Imperial College, LONDON, 12-15 SEPTEMBER, 2008 From local to global relativity Tuomo Suntola, Finland Physical Interpretations of Relativity Theory XI

Imperial College, LONDON, 12-15 SEPTEMBER, 2008

From local to global relativity

Tuomo Suntola, Finland

Physical Interpretations of Relativity Theory XI

T. Suntola, PIRT XI, London, 12-15 September, 2008 2

Newtonian space defines no limits to space or physical quantities;

- the velocity of an object increases linearly under the influence of a constant force

- the state of rest is inseparable from the state of rectilinear motion

On board of Galileo ship …


dm

dt

pF a

c velocity

time






dm

dt

pF a

2

0 0 0

½v v v

kin

d m dE d m d m v dv mv

dt dt

v xx v

c velocity

time

c velocity

kinetic energy






Relativistic space:

- increase of velocities are limited to the velocity of light which breaks the linear linkage between force to acceleration

From Galileo ship to space craft …

c velocity

time

c velocity

kinetic energy

c velocity

time 'v


Relativistic space:


c velocity


kinetic energy

dm

dt

pF a

2½kinE mv

Rectilinear 32

'1

d m

dt

pF a

2 2

2

11

1

kinE mc c mvc

c velocity

time 'v


c velocity

kinetic energy

2' 1dt dt 2' 1dr dr v c

Relativistic space:


Mathematical description (SR): redefinition of coordinate quantities:


dm

dt

pF a

2½kinE mv

Rectilinear 32

'1

d m

dt

pF a

2 2

2

11

1

kinE mc c mvc


Schwarzschild solution of GR field equations:

2

2

2ff Newton

GM

rc

2' 1dr dr

Gravitation in modified coordinates … via equivalence principle

ct’

dj

M

dsr'

dsj

r'



2

2

2ff Newton

GM

rc

2

2' 1

GMdr dr

rc 2' 1dr dr

ct’

dj

M

dsr'

dsj

r'




2

2

2ff Newton

GM

rc

22

2' 1

GMdt dt

rc 2' 1dt dt

2

2' 1

GMdr dr

rc 2' 1dr dr

ct’

dj

M

dsr'

dsj

r'


bff=vff/c



The (coordinate) velocity of free fall in Schwarzschild space goes

to zero at the critical radius 0

0.5

1

0 10 20 30 40 2

GMr

c

,ff Schwarzschild

ff Newton

2

2

2ff Newton

GM

rc

2

2c Schwd

GMr

c

22

2' 1

GMdt dt

rc 2' 1dt dt

2

2' 1

GMdr dr

rc 2' 1dr dr

ct’

dj

M

dsr'

dsj

r'


bff=vff/c


Hypothetical homogeneous space

The system of nested energy frames

local galaxy group frame

Milky Way frameSolar frame

2 20 0

0

1 1n

rest i ii

E m c

Earth frame

“Relativity of rest energy”



The system of nested energy frames

Milky Way frameSolar frame

Earth frame

20,0

0

1 1n

local i ii

f f

local galaxy group frame

“Relativity of characteristic frequencies”


Satellite in Earth gravitational frame

2 2 40 0

2,0 ,0

11 1

2 281 1 1DUf f f

2 2 4

0 02

,0 ,0

11

2 8 2

11 2 1GRf f f


Satellite in Earth gravitational frame

2 2 40 0

2,0 ,0

11 1

2 281 1 1DUf f f

2 2 4

0 02

,0 ,0

11

2 8 2

11 2 1GRf f f

0

0.2

0.4

0.6

0,8

1

0 0.2 0.4 0.6 0.8

GR DU

2 = 1

f, /f


0 0kinE c c c m m c p2kinE c m

Local relativity: Global relativity (Dynamic Universe):

Distinctive characteristics of local and global relativity

by local momentum via insert ofmass equivalence

by gravitation via tilting of space

2restE mcRest energy

Kinetic energy

0

2 20 0 1 1

rest

i ii

E c mc

m c

Description of finiteness

2' 1dt dt

2' 1dr dr

Total energyVelocity in spaceWhat is finite ?

04

"0total

GME c

R p


Velocity of mass object on board

Jump to Galileo ship …

0 00

1 shipship

v

v

v v v v

Velocity of light on board ?

characterized as a velocity frame …

0 ship c c v




0 00

1 shipship

v

v

v v v v


Momentum of mass object on board 00

1 shipvm

v

p v p


characterized as a momentum frame …

0 ship c c v




0 00

1 shipship

v

v

v v v v



1 shipvm

v

p v p

Momentum of light on board

(as reduced by the Doppler-effect)

00

0 0 0

1 1ship shipv vhm

c c

p c c p0hm

p c c


00

hc h f

p


0 ship c c v




0 00

1 shipship

v

v

v v v v



1 shipvm

v

p v p



00

0 0 0

1 1ship shipv vhm

c c

p c c p

0 0

0 0

1

1c c

T T T

0hm

p c c


00

hc h f

p


The observed

phase velocity is conserved

0 ship c c v



1 shipvm

v

p v p



00

0 0 0

1 1ship shipv vhm

c c

p c c p

0 0

0 0

1

1c c

T T T

0hm

p c ccharacterized as a momentum frame …

00

hc h f

p

Phase velocity, group velocity, observed frequencey


0 0

0 0

1

1c c

T T T


Mk

A(k+3)

A(k+2)

A(k+1)

B(k+1)

B(k+1)A(k+2)

A(k+1)

0

0 1

ˆ

1

ABAB t

ABA t B t m

j Bj k

T Tc

r

r r


0 0

0 0

1

1c c

T T T


Mk

A(k+3)

A(k+2)

A(k+1)

B(k+1)

B(k+1)A(k+2)

A(k+1)

0

0 1

ˆ

1

ABAB t

ABA t B t m

j Bj k

T Tc

r

r r

Propagation and observation of electromagnetic radiation

B

B

0 1 AAc c C

C

A

A

0 1 BBc c 0 1 CCc c

2

, 1 AA Af f

21B BBf f

, 1A A A rβ

, 1

B A

B AB

rβ

,1 BB A B A

f f rβ

,A

, , ,A A A

A

c f

c

, , ,

, 1 1

A A A A

AB A C AA A A A

c f ff f f

r rβ β

, 21

A

A

A

B

AB AA

c

c

,B Bc c

, 1

C A

C AC

rβ

C

AC AA

c

c

2

, 1 CC Cf f

,1 CC A C A

f f rβ

,C Cc c

0A t

1C t

Propagation and observation of electromagnetic radiation

C

C

A

A

00

1n

iBi

c c

,A

1

00

0

1

1

C t

A C nA t

i ri

T drc

0A t 1C t

,C A

r

1

00

0

1

1

C t

A C nA t

i ri

T drc

1/ 32 / 3

2 / 3

4

1/ 61/ 3

1/ 31 24

3 2

3 2

g

g

GI Mt

R

GI Mt

R

00

"GMc

R


0 0

0 0

1

1c c

T T T


c0

B

A

0

0 1

ˆ

1

ABAB t

ABA t B t m

j Bj k

T Tc

r

r r

0 1A Ac c

CB

A

Frequency observed

BA

0 1B Bc c 0 1C Cc c


Propagation and observation of light

Milky Way frame

Earth


00

1n

local ii

c c

Bending of light path- D is increased due to local bending of space- DT is increased (Shapiro delay)- c is reduced- l is reduced- f = c/l is conserved- p = h0 f is conserved- redshift of wavelength D l /l = DD /D

2

1

2

1

1 1 1

11 1 1

n

Bj Bj jBj kB

received A B m

Ai Ai iAi k

ff f

z

r

r

Af

Bf




0 00

1 shipship

v

v

v v v v



1 shipvm

v

p v p



0hm

p c c


00

hc h f

p


Michelson - Morley interferometer: no change in momentum in different arms – zero result guaranteed

0 ship c c v

00

0 0 0

1 1ship shipv vhm

c c

p c c p


Local relativity: Global relativity:

Distinctive characteristics of local and global relativity

2 22 2E mc pc Energy –momentum four-vector

2 2 22 2 20 0 0

2 22

0 0 0Im Re

2 22

0 0 0Im Re

m totE c c mc c p

k c k c k c

k k k

p

Planck constant h 3 20 01.1049 2h e c h c

energy

momentum

mass

wave number 2 2 2

Im Rek k k

0 kg mh

Quantum of radiation

E h 00 0 0 0

hE c c c k c c m c


Mass object as a closed energy structure (resonator)

0 0internal kp c

Internal (rest) momentum



0 0internal kp c

Internal (rest) momentum

Re 0 0 Re½ k p c

Re 0 0 Re½ k p c

Back wave

Front wave



0 0internal kp c

Internal (rest) momentum External momentum

Re 0 0 Re½ k p c

Re 0 0 Re½ k p c

Re Re 0 p p p

Back wave

Front wave



20 0 1internal k p c


Re 0 0 Re½ k p c

Re 0 0 Re½ k p c

Re Re 0 p p p

Back wave

Front wave





2

Re 0 0 Re

1½

1k

p c

2

Re 0 0 Re

1½

1k

p c

0 0 0Re Re 2 21 1

k m

p p p v v

v

Back wave- Doppler shift

Front wave+Doppler shift





v

0 0 0Re Re 2 21 1

k m

p p p v v

Unified expression of energy

2 2 00 0 0

hE N h N cc c c m c

p

Coulomb energy

A cycle of radiation

The rest energy of matter

B

q2q1

r

ic

FEM

ic

21 2 0 0c 0 0 0 c4 2

q q hE c c N c c c m c

r r

00 0rest

m

hE c c c mc

20 02h e c h c

0 kgmhIntrinsic Planck constant


0mE c p

2

0

3

2

1 1

2 1.1049 2 137.036

e

h c


2

0 0 Im 0Im 1 ComptonI k k p c

2

Re 0 0 Re

1½

1k

p c

Im

20 0 Re ReRe ½ 1 0I k p c c

2

Re 0 0 Re

1½

1k

p c


Re –

Re +

c

Re Re 0 0 Re 021k k

p p p c c c

Mass object as a closed energy structure


2

0 0 Im 0Im 1 ComptonI k k p c

2

Re 0 0 Re

1½

1k

p c

Im

20 0 Re ReRe ½ 1 0I k p c c

2

Re 0 0 Re

1½

1k

p c


Re –

Re +

c

Re Re 0 0 Re 021k k

p p p c c c

The double slit experiment

0 0external k k p c v


Mass object as a closed energy structure

Absorption pattern


Signal transmission from Earth satellite

0 0 Lv TL L LLT

c c c c

0

1 L

TT

v c

Sagnac delay

Documents

Imperial College, LONDON, 12-15 SEPTEMBER, 2008 From local to global relativity Tuomo Suntola, Finland Physical Interpretations of Relativity Theory XI