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An Unorthodox Reading: Force Determines the Density

Occasionally, however, the equation has been used the other way around, i.e. as an equation for density in a space filled with matter. A notorious example is that of Maxwell, who considered the Poisson’s equation as an equation for density, with the occasion of a construction of a system of tensions responsible for the field which creates the light and the gravitation (Williamson, 1894). This approach has nevertheless its shortcomings. Indeed, Let’s assume that we have a continuum characterized by central forces between its points, which we write in the general form

In order to reproduce the Poisson equation, which is a direct indication of the density of matter as given by the particle creating the field, we need to apply the divergence operator to this gradient. In a spherical coordinate system having as origin the material point that creates the field, this operation gives

where the prime indicates derivative with respect to r. Thus, according to the Poisson equation, the density of the continuum accepting as model this kind of material point is given by

The mass density should be always positive, so the magnitude of force should be decreasing with distance from the source faster than r^-2. Notice then that the density characterizing the forces of Newtonian type:

is always zero. This is a property that really distinguishes the Newtonian forces from among all the forces that can be obtained by a continuum theory: they are the only forces conferring to the structure of that continuum a zero Newtonian density. This was unacceptable.

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