Aberration/Doppler Shift of Light

In this essay I want to re-derive the relativistic velocity addition formula, but this time I want to devise it for something that moves at the speed of light and I want to do it in all three spatial dimensions. Let us imagine, therefore, that an observer (let's call him Stan, the stationary observer) occupies the origin of a coordinate grid and that he detects a ray of light that comes to him in his x-y plane from a direction that makes an angle θ (measured in the counter-clockwise sense) with his x-axis. He can then calculate the components of the ray's velocity as

c_{y}=cSinθ

(Eq'n 1)

and

c_{x}=cCosθ.

(Eq'n 2)

Meanwhile, Monica, the moving observer, passes
Stan as she travels along their common x-axis at the speed V in the positive
x-direction. She also detects the ray and sees it coming to her from the
direction that makes an angle θ' with the x-axis. She then calculates the
components of the ray's velocity in her inertial frame as

c_{y}'=cSinθ'

(Eq'n 3)

and

c_{x}'=cCosθ'.

(Eq'n 4)

Now we want to devise equations that will relate Monica's measurements to Stan's.

We calculate a velocity as the ratio of a
distance crossed to a time elapsed and we use differential values for the
distance and the time in order to calculate the true velocity at a given instant
rather than the average velocity over some finite elapse of time. We thus
calculate the components of a velocity as v_{x}=dx/dt and v_{y}=dy/dt.
Now if we have two inertial frames separated from each other by a velocity V
oriented parallel to their common x-axis and if the primed from moves in the
positive x-direction relative to the unprimed frame, then we can convert the
velocity of a body measured in the unprimed frame into the velocity of the same
body as measured in the primed frame by applying the Lorentz Transformation to
convert the differential coordinates from one frame's to the other's. For
motions confined to the x-y plane we have

(Eq'n 5)

(Eq'n 6)

and

(Eq'n 7)

The ratios of those equations give us the velocity transformations that we want

(Eq'n 8)

and

(Eq'n 9)

If we make the appropriate substitutions from Equations 1-4 and divide through by the speed of light, we get

(Eq'n 10)

and

(Eq'n 11)

We can test the validity of those equations by squaring them and adding them together. The result should equal one and it does.

Stan and Monica can extend Equations 10 and 11 into all three dimensions by considering a ray of light whose path lies in a plane that makes an angle φ with their common x-y plane. Again Stan writes down a description of the components of the ray's velocity in his frame:

c_{x}=cCosθ,

(Eq'n 12)

c_{y}=cSinθCosφ,

(Eq'n 13)

and

c_{z}=cSinθSinφ.

(Eq'n 14)

And Monica writes down a description of the components of the ray's velocity in her frame:

c_{x}'=cCosθ'

(Eq'n 15)

c_{y}'=cSinθ'Cosφ

(Eq'n 16)

and

c_{z}'=cSinθ'Sinφ

(Eq'n 17)

Because the angle φ represents displacements effected only in the directions perpendicular to the relative motion between the two observers, both observers will measure and use the same number to describe it, so they use the same symbol for it in their calculations. That fact makes extending Equations 10 and 11 into all three spatial dimensions perfectly straightforward: we have

(Eq'n 18)

(Eq'n 19)

and

(Eq'n 20)

But aberration doesn't only change the direction in which a ray of light flies as we go from one inertial frame to another. It also changes the linear momentum and the energy that the ray carries.

We can express the law of the conservation of
linear momentum in many ways. In one way, we can say that in any isolated system
of bodies the motion of that system's center of mass does not change so long as
the system remains isolated from other bodies. So imagine that we have a body of
mass M+m sitting motionless on the origin of our coordinate grid and imagine
that at the time t=0 we convert the mass m into energy (in accordance with E=mc^{2})
in the form of a pulse of light that we project along our negative x-axis. At
the time t=t_{1} we convert the energy in the pulse back into a body of
mass m=E/c^{2}. At that instant the body lies a distance x=ct_{1}
from the origin of our grid. But to ensure that the system's center of mass did
not move in that interval, the body of mass M must have moved a distance X in
the positive x-direction such that

MX+mx = 0

(Eq'n 21)

or

(Eq'n 22)

In order to reach the point X the body M had to move at some speed V over the interval t1, so Equation 22 becomes

(Eq'n 23)

or

MV=-E/c

(Eq'n 24)

when we divide out the time interval. We recognize MV as describing the linear momentum carried by the body M, so we must now acknowledge that E/c describes the linear momentum carried by the pulse of light.

That imaginary experiment also tells us something else. It tells us that emission of the light exerted a force upon the body M in order to give that body its linear momentum. But then, because we know that every action comes real in the company of an equal and oppositely directed reaction (another way to express the conservation law pertaining to linear momentum), we also know that the body M must have exerted a force upon the light it emitted.

Stan and Monica know that the ray that they have observed has a source, a lamp that remains effectively motionless in Stan's frame. It lies astronomically far from the origin of the grid and it emits light in very narrow pulses that each carry energy E.

In Monica's frame the lamp moves in the negative x-direction at the speed V, but because the lamp remains far from Monica the angle ' that the ray from the lamp makes with her x-axis does not change noticeably in the course of her observations of it. In this frame, because the lamp forces a momentum E'/c upon each pulse of light that it emits, the lamp does work upon the pulse in the amount

(Eq'n 25)

Thus Monica measures energy in each pulse in the amount

(Eq'n 26)

Now she wants to relate that amount to Stan's measurement.

We know that when the lamp emits a pulse of
light carrying energy E, then the lamp loses mass in the amount m=E/c^{2}.
That mass comes out of the lamp's original mass M and in both Stan's and
Monica's frames those two masses must have the same proportion to each other;

m/M = m'/M'.

(Eq'n 27)

We know that

(Eq'n 28)

so we also know that

(Eq'n 29)

Thus we can rewrite Equation 26 as

(Eq'n 30)

We calculate the magnitude of the linear momentum that the pulse carries by dividing that equation by the speed of light.

Finally, although I have not properly deduced it yet in these essays, we have in mind the fact that light possesses two more properties that aberration alters: frequency and wavelength. As an aid to the imagination in devising the appropriate equations, let's suppose that our lamp emits pulses of light at regular intervals and that, instead of confining them to a tight beam, we allow the pulses to spread uniformly in all directions. In Stan's frame, then, the pulses comprise a set of concentric spherical shells that wash over Stan's instruments at the rate (frequency) =1/t (with t representing the interval between the emission of two successive pulses. Stan calculates the spacing between successive pulses (the wavelength) as =ct=c/.

As he measures the pulses, Stan notices that
they illuminate Monica moving past his measuring rod so that he can see her as
she occupies successive positions x=x_{1}, x=x_{2}, and so on.
From those readings Stan hopes to calculate the wavelength that Monica measures.
He begins with the wavelength that he measured and subtracts from it the
distance that the pulse moved parallel to the x-axis (that is, toward Monica) in
the time interval t, calculating

x=ct-VtCosθ.

(Eq'n 31)

But Stan also knows that Monica won't measure that amount of wavelength at all, simply because relative to his measuring rod hers is shortened by the Lorentz-Fitzgerald contraction.

Imagine that Monica marks the point on Stan's measuring rod that she is passing when one pulse illuminates her and then marks how far her own measuring rod has moved past that point when the next pulse illuminates her. In Stan's frame the distance that rod moves must equal the distance calculated in Equation 31, but because the rod is contracted, the number of meters is greater than the number of meters spanned by the same distance on Stan's measuring rod, in accordance with the Lorentz factor between Stan and Monica's inertial frames. Thus, in order to calculate the wavelength that Monica measures, Stan must multiply Equation 31 by that Lorentz factor to obtain

(Eq'n 32)

In order to convert that equation into one that lets Monica use the angle that she measures in her frame, we must use the inverse of Equation 18,

(Eq'n 33)

We obtain

(Eq'n 34)

And at last we calculate the frequency that Monica would measure by way of the equation ν=c/λ and obtain

(Eq'n 35)

If we now compare that equation to the first line of Equation 30, we find a strong and perhaps suggestive similarity. When we deduce the quantum theory we will discern the reason for that similarity.

Now we have one final question to ask and to answer on this topic: Does Monica agree with that calculation?

In Monica's frame the pulses still have their spherical shape, but they do not expand away from a common center as they do in Stan's frame; each pulse expands away from the point the lamp occupied when it emitted the pulse. In that image we conceive a beautiful illustration of the second postulate of Relativity: each pulse exists as a perfectly spherical balloon of light expanding away from a point that remains motionless before us, whatever inertial frame we choose to occupy.

Monica starts her derivation by calculating the raw time interval elapsing between the emissions of two successive pulses. Using the fourth equation of the Lorentz Transformation, she obtains

(Eq'n 36)

in which she has made the distance in the temporal offset term equal to the distance the lamp moves between the two emissions (x=Vt). The apparent contraction of otherwise dilated time reflects the process that transforms dilated distance into the Lorentz-Fitzgerald contraction.

Next Monica calculates the actual time interval between the pulses as she detects them by accounting for the distance the lamp moves between the two emissions. She knows that, for all its acknowledgment of the temporal offset, Equation 36 implicitly calculates the time interval that would elapse between the arrival of the pulses in her detectors as if the lamp had not moved at all. She now wants to take the lamp's motion into account for its effect upon the arrival times of the pulses at her detectors.

Knowing that t'' is too big, Monica subtracts the time it would take light to cross the distance the lamp moved between successive emissions, taking care also to account for the angle that the light's path makes with the direction in which the lamp moves. So she subtracts from t'' the amount

(Eq'n 37)

But, again because t'' is too large, that correction subtracts too much from t'', so Monica must add back an interval of time equal to the time it takes light to cross the distance the lamp moved in that interval; that is, she must add back

(Eq'n 38)

But, of course, that correction adds back too much, so Monica must make a third correction and then a fourth and so on indefinitely. She thus obtains an infinite series;

(Eq'n 39)

We can add up that series readily and calculate at last

(Eq'n 40)

Inverting the time interval yields the frequency, so to calculate the frequency at which the pulses of light from the lamp enter her detectors, Monica inverts that equation and obtains Equation 35.

Thus we obtain the equations describing the kinematics and the dynamics of light between inertial frames.

habg