A set of equations that reduce to the Maxwell potential equations is derived in Section 4 of a different section. Dropping the time-dependent terms, the static solutions become

div grad Ψ =
Ψ c_{1}/Λ^{2}

(2 + b/s) grad(div **A**)
- curl curl **A**
= **A** c_{2}/Λ^{2}

**A** is the vector potential; Ψ
is the scalar potential. These potentials have a meaning similar to that of
the retarded potentials of the Liénard-Wiechert
[3]
equations for charge, which in turn are the same as those of the
potential representation of the Maxwell equations.

b is the bulk modulus of the vacuum, s is its shear modulus (obviously
not the same as the 3-space value). If b were equal to -s then, by a
vector identity, the
left side of the vector equation would be
∇^{2}**A**,
resulting in a perfect mathematical symmetry between the scalar
and vector equations.

If c_{1} is taken to be +1 then the static
scalar equation is the same as that of the static Proca
and (unquantized) Klein-Gordon equations [2
,4]. The vector equation is different.

The calculations of this paper use a system of units in which the scalar and vector potentials both have the units of distance. In this system Λ also has the units of distance, and represents the range of the fields. Λ is a cosmological constant.

The vector solution represents a magnetized mass. By neglecting the
magnetic field curl **A**
becomes zero, and simplifying further by the substitution
D^{2} = Λ^{2} (2 + b/s)
the vector equation reduces to

grad(div **A**)
= **A** c_{2}/D^{2}.

The cosmological term was obtained from the Lagrangian of a cosmological potential energy term. The electrostatic force on a charged sphere is repulsive, causing the electrostatic potential energy to be positive. The gravitational force is attractive, causing the gravitational potential energy to be negative. The cosmological potential energy terms also appear to be of opposite signs. c2 will be taken as -1 in these calculations.

These solutions are obtained in a computer generated file. Assuming that c1 = +1, the scalar solution is

Ψ = c4 exp(-r/Λ)/r + c3 exp(r/Λ)/r

c3 and c4 are arbitrary constants. The solution is convergent if c3=0. When r << Λ the scalar potential is quite near the q/r form of the static Coulomb field.

Assuming c_{2} = -1, a particular spherically
symmetric vector solution is

**A**_{r} =
-d^{2} D exp(-r/D)/r^{2}
- d^{2} exp(-r/D)/r

d has the units of distance, and represents the source strength. The quantity is needed for dimensional consistency. The solution is convergent, and the first two terms of the series expansion are

**A**_{r} =
-d^{2} D/r^{2}
+ d^{2}/(2 D)

Now suppose the equation

**a** = 1/2 c^{2} grad(div **A**
+ 1/c ∂Ψ/∂t)

**a** is the acceleration of a test particle, as perceived
by an observer at a great distance from the particle. If the equations
are expressed in metrical 4-vector form
then the equation represents the gradient of the expansion factor.

Substituting from the solution for **A** and requiring
that the acceleration be Newtonian leads to the solution

A_{r} = G m/c^{2}

Ψ is zero in the static gravitational solution,
but it is not zero when the mass is in motion. The potential term that
decays as 1/r^{2} represents a quadrupole. The
quadrupole makes no contribution to the acceleration, so it
has been dropped. The sign and magnitude of the coefficients in some
of these relationships can be selected in more than one self-consistent
way. All the calculations are shown in the
computer generated file.
The expansion factor,
div **A** + 1/c ∂Ψ/∂t, is
2 G m/(c^{2} r).

Although developed in series form, the equation can be viewed as
representing an approximate gauge
transform of the exponential solution. The vector potential in the
exponential solution decays approximately as 1/r^{2},
which is physically very reasonable. The gauge transformed potential
does not decay with distance, but the potentials are not directly
measurable, and the acceleration field decays approximately by the
inverse square law in either gauge.

These relationships exhibit some unusual characteristics. First,
while the exponential solution for the vector potential varies
approximately as 1/r^{2},
its divergence, which represents the expansion
factor, varies as 1/r. Differentiating a 1/r^{2}
term normally leads to a 1/r^{3}
term, so differentiation acquires some of the
characteristics of integration in this case.
Secondly, in the limit of an infinite universe the exponential terms
would decay exactly as 1/r^{2} and the
gravitational field would vanish.

In being expressed in terms of the second derivatives, the
acceleration equation
appears at first to be of higher order than the equations for
the **E** and **B** fields, but it is actually in the same order. All three
fields depend on the retarded acceleration, and none depend on the
time derivative of the retarded acceleration. An unobvious cancellation
of terms in the acceleration equation places it in the same order as
the equations of the other fields.

It is not mathematically possible to transform Newtonian
gravity to another frame of reference.
The quantity [**A**, i Ψ/c] is a 4-vector.
A 4-vector transforms in the same way as the coordinates. The potential
at the field point must be taken as existing at the advanced time
in performing the transformation. The distinction does not matter
in the frame of reference of the source, but it does matter in the
other frame of reference. That is the essential difference between
the 4-potential representation and Newtonian gravity.