Geodetic VLBI

Lecture 2Lecture 2

18 October 2010

Lecture planLecture plan

1. From measurement of space and time to measurement of space-time

2. Elements of the Special and General Relativity

18 October 2010

Lecture planLecture plan

1. From measurement of space and time to measurement of space-time

2. Elements of the Special and General Relativity

18 October 2010

From measurement of space and time to measurement of space-time

3 dimensions 1 dimension

Space Time

3 & 1

18 October 2010

From measurement of space and time to measurement of space-time

3 dimensions 1 dimension

Space Time

x(t), y(t), z(t) – equations of motion

3 & 1

18 October 2010

SpaceTime

X

Z

X’

Z’

Y

Y’

t’

t

12

222

'''

''''

ttt

zyxr

12

222

ttt

zyxr

r=r’

2

1

V

18 October 2010

Galileo transformations

18 October 2010

dt

dxv

zz

yy

vtxx

'

'

'

Hendrik Lorentz

Lorentz transformation

18 October 2010

Hermann Minkowsky

Minkowski space or Minkowski spacetime

18 October 2010

Henri Poincare

Poincaré group of symmetry transformations

18 October 2010

Albert Einstein

"On the Electrodynamics of Moving Bodies”

1905

Main postulates and final formulae

18 October 2010

Lecture planLecture plan

1. From measurement of space and time to measurement of space-time

2. Elements of the Special and General Relativity

3. Exercises

18 October 2010

Postulates of Special relativity

• The Principle of Relativity – The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other

• The Principle of Invariant Light Speed – "... light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body."

18 October 2010

Newton theory vs special relativity1. The existence of infinitely

many inertial frames. Each frame is of infinite size (covers the entire universe). Any two frames are in relative uniform motion.

2. The inertial frames move in all possible relative uniform motion.

3. There is a universal, or absolute, time.

4. Two inertial frames are related by a Galilean transformation.

5. In all inertial frames, Newton's laws, and gravity, hold.

1. Same as the Newtonian assumption.

2. Rather than allowing all relative uniform motion, the relative velocity between two inertial frames is bounded above by the speed of light.

3. Instead of universal time, each inertial frame has its own time.

4. The Galilean transformations are replaced by Lorentz transformations.

5. In all inertial frames, all laws of physics are the same (this leads to the invariance of the speed of light).

18 October 2010

SpaceTime

X

Z

X’

Z’

Y

Y’

t’

t

12

222

'''

''''

ttt

zyxr

12

222

ttt

zyxr

r = r’ ?

2

1

V

18 October 2010

Special relativity

tt

zyxzyx

'

''' 222222

2222)( zyxtcs

22222 )( dzdydxcdtds

Minkowsky metric18 October 2010

Lorentz transformation2222)( zyxtcs

zz

yyc

v

vtxx

c

v

c

xvt

t

'

'

1

'

1

'

2

2

2

2

2

X

Z

X’

Z’

Y

Y’

2

1

V

18 October 2010

Aberration

• Star γ Dra – change of position (Bradley, 1727)

18 October 2010

Aberration, not parallax

Bradley was searching for parallax, but discovered aberration of light. Parallax was found only 100 years later.

Parallax depends on distance Aberration does not depend on distance

18 October 2010

Aberration of light

c – speed of lightA

V

sinc

VA

18 October 2010

s 's

sin

cossin

coscos

s

Aberration of light

18 October 2010

sin

cossin

coscos

s

VVsssVsc

Vssc

ss )()(22

1))((

1' 2

2

Conventional group delay model

))((1

1

)2

)(1)((

1)(

2

21

)(

2

222

2

2

2

wVsc

c

sVVb

cc

wV

c

V

c

U

c

sbgrav

c

sb )',(

VVsssVsc

Vssc

ss )()(22

1))((

1' 2

2

18 October 2010

Possible modification of the conventional group delay

model

))((1

1

)2

)(1))(((

1)(

2

21

)(

2

222

2

2

2

wtaVsc

c

sVtaVb

cc

wV

c

V

c

U

c

sbgrav

Constant acceleration

18 October 2010

General relativity

Geometric theory of gravitation

Einstein, 1916

18 October 2010

Contribution to fundamental science

Geoscience Australia

18 October 2010

Test of General Relativity (deflection of the

quasar position by the Sun or planet)

First experiment (Eddington, 1919) – Solar eclipse

First VLBI experiment (Schuh et al, 1988) –

Jupiter passage near quasar

VLBI experiment for “speed of gravity” (Kopeikin,

Fomalont, 2002)

Gravitational deflection (lensing)

18 October 2010

Gravitational delay for VLBI

)(

)(ln

2

22

113 srr

srr

c

GMgrav

1r

2r

18 October 2010

Gravitational delay for VLBI

)(

)(ln

2

22

113 srr

srr

c

GMgrav

For geodetic VLBI this is a difference of the Shapiro

delays for the both ends of baseline

Sun: ~40 nsec for 1º from Sun; ~400 psec for 180º

18 October 2010

Gravitational delay for VLBI

)(

)(ln

2

22

113 srr

srr

c

GMgrav

Non-zero for all radio sources irrelevant to their

closeness to Sun

For the Earth it is ~20 psec

18 October 2010

Conventional group delay model

))((1

1

)2

)(1)((

1)(

2

21

)(

2

222

2

2

2

wVsc

c

sVVb

cc

wV

c

V

c

U

c

sbgrav

c

sb )',(

18 October 2010

Lecture planLecture plan

1. From measurement of space and time to measurement of space-time

2. Elements of the Special and General Relativity

18 October 2010