1. 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

2. 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

3. 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

4. Four Dimensional Spacetime 3+1 dimension Metric tensor

5. Proper Time Definition Four Velocity

6. Minkowskian (Galilean) Approx.

7. Lorentz Transformation 1-dimension Formula 3-dimension Formula

8. Poincare Transformation A kind of Affine Transformation Parallel Shift + Lorentz Tr. + Rotation

9. Newtonian Approximation Newtonian (Negative) Gravitational Potential: f > 0

10. Time Dilatation Newtonian Approximation Lorentzian Dilatation Gravitational Dilatation

11. Wavelength Shift Phase: Gauge Invariant 2-nd Order Lorentzian Shift Gravitational (Red) Shift

12. Post-Galilean Approximation

13. PPN Formalism C.F. Will (1981) Parametrized Post-Newtonian (PPN) PPN Parameters: ( a =1, b, g, … ) a =1 Principle of Equivalence Principle of Coincidence for GTR Einstein ’ s GTR: b = g =1, others=0 b : Non-linearlity g : Space Curvature

14. 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)

15. Acceleration and Force Four Acceleration Absolute Derivative: D Proper Mass: m Four Force

16. Geodesic Equation Principle of Equivalence “ Gravitation is not Force ” Path of Freely-Falling Bodies = Geodesic Timelike Geodesic Equation

17. Christoffel ’ s Symbol Inverse Metric: Not a Tensor = Coordinate Dependent Can be zero at a single point Analog of Gravitational Acceleration

18. Eq. of Motion of Photon Photon Path = Null Geodesic Rewriting in 3D form Newtonian Gravitational Acceleration: a Easy Solution: Successive Approximation

19. Gravitational Deflection Grav. Field = Convex Lens Deflection Angle Up to 4 Images: Einstein-Ring, -Cross Brightening = Microlensing MACHO detection S Dq E P y

20. Gravitational Delay Shapiro Effect (I.I.Shapiro 1964) Planetary Radar Bombing Pulsar Timing Observation Solar System: Sun, Jupiter, Earth,... Binary Pulsar: Companion Intermediate Stars/Galaxies: MACHO,... S P E

21. Post-Newtonian Approx. Non-linear Scalar Potential: F … b Vector (=Gravito-Magnetic) Potential: g

22. Post-Newtonian Eq. of Motion

23. Dragging of Inertial Frame Fermi Transportation 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