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THE LORENTZ TRANSFORMATION

Question: Please provide a detailed rationale of how the RS theory produces the correct answer to this “Lorentz transform” problem:

M
L <———|———> R

When L and R travel at the speed of light relative to M, Larson says the speed of R relative to L is 2 units of space divided by two units of time; thus, the velocity of R relative to L is 2/2 = l. Now suppose that L and R both travel at C/2 relative to M. If we seemingly follow the same procedure as above, it appears that the total distance involved is (½+ ½) and the total time involved is (1 + 1), so that the velocity of R relative to L should be distance/time - 1/2. Obviously somethirg is wrong. What?

Answer: Assume that the partIcles traveling with the speed C/2 are atoms. Then the rate of motion of the atom toward R relative to the motion of the atom toward L is 0.8C. The RS theory thus offers the same answer to your question as does the Lorentz transformation equation. The mode of motion of a photon (vibration) is different from that of an atom (a combination of vibration and rotation). Photons remain in the space-time locations in which they originate; atoms do not. The space-time locations of photons move at the unit rate C. Atoms do not remain in the space-time locations in which they originate. Therefore, the procedure for calculating the rate of motion of two photons going apart from each other is not applicable to calculate the motion rate of two atorns moving apart from eacil other, each at v = C/2 with respect to M. The relative motion rate of the two atorns in this case is

 
0.5C + 0.5C
  1.0C
   
u =
————— 
= ———
= 0.8C
(1)
 
1 +[(0.5C) (0.5C)/C²]
  1.25
   

The procedure, equation (1), is called the Lorentz transformation equation. How does the RS theory arrive at the Lorentz equation? How does the RS theory deduce this equation?

This question amounts to asking how does RS theory imply the transformation equation:

 
vdb + wba
 
uda =
————— 
 
1 +[(vdb) (wba)/C²]
 

This Lorentz equation or law about the composition of velocities follows from the RS theory, because the latter assumes that a light photon remains in the space-time location in which it originates and further assumes that the location progresses at unit speed or at the uniform rate of C = 3 x 105 km/sec., independent of the motion of source or detector of the photon. These assumptions are incompatible with the Newtonian-Galilean transformation equation, the Newtonian law of the composition of velocities, uda = vdb + wba ; uda = - uad.

The fact that the velocity of light is independent of the velocity of the source of the light implies that any finite velocity of the source, when added to the velocity of light, yields a resultant for the light whose magnitude equals that of the speed of light.

Now in Newtonian physics, when three particles A, B, D are moving in a straight line, and if U is the velocity of A relative to D, V is the velocity of D relative to B and W is the velocity of B relative to A, then uad + vdb + wba = 0.

However, the just stated fact and RS principle asserts that when v = C, then u = - C, whatever value w may have. This implies that the equation u+v+w = 0 is not true when velocities commensurable with that of light are involved: it works satisfactorily only when all the velocities are small compared with C.

How then to deduce the correct form of tkre law ef composition of velocities for velocities of any magnitude is now the task. Specify then that the exact relation between the three velocities is F (u,v,w) = 0. Agree that wba = - wab etc.

By permuting the three particles A, B and D note that the function F has to be a symmetric function of u, v and w.

Further, the function F has to be a linear function so that may yield a one-valued solution when solved with respect to u, v or w. Consequently, the equation assumes the form

g + h (u+v+w) + k(vw + wv) + 1 (uvw) = 0

Since when w = 0, u = - v, then g - ku² = 0 for all values of u and so g and k are zero.

Thus, the equation takes the form h(u+v+w)+1(uvw)=D. Also, u= -C when v=C, no matter what the value of w. Hence hw-1C²=0 and h=1C². Therefore,

1C²(u+v+w) +1(uvw)=0

or

u+v+w + (uvw/C²) = 0

This is the exact relation which replaces the Newtonian relation u+v+w=0. This exact relation implies that

-u-uvw/C²=v+w

and

-u[1+(vw/C²)]=v+w

-u=v+w/[1+(vw/C²)]

-uad = uda

Therefore

 
vdb + wba
 
uda =
————— 
 
1 +[(vdb) (wba)/C²]
Q.E.D.

Thus the Lorentz Law of the composition of velocities is simly the mathematically equivalent expressin of every physical theory which assumes that the speed of radiation in vacuo is independent of the mtion of the radiation source.

Reciprocity IV.1 (April 1974)


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