Universality of Newton's Law of Gravitation

Newton's gravitational law can be formulated (in normalized form):

1. $F=\nabla\phi$,
2. $\nabla\cdot F =\rho$,

that is, eliminating $F$,

• $\Delta\phi=\rho$, 

where $F$ is gravitational force, $\phi$ gravitational potential and $\rho$ is mass density, all depending on an Euclidean space coordinate $x$ and on time $t$.

Here 1 expresses that the gravitational force $F$ is conservative in the sense that the work to move a mass particle subject to the gravitational force $F$ between any two points, is independent of path. Further, 2 expresses conservation of force flux under the presence of a source.

We note that 2 is fully analogous to Gauss' law in electrostatics:

• $\nabla\cdot E =\rho$, where $E$ is the electrical field and $\rho$ charge density. 

We now ask if Newton's law can serve as a universal law connecting the gravitational potential and force field to distribution of matter or mass density? In particular we ask if there is reason to consider instead of the static $\Delta\phi =\rho$, a wave equation of the form

• $- \ddot\phi + \Delta\phi =\rho$,  where $\ddot\phi = \frac{\partial^2\phi}{\partial t^2}$? 

If we were convinced that gravitational effects propagate with finite speed, because we believe in Einstein's relativity theory, then this would be the way to go.

We would then face the problem of giving a meaning to $- \ddot\phi$ and then also to $\frac{1}{2}\dot\phi^2$ with $\dot\phi =\frac{\partial\phi }{\partial t}$, as some new form of energy. We would further face a complete lack of experimental evidence that this wave equation describes physics better than the static equation. And we would find no help from electromagnetics, where nobody has found any reason to seek an alternative to the static Gauss law $\nabla\cdot E =\rho$. 

We are thus led to the conviction that Newton's gravitational law $\Delta\phi =\rho$ is universal in the sense that we cannot find any reason to consider some more elaborate form e.g. a wave equation extension, or Einstein's equations. And according to Ockham's razor, we should not make things more complex, unless there is a reason to do so.

But what about the speed of propagation of gravitational effects? Doesn't the static equation $\Delta\phi =\rho$ require that changes in $\rho$ instantly cause changes of $\phi$ globally over any distance thus with infinite speed of propagation? Yes, if we insist that this cause-effect relation is real.

But how do we know that this the way Nature works? No, we don't know that and it has been a mystery ever since Einstein suggested that there might be gravitational waves traveling across the Universe which we might record to learn about distant invisible regions. Despite ingenious instrumentation in the LIGO project, nothing was found.

If we thus free ourselves from a preconceived cause-effect relation with matter generating gravitational force through action at distance, we may instead write Newton's law $\rho =\Delta\phi$ viewing either $\rho$ to be "created" as $\Delta\phi$ through the local operation of differentiation of $\phi$ without interaction over distance, or as an expression of "perfect harmony" between $\phi$ and $\rho$ without any need to search for any cause-effect relation.

Leibniz would say that there is no sufficient reason to brake up such a "perfect marriage"at any cost.

A universal cosmological model in the spirit of Leibniz is explored in Computational Thermodynamics Chap 32 with extension to Newtonian Matter and Antimatter.

Compare with The Speed of Gravity - What Experiments Say.