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 (nonrelativistic) 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_{e}t_{A} and x_{B} = v_{a}t_{B} into the equation for c_{s} and rearranging 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 square root quantity will be indistinguishable from unity, and we can simply use the leading factor, which is formally identical to the classical Doppler formula for both sound and light. (The identity is only formal, because the velocities here satisfy the relativistic composition law.) However, if v_{a} and/or v_{e} are fairly large (i.e., on the same order as c) and unequal, we can't neglect the square root factor. 

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 highspeed 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 10^{6} 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 ν was Doppler shifted toward the blue by some amount dν_{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, redshifted by some amount dν_{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 ϕ relative to the direct line of sight, as illustrated below. 



For two pulses of light emitted at coordinate times differing by Δt_{e}, arrival times at the receiver will differ by Δt_{a} = (1 + v_{r}) Δt where v_{r} = v cos(ϕ) is the radial component of the emitter’s velocity. Also, the proper time interval along the emitter’s worldline between the two emissions is Δτ_{e} = Δt_{e} (1 – v^{2})^{1/2}. Therefore, since the frequency of the transmissions with respect to the emitter’s rest frame is proportional to 1/Δτ_{e}, and the frequency of receptions with respect to the absorber’s rest frame is proportional to 1/Δt_{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 ϕ 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 ϕ' denote the angle with respect to the emitter's rest frame, then ϕ' is related to ϕ by the aberration equation 



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



Naturally for the "linear" cases, when ϕ = ϕ′ = 0 or ϕ = ϕ′ = π 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 velocity u (at the time of emission), an absorber moving with velocity v (at the time of reception), and a signal propagating with velocity C, all 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 = rscos(θ_{r,s}) where θ_{r,s} is the angle between the r and s vectors (with respect to the reference coordinate system), these can be rewritten 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 ν_{a}/ν_{e} = dt_{e}/dt_{a}. Therefore, the above expressions have the same form as the classical Doppler effect for arbitrarily moving emitter and receiver (with the caveat that the velocities satisfy the relativistic composition law). 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 has the same form as the classical Doppler effect, and the square root factor is a purely relativistic correction. If we interpret the angles relative to the rest frame of the emitter (at the instant of emission) we must put u = 0, resulting in Einstein’s 1905 formula, whereas if we interpret the angles relative to the rest frame of the absorber (at the instant of reception) we must put v = 0, giving the alternative version as discussed previously. 

One interesting consequence of the relativistic Doppler effect is due to the fact that the energy of a pulse of light remains proportional to the frequency under transformations from one system of inertial coordinates to another. Hence if we are approaching a source of light, the energy of a given pulse of light (relative to our rest frame) from that source is greater than if we were receding from the source, and the ratio of energies for these two cases is exactly proportional to the ratio of frequencies. (This is consistent with the quantum relation E = hn.) Now, consider an stationary object that emits two equal pulses of light in opposite directions, and then consider the amount of energy carries away by these pulses with respect to a coordinate system moving with speed v along the axis of the pulses. Classically the frequency (and hence the energy) of the forwardgoing pulse would be Doppler shifted by the factor 1 + v, and the backwardgoing pulse would be shifted by the factor 1 – v, so if each pulse carried energy ΔE/2 relative to the original stationary coordinates, for a total energy of ΔE, the energy emitted relative to the moving coordinates would be (ΔE/2)(1 + v) + (ΔE/2)(1 – v) = ΔE. Thus the energy emitted is the same. However, using the relativistic formula, the total emitted energy with respect to the moving coordinates is 



Thus the combined energy content of the emitted pulses is slightly greater with respect to the moving coordinates than with respect to the stationary coordinates. Relative to the stationary coordinates, let E denote the total energy of the object prior to the emissions, and define the parameter m_{1} such that the total energy of the object increases by (1/2)m_{1}v^{2} for an incremental velocity v. In these terms, the total energy of the object prior to the emissions is E relative to the stationary coordinates, and E + (1/2)m_{1}v^{2} relative to a system of coordinates moving (along the axis of the pulses) with an increment speed v. 

Following the emission of the pulses the total energy of the object is E  ΔE relative to the stationary coordinates, (the object remains stationary by symmetry, because the pulses are equal and opposite), and we can define a parameter m_{2} such that the total energy of the object increases by (1/2)m_{2}v^{2} for an incremental velocity v. Thus the total energy of the object following the emissions is E  ΔE + (1/2)m_{2}v^{2} relative to the moving system. Therefore, the change in energy of the object relative to the moving system of coordinates, which must equal the energy ΔE’ of the pulses relative to the moving coordinates, is 



This implies that 


Using the classical Doppler formula we have ΔE’  ΔE = 0, and so m_{1} = m_{2}, which signifies that the constant of proportionality “m” (usually called “rest mass”) between v^{2} and the change in E for incremental values of v is unaffected by the emission of the light pulses. However, the relativistic Doppler effect for two oppositelydirected pulses with incremental v gives (as noted above) 



Therefore we have 


and so 


which signifies that the “rest mass” of the object has been reduces by the amount ΔE/c^{2} due to the emission of energy ΔE. This is the argument that Einstein gave in his 1905 paper entitles “Does the Inertia of a Body Depend on its Energy Content?”. Of course, the argument assumes that energy is conserved with respect to each individual inertial coordinate system, and that the energy of an electromagnetic wave transforms from one system to another in proportion to the frequency (as implied by Maxwell’s equations as well as the relation E = hn). But even with these assumptions, it’s noteworthy that the massenergy equivalence relies crucially on the relativistic, as opposed to the classical, Doppler shift. 
