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Crash Course of Relativistic Astrometry

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.

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Crash Course of Relativistic Astrometry

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

  20. P S E 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, ...

  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

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