1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

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

11. 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

12. 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

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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 08/21/2017

20. 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

21. 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 , 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

22. 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

23. 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
SKA, Gaia, Theia
08/21/2017
The Great American Solar Eclipse Physics Conference

24. 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, , 2010
08/21/2017
The Great American Solar Eclipse Physics Conference

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

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

27. 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