2.4  Doppler Shift for Sound and Light

I was much further out than you thought

And not waving but drowning.

                              Stevie Smith, 1957

For historical reasons, some older text books present two different versions of the Doppler shift equations, one for acoustic phenomena based on traditional Newtonian kinematics, and another for optical and electromagnetic phenomena based on relativistic kinematics.  This sometimes gives the impression that relativity requires us to apply a different set of kinematical rules to the propagation of sound than to the propagation of light, but of course that is not the case.  The kinematics of relativity apply uniformly to the propagation of all kinds of signals, provided we give the exact formulae.  The traditional acoustic formulas are inexact, tacitly based on Newtonian approximations, but when they are expressed exactly we find that they are perfectly consistent with the relativistic formulas.

Consider a frame of reference in which the medium of signal propagation is assumed to be at rest, and suppose an emitter and absorber are located on the x axis, with the emitter moving to the left at a speed of v_e and the absorber moving to the right, directly away from the emitter, at a speed of v_a.  Let c_s denote the speed at which the signal propagates with respect to the medium.  Then, according to the classical (non-relativistic) treatment, the Doppler frequency shift is

(It's assumed here that v_a and v_e are less than c_s, because otherwise there may be shock waves and/or lack of communication between transmitter and receiver, in which case the Doppler effect does not apply.)  The above formula is often quoted as the Doppler effect for sound, and then another formula is given for light, suggesting that relativity arbitrarily treats sound and light signals differently.  In truth, relativity has just a single formula for the Doppler shift, which applies equally to both sound and light.  This formula can basically be read directly off the spacetime diagram shown below

If an emitter on worldline OA turns a signal ON at event O and OFF at event A, the proper duration of the signal is the magnitude of OA, and if the signal propagates with the speed of the worldline AB, then the proper duration of the pulse for a receiver on OB will equal the magnitude of OB. Thus we have

 and

Substituting x_A = -v_et_A and x_B = v_at_B into the equation for c_s and re-arranging terms gives

from which we get

Substituting this into the ratio of |OA| / |OB| gives the ratio of proper times for the signal, which is the inverse of the ratio of frequencies:

Now, if v_a and v_e are both small compared to c, it's clear that the relativistic correction factor (the square root quantity) will be indistinguishable from unity, and we can simply use the leading factor, which is the classical Doppler formula for both sound and light.  However, if v_a and/or v_e are fairly large (i.e., on the same order as c) we can't neglect the relativistic correction.

It may seem surprising that the formula for sound waves in a fixed medium with absolute speeds for the emitter and absorber is also applicable to light, but notice that as the signal propagation speed c_s goes to c, the above Doppler formula smoothly evolves into

which is very nice, because we immediately recognize the quantity inside the square root as the multiplicative form of the relativistic composition law for velocities (discussed in section 1.8).  In other words, letting u denote the composition of the speeds v_a and v_e given by the formula

it follows that

Consequently, as c_s increases to c, the absolute speeds v_e and v_a of the emitter and absorber relative to the fixed medium merge into a single relative speed u between the emitter and absorber, independent of any reference to a fixed medium, and we arrive at the relativistic Doppler formula for waves propagating at c for an emitter and absorber with a relative velocity of u:

To clarify the relation between the classical and relativistic Doppler shift equations, recall that for a classical treatment of a wave with characteristic speed c_s in a material medium the Doppler frequency shift depends on whether the emitter or the absorber is moving relative to the fixed medium. If the absorber is stationary and the emitter is receding at a speed of v (normalized so c_s = 1), then the frequency shift is given by

whereas if the emitter is stationary and the absorber is receding the frequency shift is

To the first order these are the same, but they obviously differ significantly if v is close to 1.  In contrast, the relativistic Doppler shift for light, with c_s = c, does not distinguish between emitter and absorber motion, but simply predicts a frequency shift equal to the geometric mean of the two classical formulas, i.e.,

Naturally to first order this is the same as the classical Doppler formulas, but it differs from both of them in the second order, so we should be able to check for this difference, provided we can arrange for emitters and/or absorbers to be moving with significant speeds. The Doppler effect has in fact been tested at speeds high enough to distinguish between these two formulas. The possibility of such a test, based on observing the Doppler shift for “canal rays” emitted from high-speed ions, had been considered by Stark in 1906, and Einstein published a short paper in 1907 deriving the relativistic prediction for such an experiment. However, it wasn’t until 1938 that the experiment was actually performed with enough precision to discern the second order effect. In that year, Ives and Stilwell shot hydrogen atoms down a tube, with velocities (relative to the lab) ranging from about 0.8 to 1.3 times 10⁶ m/sec. As the hydrogen atoms were in flight they emitted light in all directions.  Looking into the end of the tube (with the atoms coming toward them), Ives and Stilwell measured a prominent characteristic spectral line in the light coming forward from the hydrogen. This characteristic frequency n was Doppler shifted toward the blue by some amount dn_approach because the source was approaching them. They also placed a mirror at the opposite end of the tube, behind the hydrogen atoms, so they could look at the same light from behind, i.e., as the source was effectively moving away from them, red-shifted by some amount dn_receed. The following is a table of results from the original 1938 experiment for four different velocities of the hydrogen atom:

Ironically, although the results of their experiment brilliantly confirmed Einstein’s prediction based on the special theory of relativity, Ives and Stillwell were not advocates of relativity, and in fact gave a completely different theoretical model to account for their experimental results and the deviation from the classical prediction. This illustrates the fact that the results of an experiment can never uniquely identify the explanation. They can only split the range of available models into two groups, those that are consistent with the results and those that aren't. In this case it's clear that any model yielding the classical prediction is ruled out, while the Lorentz/Einstein model is found to be consistent with the observed results.

All the above was based on the assumption that the emitter and absorber are moving relative to each other directly along their "line of sight". More generally, we can give the Doppler shift for the case when the (inertial) motions of the emitter and absorber are at any specified angles relative to the "line of sight". Without loss of generality we can assume the absorber is stationary at the origin of inertial coordinates and the emitter is moving at a speed v and at an angle f relative to the direct line of sight, as illustrated below.

For two pulses of light emitted at coordinate times differing by Dt_e, arrival times at the receiver will differ by Dt_a = (1 + v_r) Dt where v_r = v cos(f) is the radial component of the emitter’s velocity. Also, the proper time interval along the emitter’s worldline between the two emissions is Dt_e = Dt_e (1 – v²)^1/2. Therefore, since the frequency of the transmissions with respect to the emitter’s rest frame is proportional to 1/Dt_e, and the frequency of receptions with respect to the absorber’s rest frame is proportional to 1/Dt_a, the full frequency shift is

This differs in appearance from the Doppler shift equation given in Einstein’s 1905 paper, but only because, in Einstein’s equation, the angle f is evaluated with respect to the emitter’s rest frame, whereas in our equation the angle is evaluated with respect to the absorber’s rest frame. These two angles differ because of the effect of aberration. If we let f' denote the angle with respect to the emitter's rest frame, then f' is related to f by the aberration equation

(See Section 2.5 for a derivation of this expression.) Substituting for cos(f) into the previous equation gives Einstein’s equation for the Doppler shift, i.e.,

Naturally for the "linear" cases, when f = f' = 0 or f = f' = p we have

respectively.  This highlights the symmetry between emitter and absorber that is so characteristic of relativistic physics.

Even more generally, consider an emitter moving with constant velocity u, an absorber moving with constant velocity v, and a signal propagating with velocity C in terms of an inertial coordinate system in which the signal’s speed |C| is independent of direction. This would apply to a system of coordinates at rest with respect to the medium of the signal, and it would apply to any inertial coordinate system if the signal is light in a vacuum. It would also apply to the case of a signal emitted at a fixed speed relative to the emitter, but only if we take u = 0, because in this case the speed of the signal is independent of direction only in terms of the rest frame of the emitter. We immediately have the relation

where r_e and r_a are the position vectors of the emission and absorption events at the times  t_e and t_a respectively. Differentiating both sides with respect to t_a and dividing through by 2(t_a - t_e), and noting that (r_a – r_e)/(t_a – t_e) = C, we get

where u and v are the velocity vectors of the emitter and absorber respectively. Solving for the ratio dt_e/dt_a, we arrive at the relation

Making use of the dot product identity r∙s = |r||s|cos(q_r,s) where q_r,s is the angle between the r and s vectors, these can be re-written as

The frequency of any process is inversely proportional to the duration of the period, so the frequency at the absorber relative to the emitter, projected by means of the signal, is given by n_a/n_e = dt_e/dt_a.  Therefore, the above expressions represent the classical Doppler effect for arbitrarily moving emitter and receiver. However, the elapsed proper time along a worldline moving with speed v in terms of any given inertial coordinate system differs from the elapsed coordinate time by the factor

where c is the speed of light in vacuum. Consequently, the actual ratio of proper times – and therefore proper frequencies – for the emitter and absorber is

The leading ratio is the classical Doppler effect, and the square root factor is the relativistic correction.

Return to Table of Contents