Gravitational deflection of light:  What else can the Eddington experiment tell us about gravity? Sergei Kopeikin Department of Physics and Astronomy, University of Missouri-Columbia 08/21/2017 The Great American Solar Eclipse Physics Conference The Nature of Time

The Great American Solar Eclipse Physics Conference Shakura Grishchuk Kopeikin Zeldovich Petrov Ya. B. Zeldovich and the department of relativistic astrophysics in Moscow State University (May 1986) 08/21/2017 The Great American Solar Eclipse Physics Conference

The Great American Solar Eclipse Physics Conference Abstract. Standard textbook treatment of the Eddington experiment is given in a static reference frame. However, the observations of the solar eclipse are done from the Earth which moves around the Sun. Effectively, it allows to test the Lorentz-invariance of the gravitational interaction during the solar eclipse. We discuss the mathematical framework of the Eddington experiment from the point of view of a moving observer and the results of testing of the Lorentz-invariance of the gravitational interaction in VLBI experiments with the major planets of the solar system. 08/21/2017 The Great American Solar Eclipse Physics Conference The Nature of Time

The Eddington experiment in a static frame 4-Jun-18 Blue - the stellar positions from astrometric catalogue Red - the stellar positions deflected by the solar gravity field Center of mass of the Sun measured from the gravitational deflection of light 08/21/2017 The Great American Solar Eclipse Physics Conference

What has been never discussed earlier? The Eddington experiment is conducted in a moving frame! The observed positions of stars are shifted in the sky by stellar aberration due to the orbital motion of the Earth. The optical position of Sun is shifted in the sky by stellar aberration as well. How much and in what direction in the sky are shifted the positions of stars having been deflected by the solar gravity? 08/21/2017 The Great American Solar Eclipse Physics Conference

The Eddington experiment in a moving frame 4-Jun-18 Blue - the stellar positions from astrometric catalogue Red - the stellar positions deflected by the solar gravity field Yellow – the shift of the stellar position due to the aberration of light Center of mass of the Sun measured from the gravitational deflection of light Where is the center of mass of the Sun measured from the gravitational deflection of light in a moving frame ? The answer depends on the speed of gravity. ? ? ? ? 08/21/2017 The Great American Solar Eclipse Physics Conference

The Laplace retardation of gravity effect Newtonian theory (Laplace 1825; Van Flandern 1998): Planetary orbits are unstable if the gravity force propagates with finite speed. The finite speed of gravity brings about the aberration of gravity that leads to non-conservation of the orbital angular momentum. It is not observed. The speed of gravity is infinite.     In general relativity: 08/21/2017 The Great American Solar Eclipse Physics Conference

Gravitational Position of the Sun in the Eddington experiment 4-Jun-18 Gravitational Position of the Sun in the Eddington experiment   Position of the quasar deflected by the gravity force and shifted by the aberration of light. The speed of gravity is infinite. Gravitational position of the Sun measured in the geocentric frame in case when the speed of gravity is infinite. Position of the quasar shifted by the aberration of light only. Gravitational deflection of light is switched off. Gravitational position of the Sun measured in the geocentric frame in case when the speed of gravity equals the speed of light. The aberration of gravity angle measured from the Earth

The Aberration of Gravity and its Fundamental Speed 4-Jun-18 The Aberration of Gravity and its Fundamental Speed The aberration of gravity effect is derived from solving the triangles shown in the picture   Gravitational deflection of light  =      The angular impact parameter of the light ray  = d/r r – the distance ‘lens-observer’ d – the impact parameter of the light ray  

Abstract of my APJL paper 08/21/2017 The Great American Solar Eclipse Physics Conference

Linearized General Relativity 4-Jun-18 The Harmonic (Lorentz) gauge Linearized Einstein’s Gravity Field Equations The Great American Solar Eclipse Physics Conference 08/21/2017

The Liénard-Wiechert Retarded Gravitational Potentials 4-Jun-18 The Liénard-Wiechert Retarded Gravitational Potentials 08/21/2017 The Great American Solar Eclipse Physics Conference

The Light-ray Trajectory 08/21/2017 The Great American Solar Eclipse Physics Conference

The Retardation of Gravity Experiment 4-Jun-18 The retarded time due to the finite speed of gravity Lorentz invariant equation for the bending of light Lorentz invariant equation for the electromagnetic phase delay 08/21/2017 The Great American Solar Eclipse Physics Conference

The Minkowski diagram of the interaction of gravity and light Observer Star’s world line Future gravity null cone Observer Time Future gravity null cone Future gravity null cone Space Future gravity null cone Light null cone Future gravity null cone Light null cone Observer’s world line Planet’s world line 08/21/2017 The Great American Solar Eclipse Physics Conference

The Great American Solar Eclipse Physics Conference The speed of gravity and retardation of gravity in the light-ray deflection experiment Observer and planet are at rest Planet moves uniformly relative to observer 08/21/2017 The Great American Solar Eclipse Physics Conference

The Great American Solar Eclipse Physics Conference The speed of gravity experiment (Fomalont & Kopeikin, Astrophys. J., 598, 704, 2003) Position of Jupiter taken from the JPL ephemerides (radio/optics) 1 5 Position of Jupiter as determined from the gravitational deflection of light from a quasar 4 2 3 10 microarcseconds = the width of a typical strand of human hair from a distance of 650 miles. 08/21/2017 The Great American Solar Eclipse Physics Conference

The Great American Solar Eclipse Physics Conference With Ed Fomalont in Albuquerque, NM, June 2002 Picture is taken during my visit to VLA, Socorro, NM 08/21/2017 The Great American Solar Eclipse Physics Conference

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Limitations to Positional Accuracy University of Missouri-Columbia 6/4/2018 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! 08/21/2017 The Great American Solar Eclipse Physics Conference Astrocon 2006, Princeton NJ

The Great American Solar Eclipse Physics Conference Gravitational bending of light by planetary multipoles and its measurement with microarcsecond astronomical interferometers (Kopeikin & Makarov, Phys. Rev. D75, Issue 6, id. 062002, March 15, 2007) Modeling Gravitational Field (Einstein’s theory) Modeling Propagation of Light (Maxwell’s theory) Solving Equations and Predicting the Light-Ray Deflection Patterns Discussing Fitting Parameters of the Model Interpreting Gravitational Physics 08/21/2017 The Great American Solar Eclipse Physics Conference

The deflection equations and the central inverse mapping Real ephemeris of the planet JPL ephemeris of the planet 08/21/2017 The Great American Solar Eclipse Physics Conference

Deflection patterns by gravitational multipoles Circle Cardioid the Caley’s sextic March 21, 1988 Treuhaft & Lowe DSN JPL NASA September 8, 2002 Fomalont & Kopeikin VLBA+Effelsberg 2020-2030 SKA, Gaia, Theia 08/21/2017 The Great American Solar Eclipse Physics Conference

The Great American Solar Eclipse Physics Conference Light-Deflection Experiment with Cassini and Saturn 2009 “Recent VLBA/VERA/IVS tests of general relativity” Fomalont, Kopeikin et al., IAU Symp. 261, 291-295, 2010 08/21/2017 The Great American Solar Eclipse Physics Conference

The Great American Solar Eclipse Physics Conference 08/21/2017 The Great American Solar Eclipse Physics Conference

The Great American Solar Eclipse Physics Conference 08/21/2017 The Great American Solar Eclipse Physics Conference

The Great American Solar Eclipse Physics Conference Thank You ! Happy Birthday to Thom! May the sky be clear! 08/21/2017 The Great American Solar Eclipse Physics Conference