The Doppler Twins

It’s well known that the predicted Doppler effects based on a Galilean wave theory in a given medium (in which the wave propagates at characteristic speed), either for stationary emitter and moving receiver, or for stationary receiver and moving emitter, differ from each other and from the relativistic prediction only in the second order of v/c. Now, there is a degree of ambiguity when discussing the predictions of various theories, because they make different assumptions about various ambient media, which may include both detectable and undetectable media. For example, by the end of the 19^th century most physicists (including Maxwell) had concluded that light propagates at a characteristic speed in a light-carrying medium – called the ether – but this medium was quite distinct from any ordinary palpable matter. Also, the ether was influenced (dragged along) only slightly within ordinary transparent matter, depending on the index of refraction of the matter, and there was no detectable convective dragging outside of an ordinary material substance. Still, the slight amount of dragging implied that the results of optical experiments (such as Fizeau’s) depend on the state of motion of the material substance through which the light propagates, since that was believed to slightly affect the state of motion of the ether. To eliminate this complication, optical experiments were typically performed in atmospheric air, or in evacuated chambers from which as much material substance as possible had been removed, to the extent that the index of refraction in the region was arbitrarily close to 1. In these conditions only the motion relative to the presumed ether frame had an appreciable effect. However, the ether frame was unknown, so the classical ether theories could not give unambiguous predictions. Their predictions were always contingent on a guessed ether frame.

One consequence of this ambiguity is that it affects the analysis of the differences in elapsed proper time (if any) along two different paths between two given events. Obviously while the distance between two twins is increasing they will see reduced frequencies (red shift) of signals from one to the other, and when the distance is decreasing they will see increased frequencies (blue shift). In a classical Galilean wave theory these two effects, integrated over the outbound and inbound portions of the journey, would cancel out, so that both twins would have the same number of oscillations transmitted and received (assuming identically constructed equipment). In contrast, the prediction of special relativity is that the traveling twin would actually have fewer oscillations than the stay-at-home twin, so the effects don’t exactly cancel out over the round trip.

To confirm this, consider the simple “away and back” scenario illustrated below.

To allow direct comparison, both of these are plotted in terms of space and time coordinates in which light has the characteristic speed c. For the Galilean theories this means coordinates in which the light-carrying medium is at rest. One twin remains stationary in these coordinates, progressing in time from A to C, while the other moves away and back at speed v (progressing from A to B to C). The figure on the left shows a light pulse from the traveling twin at the turn-around point to the stationary twin. The frequency of pulses emitted uniformly from the traveling twin and arriving at the stationary receiver is red-shifted during the interval from A to D, and then blue-shifted during the interval from D to C. The durations of these two intervals are L/v + L/c and L/v – L/c respectively. The amounts of redshift or blue shift depend on the theory:

To the second order these can be written as

where k = 1 for Galilean wave theory and k = 1/2 for special relativity. The total number of cycles emitted by the emitter is ν_e Δτ_e where ν_e is the emitted frequency in terms of the emitter’s measure of time, and Δτ_e is the interval of emitter’s time during which it emitted at that frequency. Likewise the total number of cycles received by the receiver is the sum of terms of the form ν_r Δτ_r where ν_r is the received frequency in terms of the receiver’s measure of time, and Δτ_r is the interval of receiver’s time during which it received at that frequency. The total number of cycles emitted and received must be equal, so we have

and therefore

The total duration of time for the stationary receiver is Δτ_r = 2L/v, so the net time dilation per unit of receiver time is

Thus for the classical Galilean wave theory with k = 1 there is no net time dilation, but special relativity with k = 1/2 predicts a net time dilation of –(1/2)(v/c)² seconds per second.

The right-hand figure above shows a light pulse propagating from the stationary twin to the traveling twin at the turn-around point. The frequency of pulses emitted uniformly from the stationary emitter and arriving at the traveling receiver is red-shifted during the interval from A to B, and then blue-shifted during the interval from B to C. The durations of coordinate time of these two intervals are both L/v. The amount of redshift or blueshift depends on the theory:

To the second order these can be written as

where k = 0 for Galilean wave theory and (again) k = 1/2 for special relativity. Again we have

and therefore

Hence we have

Thus for the Doppler effect based on classical Galilean wave theory with k = 0 there is no net time dilation, but special relativity with k = 1/2 predicts (again) a net time dilation of –(1/2)(v/c)² seconds per second. (We note that the denominator here is the traveling twin’s proper time, whereas for the opposite pulses it was the stationary twin’s proper time, but the correction involves only higher-order terms.)

As an aside, note that the classical Doppler effect for a Galilean wave theory is commonly referred to as “first-order Doppler”, and relativistic time dilation is referred to as “second-order Doppler”. This sometimes causes confusion, because the frequency ratio ν_r/ν_e in the classical Doppler effect for moving emitter and stationary receiver is 1/(1 + v/c), which has second and higher order terms in its expansion (as we saw above with k=1). However, in that case the reciprocal ratio, ν_e/ν_r , differs from 1 purely in the first order. So, in this sense, the classical Doppler effect is purely “first order”, due to the changing path length in the light-carrying medium (whatever we take that to be). The relativistic effect agrees with the classical effect in the first order, but disagrees in the second and higher orders.

In summary, the optical appearance of time differences due to the classical Doppler effect (including the second and higher order terms for the case of moving emitter) cancels out over any closed loop, resulting in no net time dilation, but the second-order effect for the relativistic Doppler shift (i.e., k = 1/2) yields the observed cumulative second-order time dilation.

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