The Lorentz Transformation of Light
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Stationary Stan and Moving Monica still don=t feel completely comfortable with their previous derivations of the relativistic Doppler shift, so they want to deduce the formulae by yet another logical path. They know that a good way to check their previous work comes out of finding an alternative way of doing the work and following it to see whether it leads to the same result. This time they want to take the sinusoidal wave function that describes the propagation of a ray of light and in some way apply the Lorentz Transformation directly to the argument of that function. They start with a very simple analysis of a plane wave propagating in their common positive x-direction.
Stan uses primed variables to represent his measurements and Monica, moving at the speed v in the positive x-direction relative to Stan, uses unprimed variables. They decide that they will track a give phase of the wave, a crest or a trough for example, over some distance and some elapse of time. Monica notes that her basic description of the wave conforms to
Because Monica can express the angular frequency that she measures as the wave passes her asω=kc, she can rewrite that description of the wave as
That equation confirms that a given phase of the wave propagates in the positive x-direction at the speed of light (in accordance with x-ct=constant). In like manner Stan describes the wave as he observes it through
For any given phase in the wave the argument of the exponential has a value that lies between zero and 2π. Since Stan and Monica have agreed to observe the same phase of the wave, they know that the statement
must stand true to Reality. To obtain the required measurements they need two events in which the wave participates so that they can measure the distance that their phase point on the wave travels and the time that elapses between the wave=s participation in those events. They might, for example, consider the wave passing two particles as defining the required events, noting that those particles can have any motion whatsoever because their locations before and after the phase point on the wave passes them have no relevance to Stan or Monica=s measurements.
Once Stan and Monica have made their measurements, Stan can check them by using the Lorentz Transformation to translate Monica=s measurements into his frame of reference:
Applying those equations to Equation 4, Stan gets
That result matches the basic relativistic Doppler shift. Note that because
we also have as true to physics
which expresses the relativistic Doppler shift more familiarly in terms of the frequency of the wave.
Having thus confirmed the validity of this approach, Stan and Monica want to extend it to light moving in directions other than the direction of the relative motion between their inertial frames. Now our observers focus their instruments upon a brief, narrow pulse of light emitted from a source that occupies their frames= common origin at the instant that x=0 and x==0 coincide. The pulse goes to a detector, which absorbs it, thereby enacting the second event that our observers= measurements require. In Stan=s frame the pulse propagates in a direction that tilts an angleθ= away from the positive x=-axis (measured in the counterclockwise direction) and in Monica=s frame the pulse=s path tilts an angle θ away from her positive x-axis.
Each observer has two ways to determine the distance that the pulse travels between the event of emission and the event of absorption; multiply the elapsed time by the speed of light or divide the distance measured only in the x-direction between the two events by the cosine of the angle that the pulse=s trajectory tilts away from the x-axis. Those distances must equal each other, so Stan writes
But now he must acknowledge explicitly the two-dimensionality of the geometry of the experiment. The Lorentz Transformation only affects the part of the pulse=s trajectory that parallels the x-axis; the part that parallels the y-axis remains unchanged between inertial frames. He wants to relate the angle that he measures between the pulse=s trajectory and his and Monica=s mutual x-axis to the angle that Monica measures, so he has to figure out a way to apply the Lorentz Transformation to that situation. He doesn=t have to put a lot of thought into that problem because he knows that the length of the pulse=s trajectory must transform in the same way that the elapsed time does, so he can use Equation 11 to work out the transformation of the cosine, from which he can calculate the angle. For convenience, he multiplies Equation 11 by the cosine and gets
Monica, of course, creates her own version of that equation. Applying the Lorentz Transformation to his equation and dividing out the Lorentz factor (because it multiplies both sides of the equation equally), Stan gets
Substituting for x from Monica=s version of Equation 12 yields
which Stan solves easily for
Monica immediately solves that equation for cosθ and finds that the solution agrees with the result of her own derivation,
Those two results display the symmetry that we expect from the theory of Relativity.
Our observers now go back to the beginning of this episode. They rewrite the wave function describing the propagation of the pulse as
Again they want to relate the arguments of their respective versions of that function so that they can work out the relationship between their measurements of the pulse=s wave number. We see the obvious choice as
but the two-dimensionality of the situation warrants caution. That equation stands true to physics as a description of the pulse=s propagation, but we cannot use it to work out the relationship between the wave numbers. As the dot product in Equation 17 indicates, the wave number represents the pulse=s propagation vector and as such it had two components, of which only the x-component,
gets caught up in the relativistic distortion of space. So we eliminate the y-components in the dot product in Equation 17 and Stan and Monica rewrite Equation 18 as
Again Stan makes the appropriate substitutions from the Lorentz Transformation (Equations 5 and 6) and the aberration law (Equation 15) and gets
Multiplying that whole equation by c+vcosθ, then gathering together the terms that don=t cancel out and factoring the resulting polynomials gives him
He then solves that equation for
Monica, carrying out her own version of the derivation from Equation 20, gets
And the relationω/k=c allows Stan and Monica to convert Equations 23 and 24 into the equivalent equations relating their measurements of the pulse=s angular frequency.
Thus Stan and Monica obtain what they call the Lorentz Transformation of Light. If a body, moving in the positive x-direction at a speed v relative to Stan=s frame of reference, emits a pulse of light in its own frame with angular frequencyω, wave number k, and in a direction that makes an angle θ with the positive x-axis, then Stan will measure the corresponding quantities in his frame in accordance with the aberration and relativistic Doppler shift equations:
in which, as usual,
the Lorentz factor. And if the emitting body stands at rest in Stan=s frame, then Monica will calculate her expected measurements of the pulse=s properties as
As Stan and Monica admire the elegance of those equations and of their derivation some wag reminds them that in the preface to his book ARelativity@, first published in 1916, Albert Einstein expressed agreement with Ludwig Boltzmann=s comment that elegance should be left to tailors and cobblers. Einstein meant the comment as a dismissal of his writing style (and not necessarily of his physics), but we can take it as indicating a good analogy: as the tailor wants to make a suit that fits a man as well as possible, so the physicist wants to make a theory that fits Reality as well as possible. As the tailor uses needle and thread to stitch together panels of cloth into a suit, so the physicist uses logic and proportion to unite equations of mathematics into a theory. And as the tailor may need to make alterations to improve the fit of his suit to his client, so the physicist may need to make corrections to improve the fit of her theory to her Reality. Einstein may object, but in the theory of Relativity we have a truly elegant mathematical theory of physics.
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