One Velocity, Two Reference Frames

According to 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 a stationary medium (similar to Michelson-Morley).   Common reformulations of this principle state that:

1) The velocity of a material object takes on the simplest formula, as seen by an observer at rest in a reference frame, no matter whether the reference frame is at rest, or moving with constant velocity, v.

2) 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 evening at dusk (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.  Also, this will not be the simplest form for the velocity of the train:

♦ v = [d / t]

To find the absolute motion of the train (material object) relative to the still air (medium at rest – Michelson-Morley), 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.  He will then blow the whistle, sending out sound waves which the caboose observer will be able to hear.  At the moment she sends the light 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.

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.

Disregarding reaction times, both observers should measure the same interval of time, t.  Since the sound wave and the speed of the train are so much slower than the speed of light, the relativistic effects of time dilation and length contraction are negligible.  The caboose moves forward 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 at rest.  The speed of the sound wave does not change, but the motion of the material object (train) is disconnected 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 reference frames moving relative to one another make this supposition mechanically plausible. They easily pass through the “conceptual walls” of the reference frames, like the ghostly spirits of a haunted house.

The sound wave and the caboose begin their journeys at the endpoints of L.  The caboose has the constant velocity, v, and the sound wave has the constant velocity, c.  To reflect the conditions under which they will meet, then 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 of the train as seen by each reference frame:

♦ v = [L / t] - c

This is obviously not the simplest formula for the velocity of the train in either reference frame.  This expression contradicts the 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 discarding the need for the addition of velocities from the Galilean transformation between references frames.