Inverse relative scaling
An analogous protocol exists for interval measurements performed by the F' chorus on clocks and rods at rest in frame F0. Figure 18.2 shows the events involved.
Event C is along the worldline of a clock at rest in frame F0 situated at the origin of spatial coordinates, so we have
Hence we may write tC = a[t'C, where a't = a_1 is the time dilation factor of moving clocks as seen by F' observers.
Event D is at one end of a ruler of length Ld at rest in frame F0 and observed at time t'D = zero in frame F'. Hence we have
Fig. 18.2 Scale changes seen in the moving frame F.
where L'D is the length of the ruler as defined by the F' chorus. The generalized transformation (18.2) gives
so the relevant contraction factor ax is given by a'x = — (a - Oft2).
The Michelson-Morley constraint
There is a hidden assumption in the derivation of the Lorentz transformation discussed in Chapter 17 that need not be made. It was pointed out by Tangherlini that no experiment, including those of Michelson and Morley [Michelson & Mor- ley, 1887], ever measures the one-way speed of light but in fact always a round-trip average [Tangherlini, 1958]. This means that the postulate that the speed of light is the same in all directions is not forced on us by the Michelson-Morley null result.
We may express this mathematically as follows. Suppose that the speed of light c in a given frame depends on position x and direction n, where n is a unit vector in a given direction. Now imagine a pulse of light is sent around a fixed, closed circuit in that frame, with points along the circuit parametrized by a real parameter X. The total length LCircuit and the total time TCircuit around the circuit are given by the closed line integrals
Then the Michelson-Morley null result is summarized by the statement
We may use this analysis to find a critical relationship between the parameters. Consider an inertial frame in 1 + 1 dimensions for which cR = cL, but one for which the null result of the Michelson-Morley experiment holds. An experiment is conducted that consists of a pulse of light sent a distance L in the positive x direction, that reflects from a mirror there and returns to the origin, the source of the light. The total time T is given by
On the other hand, the observers know that the total distance travelled is 2L. If they believe that light has constant speed c in all directions, then they will assert that
Hence for any frame for which cL = cR, the relation
must hold. Using (18.9) and (18.10) gives the condition S = у2(а - 0^2),where у is the standard Lorentz factor у = 1/^/1 - в2.
Although a path may be closed from the perspective of a given frame, from the perspective of the absolute frame it need not be closed. The question of what constitutes a closed path therefore does not have an absolute answer: the concept of ‘returning to the same position’ depends on what is mean by ‘the same position’.