More on New View of Motion under Gravitation

        Combined gravitational potential $\phi (x(t),t)$ for Earth--Moon system with Lagrange points.

Motion under gravitation is according to the New View described by the following set of equations in terms of the gravitational potential $\phi (x,t)$ and immaterial free fall trajectories $x(t)$;

1. $\Delta\dot\phi +\nabla\cdot (u\Delta\phi ) = 0$,                         (mass conservation)
2. $u(x(t),t)=\dot x(t)$,
3. $\ddot x(t) +\nabla\phi (x(t),t) = 0$,                          (Newton's 2nd Law)

where the mass density $\rho (x,t)$ does not appear, but is instead tied to $\phi (x,t)$ by the relation $\rho (x,t)=\Delta\phi (x,t)$ by differentiation as instant local action in space.  Notice that the immaterial trajectories cover space even where $\Delta\phi (x,t)=0$ expressing absence of matter. The differential equations 1-3 are complemented by initial conditions $\phi (x,0)$ and $x(0)$ and $\dot x(0)$.

To avoid local gravitational collapse from the tendency of gravitational motion to concentrate matter, a pressure force $-\nabla p$ with $p$ a pressure is needed to balance the gravitational force $\rho\nabla\phi$ locally. With such a pressure force present the basic case to consider is a two-body problem as the motion of two bodies 1 and 2 as solid spheres of mass $m_1$ and $m_2$ under gravitation. The corresponding gravitational potential can be expressed as a smoothed version of the potential of the corresponding point masses (in normalized form):

• $\phi (x,t) =\phi_1(x,t) +\phi_2(x,t)\equiv \frac{m_1}{\vert x - x_1(t)\vert} + \frac{m_2}{\vert x - x_2(t)\vert}$

where $x_1(t)$ is the position of the center of body 1 at time $t$ and $x_2(t)$ that of body 2. The net gravitational force $F_{12}(t)$ acting on body 1 will then be given by

• $F_{12}(t)=m_1\nabla\phi_2(x_1(t),t) = \frac{m_1m_2}{\vert x_1(t) - x_2(t)\vert^2}e_{12}$,

where $e_{12}$ is a unit vector in the direction $x_2(t) - x_1(t)$, because the the force from $\phi_1$ will be balanced by a local pressure. Likewise the net gravitational force $F_{21}$ acting on body 2 is given by

• $F_{21}(t)=m_2\nabla\phi_1(x_2(t),t) = \frac{m_1m_2}{\vert x_2(t) - x_1(t)\vert^2}e_{21}= -F_{12}(t)$.

We thus recover Newton's law of gravitation, which is no surprise of course, but the important aspect is that the full gravitational potential $\phi (x,t)$ is the sum of the gravitational potentials $\phi_1(x,t)$ and $\phi_2(x,t)$, both changing with time, and that the self-gravitational force with $\nabla\phi_1$ acting on body 1 is balanced by a local pressure and thus effectively is canceled. This means that the trajectory $x_1(t)$ is determined by Newton's 2nd law with $\phi =\phi_2$.

The whole mystery of motion under gravitation is now expressed in the equation $\Delta\dot\phi +\nabla\cdot (u\Delta\phi ) = 0$, which is no mystery since it expresses  mass conservation.