Relativistic Aspects of SLR/GPS Geodesy · Solar system (including Earth-Moon system, space geodesy, satellite navigation) is a unique laboratory for testing GR as we have direct

Relativistic Aspects of SLR/LLR Geodesy

Alexander Karpik

Elena Mazurova

Sergei Kopeikin

October 28, 2014 19th International Workshop

on Laser Ranging (Annapolis, MD) 1



Acknowledgement The present work has been supported by the grant of the Russian Scientific Foundation 14-27-00068

http://grant.rscf.ru/prjcard?rid=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





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



http://en.wikipedia.org/wiki/Topological_space

http://en.wikipedia.org/wiki/Euclidean_space

October 28, 2014 6

From Galileo to Einstein: Gravitation is not a Scalar Field!



Building Blocks of Relativity

uuT

RgRG

R

gggg

gggg

gggg

gggg

gggg

g

ji

i

Tensor Energy -Stress Matter ofDensity

2

1 Tensor Einstein Operator sLaplace'

- Tensor Curvature Force Tidal

2

1 Connection Affine Force nalGravitatio

Tensor Metric FieldScalar

,,

,,,

33323130

23222120

13121110

03020100



Einstein’s Field Equations and Gauge Freedom

Tc

G

tc

g

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Solving Einstein’s Equations

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RL

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



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.


on Laser Ranging (Annapolis, MD)



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


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

E R H V E

v v N v

N V

3

3 3 3 3 3 3,

3 3 2 2,

2 2

2 2 11 2 2 3 4 2

2

1 2 3 3

2

CB

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

C

C

C

C

C

B

GMGM

R R

GM RR R R R R R R R R R R R

GMR R R R R R

R R



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



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)



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

2

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



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)



Documents

Relativistic Aspects of SLR/GPS Geodesy · Solar system (including Earth-Moon system, space geodesy, satellite navigation) is a unique laboratory for testing GR as we have direct