Consider the rubber-sheet analogy. If you place an iron ball on a rubber sheet, you will see the ball depress and curve the rubber sheet. If you roll the ball accross the sheet, you will see the sheet flatten out at the ball's previous position and the sheet will begin to curve at the ball's current position. Over a given distance, it takes time for the curve to flatten and reform. We can calculate the speed of this process by dividing the distance by the time. One might assume we can calculate the speed of gravity in an analogous manner. In fact, I had an email exchange with Sergei Kopeikin who claimed that during the Jovian Deflection Experiment he observed Jupiter's gravity fading and reforming as Jupiter moved through space. The speed of this process turned out to be the speed of light or close to it. He also informed me that the gravity was stronger at Jupiter's previous (retarded) position and weaker at Jupiter's current position due to the light-time delay.
That got me thinking. If Jupiter moved at light speed, there would be zero gravity at its instant position, since if takes time for the spacetime to curve and when it does, Jupiter has moved to its next position. Its maximum gravity would be at a previous position. This is true even if Jupiter moves much slower than light speed. And so ... Houstin, we have a problem: Newton and Einstein created equations that assume Jupiter's maximum gravity is located at Jupiter's current position, not its previous position. If we plug in zero (or minimum gravity) at Jupiter's current position, or, plug in maximum gravity where there is no Jupiter (mass Mj), the equations break down and become inequalities:
Additionally, the Jovian observations are inconsistent with observations of the solar system orbiting the center of the Milky Way galaxy. According to Ethan Siegel (see reference below), the sun and the planets orbit the galaxy's center on the same plane. This implies that the solar system's curved spacetime moves in sync with the sun and planets. If the sun were to get ahead of its gravity (like Jupiter), the planets would lag behind the sun and form what looks like a vortex.
There is also a physical and thought experiment that can verify whether or not curved spacetime lags behind a planet's motion: Imagine two sky divers (Alice and Bob) jumping from a jet. Alice holds a target and Bob holds and aims a paintball gun. Bob has perfect aim and takes aim and fires along the horizontal axis (x). The paintball accelerates and hits the bullseye. This would not be possible if Alice, Bob, the target and the paintball were not falling at the same rate. Here is a crude illustration of what has happened so far:
If the paintball's acceleration vector lagged behind, it would have followed the path of the dotted line. It didn't; it followed the path of Earth's spacetime curvature along with Alice and Bob. Not only does gravity pull down matter, it also pulls down acceleration vectors. Additionally, the fact the paintball hit the bullseye implies that Earth's curved spacetime vector, along with everything else, follows the sun's curved spacetime as the Earth orbits the sun. Imagine the sun's gravity pulling everything forward along the z axis:
From this experiment we can infer that the solar system orbits the galaxy center on the same plane because the solar system's curved spacetime vectors follow the galaxy's curved spacetime. Or, another way to put it, the solar system's gravity falls at the same rate as the solar system. This means where there is mass, there is gravity and vice versa. Gravity does not lag behind a mass's movement. This is consistent with gravity equations, but not consistent with the rubber-sheet analogy or the Jovian experiment.
OK, so the Jovian experiment is called into question. So what? Surely LIGO's discovery of gravitational waves clinches the notion that scientists have successfully measured the speed of gravity. There's even a nice quadrupole-moment equation that gives the strain or amplitude (h) of such waves:
As the black holes' orbits decay, gravitational waves carry away energy and momentum. Unfortunately, these quantities are conserved. Why is this unfortunate? It is the hope of many physicists that gravitons make up gravitational waves. It is believed that when gravitons interact with matter, this interaction will be indistinguishable from gravity, but gravity does not appear to conserve force and momentum. For example, if you consider falling objects at rest and the earth accelerating to them, the earth accelerates more if it gains mass and accelerates less if it loses mass. Or, consider the earth at rest and drop any two objects with different masses in a vacuum chamber and they will appear to have virtually the same velocity at any point in spacetime:
Notice at equations 6 and 6a there's squared momentums in the numerators and they are not conserved because the velocity c is constant. By contrast, if gravitational waves interact with masses M and M', we have the following:
Since the momentum is conserved, we can expect the strain h to change when the waves interact with different masses. If the strain is gravitational, it should not change at all. Gravitational waves behave somewhat like a Newtonian force. If there is just enough lost energy to move a feather, that energy will not move a mack truck. Gravity has no problem moving both the feather and the truck.
Additionally, any quadrupole-moment force can cause gravitational waves! Let me demonstrate. Take equation 4 and make some substitutions:
Equation 16 shows that any force with a quadrupole moment can cause gravitational waves. An example would be a rotating dumbbell powered by an electric motor. Perhaps such waves should be relabeled "vacuum waves." It is highly doubtful they are made up of gravitons. If they were, there would be a strong correlation between gravitational waves and the strength of gravity. Earth is the strongest source of gravity we experience; yet, its gravitational waves are nil. By contrast, the gravity we experience from black holes lightyears away is nil, but their gravitational waves are significant. It is also highly doubtful the speed of gravity was successfully measured. However, we can sate with confidence that vacuum waves propagate at or close to the speed of light.
References:
1. Ibison, Michael, Puthoff, Harold E., Little, Scott. The Speed of Gravity Revisited.
2. Kopeikin, Sergei, Fomalont, Edward B. 27 Mar 2006. Aberration and the Fundamental Speed of Gravity in the Jovian Deflection Experiment.
3. Flanagan, Eanna. Hughes, Scott A. 2005. The Basics of Gravitational Wave Theory. New Journal of Physics.
4. Carlip, S. Aberration and the Speed of Gravity. December 1999.
5. Van Flandern, T. 1999. The Speed of Gravity What the Experiments Say. Meta Research University of Maryland Physics Army Research Lab.
6. Siegel, Ethan. August 30, 2018. Our Motion Through Space Isn't A Vortex, But Something Far More Interesting. Forbes
7. Galileo's Leaning Tower of Pisa experiment. Wikipedia.
8. David Scott does the feather hammer experiment on the moon | Science News. Youtube.com
9. Tzortzakakis, Filippos, LIGO Analysis: Direct Detection of Gravitational Waves. Journal of Research Progress Vol. 1.
