Green can correlate his t2 with the t1 given in Red's letters, and Red can correlate his t3 with the t2 given in Green's letters. t2 = f( t1 ) and t3 = g(t2 ). In general f(w) and g(w) will not be the same function. We therefore "recalibrate" both Red's and Green's measurement of time, so as to have the two functions the same. We find the "functional square root" of f(w) and g(w), that is a function h( ) such that h(h(w)) = g(f(w)). We cannot do this directly, but we can find functions h( ) and k( ), such that for any variable x,

k-1[h(h(k(x)))] = g(f(x)).

Red and Green agree to recalibrate their clocks, so that henceforward, a message dispatched by Red at t will be received by Green at t where t = h(t) and vice versa.
Red now will not be able to say My messages travel faster than yours do, Green. Nor Green to say My messages travel faster than yours do, Red.

I

Radar Rules

(i) t2 = 1/2 (t1 + t3 )
(ii) l2 = 1/2 c (t1 - t3 ).

II

Radio Rules

(i) t = h(t)
(ii) t = h(t).

By radar rule I (i) t2 = 1/2 ( t1 + t3)
By radio rule II (i) t1 = h-1( t2)
By radio rule II (ii) t3 = h( t2)

Therefore Q( t2) = t2 = 1/2 [h-1 (t2) + h(t2)]. (1)
Also l2 = vt2.
t2 = l2/v = 1/2 c (t3 - t1)/v
and hence

Q( t2) = 1/2 c [h(t2) - h-1 (t2)]/v ;

therefore

Q( t2)v/c = 1/2 [h(t2) - h-1 (t2)] (2)


Adding (1) and (2)

Q( t2) + Q(t2)v/c = h(t2)
(1 + v/c) Q( t2) = h(t2) (3)

Subtracting (2) from (1)

Q( t2) - Q( t2)v/c = h-1 (t2)
(1 - v/c) Q( t2) = h-1 (t2) (4)

Dividing (3) by (4)

(1 + v/c) / (1 - v/c) = h(t2) / h-1 (t2) (5)

Substituting w for t2, we have

(1 + v/c) / (1 - v/c) = h(w) / h-1 (w)
= h(h(t2)) / h-1 (h(t2)) = h(h(t2)) / t2 (6)
h(h(t2)) = t2 ((1 + v/c) / (1 - v/c)) (7)


Equation (7) has

h(t2) = t2 \/ ((1 + v/c) / (1 - v/c))

as its solution. From it we can obtain

h-1 (t2), and so Q(t2).
h-1 (t2) = t2 \/ ((1 - v/c) / (1 + v/c))

Thus, by (1)

2Q( t2 ) / t2 = 1 / \/(1-v2 /c2)

Q(t2) is a special case of

L(iv) t = (1 - v2/ c2)-1/2 ( t - vx/ c2 ).

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