The following calculations are exploratory. They are not a part of any established theory. There are still some fundamental inconsistencies in these calculations, and the investigation represents an on-going effort.
The behavior of light rays in the Schwarzschild gravitational solution can be represented by a refractive index model . The solution is only valid from the perspective of a distant observer, which corresponds to the perspective of a potential theory. The speed of light is rarely c in such a representation. The perspective is that of the space of the potentials, and the potentials do not correspond directly to real space.
In the most general case mathematically possible, the distance between two infinitesimally spaced points has both symmetric and antisymmetric components. Simulate the antisymmetry by starting with two synchronized clocks, then displace one of the clocks in time. The indicated propagation velocity in one direction is now greater than c, while the indicated velocity in the other direction is less than c. One part is symmetric and one part is antisymmetric, and neither part is c.
In the calculations the scalar potential of the Maxwell equations is treated as a displacement in time. The vector potential is treated as a displacement in space. These equations contain terms quadratic in field strength, but those terms can not be carried in deriving the Maxwell equations, which are linear equations.
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 the symmetric and antisymmetric parts.
The symmetric part includes the expansion factor, a shear term, and a space rotation. The expansion factor is not zero within the interior region of a spherical mass shell, yet it is not locally detectable there. These are potential equations. They do not correspond well to the locally measurable relationships of space and time. The calculations are developed as though the displacements were locally measurable, but they actually represent the perspective of a distant observer.
The antisymmetric part has the units of seconds/volt. The magnitude of the effect is developed in the pulsar section. The effect is very small, about 10-27 sec/volt, so direct laboratory evaluation is probably not possible. An electrostatic field strong enough to produce a measurable effect would demolish the apparatus.
The Maxwell equations are not in need of rederivation. The primary
objective of these calculations is to provide some justification for the
perspective, then to proceed on to the nonlinear terms. The nonlinear
terms become important in the intense field, and they also lead to
a coupling of the electrostatic and gravitational fields. Further,
the cosmological term in these solutions appears to represent the weak
|Section 1||Extrapolating the 4-potential|
|Section 2||The second rank contravariant tensor|
|Section 3||The decomposition products of the third rank contravariant tensor in 3+1 space|
|Section 4||The Euler-Lagrange equations of the second rank tensor in 4-vector form|
|Section 5||The Euler-Lagrange equations of the second rank tensor in 3+1 space|
|Section 6||The transformation of a time warp|
Other online material:
An Oscillatory Pulsar Model
A Study of the Retarded Gravitational Field
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 27 Jan 2001