Clarifying the Motion Transformation for the Three-Dimensional Translation

By RE CASTEL

To help clarify the motion transformation for the three-dimensional translation of the phenomenon (of light) in nature, let us consider the following.

Consider a flash of light with its center at the origin of an x-y-z spatial coordinate system and its transformation to the x'-y'-z' spatial coordinate system for a given elapsed time with the temporal coordinates from t to t'. Since the propagation of light is three-dimensional, the propagation of the flash of light will be a spherical radiation with a radiant velocity c. The spherical radiation can be represented by the following transformation equations according to an elapsed time t with v=0:

x'=tc                 or         x'=x                 such that           x=tc

y'=tc                 or         y'=y                 such that           y=tc

z'=tc                 or         z'=z                 such that           z=tc

 t'=tc/c              or          t'=t                  such that           t=tc/c

Consider now a concentric velocity v due to gravity that interacts with the velocity c. The velocity c of the spherical electromagnetic radiation will now therefore be effected the concentric velocity v. The resultant velocity of the spherical radiation becomes c–v. Since x=tc, y=tc and z=tc, we have the following transformation equations according to an elapsed time t with v>0:

x'=t(c–v)            or         x'=x–tv              which is            x'=x(1–v/c)

y'=t(c–v)            or         y'=y–tv              which is            y'=y(1–v/c)

z'=t(c–v)            or         z'=z–tv              which is            z'=z(1–v/c)

and for t', since x=tc and t=tc/c,  we have

 t'=t(c–v)/c         or          t'=(x–tv)/c

such that                                  t'=(tc–tcv/c)/c

and such that                           t'=t–vx/c2          which is             t'=t(1–v/c)

In order to preserve the velocity c as the same in both the unprimed and primed frames of reference according to the idea of the constant velocity of light with c=λƒ, an invariance factor becomes necessary. Thus, the resultant motion transformation equations are:

x'=x(1–v/c) b

y'=y(1–v/c) b

z'=z(1–v/c) b

 t'=t(1–v/c) b

The obvious candidate for the invariance factor b is (1–v/c)–1 because it is the reciprocal of (1–v/c). With the approximation for b, the resultant transformation factor is (1–v2/c2)–1.

Now, in order to reflect the physical change brought forth by the three-dimensional motion transformations according to the interaction of the c and v velocities, we fix the space dimension and the time dimension and introduce by substitution the variable E or the variable m to show the change effected in terms of the change in the mass m or the change in the energy E. Thus, with the transformation factor equal to (1–v2/c2)–1, we have —

E'=E(1–v2/c2)–1

or

m'=m(1–v2/c2)–1

This transformation equation now allows the derivation of the genesis formula with the connection to the three-dimensional contraction effected by gravity wherein ∆v=At and where A is the acceleration due to gravity. We use the familiar mo instead of m' in the following.

m=mo(1+v2/c2)

m=mo(c2/c2+v2/c2)

mc2=moc2+mov2

E=Eo+mov2

E–Eo=mo[v–0]2

∆E=mo[∆v]2


In comparison, Einstein's derivation of the Lorentz transformation equations essentially produced the following:

x'=(x–vt) b

y'=y(1–v2/c2)½ b

z'=z(1–v2/c2)½ b

t'=(t–vx/c2) b

These suggested that the invariance factor b is (1–v2/c2)–½ because it is the reciprocal of (1–v2/c2)½. But note that with b=(1–v2/c2)–½ the simplification after the multiplication of the reciprocals is only for the two-dimensional translation since there are no translations along the y and z axes with y'=y=0 and z'=z=0.

Einstein also used b=(1–v2/c2)–½ as the transformation factor. The rationale for Einstein's application of the transformation factor b=(1–v2/c2)–½ is a bit difficult to understand. But Einstein also introduced the variable m to show the change effected in terms of the change in the mass m. Essentially, he substituted m' for x' (or for t') and m for (x–vt) (or for (t–vx/c2)). This allowed the transformation factor b=(1–v2/c2)–½ to show  —

m'=m(1–v2/c2)–½

This allowed the connection to the classical K.E.=½mv2 that led Einstein to the conclusion that E=mc2 – which apparently could not have been done in any other way except through the connection with the classical idea.

But it is obvious that Einstein's formulation for the change in mass is only for a two-dimensional translation of the mass m'. Clearly, the m'=m(1–v2/c2)–½ does not relate a resolved three-dimensional translation effected by two pairs of c and v, since the Lorentz equations indicate y'=y=0 and z'=z=0.


For the more extensive treatment regarding Einstein's derivation of the Lorentz transformation equations, try the following:

Kinematic subtleties in Einstein’s first derivation of the Lorentz transformations – by Alberto A. Martinez
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