Modified Newton Law of Gravitation

Newton's inverse square law of gravitation was by Laplace formulated as the Poisson equation

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

where $\phi (x,t)$ is gravitational potential, $\rho (x,t)$ is mass density,  $x$ is a Euclidean space coordinate, $t$ a time coordinate and $\Delta$ is the Laplacian differential operator with respect to $x$. The gravitational force is given by the potential gradient $-\nabla\phi (x,t)$. The trajectory $x(t)$ of a test particle can be computed from Newton's 2nd Law

• $\dot v (x(t),t) =-\nabla\phi (x(t),t)\equiv f(x(t),t)$        (2)

where $\dot v=\frac{dx}{dt}$ is the velocity of the particle acted upon by the gravitational force $f(x(t),t)$. This model describes the motion of a Universe subject to gravitation, and so represents a formidable achievement of mathematical physics. Test the model here.

Since the same time coordinate appears on both sides of (1), it appears that Newton's law of gravitation involves hard-to-explain instant action at distance and so invites to alternatives to (1) with finite speed of propagation of effects.              

One possibility is to relax the Poisson equation (1) into a wave equation

• $\epsilon\ddot\phi (x,t)-\Delta\phi (x,t) =-\rho (x,t)$,       (3)

supporting gravitational waves with finite speed of propagation. Another is relaxation into a heat equation

• $\epsilon\dot\phi (x,t) - \Delta\phi (x,t) =-\rho (x,t)$,        (4)

where $\epsilon$ is a small positive constant, with effectively finite speed of propagation (scaling with $\frac{1}{\epsilon}$). since only vanishingly small effects propagate with unlimited speed. 

With small $\epsilon$ solutions to (4) stay close to those of (2), while wave solutions of (3) in general do not. We therefore focus on the heat equation (4), which has received little attention in the literature. 

The relaxation in (4) corresponds to a delay of the action the effect of the gravitational force. The delay effect comes to expression in computing a particle trajectory $x(t)$ by Explicit Euler time-stepping with time step $dt$, where $x(t+dt)=x(t)+dx$ and $v(t+dt)=v(t)+dv$ are computed/predicted from $x(t)$ and $v(t)$ by Dumb Euler as position first:

• dx = v(t)*dt, 
• dv =f(x(t+dt),t)*dt, 

or Smart Euler as velocity first:

• dv = f(x(t),t)*dt, 
• dx = v(t+dt)*dt. 

Compare yourself Dumb Euler with Smart Euler and see a big difference in the delay effect. We see that in  Smart Euler velocity is updated from the force at old position, while position is updated from new velocity, and in Dumb Euler it is the other way around. 

The delay effect from replacing (2) by (4) thus comes to expression in Explicit Euler time stepping which in the form of Smart Euler is remarkably small. 

We thus find support to an idea of Modified Newton Gravitation according to the heat equation (4) with effectively a finite speed of propagation of gravitational effects, which is not critically depending on the relaxation parameter $\epsilon$. It is then natural to speculate about the possible physicality of Smart Euler with a delay effect from explicit time stepping not asking for instant action at distance. 

More posts on associated New Newtonian Gravitation with (1) updated according to (2) with Explicit Euler with effectively finite speed of propagation of gravitational effects/force/potential. Hopefully it can help to resurrect Newton's theory of gravitation and avoid the black hole of Einstein's General Relativity. 

PS We thus see a formal connection between temperature as measure of heat energy and gravitational potential as measure of gravitational energy with the connection:

• temperature $T$ $\Longleftrightarrow$  potential $\phi$ 
• heat flux $Q=-\nabla T$ $\Longleftrightarrow$ gravitational force $f=-\nabla\phi$
• heat sink $-F$ $\Longleftrightarrow$ mass density $\rho$
• heat capacity $\kappa$ $\Longleftrightarrow$ ?? $\epsilon$

both described by the same heat equation (4) expressing conservation of energy 

• $\kappa\dot T +\nabla\cdot Q = F$.

Newton's theory of gravitation is thus based on a principle of conservation of energy, which may be hard to dispute to motivate a need of Einstein's theory: It is unthinkable that Newton's inverse square law is incorrect, unless your thinking is comparable to Einstein's thinking...