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The Potential and the Matter Density

The existence of potential means more than the conservation law. It opens the way to speculations about the structure of matter and the characterization of continuum from mechanical point of view. In order to see what this means, notice that we can write the Newtonian force (1) as

where Ñ is the gradient operator. Considering the force acting upon unit mass

it is a characteristic of the space around the mass exerting it, which came to be called the gravitational field. This force exists in each point of the space, regardless of the properties of this point: it can be simply a position in space or the location of a material particle. Therefore the force is a continuous function of the position in space and can very well be the characteristic of a continuum. If we calculate its flux through a sphere of radius r, according to the formula

where is the unit normal to sphere and dA is its elementary area, we get an interesting result. As the unit normal to the sphere is just the unit vector (the versor) of the position vector, and dA = r²sinθdθdφ we have

Therefore the flux of the gravitational force is, up to a universal constant, the mass of the material point creating the force. Now, the mass of the source of field can be written in the form of a triple integral involving the Newtonian density of matter

where ρ ( ) denotes the density of matter at the location and dV is the volume element of space at the very same location. Inserting equation (5) into (4) and using equation (1) gives

Now use the Gauss theorem for the left hand side of this equation, and we finally get

which means that the integrand is zero and so we have the Poisson equation for the potential

The potential is usually taken without the gravitational constant, which amounts to a simple redefinition of the potential: V₁ = GV. The Poisson equation then becomes

From the perspective of General Relativity this equation came to be taken as the fundamental equation of the Mechanics. It is usually considered not as such, but in conjunction with the idea that we know the density of the matter. Thus, it is usually an equation giving us the forces.

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