A 4-potential representation of Newtonian gravity

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 c12
(2 + b/s) grad(div A) - curl curl A = A c22

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 ∇2A, resulting in a perfect mathematical symmetry between the scalar and vector equations.

If c1 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 D2 = Λ2 (2 + b/s) the vector equation reduces to

grad(div A) = A c2/D2.

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 c2 = -1, a particular spherically symmetric vector solution is

Ar = -d2 D exp(-r/D)/r2 - d2 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

Ar = -d2 D/r2 + d2/(2 D)

Now suppose the equation

a = 1/2 c2 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

Ar = G m/c2

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/r2 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/(c2 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/r2, 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/r2, its divergence, which represents the expansion factor, varies as 1/r. Differentiating a 1/r2 term normally leads to a 1/r3 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/r2 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.

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