The Photodynamics of Moving Bodies
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In the quantum theory we associate with a photon two kinematic properties that correspond to two dynamic properties. We associate the photon=s energy with an angular frequency,
and we associate the photon=s linear momentum with a wave number,
Those kinematic properties come to us as the reciprocals of the elapsed time and the spanned distance that we associate with one radian of one cycle of the electromagnetic wave of which the photon is a curd (ah, yes, the cottage cheese theory of light). I now want to work out the equations that will transform those properties as one observer would measure them into the same properties as another observer, moving relative to the first along their common x-axis, would measure them. In doing so I will obtain something like a reciprocal Lorentz Transformation.
As usual, we will use a series of imaginary experiments to work out the equations that we want. As we do with real experiments, we so design our imaginary experiments that they will produce events that will reveal the information we want. Our imaginary measurements of those events will have missing pieces whose mathematical form we infer from the requirement that they be consistent with the known laws of physics and of logic.
We begin with Observer Stationary Stan, who has a body of mass M floating at the origin of a coordinate grid that he has etched into the phlogistonic emulsion embedded in the Šther filling the region in which he will conduct his experiment. We can use non-existent fantasy materials, such as phlogiston and Šther, in these experiments as aids to the imagination so long as they do not interfere with the action of the experiment: in this case we use the phlogistonic emulsion as a medium on which we can draw and make measurements without in any way changing the properties of the experimental body or of any photons involved in Stan=s experiments.
Stan contrives to make his experimental body emit two identical photons in opposite directions in the x-y plane (I will ignore the z-direction in this essay for convenience, since whatever we infer for the y-direction applies also to the z-direction). Emission of those photons reduces the mass of the emitting body byΔM, so each photon carries energy
Because photons possess no rest mass, they have the relationship E=pA c between their energies and their linear momenta. Thus each of Stan=s photons carries linear momentum
in which the vector c designates the fact that the photon moves at the speed of light in a specific direction. Because the two photons go in opposite directions, their linear momenta cancel out and the linear momentum of the emitting body remains equal to zero.
Stan has put three clocks into his setup. The clock on the emitting body stops at time t0 when the body emits the two photons. The two other clocks, one at y=a, x=b and the other at y=-a, x=-b, stop at time t1 when they absorb the photons. As the photons travel they etch a trace of their passage into the phlogistonic emulsion, thereby drawing straight lines that make an angleθ with the x-axis such that
Now let=s turn our attention to Observer Mobile Monica. She moves at speed V in the negative direction parallel to Stan=s x-axis and she has contrived for her x=-axis to coincide with Stan= s x-axis for convenience. She has also established her y=-axis parallel to Stan= s y-axis. Thus, as Monica observes them, Stan, his coordinate grid, and his apparatus move at the speed V in her positive x= -direction. If Stan measures between two events the distances and duration x, y, and t, then Monica will measure between those same two events the distances and duration x=, y=, and t=, which she can relate to Stan=s measurements through the Lorentz Transformation:
1. In Monica=s frame Stan=s frame moves a distance Vt= between the two events, so Monica must add that distance to the dilated version of the distance that Stan measures between the events in the x-direction. Further, because Stan=s clocks tick off time that has dilated in Monica=s frame to fill more time ticked off Monica=s clocks, Monica must account for that dilation in her calculation;
2. Oriented perpendicular to the direction of relative motion, the y-direction doesn=t change for either observer;
3. As noted, Monica must account for the dilation of time elapsed on Stan=s clocks. If Stan=s clocks are separated from each other by some distance in the x-direction, Monica must add a temporal offset to the dilated time to compensate the fact that in her frame Stan=s fore clock appears pushed into the past relative to his aft clock;
Monica also knows that in her frame Stan=s emitting body initially ponders a mass
That fact implies that the photons carry away an energy that corresponds to
but Monica wants a proof. Conservation of energy should give her a good enough proof, but she wants to augment it with a proof that involves a different conservation law.
Imagine that a rod made of massless, transparent unobtainium extends in the negative y-direction from Stan=s emitting body. The body emits a burst of photons into the rod, they propagate to its far end, and there they condense into a body of massμ (the physical laws that process would violate are irrelevant to the subject of our imaginary experiment, so we can ignore them just as we ignore other irrelevant details, such as the color of the emitting body). Because of Newton=s third law of motion, the emitting body recoiled when it emitted the photons, so it moved a small distance y1 from the origin of Stan=s coordinate grid as the photons moved a distance y2 before condensing into a small body. No external forces acted on that little system, so it remains balanced on the origin of Stan=s coordinate grid; that is, the system=s center of mass remains on the grid=s origin. That fact means that if Stan puts a needle on his x-axis and pushes it against the unobtainium rod where it crosses the grid=s origin, the compound body will accelerate but it will not rotate. From the counterbalancing of the inertially instigated torques in the composite body Stan can then calculate
Reality has the fundamental property that any event that exists for one observer necessarily exists for all observers (even if they don=t actually observe it). Thus Monica will also observe Stan=s composite body accelerating (though at a rate different from the one that Stan calculates from his measurements) and not rotating. That latter observation tells Monica that the composite body is balanced in her frame, that the inertially induced torques acting on the component bodies counterbalance each other, so she writes her own version of Equation 11 as
Monica knows that y=2=y2 and y=1=y1. She also has Equation 9, so now she knows that the law of conservation of angular momentum necessitates that Equation 10 stand true to Reality.
Referring back to Stan=s original experiment, Monica now knows for certain that the two photons each carry an amount of energy equal to
By way of Equation 1 and Stan=s measurement of each photon=s angular frequency, Monica also knows that
This puzzles Monica because she knows that frequency corresponds to the reciprocal of an elapsed time, so time dilation should make the frequencies of the photons in her frame smaller, not larger. For the moment she leaves it as a paradox that she will resolve later and turns her attention to the photons= linear momenta.
In Stan=s frame the emitting body doesn=t move, so it suffers no change of linear momentum when it emits the photons. But in Monica=s frame the body initially carries momentum M=V, so when it emits the two photons it loses an amount of linear momentum equal toΔM=V. To satisfy the laws of conservation of linear momentum, the photons must have taken that momentum with them, so now Monica knows that each photon took
to add to the linear momentum conferred upon it by its emission.
She must add that amount to the x-component of the basic momentum of the photon as it appears in her frame. If she and Stan were to observe a photon propagating in the x-direction only and if, at some instant, they could take a picture of it by way of some Štherial camera, they would find that the photon spans the distance between two points. But that distance is shorter for Monica than it is for Stan, so the wave number of the photon and, therefore, its momentum must be correspondingly larger; that is, Monica has
When she adds in the momentum from Equation 15, she gets
She knows right away that the component of the momentum in the y-direction cannot differ between her frame and Stan=s, so she has
And lastly Monica looks at the photon=s energy and associated angular frequency. She already has Equation 14, which tells her how the photon=s fundamental energy differs in her frame from its value in Stan=s frame. But in her frame another phenomenon comes into play. Whenever a body absorbs or emits light, that light exerts a force upon the body as it enacts the transfer of momentum, thereby obliging the body to exert the necessary equal and oppositely directed force upon the light. If the body is moving, then the latter force does work upon the light, changing its energy. For a single photon emitted in the direction of the body=s motion (or for the component of a photon=s momentum parallel to the direction of motion) the net work done on the photon corresponds to
Adding that result to Equation 14 gives Monica
In accordance with Equations 1 and 2, Monica divides Equations 17, 18, and 20 by Planck=s constant (S) and obtains the wave-parameter analogue of the Lorentz Transformation:
With those equations she conducts a simple test. She calculates
which gives her the wave-parameter analogue of the Minkowski metric.
Now she can go back and resolve a minor paradox that she discerned earlier. She takes the case in which k=x=0, which represents a photon moving only in the y-direction in her frame, and calculates from Equation 21 that
Substituting that into Equation 23 to calculate the corresponding frequency gives her
which displays what she originally anticipated based on time dilation acting on the emitted photons. She now has the wave-parameter analogue of the Lorentz-Fitzgerald contraction.
And finally Monica substitutes kx=kCosθ and ω/c=k into Equation 21 to get
From her previous study she knows that Stan=s photons trace in her frame paths that make an angleθ= away from her x=-axis in accordance with
Monica substitutes that into Equation 27, divides the equation by Cosθ+V/c, and multiplies the equation by 1+Cosθ(V/c) to get
which is the reciprocal, to within a constant factor, of the relativistically Doppler shifted wavelength of the photon. Substituting kx=Cosθ(ω/c) into Equation 23 gives her
which describes the relativistic Doppler shift of the photon=s frequency.
Now Stan and Monica have a version of the Lorentz Transformation that they will find useful in the development of the quantum theory, among other things.
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