0,01) 10 Hz/s

dobs

obs t

Conclusion IThe conformal transformation of time

leads to the following consequences:

The cosmological location distance can be determined as an explicit function of red shift R(z).

The combination of this function with the SR expression for the longitudinal Doppler-effect V(z) gives the explicit analytic expression for the ratio V(z) / H0 R(z).

The ratio coincides with the Hubble law in the limit of ,and possesses a maximum at

The appearance of this maximum is a pure kinematic manifestation of the time inhomogeneity and does not need any special gravitational sources (like the dark energy).

0

.1 /

tt

t t 1

0 02t H

1z0,475.z

Conclusion II

2. In the location-type experiments uniform blue-shifted frequency drift appears, which mimics constant acceleration towards the observer

The Pioneer Anomaly is the first really observed effect of that kind. The observed drift can be used for local experimental determination of Hubble constant.

This effect can be observed in principle at any frequency even between mutually moveless emitter and receiver in the absence of any gravitating centers.

0 0.w cH

Thank You for Your attention.