I used to wonder whether one could find a simple physical or geometric model to illustrate how the energy distribution within the universe could give rise to a local gravitational potential of the form sum(m/r), or more generally to a metric whose terms are functions of a similar sum. It recently struck me that there's a very simple model related to the concept of QM phase that naturally gives rise to a local quantity proportional to m/r, as follows:(adsbygoogle = window.adsbygoogle || []).push({});

Suppose that all objects emit spherical scalar complex waves, propagating at [itex]c[/itex], whose frequency is determined by the object's energy, and which multiply together, so that the phases add. If the frequency of object [itex]n[/itex] is [itex]\omega_n[/itex] and the effective distance to it is [itex]r_n[/itex], then the overall phase factor relative to the phase at time 0 is as follows:

[tex]

\Psi = exp(\Sigma \, i \omega_n(t - r_n/c))

[/tex]

Now consider some of these waves passing an observation point, and compare the phase at that central point with the phase at two points in a straight line on either side of the observation point. As the wave fronts are curved, the phase at the central point is slightly ahead of the phase on either side. The line between the points on either side can be rotated in any direction; if it lies along the line of propagation of the wave, then the average of the phase at those points is equal to the phase at the central observation point, but for any other direction the average is slightly behind. The amount by which the phase on either side lags behind the central phase for a given distance away from the centre is proportional to the frequency of the wave and to the curvature of the wave front (which is inversely proportional to the wave front radius).

Mathematically, this process is equivalent to taking the divergence of the gradient (that is, the Laplacian) of the original phase:

[tex]

\nabla^2 (\Sigma \, i \omega_n(t - r_n/c)) =

\Sigma \, \frac{-2 i \omega_n}{r_n c}

[/tex]

This quantity is derived entirely from the distribution of a hypothetical local physical scalar quantity (the "total scalar phase" for all the objects being considered), yet is proportional to the sum of m/r for the relevant objects and can be theoretically extended to include all sources in the universe.

Whether the above formula is just an illustrative toy model or whether it might have some relationship to the underlying physics behind GR is beyond my ability to tell at this point, and any speculation on such questions would presumably be likely to stray outside the scope of this forum. The formula does obviously hypothesize concepts which don't form any official part of GR, in that for example if this "total scalar phase" were physically real, it would mean that the effective total energy of the universe would have a well-defined local value as seen at any point (it would simply be the time derivative of the total scalar phase).

If this is a known model, I'd be very interested to hear more about it. A similar model also works for the electromagnetic potential (but in that case instead of a phase we have something like a rotation which can be in either sense).

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# Gravitational potential from QM-like phase

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