Recall the previous post and the basic model of Neo-Newtonian Cosmology in terms of gravitational potential $\phi$ as primordial, mass density $\rho$, momentum $m$ and $u=\frac{m}{\rho}$ material velocity all depending on a Euclidean space coordinate $x$ and time coordinate $t$ with the dot on top representing differentiation with respect to time, as a model describing all of celestial mechanics:
- $\rho = \Delta\phi$ (inverse square law in differential form)
- $\dot\rho +\nabla\cdot m=0$ (conservation of mass)
- $\dot m +\nabla\cdot (um)+\rho\nabla\phi=0$ (conservation of momentum: Newton's 2nd Law)
We compare with the basic model of all of (inviscid) incompressible turbulent fluid flow as Euler's equations:
- $\nabla\cdot u=0$ (incompressibility)
- $\dot\rho +\nabla\cdot m=0$ (conservation of mass)
- $\dot m +\nabla\cdot (um)+\nabla p=0$ (conservation of momentum: Newton's 2nd Law)
where $p$ is pressure. Here incompressibility is imposed as a stipulation rather than law of physics, which can be realised through the physical law:
- $\Delta p = -C\nabla\cdot u$ (pressure law)
where $C$ is a large positive constant forcing $\nabla\cdot u$ to be small. We see that pressure $p$ and gravitational potential $\phi$ serve similar roles in Newton's 2nd Law steering the dynamics, and that both appear as solutions to a Poisson equation with infinite speed of propagation of effects from source to solution. We see that the pressure has the double role to drive the flow while securing incompressibility.
We thus see that both $p$ and $\phi$ act as omnipresent immaterial controls of dynamics with instant connection to a whole material universe, as solutions to Poisson's equation with source of material form, thus as connection immaterial-material or mind-body in the setting of consciousness in recent posts.
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