In Search of Intermediary Motion

The question I want to ask is: can the following thought experiment detect absolute motion, or does a sort of intermediary motion emerge which is neither absolute motion, nor absolute rest? It is like Einstein relative motion, but without using a Galilean or Lorentz transformation between reference frames. It comes about when sound waves are used to investigate the motion of material objects through a stationary or moving medium.

According to Galileo, Newton, and Einstein, the principle of relativity states that no mechanical experiment can be done to detect absolute motion, or motion of a material object relative to an everywhere stationary medium (similar to Michelson-Morley aether).   A common reformulation of this principle state that:

♦The same formula is not used for the constant velocity v of a material object as seen by an observer in a reference frame in which the object is viewed as being at rest; or, as seen by an observer in a reference frame in which the object is viewed as being in motion (Galilean addition of velocities).

On a windless night (air molecules at rest relative to the earth), a train of length L is traveling at the constant velocity v along a flat straight section of train track.  There is an observer in the caboose (train reference frame) as well as another observer on the station platform near the track (earth reference frame).  They each have identical clocks with which to conduct the following thought experiment.  They will attempt to detect absolute motion, or at least test a common reformulation of the classical principle of relativity.  That is, to show that two observers can measure the same value for the velocity v of the train using the same formula, without a Galilean transformation, although these two references frames are moving relative to one another.

The observer in the caboose has a light source with which she will send a signal to the engineer at the front of the train.   She will lean through an open window to do this.  Outside the window, the still air does not have the velocity of the train, thus the air velocity will have either some or no effect on the velocity of the sound wave, but it will yield the same result for both observers.  Upon seeing the light signal the engineer will blow the train’s whistle, sending out sound waves which the caboose observer will be able to hear.  At the moment she sends the signal she starts the single clock that she has.  The platform observer will also see this signal and he will start his single clock at the same moment.  Thus, their clocks have essentially been synchronized.  The two observers will then be in a position to find the motion of the train (material object) relative to the still air (medium at rest, Michelson-Morley).

A sort of intermediary motion emerges from the mist

Over this short distance the light signal is effectively instantaneous, so that the time t she measures is essentially the time for the sound wave to travel the length L to her ear.  When she hears the whistle sound she stops her clock and then once again flashes her light.  The platform observer also stops his clock upon seeing this second flash.  The time interval between the two light flashes therefore represents the time interval between the departure and arrival events of the sound wave, as seen by the observer in either reference frame.  Disregarding reaction times, both observers should measure approximately the same interval of time t.  Since the speeds of the sound wave and the train are so much slower than the speed of light, the Special Relativistic effects of time dilation and length contraction are negligible.

As the experiment proceeds, the caboose moves in the forward direction, at the speed of the train, to meet the rearward travelling sound wave, so the sound wave will travel a distance that is less than L, the length of the train measured at rest.  The speed of the sound wave is not altered by the speed of the source, but the motion of the material object (train) is disengaged from the medium (still air).  This should lead to, approximately, identical time interval measurements by the observer in each reference frame.  The air molecules freely flowing between the two reference frames moving relative to one another make this supposition mechanically plausible.  The mechanical disengagement of the physical train from the still air permits the easy mathematical passage between reference frames, that is the critical premise that underlies this thought experiment.  The airy particles pass easily through the porous conceptual walls of the reference frames, like the ghostly spirits of a haunted house.

The caboose and the train engine are at a fixed distance apart.  They have formed a tandem which is moving through the air (medium), both at a single velocity, maintaining this distance of separation.  The sound wave and the caboose, having begun their journeys at the endpoints of L, will meet at the same location in space as seen by either reference frame.  The caboose will have the constant velocity v, and the sound wave will have the constant velocity c.  In the same duration of time t, they will have, taken together, traversed the distance L.  That is, the distance the sound wave has travelled rearward ct, added to the distance the caboose has travelled forward vt (distance = speed × time), should equal L.  Thus, all the variable values are available to each observer within all the adjacent reference frames.  To reflect the conditions of their meeting, somewhere within the length L, the following equation can be set up:

♦ L = ct + vt

If they have measured the same interval of time in both reference frames, then this formula can be solved for v, the velocity (motion) of the train as seen by either reference frame:

♦ v = [L / t ] – c

(A similar argument can be made if the train is moving in the reverse direction)

Let, L = 1000 meters; c = 343 meters/second; assume v = 30 meters/second:

♦not, t = L / c (measured when train is at rest) = [1000 m] / [343 m/s] = 2.92 s

♦but, t = L / (c +v ), (train in motion) = [1000 m] / ([343 m/s] + [30 m/s]) = 2.68 s

♦v = [L / t ] – c = [1000 m / 2.68 s] – 343 m/s = 30.1 m/s

This expression contradicts the Galilean, Newtonian, and Einsteinian principle of relativity in that although the two reference frames are moving relative to each other, they can each use one and the same formula to find the velocity of the train as seen from either reference frame.  This results in bypassing the need for the addition of velocities from the Galilean transformation between references frames, when sound waves are used to investigate the motion of a material object through still air.  Thus, a sort of intermediary motion emerges from the mist amongst the reference frames.