1. Spacetime astrometry and gravitational experiments in the solar system Sergei Kopeikin University of Missouri October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 1

2. Abstract Astrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The main goal of spacetime astrometry is to build the inertial coordinate system in the sky and to test general theory of relativity as well as other fundamental theories. Modern astrometry uses the sophisticated technologies and techniques including the satellites in deep space, ultra- precise atomic clocks, very long baseline interferometry (VLBI) and Doppler tracking. We overview the current astrometric space missions and discuss the theoretical principles of the gravitational experiments utilizing the light propagation through the gravitational field of the massive bodies in the solar system. We pay a special attention to the goals and results of the light-propagation experiments in time-dependent gravitational field of planets and Sun which were conducted in the last decade. We will also touch upon a possibility of the local measurement of the Hubble constant with spacecraft’s Doppler tracking without making a direct observation of cosmological objects (quasars, supernova).astronomystarscelestial bodies October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 2

3. Contents 1.Astrometric Experiments 2.Gravitational Field Model 3.Light-ray Propagation 4.Light-ray Deflection Angle 5.Gravitomagnetism and the speed of gravity 6.Gravitational Time Delay 7.The idea of the speed-of-gravity experiment 8.Jovian 2002 and Cronian 2009 experiments 9.Cassini gravitomagnetic experiment 10.“Pioneer anomaly” - Local measurement of the Hubble constant? October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 3

4. Astrometry in Space October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 4

5. 5 SIM SIM PlanetQuest has been designed as a space-based 9-m baseline optical Michelson interferometer operating in the visible waveband. This mission might open up many areas of astrophysics, via astrometry with unprecedented accuracy. Over a narrow field of view (1°), SIM aimed to achieve an accuracy of 1 µas in a single measurement! October 14, 2014

6. Colloquium at the University of Mississippi, Oxford, USA 6 GAIA Gaia: was launched in 2013. It scans the sky continuously according to a pre-defined pattern. The satellite rotates around its spin axis at a rate of 60 arcsec/s, equivalent to a spin period of 6 hours. The spin axis itself precesses at a fixed angle of 45 degrees to the Sun. The line of sight of the two astrometric instruments are separated by the 'basic angle', which is 106.5 degrees. Astrometric precision 10 μas. October 14, 2014

7. Colloquium at the University of Mississippi, Oxford, USA 7 JASMINE = Japan Astrometry Satellite Mission for INfrared Exploration. It will survey the Milky Way and its bulge in the infrared band around 1 milli-micron, measure positions, distances, and proper motion of several hundred million stars at high accuracy approaching 10 μas. Launch date: 2020÷24. October 14, 2014

8. Colloquium at the University of Mississippi, Oxford, USA 8

9. 9 Square Kilometer Array (SKA) The SKA will be an interferometric array of individual antenna stations, synthesizing an aperture with a diameter of up to several thousand kilometers. The SKA is a new generation radio telescope that will be 100 times as sensitive as the best present-day instruments. It will unlock information from the very early Universe and, using novel capabilities, be able to undertake entirely new classes of observation including VLBI with a micro-arcsecond resolution. October 14, 2014

10. Colloquium at the University of Mississippi, Oxford, USA 10

11. Colloquium at the University of Mississippi, Oxford, USA 11 VERA VLBI Exploration of Radio Astrometry is the first VLBI array dedicated to phase-referencing micro-arcsecond astrometry. S269 (Sharpless 269) is a massive star forming region toward constellation Orion. VERA has successfully measured its trigonometric parallax of 189 +/- 8 micro-arcsecond. This is the smallest parallax ever measured, corresponding to a source distance to 17,250 light year (~ 5.3 Kpc). October 14, 2014

12. Gravitational Field Model October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 12

13. Existing and incoming astrometric facilities demand new approach in theoretical understanding of light propagation through the variable gravitational fields generated by moving, oscillating, and rotating massive bodies as well as the field of gravitational waves. October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 13

14. Colloquium at the University of Mississippi, Oxford, USA 14 1.Linearized general relativity 2.The harmonic gauge 3.The gravity field equation (c = 1) October 14, 2014

15. Colloquium at the University of Mississippi, Oxford, USA 15 Retarded gravitational potentials the retarded time: October 14, 2014

16. Light-ray Propagation October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 16

17. Colloquium at the University of Mississippi, Oxford, USA 17 The light-ray perturbation The unperturbed equation of light ray The perturbed equation of light ray The Christoffel symbols The wave vector decomposition The light-ray geodesic October 14, 2014

18. Colloquium at the University of Mississippi, Oxford, USA 18 The unperturbed light-ray trajectory October 14, 2014

19. Light-ray Deflection Angle October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 19

20. Colloquium at the University of Mississippi, Oxford, USA 20 The light-ray deflection angle Time argument is the retarded time: s = t - r Gravitational field of a moving planet is localized on null cone and interacts with light with retardation. October 14, 2014

21. Colloquium at the University of Mississippi, Oxford, USA 21 The deflection equations and the central inverse mapping October 14, 2014

22. Colloquium at the University of Mississippi, Oxford, USA 22 Snapshot deflection patterns Monopole Dipole Quadrupole October 14, 2014

23. Colloquium at the University of Mississippi, Oxford, USA 23 Dynamic deflection patterns Circle Cardioid Cayley’s sextic March 21, 1988 Treuhaft & Lowe DSN JPL NASA September 8, 2002 Fomalont & Kopeikin VLBA+MPfRA Not measured yet (SIM, SKA, Gaia, JASMINE, VERA?) October 14, 2014

24. Gravitomagnetism and the speed of gravity October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 24

25. Colloquium at the University of Mississippi, Oxford, USA Gravitomagnetism GRAVITOMAGNETIC FIELD arises from moving masses just as a magnetic field arises from moving electric charges. The gravitoelectric potential The leading term is U=GM/r. The gravitomagnetic potential The leading term is (v/c)U. The metric tensor October 14, 201425

26. Two types of gravitomagnetic field Extrinsic (Lorentz-Einstein): caused by translational currents of matter induced by motion of massive bodies in space with respect to observer October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 26 Intrinsic (Lense-Thirring): caused by rotating currents of matter induced by angular momentum of the massive body

27. Post-Newtonian parameter  labels time- dependent gravitational effects and characterizes the speed of the respond of the gravitational field to the positional changes of a massive body. We call it the “speed of gravity” parameter Hence, The speed of gravity is “the speed of light” entering the gravity sector of the fundamental interactions. Colloquium at the University of Mississippi, Oxford, USA Speed-of-gravity Parameterization of Gravitomagnetism Gravity Fields Gauge condition Einstein’s Field Equations October 14, 201427

28. Gravitational Time Delay October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 28

29. Colloquium at the University of Mississippi, Oxford, USA Gravitational Time Delay October 14, 201429

30. Extrinsic gravitomagnetic force on a test particle October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 30

31. Colloquium at the University of Mississippi, Oxford, USA Parameterized Time Delay Equation Kopeikin S. (2004) Class. Quant. Grav., 21, 3251 Kopeikin S. (2006) Int. J. Mod. Phys. D, 15, 305 Kopeikin S. & Fomalont E. (2006) Found. Phys., No. 1, pp. 1 - 42 Kopeikin & Makarov (2007) Phys. Rev. D, 75, 062002 October 14, 201431

32. Gravitational Time Delay by a moving body October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 32 Look like a retarded time

33. The idea of the speed-of-gravity experiment October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 33

34. Colloquium at the University of Mississippi, Oxford, USA 34 The Minkowski diagram of the light-gravity field interaction Leonid observes. Leonid’s world line Kip’s world line Planet’s world line Future gravity null cone Light null cone Kip emits light October 14, 2014

35. Colloquium at the University of Mississippi, Oxford, USA 35 The null cones for gravitational field and light Observer and planet are at rest Planet moves uniformly relative to observer October 14, 2014

36. Jovian 2002 and Cronian 2009 experiments October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 36

37. Colloquium at the University of Mississippi, Oxford, USA 37 The Jovian 2002 experiment Position of Jupiter taken from the JPL ephemerides Position of Jupiter determined from the gravitational deflection of light by Jupiter The retardation effect was measured with 20% of accuracy, thus, proving that the null cone for gravity and light coincides (Fomalont & Kopeikin 2003) 10 microarcseconds = the width of a typical strand of a human hair from a distance of 650 miles!!! October 14, 2014

38. Edward B. Fomalont (observation, data processing) Sergei M. Kopeikin (theory, interpretation) The speed-of-gravity experiment (2002) VLBA support: NRAO and MPIfR (Bonn) October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 38 Albuquerque 2002

39. 39 Basic Interferometry (in one minute) October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA

40. 40 Limitations to Positional Accuracy Location of Radio Telescope Position on earth (1 cm) Earth Rotation and orientation (5 cm) Time synchronization (50 psec) Array stability (5 cm) Propagation in troposphere and ionosphere Very variable in time and space (5 cm in 10 min) CONVERSION FACTORS for astrometry: 1 cm = 30 psec = 300 microarcsec 0.03cm = 1 psec = 10 microarcsec Phase-referencing VLBI technique can achieve 10 microarcsec! October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA

41. Interpreting the speed-of-gravity experiment October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 41 Kopeikin & Fomalont - gravity sector of GR is compatible with SR speed of gravity = speed of light [  = 1 ] gravitomagnetic (velocity-induced) field of moving Jupiter 1.Will – aberration of light (radiowaves) from the quasar 2.Asada, Carlip – speed of light (radiowaves) from the quasar 3.Nordtvedt – retardation of radio waves from the quasar in Jovian’s magnetosphere 4.Pascual-Sanchez – the Römer delay of light (already known since 1676) 5.Samuel – retardation of radio waves emitted by Jupiter itself 6.Van Flandern – the quantity measured was already known to propagate at the speed of light

42. Light Deflection Experiment with Saturn and Cassini spacecraft as a calibrator (Proc. IAU Symp. 261, 2009) October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 42

43. Cassini Gravitomagnetic Experiment October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 43

44. Colloquium at the University of Mississippi, Oxford, USA 44 Gravitomagnetic Field in the Cassini Experiment (Kopeikin et al., Phys. Lett. A, 2007)Kopeikin et al., Phys. Lett. A, 2007) Gravitomagnetic Doppler shift due to the orbital motion of the Sun Bertotti-Iess-Tortora, Nature, 2004 However, the gravitomagnetic contribution was not analyzed October 14, 2014

45. Gravitational time delay in the ODP code October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 45

46. 1.Cassini solar conjunction experiment has a potential to detect the gravitomagnetic field of the moving Sun directly! 2.It requires re-processing of the data 3.The announced value for is based on the implicit assumption that the gravitomagnetic deflection of light agrees with GR, but this assumption was not tested. Colloquium at the University of Mississippi, Oxford, USA Numerical Estimates for Cassini Doppler Shift The peak value of the Doppler shift is caused by orbital motion of Earth and reaches. R.M.S. error of the measurements is Doppler shift due to the orbital motion of Sun is The value of (  -1) would be affected by the solar motion by the amount if the gravitomagnetic deflection of light were not in accordance with GR October 14, 201446 Conclusions

47. PROGRESS IN MEASUREMENTS OF THE GRAVITATIONAL BENDING OF RADIO WAVES USING THE VLBA October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 47 E. Fomalont, S. Kopeikin, G. Lanyi, and J. Benson The Astrophysical Journal, 699, 1395 (2009) γ = 0.9998 ± 0.0003 October 2005

48. Pioneer Anomaly: Local measurement of the Hubble constant? October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 48

49. October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 49

50. October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 50 Heat recoil explanation of the Pioneer anomaly

51. Background metric October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 51 Standard assumption of gravitational experimental physics is that spacetime is asymptotically flat where t is the proper time measured by static observers. In fact, we live in the expanding universe described on all scales by the Robertson-Walker metric where t is the proper time measured by the Hubble observers.

52. Local Diffeomorphism October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 52

53. Special Conformal Transformation October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 53

54. Local Minkowski Coordinates October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 54

55. Einstein’s principle of equivalence October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 55

56. Motion of light in local coordinates October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 56

57. Doppler shift October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 57 Emitter’s world line Receiver’s world line

58. Doppler shift October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 58 Frequency of radio waves: Doppler shift: Light-ray trajectory: Observer’s proper time:

59. Time derivatives October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 59 Relation of the proper time of moving clocks to the cosmic time: Light-ray path: Relation of the cosmic time at the point of emission to that at the point of observation

60. Doppler tracking experiment October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 60 Doppler shift equation: predicts gravitational blue shift of frequency for static observers in cosmology: Pioneer anomaly may have a cosmological explanation! + _ Doppler shift for distant quasarsDoppler shift for local (static) observers Integrated Doppler shift: has the same sign and magnitude as the Pioneer anomaly.

61. Thank you! October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 61