New View on Motion under Gravitation

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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.

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