1.  A non-symmetric metric

Crab Unenhanced optical Image of the Crab Nebula. Home of the 950 year old 30 Hz pulsar PSR B0531+21.

Begin with two coincident synchronized clocks, then move them slowly apart in an unspecified field. At a scheduled time, as indicated by the transmitting clock, transmit a bidirectional pulse. The arrival times at the other clock, as indicated by the receiving clock, can be decomposed into a symmetric part and an antisymmetric part. A tensor is easily derived from this metric. It exists in a real-valued 3x3 structure.

The Maxwell equations can be directly derived from the linearized form of this metric. The basis of the relationship is that the symmetric part represents the first difference of the vector potential, and the antisymmetric part represents the first difference of the scalar potential. Space-time cross terms appear when the potentials change with time. The derivation utilizes the Lagrangian, and is a direct extension of the methods utilized in the study of elastic media. The antisymmetric part of the tensor replaces the kinetic energy term.

These relationships imply that the scalar potential of the Maxwell equations can be viewed as a displacement in time, and that the vector potential can be viewed as a displacement in space. The Liénard-Wiechert retardation equations can be derived from the assumption that the scalar potential represents a displacement in time. These displacements represent a 4-space that is non-flat in the most general way. When working with linear equations this alternative point of view makes no difference, but when the fields become intense the metrical representation predicts the existence of nonlinear terms. For the electrical solutions terms quadratic in field strength occur. Electrostatic-gravitational cross terms also occur.

In being symmetric in static solutions, the first difference of the vector potential is evidently related to the expansion factor of the gravitational field, which is known to behave similarly to the Newtonian gravitational potential. Potentials are not locally measurable, so the expansion factor is a quasi-potential rather than a field. In other words, in the gravitational field the locally-measured propagation velocity is c even in the presence of the expansion factor, because the expansion factor is not a field, and is not locally detectable. The clocks in the model express potential relationships that cannot be locally evaluated. The first difference of the scalar potential is locally measurable, so it is a field.

The magnetic field also behaves like a quasi-potential, because the only way to statically measure it at a single point is with a magnetic monopole, and none are available. When represented by the equations of the antisymmetric metric, the magnetic field is a rotation. If the angle of rotation is constant over an extended region then the rotation angle is not locally detectable. The magnetic field will be referred to as a field in these calculations, but it behaves like a quasi-potential.

Ψ is the scalar potential and A is the vector potential. The electrical fields are obtained from the potential solutions by the relationships B = curl AE = -grad Ψ - 1/c ∂A/ ∂t. The symmetric terms are investigated on another page. The quantities ∂Ψ/∂t and div A are never observationally distinguishable, nor are grad Ψ and ∂A/ ∂t.

The calculations shown on the other web page suggest some interesting relationships, but they are linearized, so they can't be used in developing the pulsar model. A simpler method will be developed.


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