crash course of relativistic astrometry four dimensional spacetime poincare transformation time...
TRANSCRIPT
![Page 1: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/1.jpg)
Crash Course of Relativistic Astrometry
Four Dimensional SpacetimePoincare TransformationTime DilatationWavelength ShiftGravitational Deflection of LightGravitational Delay of LightPost-Newtonian Equation of MotionDragging of Inertial Frame
![Page 2: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/2.jpg)
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
![Page 3: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/3.jpg)
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
![Page 4: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/4.jpg)
Four Dimensional Spacetime
3+1 dimension
Metric tensor
0,1,2,3 x
3
0,
2
dxdxgds
ctx 0
![Page 5: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/5.jpg)
Proper Time
222 dsdc Definition
Four Velocity
d
dxu
![Page 6: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/6.jpg)
Minkowskian (Galilean) Approx.
1000
0100
0010
0001
g
I0
0THG
1
![Page 7: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/7.jpg)
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
![Page 8: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/8.jpg)
Poincare Transformation
A kind of Affine Transformation
Parallel Shift + Lorentz Tr. + Rotation
xPxxx Oˆˆˆ
R0
01RP LR
![Page 9: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/9.jpg)
Newtonian Approximation
I0
0TcG 2
21
Newtonian (Negative) Gravitational Potential: > 0
![Page 10: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/10.jpg)
Time Dilatation
Newtonian Approximation
Lorentzian DilatationGravitational Dilatation
2eff
2
21
2
11
c
v
cdt
d
![Page 11: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/11.jpg)
Wavelength Shift
Phase: Gauge Invariant
2-nd Order Lorentzian ShiftGravitational (Red) Shift
f
f0
![Page 12: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/12.jpg)
Post-Galilean Approximation
I12
2
2
21
c
cG
T
0
0
![Page 13: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/13.jpg)
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
![Page 14: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/14.jpg)
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)
![Page 15: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/15.jpg)
Acceleration and Force
Four Acceleration
Absolute Derivative: DProper Mass: mFour Force
uuΓ
d
du
d
Dua
maf
![Page 16: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/16.jpg)
Geodesic Equation
Principle of Equivalence “Gravitation is not Force”
Path of Freely-Falling Bodies = Geodesic
Timelike Geodesic Equation
0
uuΓ
d
dua
0f
![Page 17: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/17.jpg)
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
![Page 18: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/18.jpg)
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
![Page 19: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/19.jpg)
Gravitational Deflection
Grav. Field = Convex LensDeflection Angle
Up to 4 Images: Einstein-Ring, -CrossBrightening = Microlensing MACHO detection
2
tan1
2
SErc
S
E
P
![Page 20: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/20.jpg)
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
![Page 21: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/21.jpg)
Post-Newtonian Approx.
I123
342
2
221
cc
cc
Φ
cG
T
g
g
Non-linear Scalar Potential: … Vector (=Gravito-Magnetic) Potential: g
![Page 22: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/22.jpg)
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
![Page 23: Crash Course of Relativistic Astrometry Four Dimensional Spacetime Poincare Transformation Time Dilatation Wavelength Shift Gravitational Deflection of](https://reader035.vdocuments.net/reader035/viewer/2022062517/56649f005503460f94c15b54/html5/thumbnails/23.jpg)
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