THE LORENTZ TRANSFORMATION
M 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?
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 This question amounts to asking how does RS theory imply the transformation equation:
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 10 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 u 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 w 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
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. — |