Crash Course of Relativistic Astrometry

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)
• Einsteins 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
• Einsteins GTR
• Principle of General Relativity
• Principle of Equivalence
• Principle of Coincidence for GTR

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Four Dimensional Spacetime

• 31 dimension
• Metric tensor

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Proper Time

• Definition
• Four Velocity

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Minkowskian (Galilean) Approx.
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Lorentz Transformation

• 1-dimension Formula
• 3-dimension Formula

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Poincare Transformation

• A kind of Affine Transformation
• Parallel Shift Lorentz Tr. Rotation

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Newtonian Approximation

• Newtonian (Negative) Gravitational Potential f gt
  0

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Time Dilatation

• Newtonian Approximation
• Lorentzian Dilatation
• Gravitational Dilatation

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Wavelength Shift

• Phase Gauge Invariant
• 2-nd Order Lorentzian Shift
• Gravitational (Red) Shift

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Post-Galilean Approximation
13
PPN Formalism

• C.F. Will (1981)
• Parametrized Post-Newtonian (PPN)
• PPN Parameters (a1, b, g, )
• a1
• Principle of Equivalence
• Principle of Coincidence for GTR
• Einsteins GTR bg1, others0
• b Non-linearlity
• g Space Curvature

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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)

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Acceleration and Force

• Four Acceleration
• Absolute Derivative D
• Proper Mass m
• Four Force

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Geodesic Equation

• Principle of Equivalence
• Gravitation is not Force
• Path of Freely-Falling Bodies
• Geodesic
• Timelike Geodesic Equation

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Christoffels Symbol

• Inverse Metric
• Not a Tensor Coordinate Dependent
• Can be zero at a single point
• Analog of Gravitational Acceleration

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Eq. of Motion of Photon

• Photon Path Null Geodesic
• Rewriting in 3D form
• Newtonian Gravitational Acceleration a
• Easy Solution Successive Approximation

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Gravitational Deflection

• Grav. Field Convex Lens
• Deflection Angle
• Up to 4 Images Einstein-Ring, -Cross
• Brightening Microlensing
• MACHO detection

S
Dq
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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, ...

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Post-Newtonian Approx.

• Non-linear Scalar Potential F b
• Vector (Gravito-Magnetic) Potential g

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Post-Newtonian Eq. of Motion
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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