I introduce a modification to my thought experiment; it will expand the use of the lantern as a light signal by the caboose observer. This modification will pull the platform observer into the midst of the experiment.

There is once again a train of length, L, moving at a constant velocity, v, along a level straight section of track on a windless day. Also, there is again an operator in the engine car and an observer in the caboose car. The platform observer will now take on a more significant role.

The air molecules, because it is a windless day, along with the platform and the Earth, are at rest relative to the moving train. There is a reference frame attached to the train and a reference frame attached to the Earth thus creating two coordinates systems moving relative to one another. Since the outside air / medium is disconnected from the motion of the train, the method outlined in this thought experiment makes it possible to calculate the velocity of the train. If the experiment were conducted within a single enclosed car, then the air molecules would follow the motion of the train and it would be impossible to find the velocity of the train by the method I have presented here.

The thought experiment begins with the caboose observer flashing her light signal to the engine car; at the same moment she begins her clock. The operator blows the train whistle at the moment he receives the light signal. Over this short distance the light signal is effectively instantaneous. She is prepared to measure the time, t, for the sound wave from the whistle to reach her.

When the she hears the sound wave travel the length of the train, from the engine to the caboose, she flashes her lantern once more, and stops her clock. Now this is where my modification enters the experiment. The platform observer also has a clock and he is able to see the flashes of light from the lantern. So, at the first flash he begins his clock, and at the second flash he stops his clock, thus also measuring the flight time, t, of the sound wave.

Now the physics question becomes, will both observers measure the same time, t? Since the two observers are moving at a constant rectilinear velocity relative to one another, by the principle of relativity they should find different velocities as viewed from the other’s reference frame. This applies to a material object flying through space, because they are in reference frames moving relative to each other. However, this does not apply to sound waves because of their violation of invariance, a concept known to science.

Her goal is, once again, to find the velocity of the train entirely from within the reference frame attached to the train. The principle of relativity says this not possible, but she imagines herself to be a clever science girl. She ponders upon the problem and imagines that a sound wave would be a solution to her problem; but may open a Pandora’s Box, of which she knows not the contents. Nonetheless, she proceeds.

She sets her equations as I have shown before. For the train moving forward, the caboose meets the rearward traveling sound wave within the distance L = ct - vt, with c representing the known speed of sound; the sound wave and the caboose start at the endpoints of L. If the train were to go in reverse, the sound wave from the whistle at the engine would have to overtake the rearward going caboose, so then, a similar type of formula would be applied: L - vt = ct. Both the sound wave and the caboose start at the endpoints of L, with each moving in the same direction. Each of the above formulas can be solved for the velocity of the train, v.

If her algebra is correct, what are the implications? She has found the velocity of the train, she thinks, but she is also aware that this seems to contradict the principle of relativity. The two observers have measured the same velocity for the train though each is in a reference frame moving relative to the other. The train length, L, is found from the technical specifications. The speed of sound, c, can be found in any science text. Thus the platform observer and the caboose observer can use the same equation, L = ct - vt. If they measure the same time, t, as implied by the formulas, then they will each find the same velocity, v, for the train. If they measure the same time, t, as implied by the formulas, then they will each find the same velocity, v, for the train.

The caboose and platform observer could also find the velocity, v, of the train by another means. By noting the landmarks immediately opposite to the flashes, relative to the embankment, and by measuring the distance between the landmarks by some device, then v = d / t could be found. But the landmark method cannot be extended to find a definition for simultaneity, or make use of Doppler to find a deeper interpretation of the motion of a material object through space. Even more, only sound waves are a mechanical means that can be done from within the train’s reference frame to find v in as advantageous a way, as by sound waves.

A scenario that is similar, but not the same, is by throwing a lump of coal rearward (instead of a sound wave) from the steam engine with a strong arm. Neglecting air resistance and gravity, it should fly in a level and straight line. From this thrown material object or any other similar type of mechanical experiment, she cannot find the velocity of the train, while riding upon the train. But by a sound wave, she can perform an experiment that allows her to find the train’s velocity. She has uncovered another of reality’s many paradoxes.

If the platform observer threw a lump of coal, with the same arm strength as the train operator, to another person on the platform, then the platform observer would measure the same velocity (v = d / t) as the train observer for the velocity of the of lump of coal between the engine and caboose, though the train is moving and the platform is at rest. The outsider, by addition of velocities, measures a different velocity for the lump of coal, but he cannot communicate the illusion of the caboose observer’s measurement to her. He sees the lump of coal on the train travel a shorter distance and thus a shorter time (by Galilean transformation). But she is trapped in her illusions, with no mathematical way to clear the shadows of her blindness. She has no way to find the velocity of the train.

Until a sound wave is applied to the problem. That sound waves violate invariance is already well-known to physicists. Exploiting the phenomenon that sound waves do not gain any addition of velocity (v_x = v'_x - v₀) from transfer of momentum, then this thought experiment makes it possible to measure the same velocity, distance, and time, across reference frames moving relative to each other. The seemingly paradoxical statements can both be true with a little cleverness. That the principle of relativity reflects reality and does not reflect reality seems an inescapable trap. The observers in two reference frames moving relative to one another can both measure different velocities for an object and the same velocity for that object, namely the velocity of the train.

An intermediary motion arises from betwixt the reference frames. It is different from the transfer of momentum imposed upon a material object by its initial cause of motion. Whether the object or reference frame is already in motion, or at rest, the law describing the object’s motion will be the same simple law (v = d / t), as viewed from within the reference frame. A sound wave leaps the hedge between reference frames; its velocity is not altered by the state of motion, or by the state of rest of the source. And by lifting the veil from this intermediary motion, she has found the velocity, v, of the train.