Only the Thomas terms are shown here.

	Section 1	Program listing with HTML links to library functions
	Section 2	Program listing without HTML links
	Section 3	Calculations

The later and more complete calculations shown at /som2 extend the retardation equations for charge to the solutions associated with the tensor of the fourth rank.

This section shows more detailed calculations for the static solutions of the field equations.

The gravitational potential solution in spherical coordinates is

Ar = -k^2 exp(-r/rr)/( r^2/rr) - k^2 exp(-r/rr)/r

k^2 is the source strength and rr is the range of the field. k^2 has the units of distance squared. This solution was obtained by first obtaining the static scalar solution with a computer algebra program, then, using its form as a guide, discovering the vector solution by a trial and error process. There are therefore no assurances that the solution is unique.

More recent and more complete calculations for this equation are shown at www.s-4.com/mass

The source strength can be represented as b rr instead of k^2 in all the gravitational equations. b has the same form as the Schwarzchild radius, and it represents the mass of the source, but the scaling relationships are different in non-metrical equations. As discussed in the paper, the scalar and vector potentials in these equations have the units of distance, but they are in a cosmological system of units rather than the MKS system. The gravitational MKS scaling relationships in the nearby region follow from Newtonian gravity. The MKS electrical scaling relationships follow from the equation for the E field of a charged particle.

The equation is analytically awkward to work with. Although inelegant, numerical substitutions are a simple and effective way of evaluating its behavior.

Ar= -9999.5 k^2/rr      // r=0.01 rr
Ar= -999999.5 k^2/rr      // r=.001 rr
Ar= -99999999.5 k^2/rr      // r=.0001 rr

An approximate solution is

Ar=-k^2 rr/r^2

Substituting numerical values

Ar= -10000 k^2/rr      // r=0.01 rr
Ar= -1000000 k^2/rr      // r=0.001 rr
Ar= -100000000 k^2/rr      // r=0.0001 rr

The approximate solution is clearly approaching the same limit as the exact equation. However, the derivatives of the approximate equation are totally wrong. The full equation has to be used for the differentiations.

div(A) is

k^2/(exp(r/rr) r rr)

exp(r/rr) approaches 1 when r << rr, so the approximate solution is easily obtained. The potential decays as 1/r^2 but the first derivative decays as 1/r. The behavior is not possible without exponential terms. The roles are reversed for the static scalar potential, with the potential decaying as 1/r and the first derivative decaying as 1/r^2.

The second derivative, grad(div(A)), is

ar=-k^2/(r^2 rr) -k^2/(r rr^2)

Substituting numbers

ar= -9999.5 k^2/rr^3      // r = 0.01 rr
ar= -999999.5 k^2/rr^3      // r = 0.001 rr
ar= -99999999.5 k^2/rr^3      // r = 0.0001 rr

An approximate solution when r << rr is

ar= -k^2/(rr r^2)

Numerically

ar= -10000 k^2/rr^3      // r = 0.01 rr
ar= -1000000 k^2/rr^3      // r = 0.001 rr
ar= -100000000 k^2/rr^3      // r = 0.0001 rr

The third derivative of the static gravitational solution is

div(grad(div(A))) = div(a) = k^2 exp(-r/rr)/(r rr^3)

a is the acceleration. div(a) is zero in Newtonian gravity.

The third derivative of the static scalar solution is

grad(div(grad(Ψ)))=-grad(div(E)) = -q exp(-r/rr)/(r^2 rr^2 + r rr^3)

The solutions show that the cosmological influence is not zero in the infinitesimal of the static solutions of the tensor of the fourth rank, although it is much too small to detect in the nearby region.

If you find technical or conceptual error in any of the material at this site then please email me the details of it.

Last update 22 Oct 2014      Revision History