New Transformation Equations and the Electric Field Four-vector

6. The Transformation Equations

We wish to find the most general set of coordinate transformation equations that assure the covariance of the laws of physics. Since the spacetime interval is described by a Euclidean metric, in this theory, the transformation matrix we will use must be orthogonal. We are making a transformation, initially, from a stationary unprimed frame of reference to a uniformly moving primed frame, so we assume that the components of the transformation matrix contain the components of the four-velocity, , of the moving frame. But, the components must be dimensionless so that the transformed quantity has the same units as the original quantity. Therefore, we divide each component of the transformation matrix by the invariant speed of light, . We also suspect that each component of the transformed quantity should depend on all of the components of the original quantity. With this in mind, we choose the most simple combination of components that satisfy these requirements for the transformation matrix, ,

where the are the components of the four-velocity of the primed frame relative to the unprimed frame. The "" and "" signs preceding some of the terms in (6.1) imply that these terms can be either plus or minus without affecting the orthogonality. That is, the signs of these terms may be taken as the top sign in one transformation and the bottom sign in another transformation, while the terms preceded by "+" or "-" signs remain fixed. However, the choice of sign must be consistent, in other words, we must take either the top sign, throughout, or the bottom sign, throughout. For example, if we take the "+" part of the component in (6.1), we must take the "-" part of , the "-" part of , the "+" part of , etc..

Due to the seemingly indeterminate nature of these terms, we suspect that they exist as a kind of mixture, or superposition, of both signs, simultaneously. It is only when the sign of a term needs to be specified, that the term becomes associated with one sign or the other. Therefore, these signs might, more properly, be called plus and minus, rather than plus or minus signs, until one or the other sign is specified. However, even though both of these signs (plus and minus, and minus and plus) are combinations of both plus and minus, they retain their opposite nature. In addition, the specification of the sign of one of these indeterminate terms immediately establishes the signs of all the other indeterminate terms completely.

The "" and "" (this sign is "minus/minus", not "equals", in this context) signs preceeding some terms in (6.1) can be taken as single "+" and "-" signs, although as with the "" and "" signs, they are a combination or superposition of both signs. Multiplication of combination signs is carried out by multiplying the corresponding parts of each sign. That is, the top parts are multiplied together, separately, and the bottom parts are multiplied together, separately. For example, if we wish to multiply "" and "", we get "". From this point, for simplicity, all operations involving (6.1) will use only one of the combination signs at a time. However, the combination sign associated with each term of (6.1) will still be implied in any mathematical operations carried out.

The Lorentz transformation matrix is REPLACED by (6.1).

The coordinate transformation equations from the unprimed frame to the primed frame associated with the matrix, , are

where

For a velocity in the x-direction, we have a rotation of the x'-t' plane relative to the x-t plane, but we also have, contrary to relativity, a rotation of the y'-z' plane in the y-z plane. The axes are rotated by the angle . Looking from the unprimed frame in the direction of the positive x-axis, the rotation of the y'-z' plane relative to the y-z plane, due to the indeterminate terms in (6.1), is in both the clockwise and counter-clockwise directions, simultaneously, until one direction or the other is determined by additional conditions. These rotations are not necessarily observable as rotations. They may be manifested as seemingly unrelated properties of bodies in motion, such as precession, time dilation, length contraction, etc.

The inverse transformation equations from the primed frame to the unprimed frame are

The transformation equations in (6.2) and (6.4) REPLACE the Lorentz transformation equations.

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