Relativistic Aspects of SLR/LLR

Sergei Kopeikin

Elena Mazurova

Alexander Karpik

19th International Workshop October 28, 2014 1 on Laser Ranging (Annapolis, MD) Acknowledgement

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

19th International Workshop October 28, 2014 2 on Laser Ranging (Annapolis, MD) 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

19th International Workshop October 28, 2014 3 on Laser Ranging (Annapolis, MD) 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

19th International Workshop October 28, 2014 4 on Laser Ranging (Annapolis, MD) 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

19th International Workshop October 28, 2014 5 on Laser Ranging (Annapolis, MD) From Galileo to Einstein: Gravitation is not a Scalar Field!

October 28, 2014 6 Building Blocks of Relativity  g g g g   00 01 02 03   g10 g11 g12 g13  Scalar Field   Metric Tensor g    g g g g   20 21 22 23     g30 g31 g32 g33 

1 Gravitational Force    Affine Connection   g  g  g  g  i  2  ,  ,  ,

       Tidal Force i j  Curvature Tensor R    , -  ,      

1 Laplace's Operator   Einstein Tensor G  R  g R   2 

Density of Matter   Stress - Energy Tensor T  u u 

19th International Workshop October 28, 2014 7 on Laser Ranging (Annapolis, MD) Einstein’s Field Equations and Gauge Freedom

 1  8G   1   R  g R  T (R  g R)  0  T ;  0 2 c4 2 ; Four Bianchi identities indicates the existence of the gauge freedom in the choice of the metric tensor. More specifically, any four out of ten components of the metric tensor can be chosen arbitrary. It is equivalent to the choice of a specific class of coordinate systems (CS).  ( - gg ),  0 the harmonic gauge is one of the most convenient CS. It was recommended for use by GA of the IAU2000.    - gg  

 1  2  16G       T     2 2  4    c t  c

19th International Workshop October 28, 2014 8 on Laser Ranging (Annapolis, MD) Solving Einstein’s Equations

Small parameters:   (size of the body)/(distance between the bodies)  L / R   (speed of matter)/(speed of )  v / c   (gravitational radius of the body)/(size of the body)  GM / c2 L

Post - Newtonian Approximations (non - analytic expansion, elliptic equations) :   2  3  4  5  6  7  8    1    2    3    4    5    6    7   ln  8 ...

LAGEOS, LARES LLR LLR, GNSS, VLBI Post - Minkowskian Approximations (analytic expansion, hyperbolic equations) :   2  3    1   2   3 ...

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. 19th International Workshop October 28, 2014 9 on Laser Ranging (Annapolis, MD) 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

ww g()() x G w G ()()wO   2  xx  ,,   

Specific choice of coordinates is determined by the boundary conditions imposed on the metric tensor components. We - Einstein’s followers - distinguish between the global and local coordinates in the sense of applicability of the Einstein principle of equivalence. 19th International Workshop October 28, 2014 10 on Laser Ranging (Annapolis, MD) 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. 19th International Workshop October 28, 2014 11 on Laser Ranging (Annapolis, MD)

Global and Local Coordinates (IAU 2000 Resolutions)

푡, 푥 - barycentric coordinates

푢, 푤 - geocentric coordinates

푇, 푋 - observer’ coordinates

Nonlinear coordinate transformations

u  u(t, x) T T(,) u w wi  wi (t, x) Xii X(,) u w R L  rg

19th International Workshop October 28, 2014 12 on Laser Ranging (Annapolis, MD) The gauge freedom in SLR/LLR

푐4 휈퐵

Here 휈퐵 and 휆퐵 are constant coordinate parameters which choice defines the class of a barycentric coordinate system used in SLR/LLR data processing software

휈퐵 = 휆퐵 = 0 harmonic coordinates

휈퐵 = 0 ; 휆퐵 = 1 + 훾 Painlevé coordinates

19th International Workshop October 28, 2014 13 on Laser Ranging (Annapolis, MD) EIH equations of motion in the barycentric coordinates Kopeikin, PRL, 98, Issue 22, id. 229001 (2007); Kopeikin & Yi, CMDA, 108, 245-263 (2010)

ii2 2  2CCii 1  2  2 aB  E BC  v B H BC   v C H BC  CB cc

iGM C i BC i 1 i ERHVEBC 32 BC1  BC    BC  BC  RcBC  c Gravito-electric force Gravito-magnetic (orbital motion-induced) force

3 2   1  2  3vv22  1  2  6v v   3    N v  BCCCC B B C C2 BC C

2 GM B GM C  3C  NVBBCC   1  2  2 22B    C   RRBC BC 1 2  22   2   1   3  4  2 GM R3 DD D  C C   D BC 3 3 3 3 3 3 DBC , RRRRRRRRRRRRCD BC BD BC BC CD BC BD CD BD2 BD CD 12  33   GM RR C C DD   D BC BD  3 3 2 2 DBC ,  2RRRRRRCD BD BD BC CD BC  19th International Workshop October 28, 2014 14 on Laser Ranging (Annapolis, MD) (1 + γ) GMRRvv()() t t  BBBB2 2 1B 1  B 24  B c푐  R21B R B

풕ퟏ - time of photon’s emission at point 풙ퟏ

풕ퟐ - time of photon’s reception at point 풙ퟐ

 The gauge-fixing parameters B and  B 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. 19th International Workshop October 28, 2014 15 on Laser Ranging (Annapolis, MD) Toward a better SLR/LLR relativistic model 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) 19th International Workshop October 28, 2014 16 on Laser Ranging (Annapolis, MD) Radial (synodic) relativistic effects in the orbital motion of satellite/Moon GM Schwarschild ~ 1 cm cm c2

R v Lense-Thirring R ~~ 2 .1/ 0.30.3 mm mm cc ~ 2.1/0.3 mm

2 −2 −4 GM R ~ 10 / 10 4 mm PN Quadrupole 2 J2 ~ 2 10 mm c r v v ~ from from a a few few meters meters Gauge-dependent terms r  ... c c down down to to a a few fe w cm mm

2 n푛푆 v v PNTidal Gravitomagnetic gravito-magnetic   r ~ ~0 .1 / a 1few mm mm n c c

2 푛 2 n 푆 v PNTidal Gravitoelectric gravito-electric r ~ a a fewfew cmcm nc 

2 n푛푆 GM  Non-linearity of gravity  2 ~ 0.10.1 mm nc October 28, 2014  17 Relativistic Geoid Kopeikin S., Manuscripta Geodaetica, vol. 16, 301 - 312 (1991) (theory in progress)

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, 휏, of an ideal clock carried out by static observers with fixed geodetic coordinates 푟, 휃, 휙, is constant.

19th International Workshop October 28, 2014 18 on Laser Ranging (Annapolis, MD) 19th International Workshop October 28, 2014 19 on Laser Ranging (Annapolis, MD)