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