Cosmology: The Illusion of Instant Action at Distance

Distribution of matter/mass in the Mira simulation of the Universe with 1 trillion particles.

Let us return to the New Newtonian Cosmological Model in the form of Euler's equations for a compressible gas subject to Newtonian gravitation: Find $(\phi ,m,e,p)$ depending on a Euclidean space coordinate $x$ and time $t$, such that for all $(x,t)$:

• $\vert\dot\rho\vert + \nabla\cdot m =0$                                                           (1)
• $\dot m +\nabla\cdot (mu) +\nabla p + \rho\nabla\phi =0$                             (2)
• $\dot e +\nabla\cdot (eu) +p\nabla\cdot u =0$,                                        (3)
• $\rho =\Delta\phi$                                                                           (4)

where $\phi$ is gravitational potential, $\rho$ is mass density, $m$ is momentum, $u=\frac{m}{\vert\rho\vert}$ is matter velocity, $e=\rho T$ is internal heat energy with $T$ temperature, $p=\gamma e$ is pressure with $0<\gamma <1$ a gas constant and the dot indicates time differentiation, cf. Computational Thermodynamics Chap 32. The equations (1)-(4) form a dynamical system as a system of partial differential equations depending on time, which evolves from one time instant to the next. 

Here $x$ is space coordinate in a fixed Euclidean coordinate system, and $t$ is a time coordinate as measured by the same standard clock for all $x$. We may view (1)-(4) to be a complete cosmological model including effects of gravitation and gas dynamics, but not electro-magnetics, radiation/light and atomic physics, as a continuum model without smallest scale, see picture above. 

The equations (1)-(4) express:

1. Conservation of mass/matter
2. Newton's 2nd Law.
3. Conservation of (internal) energy. 
4. Newton's Law of Gravitation connecting mass density to gravitational potential. 

The equations (1)-(4) are solved by time-stepping where (1)-(3) are used to update $\rho (t)$, $m(t)$ and $e(t)$ from time $t$ to $t+dt$ with $dt$ is a small time step, while (4) acts as a side condition, which can be relaxed into the following dynamical equation in $\phi$ with $\epsilon$ a vanishingly small positive constant:

• $\epsilon\dot\phi -\Delta\phi = -\rho$,                                                            (5)  

which can also be solved by time-stepping.  With this modification all equations can be solved by time-stepping with each equation of the form $\dot u=A(u)$ where $A(u)$ involves differentiation of $u(x,t)$ as a local operation in space. The time step $dt$ restriction for explicit time stepping of (5) has the form $dt <\epsilon dx^2$ with an effective speed of spreading of effects of $\frac{1}{\epsilon dx}$ with $dx$ a space step. 

While (4) as an equation for $\phi$ formally involves instant action at distance in the sense that the gravitational force from a mass at one location is felt instantly at all other locations, time stepping of the  modified equation only involves local action as differentiation effectively giving finite speed of propagation of effects.  

This connects to The World as Computation describing physics as time stepping of dynamical systems requiring only local instant action. 

The key is the relaxation of (4) into (5), which replaces a formal instant action at distance by a reality of instant local action. Even if it appears that the presence of mass at one location is instantly felt at any distance, it is not necessary to insist that this is a reality. In the equation (5), the relaxation term $\epsilon\dot\phi$ will remain vanishingly small, as long as nothing very dramatic is imposed, and so (4) will hold even without instant action at distance. 

This gives an alternative to the previous idea of viewing (4) in the form $\rho =\Delta\phi$ with $\rho (x)$ given by instant local action as differentiation of $\phi$ with thus $\phi$ primordial. By relaxation into (5) the role of primordial is relaxed into what Leibniz describes as pre-established harmony: Gravitational potential and mass together in full harmony evolve in time so as to satisfy (4), which is effectuated by time stepping of (5).

Let us now compare (5) with the wave equation 

• $\epsilon^2\ddot\phi -\Delta\phi = -\rho$,                                    (6)

which carries different physics in the form of gravitational waves spreading with speed $\frac{1}{\epsilon}$. 

We see that (5) and (6) are conceptually different with (5) acting like a relaxation with vanishing effect so as to give appearance of instant action at distance, while (6) is a wave equation with non-vanishing effect. 

Since gravitational waves have shown to be exceedingly difficult to detect, (5) appears to capture physics better than (6) as a representation of a Relaxed Universe. 

You can play with the above model in Leibniz World of Math Model Shop 21 Cosmology.