### 2.  The electrical field limits of the vacuum Supernova [1987a]. The outer rings are due to an earlier outburst. Mainstream pulsars tend to be found near the a supernova remnant.

The scalar potential represents a displacement in time, so if the E field between two capacitor plates is Em = 1/(ξ c) then a pulse transmitted from one plate to the other arrives at the same time it was transmitted. At higher voltages it arrives before it was transmitted, which is an unlikely circumstance. A field limit is implied. The presence of quadratic terms in the equations also implies the existence of field limits.

For weak static fields the round trip velocity between the capacitor plates is c. There may be corrections which cause the round trip velocity to remain at c in intense static fields, but it will not be necessary to consider that possibility just yet. Consider also that formulations based on quadratic forms are interpretable as representing the product of bidirectional quantities, and that the one-way constraints assumed to exist within them are not necessarily there.

The limiting condition has the meaning of a dimensioned physical constant rather than of a geometrical relationship, so it becomes necessary to make an identification. Beginning with the method of dimensional analysis, write

Bm = b c2 me2/(hbar qe) = b x 4.41 x 109 tesla = b x 4.41 x 1013 gauss

hbar = h/(2 π), h=Planck's constant. qe=charge of electron, me=mass of electron. b is an undetermined coefficient with a magnitude that is probably near unity. All equations are in SI units. Some MKS electrical equations are difficult to convert to the CGS system, and conversely .

The identity α = μ0 c qe2/(2 h) can be used to parameterize the equation in terms of the fine structure constant instead of Planck's constant. This form is more appropriate for comparisons to the classical electron. μ0 is the vacuum magnetic permeability.

The only observationally established field limits are the electrical limits of the Dirac equations and the gravitational limit at the Schwarzschild radius. These two limits are of different forms, but charge and mass are of different symmetries, and it often works out that way.

Shabad and Usov  derived a critical field strength where a resonance condition affects the conversion of gamma rays to electron-positron pairs. The critical field does not represent a precisely defined field limit, but it is suggestive of a limiting condition. The equation for the critical field is obtained by setting b to 1 in the above equation.

In interpreting such solutions to the Dirac equations there is the important consideration that the scaling relationship needed in the metrical equations is that of a weak field and linear extrapolation to a point that looks like a limit, but without implying that the equations themselves would remain valid at that limit.

The calculations of this section can be viewed in a computer generated file.

The E and B fields carry equal energies in electromagnetic radiation, implying the equation 1/2 ε0 Em2 = 1/2 Bm20. Solving for Em and applying the identity c = 1/(ε0 μ0)1/2

Em = b c3 me2/(hbar qe) = b x 1.32 x 1018 volts/meter

It follows from the opening paragraph that

1/(Em c) = ξ = hbar qe/(b c4 me2) = b-1 x 2.52 x 10-27 second/volt

The sign of ξ is undetermined.

For a quantum of charge the minimum radius is  rc/(b α)1/2, where rc is the classical radius, μ0 qe2/(4 π me). α is the fine structure constant. If b were 1 the metrical radius would be greater than the classical radius by a factor of about 12. This method of calculation obviously breaks down altogether in the vicinity of a proton, meaning that the Maxwell equations are not even a useful approximation in that region.

The Einstein electrical field limits for a quantum of charge would be obtained by shrinking a classical electron to the point of gravitational collapse. The field limits obtained in that way are enormously greater than those of the Dirac equations, meaning that the Dirac equations predict that the nonlinear electrical interactions are far stronger than those of the Einstein theory. (The linear relationships are just those of the Maxwell equations in the asymptotic limits of any theory). The electrical solutions of the general theory have no observational confirmation.

The electrical scaling relationships of the general theory are based on the linear relationships of the Maxwell and Newton equations. There are additional physical constants in the overall system. The only way to avoid the problem of linear dependencies is to utilize both linear and nonlinear terms in determining the scaling relationship. Just because the Newtonian gravitational energy and the Maxwellian field energy are computable, it is in no way implied that the metrical scaling relationship between the two fields is known. The classical electrical and gravitational systems of units form two distinct and orthogonal systems. No existing theory or system of units consistently connects the gravitational and electrical fields. The general theory contains no electrical arguments or principles to bridge this gap. Gravity stands alone. Refer to any textbook on the general theory of relativity for another perspective.

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