Report

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 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, ultraprecise 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). October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 2 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 Astrometry in Space October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 4 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 Colloquium at the University of Mississippi, Oxford, USA 5 GAIA October 14, 2014 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. Colloquium at the University of Mississippi, Oxford, USA 6 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 Colloquium at the University of Mississippi, Oxford, USA 7 October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 8 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. Colloquium at the University of October 14, 2014 Mississippi, Oxford, USA 9 Mauna Kea Hawaii at the University of Mississippi, Oxford, USA October 14, 2014 Kitt Peak Arizona Owens Valley California Brewster Washington North Liberty Iowa Hancock New Hampshire Pie Town New Mexico Fort Davis Texas Los Alamos New Mexico St. Croix Virgin Islands 10 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 Colloquium at the University of Mississippi, Oxford, USA 11 Gravitational Field Model October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 12 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 1. Linearized general relativity g h 2. The harmonic gauge h 1 2 h 0 3. The gravity field equation (c = 1) 2 2 2 h 0 t October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 14 Retarded gravitational potentials h00 i 2 I ij ( s ) 2 I (s) ... i i j x r x x r 2M r ij 4 I (s) 2I (s) ... j r x r i h0 i 2I hij ij h00 ij I (s) M xP (s) October 14, 2014 ... r the retarded time: i (s) i str I ij i j ( s) M xP ( s) xP ( s) J Colloquium at the University of Mississippi, Oxford, USA ij (s) 15 Light-ray Propagation October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 16 The light-ray perturbation dK The light-ray geodesic The unperturbed equation of light ray The perturbed equation of light ray October 14, 2014 K dk dx 0 d 1 k 2 p ertu rb atio n u n p ertru rb ed n u ll vecto r h h h x x x d d K K d The wave vector decomposition The Christoffel symbols d 0 h Colloquium at the University of Mississippi, Oxford, USA k 1 2 x h k k 17 The unperturbed light-ray trajectory x N ( ) k i r October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA i d 2 i 2 18 Light-ray Deflection Angle October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 19 The light-ray deflection angle 2 d x d i 2 1 h k k 2 k k j i i j i d 1 i 1 i j p k h i k h 00 k k k h jp d 2 2 i j dx d i Sun M D Q i i M i i 4M i i 1 n i d j i D d Q i 4 I (s) 4I 2 jp d 3 n n (s) i j m m n n October 14, 2014 i i j j j j 4k I (s) n Time argument is the retarded time: s = t - r i d n n m m m m n m m n p i j p i j p i p j Gravitational field of a moving planet is localized on null cone and interacts with light with retardation. Colloquium at the University of Mississippi, Oxford, USA 20 The deflection equations and the central inverse mapping M 1 cos n D L d Q J 2 ( z n ) n ( z m ) m R 2 d 2 L 2 d 2 ( s n ) ( z n ) 2 2 ( s m ) n 2 ( s n )( s m ) m 2 ( z m ) n 2 ( z n )( z m ) m 2 R limb d limb 4 1 k v P October 14, 2014 MR Colloquium at the University of Mississippi, Oxford, USA 21 Snapshot deflection patterns Monopole Dipole Quadrupole October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 22 Dynamic deflection patterns Circle 2 r cos r 2M X0 March 21, 1988 Treuhaft & Lowe DSN JPL NASA October 14, 2014 Cardioid p 1 cos 2 pr L X0 September 8, 2002 Fomalont & Kopeikin VLBA+MPfRA Colloquium at the University of Mississippi, Oxford, USA Cayley’s sextic q L cos 3 3 cos r L q L 2 X 0 2 Not measured yet (SIM, SKA, Gaia, JASMINE, VERA?) 23 Gravitomagnetism and the speed of gravity October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 24 Gravitomagnetism GRAVITOMAGNETIC FIELD arises from moving masses just as a magnetic field arises from moving electric charges. g h c 2 h 00 2 Ai October 14, 2014 c 2 4 h0 i The metric tensor The gravitoelectric potential The leading term is U=GM/r. The gravitomagnetic potential The leading term is (v/c)U. Colloquium at the University of Mississippi, Oxford, USA 25 Two types of gravitomagnetic field Intrinsic (Lense-Thirring): caused by rotating currents of matter induced by angular momentum of the massive body October 14, 2014 Extrinsic (Lorentz-Einstein): caused by translational currents of matter induced by motion of massive bodies in space with respect to observer Colloquium at the University of Mississippi, Oxford, USA 26 Speed-of-gravity Parameterization of Gravitomagnetism Post-Newtonian parameter labels timedependent 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 c g c / ε Gravity Fields Gauge condition Hence, c cg The speed of gravity is “the speed of light” entering the gravity sector of the fundamental interactions. Einstein’s Field Equations October 14, 2014 ε Colloquium at the University of Mississippi, Oxford, USA 27 Gravitational Time Delay October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 28 Gravitational Time Delay October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 29 Extrinsic gravitomagnetic force on a test particle 2 v 4 1 2 2 v v dt c c dv ex trin sic Fg m 4 c ex trin sic Fg m F n o ise v A 2 2 v 1 1 v v v 3 2 A 4 2 2 c t c t c c c c 2 c t 4 A th ese term s van ish in th e field o f a ro tatin g m ass b ein g at rest M assive b o d y m u st m o ve w rt o b server to g en erate th e ex trin sic G M . H o w to m easu re it? USE PHOTONS ! c dk dt F o r p h o to n s v c k th at am p lifies th e P N term s d ep en d in g o n v/c = O (1 ) 2 4k k ex trin sic Fg m F n o ise "N ew to n ian " fo rce F ex trin sic gm 4k A 2 k 4 k c t p o st-N ew to n ian fo rce o f th e o rd er o f V / c October 14, 2014 k A 4 A c t 1 2c 2 2 t 2 p o st-N ew to n ian fo rce o f th e o rd er o f V Colloquium at the University of Mississippi, Oxford, USA 2 /c 2 30 Parameterized Time Delay Equation t1 t 0 1 c ( t1 , t 0 ) | x 1 x 0 | ( t1 , t 0 ) 1 2 t1 t0 x N (t ) x 0 c k (t t0 ) h ( , x ) t dt k k h ( t , x N ( t )) 1 d k k t0 x x N ( ) 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, 2014 Colloquium at the University of Mississippi, Oxford, USA 31 Gravitational Time Delay by a moving body h00 2G M hij | x z (t ) | photon: x 2 G M ij | x z (t ) | x N (t ) x 0 ck (t t0 ) GM 1 ( t1 , t 0 ) 2 3 1 k v ln c cg m assive bo dy: z ( t ) z 0 v ( t t 0 ) | x 1 z ( s1 ) | k x 1 z ( s1 ) | x z ( s ) | k x z ( s ) 0 0 0 0 v2 z ( s1 ) z ( t1 ) | x 1 z ( t1 ) | O 2 c cg g v h0 i v | x z ( t ) | c g 4G M v2 z ( s0 ) z (t0 ) | x 0 z (t0 ) | O 2 c cg g v Look like a retarded time s1 t1 1 cg October 14, 2014 | x 1 z ( t1 ) | s0 t0 Colloquium at the University of Mississippi, Oxford, USA 1 cg | x 0 z (t0 ) | 32 The idea of the speed-of-gravity experiment October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 33 The Minkowski diagram of the light-gravity field interaction Leonid observes. Kip’s world line Future gravity null cone Future gravity null cone Future gravity null cone Future gravity null cone Future gravity null cone Kip emits light October 14, 2014 Planet’s world line Colloquium at the University of Mississippi, Oxford, USA Leonid’s world line 34 The null cones for gravitational field and light Observer and planet are at rest October 14, 2014 Planet moves uniformly relative to observer Colloquium at the University of Mississippi, Oxford, USA 35 Jovian 2002 and Cronian 2009 experiments October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 36 The Jovian 2002 experiment Position of Jupiter taken from the JPL ephemerides Position of Jupiter determined from the gravitational deflection of light by Jupiter 10 microarcseconds = the width of a typical strand of a human hair from a distance of 650 miles!!! October 14, 2014 The retardation effect was measured with 20% of accuracy, thus, proving that the null cone for gravity and light coincides (Fomalont & Kopeikin 2003) Colloquium at the University of Mississippi, Oxford, USA 37 The speed-of-gravity experiment (2002) Edward B. Fomalont (observation, data processing) Sergei M. Kopeikin (theory, interpretation) October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA Albuquerque 2002 VLBA support: NRAO and MPIfR (Bonn) 38 Basic Interferometry (in one minute) October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 39 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 40 Interpreting the speed-of-gravity experiment 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 October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 41 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 Cassini Gravitomagnetic Experiment October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 43 Gravitomagnetic Field in the Cassini Experiment (Kopeikin et al., Phys. Lett. A, 2007) Gravitomagnetic Doppler shift due to the orbital motion of the Sun Bertotti-Iess-Tortora, Nature, 2004 1 (2.1 2.3) 10 5 However, the gravitomagnetic contribution was not analyzed October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 44 Gravitational time delay in the ODP code T h e lin earized w .r.t. v/c tim e d elay eq u atio n can b e re-fo rm u lated as fo llo w s ( K o p eikin arX iv:0 8 0 9 .3 4 3 3) C assin i-E arth R1 R 2 R1 2 GM 1 2 1 k v ln 3 c R R R c 2 12 1 R 1 x 1 z ( t1 ) R 2 x 2 z (t2 ) z ( t1 ) z 0 v ( t1 t 0 ) R1 2 = | R 1 R 2 | z ( t2 ) z 0 v ( t2 t0 ) N o tice th at velo city v o f th e lig h t-ray d eflectin g b o d y en ters th e arg u m en t o f th e lo g arith m in th e tim e d elay. October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 45 • Numerical Estimates for Cassini Doppler Shift The peak value of the Doppler shift is caused by orbital motion of Earth and reaches 6 10 . R.M.S. error of the measurements is 1 10 Doppler shift due to the orbital motion of Sun is 2 .9 10 The value of (-1) would be affected by the solar motion by the amount 1 . 2 10 if the gravitomagnetic deflection of light were not in accordance with GR 10 • • • 14 4 Conclusions 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 1 (2.1 2.3) 10 is based on the implicit assumption that the gravitomagnetic deflection of light agrees with GR, but this assumption was not tested. 5 October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 46 13 PROGRESS IN MEASUREMENTS OF THE GRAVITATIONAL BENDING OF RADIO WAVES USING THE VLBA E. Fomalont, S. Kopeikin, G. Lanyi, and J. Benson The Astrophysical Journal, 699, 1395 (2009) γ = 0.9998 ± 0.0003 October 2005 October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 47 Pioneer Anomaly: Local measurement of the Hubble constant? October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 48 October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 49 Heat recoil explanation of the Pioneer anomaly October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 50 Background metric 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. October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 51 Local Diffeomorphism We introduce the conformal time: where ≡ . It reduces the RW metric to the conformally-flat form: Now, we look for a local diffeomorphism reducing the RW metric to the Minkowski metric: which means October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 52 Special Conformal Transformation October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 53 Local Minkowski Coordinates Expand the scale factor, and substitute it to the local diffeomorphism . Compare with the Taylor expansion of the special conformal transformation w.r.t. vector . It yields Local Minkowski coordinates are defined by the special conformal transformation where t is the proper time measured by the Hubble observer. The Minkowski time coordinate is not the proper time except for the timelike world line = = . October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 54 Einstein’s principle of equivalence The Christoffel symbols are nil in the local Minkowski coordinates. According to EEP any test particle moves along a geodesic which are straight lines One can prove that = 0 on photon’s worldline (but remember that 0 is not a proper time of observer). We want to parameterize the geodesic with the proper time t measured by the observer along her/his worldline: October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 55 Motion of light in local coordinates EEP, applied to a conformal manifold, tells us that a freely-moving particle experiences a geometric (Finsler-type) force because for a particle moving with the velocity v 1 0 = + 2 2 2 In particular, equation of motion of photons in the local coordinates in cosmology Light (in local coordinates) moves non-uniformly! October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 56 Doppler shift 2 Emitter’s world line 2 1 1 Receiver’s world line 0 October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 57 Doppler shift Frequency of radio waves: Doppler shift: Light-ray trajectory: Observer’s proper time: October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 58 Time derivatives 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 October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 59 Doppler tracking experiment Doppler shift equation: predicts gravitational blue shift of frequency for static observers in cosmology: _ + Doppler shift for local (static) observers Integrated Doppler shift: Δ = 1 Doppler shift for distant quasars = − 1 =1 1 has the same sign and magnitude as the Pioneer anomaly. Pioneer anomaly may have a cosmological explanation! October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 60 Thank you! October 14, 2014 Colloquium at the University of Mississippi, Oxford, USA 61