relativistic aspects of slr/gps geodesy · solar system (including earth-moon system, space...
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Relativistic Aspects of SLR/LLR Geodesy
Alexander Karpik
Elena Mazurova
Sergei Kopeikin
October 28, 2014 19th International Workshop
on Laser Ranging (Annapolis, MD) 1
October 28, 2014 19th International Workshop
on Laser Ranging (Annapolis, MD) 2
Acknowledgement The present work has been supported by the grant of the Russian Scientific Foundation 14-27-00068
Content:
• Introduction to relativity
• Solving Einstein’s equations
• Gauge freedom
• Global and local coordinates
• Gauge freedom of EIH equations
• Toward better SLR/LLR relativistic modelling
• Relativistic geoid
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on Laser Ranging (Annapolis, MD) 3
October 28, 2014 19th International Workshop
on Laser Ranging (Annapolis, MD) 4
Relativity for a Layman
Put your hand on a hot stove for a
minute, and it seems like an hour.
Sit with a pretty girl for an hour, and it
seems like a minute. That's relativity!
A. Einstein
Space-time Manifold
• A manifold is a topological space that resembles Euclidean space near each point.
• Although a manifold resembles Euclidean space near each point, globally it may not.
• Spacetime manifold in the solar system is not like Euclidean space.
• Conclusions
– Do not impose the Newtonian concepts in testing GR
– Be as much close to the Newtonian concepts as possible but not closer
October 28, 2014 19th International Workshop
on Laser Ranging (Annapolis, MD) 5
October 28, 2014 6
From Galileo to Einstein: Gravitation is not a Scalar Field!
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on Laser Ranging (Annapolis, MD) 7
Building Blocks of Relativity
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RgRG
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Tensor Energy -Stress Matter ofDensity
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- Tensor Curvature Force Tidal
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on Laser Ranging (Annapolis, MD) 8
Einstein’s Field Equations and Gauge Freedom
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October 28, 2014 19th International Workshop
on Laser Ranging (Annapolis, MD) 9
Solving Einstein’s Equations
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:equations) hyperbolic expansion, (analytic ionsApproximatn Minkowskia-Post
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Solar system (including Earth-Moon system, space geodesy, satellite navigation) is a unique laboratory for testing GR as we have direct access and can measure all geometric and relativistic parameters from a set of independent observations and space missions.
LAGEOS, LARES LLR LLR, GNSS, VLBI
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on Laser Ranging (Annapolis, MD) 10
The Residual Gauge Freedom and Coordinates
The gauge conditions simplify Einstein's equations
but the residual gauge freedom remains. It allows us
to perform the post-Newtonian coordinate transformations:
( )
( ) ( )
w x x
w wg x G w G
x x
2
, ,( ) ( )
Specific choice of coordinates is determined by the boundary
conditions imposed on the metric tensor components.
w O
We - Einstein’s followers - distinguish between the global and local coordinates in the sense of applicability of the Einstein principle of equivalence.
11
Why to introduce the local coordinates ? • Earth-satellite/Moon system is a binary system residing on a
curved space-time manifold of the solar system
• Motion of satellites are described in the most elegant way by the equation of deviation of geodesics in the presence of the (more strong) gravitational attraction of Earth.
• N-body equations of motion have enormous gauge freedom leading to the appearance of spurious, gauge-dependent forces having no direct physical meaning
• Introduction of the local coordinates is
– to remove all gauge modes,
– to construct and to match reference frames in the Earth-satellite/Moon system down to a millimeter precision,
– to ensure that the observed geophysical, geodetic and orbital parameters are physically meaningful and make sense.
October 28, 2014 19th International Workshop
on Laser Ranging (Annapolis, MD)
October 28, 2014 19th International Workshop
on Laser Ranging (Annapolis, MD) 12
Global and Local Coordinates (IAU 2000 Resolutions)
L R
gr
( , )
( , )i i
T T u w
X X u w
),(
),(
xtww
xtuu
ii
𝑡, 𝑥 - barycentric coordinates
𝑢, 𝑤 - geocentric coordinates
𝑇, 𝑋 - observer’ coordinates
Nonlinear coordinate transformations
The gauge freedom in SLR/LLR
13 October 28, 2014 19th International Workshop
on Laser Ranging (Annapolis, MD)
Here 𝜈𝐵 and 𝜆𝐵 are constant coordinate parameters which choice defines the class of a barycentric coordinate system used in SLR/LLR data processing software 𝜈𝐵 = 𝜆𝐵 = 0 harmonic coordinates 𝜈𝐵 = 0 ; 𝜆𝐵 = 1 + 𝛾 Painlevé coordinates
𝑐4 𝜈𝐵
14
EIH equations of motion in the barycentric coordinates Kopeikin, PRL, 98, Issue 22, id. 229001 (2007); Kopeikin & Yi, CMDA, 108, 245-263 (2010)
3 2
22 2
2
2 2 2 1 2 2
11
31 2 3 1 2 6 3
2
3 1 2 2
i ii i
B BC B BC C BC
C B
ii i iC BC
BC BC BC BC
C C
C C C
BC
BC
BC B B C C BC
B BC
C
C C
c c
GM
cR c
v v
a E v H v H
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v v N v
N V
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3 3 3 3 3 3,
3 3 2 2,
2 2
2 2 11 2 2 3 4 2
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BC BC
D BC
D B C CD BC BD BC BC CD BC BD CD BD BD CD
D BC BD
D B C CD B
DD D
D D
D BD BC CD B
C
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GMGM
R R
GM RR R R R R R R R R R R R
GMR R R R R R
R R
October 28, 2014 19th International Workshop
on Laser Ranging (Annapolis, MD)
Gravito-electric force Gravito-magnetic (orbital motion-induced) force
15
The gauge-fixing parameters and enters both the N-body equations of motion and the equations of light propagation. All together it makes the procedure of fitting the measured parameters to SLR/LLR data gauge-invariant.
B B
2 2 1 1
2
2 1
( ) ( )B B B B
B B
BB
B
GM t t
c R R
R Rv v
October 28, 2014 19th International Workshop
on Laser Ranging (Annapolis, MD)
𝒕𝟐 - time of photon’s reception at point 𝒙𝟐
𝒕𝟏 - time of photon’s emission at point 𝒙𝟏
(1 + γ)
𝑐4
LLR test of General Relativity is far from being completed as the currently employed data processing algorithm does not distinguish between the spurious coordinate-dependent forces and the true (curvature related) gravitational forces.
To separate the spurious forces, being dependent on the choice of coordinates, the relativistic theory of local frames must be employed (see the textbook by Kopeikin, Efroimsky, Kaplan “Relativistic Celestial Mechanics of the Solar System” Wiley, 2011)
There are other problems with the interpretation of the measurement of SEP and/or Gdot as we need a much more consistent theory of these violations (see “Frontiers in Relativistic Celestial Mechanics” ed. S. Kopeikin, De Gruyter, 2014)
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on Laser Ranging (Annapolis, MD) 16
Toward a better SLR/LLR relativistic model
Radial (synodic) relativistic effects in the orbital motion of satellite/Moon
2
2
4
22
Schwarschild 1 cm
Lense-Thirring 0.3 mm
PN Quadrupole
from a few meters Gauge-dependent terms ...
2 10 mm
GM
c
R vR
c c
GM RJ
vvr
c
c
c
r
2
2 2
2
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PN Gravitomagnetic a few mm
PN Gravitoelectric a fe
w cm
Non-linear
ity of gravit
down to a f w
y
e mm
n vvr
n c c
n vr
n c
n GM
n c
0.1 mm
~ 1 cm
~
~
~
~ a few cm
~ 0.1 mm
~ from a few meters down to a few cm
~ 2.1/0.3 mm
~ 10−2/ 10−4 mm
~ 2.1/0.3 mm
𝑛𝑆
𝑛𝑆
𝑛𝑆
Tidal gravito-magnetic
Tidal gravito-electric
October 28, 2014 17
~ 0.1 / 1 mm
Relativistic Geoid
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on Laser Ranging (Annapolis, MD) 18
Definition 1: The relativistic a-geoid represents a two-dimensional surface at any point of which the direction of plumb line measured by a static observer is orthogonal to the tangent plane of the geoid's surface. Definition 2: The relativistic p-geoid represents a two-dimensional level surface of a constant pressure of the rigidly rotating perfect fluid. Definition 3: The relativistic u-geoid represents a two-dimensional surface at any point of which the rate of the proper time, 𝜏, of an ideal clock carried out by static observers with fixed geodetic coordinates 𝑟, 𝜃, 𝜙, is constant.
Kopeikin S., Manuscripta Geodaetica, vol. 16, 301 - 312 (1991) (theory in progress)
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on Laser Ranging (Annapolis, MD) 19