Crash Course of Relativistic Astrometry
Four Dimensional SpacetimePoincare TransformationTime DilatationWavelength ShiftGravitational Deflection of LightGravitational Delay of LightPost-Newtonian Equation of MotionDragging of Inertial Frame
TheoriesSpecial 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) Formali
sm
PrinciplesSpecial 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
Metric tensor
0,1,2,3 x
3
0,
2
dxdxgds
ctx 0
Proper Time
222 dsdc Definition
Four Velocity
d
dxu
Minkowskian (Galilean) Approx.
1000
0100
0010
0001
g
I0
0THG
1
Lorentz Transformation
nnn
n
cosh sinh
sinhcosh
T
L
1-dimension Formula
3-dimension Formula
x
tc
x
tc
coshsinh
sinhcosh
ˆ
ˆ
c
v
v
vn
Poincare Transformation
A kind of Affine Transformation
Parallel Shift + Lorentz Tr. + Rotation
xPxxx Oˆˆˆ
R0
01RP LR
Newtonian Approximation
I0
0TcG 2
21
Newtonian (Negative) Gravitational Potential: > 0
Time Dilatation
Newtonian Approximation
Lorentzian DilatationGravitational Dilatation
2eff
2
21
2
11
c
v
cdt
d
Wavelength Shift
Phase: Gauge Invariant
2-nd Order Lorentzian ShiftGravitational (Red) Shift
f
f0
Post-Galilean Approximation
I12
2
2
21
c
cG
T
0
0
PPN FormalismC.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
GeodesicExtension of “Straight” LineForce-free pathTime-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
Absolute Derivative: DProper Mass: mFour Force
uuΓ
d
du
d
Dua
maf
Geodesic Equation
Principle of Equivalence “Gravitation is not Force”
Path of Freely-Falling Bodies = Geodesic
Timelike Geodesic Equation
0
uuΓ
d
dua
0f
Christoffel’s Symbol
x
g
x
g
x
ggΓ
2
1
Inverse Metric:Not a Tensor = Coordinate DependentCan be zero at a single pointAnalog of Gravitational Acceleration
gg
Eq. of Motion of PhotonPhoton Path = Null Geodesic
Rewriting in 3D form
Newtonian Gravitational Acceleration: aEasy Solution: Successive Approximation
0
kkΓ
d
dk
d
Dk
22
1
d
d
cct
vvaa0
v
Gravitational Deflection
Grav. Field = Convex LensDeflection Angle
Up to 4 Images: Einstein-Ring, -CrossBrightening = Microlensing MACHO detection
2
tan1
2
SErc
S
E
P
Gravitational DelayShapiro Effect (I.I.Shapiro 1964)
Planetary Radar BombingPulsar Timing ObservationSolar System: Sun, Jupiter, Earth, ...Binary Pulsar: CompanionIntermediate Stars/Galaxies: MACHO, ...
S
P
E
PESPSE
PESPSE
rrr
rrr
clog
12
Post-Newtonian Approx.
I123
342
2
221
cc
cc
Φ
cG
T
g
g
Non-linear Scalar Potential: … Vector (=Gravito-Magnetic) Potential: g
Post-Newtonian Eq. of Motion
JKJKJK
JJK
JK
JJKKJJ
KKL JL JL
L
KL
LJK
KJJKKJJKKJ JK
JKJK
B
r
rrA
r
vvr
arvrvvv
v
vvvrrrr
a
2122
,22
3121
122
, , ,
2
2
2
3
KJJ
JK
JKJKJKJK
JK
JK
K
r
BA
rcta
vra
v 43
1
d
d22
Dragging of Inertial Frame
Fermi Transportation Extension of “Parallel” Transportation
Locally Parallel Globally Non-RotatingNo Coriolis Force Rest to Quasars STR: Thomas PrecessionGTR: Geodesic Precession: ~2”/cyLense-Thirring Effect: rot g
3c
av