Gravitation and Continuum Models

In the CNPS talk on Febr 3 I tried to expose the virtues of a continuum as a spatial 3d Euclidean x-coordinate system without smallest scale as the reference system of continuum mechanics in Eulerian form.  As a basic example let us consider the Euler equations for incompressible flow expressing balance of momentum (Newton's 2nd Law) combined with incompressibility in the form 

• $\nabla\cdot u = 0$        (1)

stating that divergence of velocity field $u(x,t)$ vanishes for all $x$ and time $t$. Here (1) appears as a stipulation or side condition for which the Lagrange multiplier is the pressure $p$, which appears as a pressure force $\nabla p$ in the momentum equation with connection through Gauss Law:

• $\int p\nabla\cdot u\, dx = -\int \nabla p\cdot u\, dx$.

The bottom line is that $\nabla p$ appears in the momentum equation as a force effectively imposing (1) while not specifying the physical nature of the force in a pressure law. The beauty is now that solving the Euler equations computationally gives full information about incompressible flow with vanishingly small viscosity, as shown in this book and this book. The divergence zero condition (1) is in computation replaced by an effective computational pressure law of the form 

• $-\Delta p = \frac{\nabla\cdot u}{\delta}$,     (2)  

where $\delta $ is a small parameter scaling with the mesh size, for which true physics is not needed. The Euler equations as a continuum model thus in computational form constructs a pressure law imposing near incompressibility. The continuum model in computational form thus invents physics which shows to describe reality in the form of physics as computation. 

We compare the continuum model with a particle model of a fluid asking for full specification of force between particles, and understand that a computational continuum model relieves us from a very difficult if not impossible task coming with a particle model. 

We now turn to Newtonian gravitation where the analog of (1) is Newton's Law of Gravitation in the form 

• $-\Delta \phi = \rho$       (3) 

connecting gravitational potential $\phi$ to mass density $\rho$ by the Laplacian differential operator $\Delta$. The corresponding Lagrange multiplier appears in the momentum equation as 

• $\rho\nabla\phi$                (4)

interpreted as gravitational force analogous to the pressure force connected to (1). Computationally (3) may take the following form allowing time-stepping:

• $\frac{\dot\phi}{C}-\Delta \phi = \rho$           
• $\frac{\ddot\phi}{C^2}-\Delta \phi = \rho$     (5)

where $C$ is a large constant representing effective speed of propagation, and the dot signifies differentiation with respect to time. Comparing computations with observation indicates that $C$ is much larger than the speed of light. 

Recall that it is well understood by everybody, except Einstein and his followers, that (3) expresses that (i) gravitational force $F$ is conservative, thus given by a potential $\phi$ as $F=\nabla\phi$,  and that (ii) $F$ is conserved in the sense of Gauss Law with $\nabla\cdot F = 0$ where there is no mass. To question (3) lacks rationale as it would violate (i) or (ii). In fact (3) is the prime jewel of all of physics, and to seek to modify it makes no sense. 

The beauty is here that the Euler equations augmented by gravitation in the form (3) and (4) (see this book) appears to describe a very rich world on a very wide range of scales, without having to specify the exact nature of the real physics of gravitation, which is still hidden, thus following the spirit of Newton.

The beauty is enhanced by realising that also quantum mechanics can be captured as a continuum model over a 3d Euclidean coordinate system without smallest scale allowing microscopics and macroscopics to have the same seamless conceptual form as shown in Real Quantum Mechanics.  This is shocking to modern physicists educated to view microscopics beyond comprehension for humans with only macroscopic experience.

Continuum models like the Euler equations thus appear as realisations of physics as computation expressing physics in possibly new forms open to understanding. 

PS1 The total energy for incompressible flow based on (2) includes a positive contribution of the form

• $\delta\int\vert \nabla p\vert^2dx$ 

and similarly total energy balance with gravitation in the form (4) contributes (with details here)

• $\int\vert\nabla\phi\vert^2dx$     

 as a natural expression of gravitational energy (as a source of kinetic energy) and in the form (5):

• $\int\vert\nabla\phi\vert^2dx+\frac{1}{C^2}\int\dot\phi^2dx$,

where the real physics of the second term with the time derivative $\dot\phi$ is less clear, and so may be interpreted rather as computational artefact allowing time-stepping. Recall that the presence of a time derivate in an energy expression represents kinetic energy from motion of matter, which is not an aspect of $\phi (x,t)$ expressing spatial presence of gravitational potential/force.   

PS2 Multiplying (3) by $\phi$ and integrating gives:

• $\int\vert\nabla\phi\vert^2dx = \int\rho\phi dx$           

where the right hand side commonly is referred to as gravitational potential energy. We see that the left hand side includes only the gravitational potential $\phi$, which connects to viewing $\phi$ as primary, as suggested in previous posts on New Newtonian gravitation. 

PS3 We may compare (3) with a law of the form 

• $\phi = \rho$

which expresses instant local action and  connects to gas law of (isothermal) compressible flow of the form $p=\rho$ with $p$ pressure, with $\nabla\phi$ corresponding to $-\nabla p$.