Crash Course of Relativistic
Astrometry
Four Dimensional Spacetime
Poincare Transformation
Time Dilatation
Wavelength Shift
Gravitational Deflection of Light
Gravitational Delay of Light
Post-Newtonian Equation of Motion
Dragging of Inertial Frame
Theories
Special Theory of Relativity (STR)
Einstein’s General Theory of Relativity
(GTR)
General Relativistic Theories



Brans-Dicke,Nordvegt,…
Scalar-Vector, Scalar-Tensor, …
Parametrized Post-Newtonian (PPN)
Formalism
Principles
Special Relativity



Principle of Special Relativity
Principle of Constant Speed of Light
Principle of Coincidence for STR
Einstein’s GTR



Principle of General Relativity
Principle of Equivalence
Principle of Coincidence for GTR
Four Dimensional Spacetime
3+1 dimension
x
  0,1,2,3

x  ct
0
Metric tensor
ds
2

3
g  dx dx


, 0


Proper Time
Definition
c d   ds
2
2
Four Velocity
2

dx
u 
d

Minkowskian (Galilean) Approx.
g   
 1

0

0

0

0
1
0
0
0
0
1
0
 1 0 

G  H  

I 
0
T
0

0

0


1
Lorentz Transformation
1-dimension Formula
 ctˆ   cosh 

  
 xˆ   sinh 
sinh   ct 
v

  
c
cosh   x 
3-dimension Formula
 cosh 
L  
 sinh   n
sinh   n 
v
n

cosh   n  n 
v
T
Poincare Transformation
A kind of Affine Transformation
ˆ
 

ˆ
x x x
O
ˆ

P x

Parallel Shift + Lorentz Tr. + Rotation
P  LR
1 0 

R  
0 R
Newtonian Approximation
Newtonian (Negative) Gravitational
Potential:  > 0
2

1 2

G
c

0


0 

I 
T
Time Dilatation
Newtonian Approximation
d
1
v
 1  2  
dt
c 
2
2
Lorentzian Dilatation
Gravitational Dilatation

eff
  1  2
c

Wavelength Shift
Phase: Gauge Invariant

f  
  0 



f

2-nd Order Lorentzian Shift
Gravitational (Red) Shift
Post-Galilean Approximation
2

 1 2
c
G
 0



0


 2  
1  2 I 
c  

T
PPN Formalism
C.F. Will (1981)
Parametrized Post-Newtonian (PPN)
PPN Parameters: (=1, , , …)
=1


Principle of Equivalence
Principle of Coincidence for GTR
Einstein’s GTR: ==1, others=0
: Non-linearlity
: Space Curvature
Geodesic
Extension of “Straight” Line
Force-free path
Time-like: Path of Mass Particle

Baryon, Lepton, …
Null: Path of Massless Particle

Photon, Graviton, …
Space-like: Space Coordinate Grid

Path of Virtual Particle (Tachyon)
Acceleration and Force
Four Acceleration


Du
du
  
a 

 Γ u u
d
d

Absolute Derivative: D
Proper Mass: m
Four Force


f  ma
Geodesic Equation
Principle of Equivalence

“Gravitation is not Force”
Path of Freely-Falling Bodies
= Geodesic
Timelike Geodesic Equation

f
du
  
a 
 Γ u u  0
d


0
Christoffel’s Symbol
Γ


1   g  g  g 
 g      
2
x
x
 x






Inverse Metric: g g   
Not a Tensor = Coordinate Dependent
Can be zero at a single point
Analog of Gravitational Acceleration
Eq. of Motion of Photon
Photon Path = Null Geodesic


Dk
dk
  

 Γ k k  0
d
d
Rewriting in 3D form
dv
1 
 0 2
dt
c
 a  v v 
a

2


c 
Newtonian Gravitational Acceleration: a
Easy Solution: Successive Approximation
P
Gravitational Deflection
Grav. Field = Convex Lens
Deflection Angle

1   

 
tan
2
c rSE
2
S


E
Up to 4 Images: Einstein-Ring, -Cross
Brightening = Microlensing

MACHO detection
Gravitational Delay
P
Shapiro Effect (I.I.Shapiro 1964)
 rSE  rSP  rPE 
1 

  2 log
c
 rSE  rSP  rPE 
S
Planetary Radar Bombing
Pulsar Timing Observation
Solar System: Sun, Jupiter, Earth, ...
Binary Pulsar: Companion
Intermediate Stars/Galaxies: MACHO, ...
E
Post-Newtonian Approx.

2 2Φ
 1  2  4
c
c

G

g

3
c


g

3
c

 2  
1  2 I 
c 

T
Non-linear Scalar Potential: F … 
Vector (=Gravito-Magnetic) Potential: g
Post-Newtonian Eq. of Motion
dv K
1
 aK  2
dt
c
  J   AJK rJK  BJK v JK

  
 3  4 a J 

2
rJK
J  K  rJK  

 J rJK
a K   3 , rJK  rJ  rK , v JK  v J  v K ,
J  K rJK
L
L
AJK  2   
 2   1
 v 2K
L  K rKL
L  J rJL
3  rJK  v J
 1   v  21   v J  v K  
2  rJK
2
J
BJK  rJK  2  2 v K  1  2 v J 
2
 rJK  a J
 
,
2

Dragging of Inertial Frame v  a
Fermi Transportation

c
3
Extension of “Parallel” Transportation
Locally Parallel  Globally Non-Rotating
No Coriolis Force  Rest to Quasars
STR: Thomas Precession
GTR: Geodesic Precession: ~2”/cy
Lense-Thirring Effect: rot g