Acceleration and the Speed of Light

We know that a ray of light passes all observers at the same speed, regardless of how those observers move relative to each other, so long as those observers occupy inertial frames of reference. Can we remove that last proviso from that statement? Can we modify the statement to include observers occupying non-inertial (i.e. accelerating) frames of reference?

Let's assume that we cannot. Let's take as our hypothesis the statement that in an accelerating frame of reference the speed of light differs from 299,792.458 km/sec (186,282.709 miles/sec.). That statement gives us two options when an accelerating observer stops accelerating and takes up occupancy of an inertial frame:

A) light continues to move past the observer at its altered speed, or

B) the speed of light reverts to its proper value.

Option A violates the second postulate of Relativity, which we have deduced as a theorem. Further, were it to come true to Reality, we could have two observers standing motionless relative to each other in an inertial frame and measuring different velocities from the same ray of light. We recognize that possibility as an absurdity, dismiss Option A from consideration, and accept Option B as the legitimate extension of our hypothesis.

If we have an observer who suffers a period of acceleration in order to go from occupying one inertial frame to occupying another inertial frame a substantial velocity away from the first, when does the speed of light change for that observer? Again, we have two options:

A') the speed of light changes during the period of uniform acceleration, or

B') the speed of light changes at the ends of that period, when the acceleration itself changes.

Option A' requires of Reality that a uniform acceleration increase/decrease the speed of light from its proper value and then decrease/increase it back to its proper value in time for the acceleration to end. We know that a single phenomenon cannot yield two different (indeed, opposing) effects. Further, we know that it takes a difference to make a difference and yet we have nothing in a uniform acceleration sufficiently different to instigate a change from increasing the speed of light to decreasing it (or vice versa). Thus, we dismiss Option A' as false to our hypothesis and accept Option B' as true to the hypothesis.

For any ray of light, then, we describe its motion through a frame accelerating at the rate as

c=c_{0}+Δc,

(Eq'n 1)

in which c_{0}=299,792.458 km/sec and Δc=Δc(α)
represents the change in the speed of light due to the acceleration of the
frame. But yet again we have two options for what Reality does in accordance
with our hypothesis:

A'') Δc represents a change in the velocity of a light ray, or

B'') Δc represents a change in the speed of a light ray.

Before we examine those options in more detail we should note the fact that Option A'' gives us a vector addition and Option B'' gives us a scalar addition. In Option A'' Δc represents a vector oriented in the direction parallel to the acceleration: only that orientation gives us a unique augmentation of the motion of light in an accelerating frame. Option B'', on the other hand, takes the opposite tack and adds the same increment to the speed at which all rays of light fly, regardless of the direction in which those rays move, thereby imputing to Δc no unique preferred direction.

Let's assume that Reality manifests Option A''.
We have an observer O-1 riding in an elevator that moves in the negative
y-direction and accelerates in the positive y-direction. At some time t_{0}
the elevator comes to rest in an inertial frame occupied by a second observer
O-2 and begins to move in the positive y-direction. Two small holes penetrate
the elevator's walls in such a way that a straight line passing through their
centers lies parallel to the observers' temporarily common x-axis and
perpendicular to their common y-axis.

Observer O-2 projects two pulses of light in
opposite directions parallel to their x-axis in such a way that at the time t_{1}=t_{0}-Δt
the pulses pass through the opposing holes and enter the elevator. At t_{0},
when the elevator comes to rest, the pulses have traveled halfway across the
elevator's width and pass through each other. And at t_{2}=t_{0}+Δt
the pulses pass through the holes and exit the elevator. O-2 knows that will
happen because, with the elevator under uniform acceleration, from t_{1}
to t_{0} the elevator descends a certain distance and then from t_{0}
to t_{2} it ascends the same distance, bringing the holes back up to the
line along which the pulses fly.

For O-1 each pulse enters the elevator with a small component of velocity in the positive y-direction (due to aberration of the light moving horizontally in O-2's frame) plus Δc. After crossing the elevator the pulse has a y-ward component of velocity in the negative y-direction (again, due to aberration) plus Δc, which has not changed. Over the course of that traverse the contributions to each pulse's y-ward motion due to aberration cancel out and those due to Δc accumulate. Thus, in O-1's frame the pulses would miss the holes and strike the elevator's walls rather than exit the elevator.

Reality will not manifest any phenomenon that presents mutually contradictory aspects to different observers, so we must exclude from our description of Reality any proposition that leads to a description of such a phenomenon. Thus we dismiss Option A'' from our consideration.

Thus we infer that Option B'' correctly describes Reality. But what kind of Reality does it describe?

In O-1's frame the pulses will still miss the exit holes in the elevator walls. Because the pulses travel faster than the standard speed of light, they will reach the opposite walls of the elevator too early and will not have descended enough to meet the holes. However, if all lengths in the accelerating frame dilate in the proportion

γ=1+Δc/c_{0},

(Eq'n 2)

then the pulses will take a longer elapse of time in just the right proportion to reach the walls when the exit holes have come to the right position for the pulses to pass through them. That dilation of distances must appear to O-1 as a contraction of all distances in any inertial frame that coincides with the accelerating frame at some given instant; the elevator must appear to have expanded relative to an identical elevator at rest in some inertial frame.

As the elevator passes a ruler mounted beside its track, both O-1 and O-2 must see its sides pass by the same marks on that ruler. Thus O-2 will see the elevator expanded by the factor γ. But in O-2's inertial frame the light pulses travel at the standard speed, so they will arrive at the opposite walls of the elevator too late to meet the exit holes. Thus we cannot mandate an expansion of lengths in accelerating frames.

Do we have any other possibilities before us that would, to use Ptolemy's term, save the appearances? In order for the pulses to pass through the exit holes for both observers we must have as true to Reality

c't'=c_{0}t,

(Eq'n 3)

which necessitates that clocks in the accelerating frame count time more slowly than clocks do in any inertial frame with which the accelerating frame instantaneously coincides; that is,

t'=t/γ

(Eq'n 4)

This accelerative time dilation effect seems to solve our observers' problem quite handily.

So now O-1 lets a small body drop inside the elevator and measures its apparent acceleration. When the observers complete and compare their calculations, they find that the acceleration that O-2 measures of the elevator differs from the apparent acceleration that O-1 measures of the dropped body in accordance with

a'=aγ^{2}.

(Eq'n 5)

But that spoils the agreement in Equation 3. O-1 must measure the same acceleration of the light pulses and that too-large acceleration makes the pulses miss the exit holes again.

We can fix that problem by asserting that

t'=t/√γ.

(Eq'n 6)

In that case Equations 3 and 5 become

c't'=c_{0}t/√γ

(Eq'n 7)

and

a'=aγ

(Eq'n 8)

which gives us

a't'=at√γ.

(Eq'n 9)

Thus we seem to have solved our problem: the distance that the pulses fly in the two reference frames has the same proportionality to the distance that they fall in one of those frames and the distance the elevator moves in the other. But Equation 7 necessitates that distances in the accelerating frame contract in proportion to 1/√γ, so in O-2's frame the pulses will arrive at the opposite walls too early to pass through the holes. In that case the pulses would pass through the exit holes for O-1 and not for O-2.

At last we come to the only solution that works
for both observers. Both O-1 and O-2 will see the pulses pass through the exit
holes if and only if Δc=0. Now we can augment Einstein's second postulate to
read: a given ray of light passes all observers at the same speed (299,792.458
kilometers per second), regardless of any differences in position, orientation,
velocity, or acceleration that we may find between any two of those observers.

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