Speed of Gravity? Newton or Einstein?

In Newtonian mechanics gravitational potential $\phi (x,t)$ is connected to mass density $\rho (x,t)$ by the Laplacian differential operator $\Delta$ through the equation 

• $\Delta\phi (x,t) = \rho (x,t)$                       (1)

where $x$ is a Euclidean coordinate and $t$ a time coordinate. 

The standard way is to view the gravitational potential $\phi (x,t)$ and gravitational force $\nabla\phi (x,t)$ at some space-time coordinate $(x,t)$ as somehow being generated from the distribution of mass density $\rho (y,t)$ for all $y$ different from $x$ in an apparent instant action at distance at time $t$ as if gravitational force is propagated with infinite speed.  Solving the differential equation in terms of $\phi$ is a global operation of integration/summation. 

But instant action at distance is inexplicable and so in modern physics Newton's mechanics has been replaced by Einstein's General Theory of Relativity GR, where gravitational force is propagated with the finite speed of light c. 

On the other hand, if we in (1) view the gravitational potential $\phi (x,t)$ as primary from which mass density $\rho (x,t)$ is generated by the action of the differential operator $\Delta$ as 

• $\rho (x,t) =\Delta\phi (x,t)$                      (2)

which is a local operation at $x$ of differentiation, which is possible to view to be instant. Mass density is here generated by instant local action from gravitational potential, and then the problem of instant action at distance evaporates and it is no longer necessary to replace Newton by Einstein. 

Newtonian mechanics describes celestial dynamics in the form an initial value problem 

• $\dot x(t) = f(x(t))$ for $t>0$ with $x(0)$ given  

where $x(t)$ represents positions of celestial bodies at time $t$, and $f(x)$ is a given function of $x$ including Newton's law of gravitation. The  differential equation can be solved by explicit time stepping of the form

• $x(t) = x(t-dt) + dt*f(x(t-dt))$                      

where $dt>0$ is a time step. The position $x(t)$ at time $t$ is thus computed from previous position $x(t-dt)$ with a correction $dt*f(x(t-dt))$ determined at the previous time $t-dt$, thus with a time delay of $dt$. We can view the time delay $dt$ as an expression of finite speed of propagation $C$, and we now ask if observations can give information about the size of $C$. 

We do this in the simplest case of one small body (Earth) orbiting a big body (Sun) as expressed in this code, where we can test the effect of different time steps $dt$. By normalisation we can connect $dt$ to $\frac{1}{C}$ as the time required for a gravitational signal from the Sun to reach the Earth. The effect on position $x(T)$ at time $T>>0$ of explicit time stepping with time step $dt$ at best scales with $T*dt$ and if we ask for a precision of $\epsilon$, we have 

• $dt < \frac{\epsilon}{T}$, that is $C>\frac{T}{\epsilon}.$                              

Relevant values may be $T>10^4$ and $\epsilon <10^{-4}$, that is 

• $C>10^8$.

If we put this number in the perspective of the Sun-Earth system with the speed of the Earth about $0.0001$ times the speed of light $c$, we get $C>10^{12}$ to be compared with $c=3\times 10^8$ meter/second with thus a factor at least $10^3$. We can compare with the estimate $10^7$ made by Laplace and even $10^{10}$ from the PS below.  

In any case, observations indicate that the required speed of propagation of gravitational effects in (1) is several orders of magnitude bigger than the speed of light.

A planetary system based on (1) with a time delay from finite speed of gravity equal to the speed of light would not persist over time. 

Newtonian mechanics describes celestial/planetary motion very accurately over long time with a from observations apparent speed of gravity much bigger than the speed of light. If (1) is viewed to express Newton's law of gravitation this essentially requires instant action at distance, which is unthinkable.

But changing view to (2) replaces instant action at distance by instant local action, which is thinkable.  

Since in GR the speed of gravity is finite, one would expect to see effects in GR of time delay, but that would contradict observations where no time delay can be detected. The situation is handled by claiming that  there is a very subtle strange effect of cancellation in GR, which means that in the end the effective speed of gravity is infinite. So GR says the the speed of gravity is finite equal to the speed of light, but the effect of finite speed is cancelled and so the net effect is zero as if the speed of gravity in fact is infinite as in Newtonian gravity. Do you buy this argument? 

Recall that Einstein when claiming that Newtonian gravitation must be replaced by GR, could not refrain from expressing "Newton, forgive me." as if he had committed a scientific crime.

It is also possible consider a potential-mass connection of the form  

• $\Delta\phi - \rho = 0$                          (3)

where the cause-effect is not indicated. Here (3) appears rather as a side condition expressing a balance of potential and mass without worrying about casual connection, see this book and this computation exploring the Euler equations for fluid flow with gravitation.

Sum up: There is no reason to replace Newton by Einstein, and anyway doing so leads to a quagmire of mysteries. It is not necessary to view mass as primary from which gravitational potential/force is formed by apparent instant action at distance as in (1), which is unthinkable. We may as well turn (1) around into (2) viewing gravitational potential to be primary from which mass is formed by instant local action, which is not unthinkable. 

PS1 Tom van Flandern states in The speed of gravity - What experiments say:

• Laboratory, solar system, and astrophysical experiments for the “speed of gravity” yield a lower limit of $2\times 10^{10}c$.
• But mediation requires propagation, and finite bodies should be incapable of propagating at infinite speeds since that would require infinite energy. So instantaneous gravity seemed to have an element of magic to it.
• We will examine the explanations offered by GR for these phenomena, and conclude that in the most widely taught curved space-time interpretation of GR the acceleration of bodies through space lacks a causal connection to the source of gravity. And we will confront the dilemma that remains when we are through: whether to modify our existing interpretation of GR, or give up the principle of causality.

It thus appears that GR assuming that the speed of gravity is finite equal to the speed of light, is incompatible with experiments showing an infinite speed of gravity, which asks for modification of GR.

This modification may simply be a return to Newtonian gravitation with a law of gravitation of the form (2) with instant local action, which is not in contradiction to causality.  

The reason that mass traditionally is viewed to be primary and potential a derived quantity as in (1), is (probably) that mass may be directly visible and gravitational potential/force is not. On the other hand, all bodies directly "feel" gravitational force, and so gravitational potential/force is very present although not directly visible.   

PS2 Further evidence of speed of gravity being much larger than the speed of light from observation of satellite motion, is given here.

PS3 Maxwell's wave equations for electromagnetics describe propagation of light of all frequencies at the same speed = speed of light = c. Augmented with an Abraham-Lorentz radiation force Maxwell's wave equations also describe radiative heat transfer as Computational Blackbody Radiation as a one-way transfer of energy from warm to cold mediated by two-way electromagnetic waves, more precisely as a resonance phenomenon. The speed of radiation thus in principle can be viewed to be equal to c, even if in reality effective transfer of energy from resonance may change at a slower speed. 

Finite speed of gravity = C requires augmentation of (1) into a wave equation $-\frac{\ddot\phi}{C^2}+\Delta\phi =\rho$ with the dot signifying differentiation in time, and transfer of energy by gravitational waves requires some form of Abraham-Lorentz force. Observations show that C is much bigger than c, and so both theory and observation supporting C=c, is lacking. 

While radiative wave energy transfer is a reality, there seems to be little evidence that gravitational wave energy transfer is real. The proclaimed experimental detection of very faint gravitational wave energy transfer from distant merging black holes suffers from the difficulty of finding a needle in a haystack.