LLR and gravito-magnetism M.Soffel, S.Klioner (TU Dresden) J.Müller and L.Biskupek (Uni Hannover)

1. LLR and gravito-magnetism M.Soffel, S.Klioner (TU Dresden) J.Müller and L.Biskupek (Uni Hannover)

3. EIH equations of motion and gravito-magnetism • The DSX-framework uses N+1 different coordinate systems • for the description of the gravitational N-body problem • One global system (ct, x) (BCRS) • (for the description of the overall motion) • One local system (cT,X) for each of the N bodies, co-moving with • one of them (e.g., the GCRS) • (for the description of the local physics)

4. The global metric The local metric: same form, but potentials W,W instead of w, w a i

5. Gauge transformations This form of the metric tensor is invariant under special gauge transformations of the kind or

6. DSX: derivation of the EIH- equations or, transforming into the global system:

7. DSX: derivation of the EIH- equations EIH.equation for body A: geodetic equation in extermal metric induced by w and w or, transforming into the global system: i ext ext

8. Introduction of ‚gauge-invariant‘ E and B fields

9. We can then write the EIH- equations of motion with PPN parameters and Δ = 2γ + 2 + α1/2 in the PPN framework

11. Δ-terms

12. 2. Gravito-magnetism and LLR • Two standpoints taken in the paper • Murphy‘s standpoint: one singles out a special reference • system (BCRS), where gravito-magnetism is studied • Fit • to LLR-data 2.

13. 2. Will‘s standpoint: LLI necessarily requires and the appearance of gravito-magnetism as in GRT A corresponding violation of LLI-invariance can be described with α1 (a preferred frame parameter) To be conservative: choose the BCRS as preferred frame instead of the cosmic rest frame Fit α1 to PPN-data = 0

14. 3. Determination of ηG and α1 from LLR-data The full data set of LLR normal points between 1970 and 2007 was used Data from 7 different sites on Earth to 4 retroreflector arrays on the lunar surface; LLR-analysis procedure described in: 1. 2. 3. 4.

15. Fitting for ηG (together with ~ 170 other parameters) formal 1σ-error

16. A lot of work was done to assess a realistic error for ηG e.g., various correlations were carefully studied (e.g., with orbital parameters, the Nordtvedt parameter) Realistic uncertainty: ηG = 1.5 x 10**(-3) For that value:

17. Fitting for α1 from LLR data with BCRS as preferred frame • Choosing the cosmic rest frame as the preferred one • might be natural; should be motivated further • We took the very conservative standpoint • and chose the BCRS as preferred frame formal error

18. A realistic error for α1 Investigations similar to the one for Δ gave: 4 x 10**(-3) First determination of α1 with BCRS as preferred frame!

19. To understand the complex of problems further we also followed a perturbation theoretical approach Lunar motion about the Earth

20. Various perturbations related with gravito-magnetism • 1. ‚Lorentz-contraction‘ terms • 2. Nonlinear gravity terms • 3. Motional effects • Geodetic precession is independent of gravito-magnetism:

21. RESULTS Semi-monthly terms Monthly terms

22. Semi-monthly terms Monthly terms Δ = 2γ + 2 + α /2 PPN: 1 In Einstein‘s theory of gravity all gravito-magnetic terms are exactly cancelled by corresponding gravito-electric terms already in the BCRS In the usual PPN framework only α1 – terms remain

23. CONCLUSIONS • Due to LLI-invariance gravito-magnetic terms appear • necessarily • In GRT the GM terms in lunar motion are nullified by GE terms • already in the BCRS • There are many possibilities for a corresponding test theory • We have pursued two standpoint (Murphy & Will) and fitted • corresponding parameters to LLR data • The derived upper limits confirm GRT