The principle of relativity as annunciated by Galileo, Newton, Einstein, and others, states that no mechanical experiment can be done in an enclosed room that could detect the absolute motion of the room. There may not be anything such as absolute motion, only statements of relative motion expressed by mathematical formulas; but I hypothesize that the following thought experiment outlines a way of defining an intermediary motion that arises from some Aether, that resides between absolute motion and relative motion. This is in contradiction to any form of Relativity that has been expressed by modern science.

Suppose a long train with several cars, a caboose, and a locomotive engine, travels down a long level section of track at a constant velocity, v. There is an observer in the caboose and an engineer in the locomotive car. It is a windless day (still air, motionless medium). This observer might ask: can I find the speed of the train relative to the air, or another observer at rest on a nearby platform (both at rest relative to train)?

She has a light source (a lantern, maybe) to send a signal to the locomotive car and engineer. This light signal is effectively instantaneous over this short distance. He blows the whistle when he receives the signal (disregarding reaction time). If she starts her clock when she sends the light signal, then she can measure the time, t, for her to hear the returning sound signal (speed of sound, c; c is a symbol for sound and light, and represents a constant, no addition of velocities).

Thus, during the same time that the train is in forward motion, the whistle sound wave is in rearward motion. She speculates that she will meet the sound pulse somewhere within the distance, D, from the engine to the caboose. By algebra, with the opposite endpoints of D as the starting places of the caboose’s motion, and the sound wave’s motion:

♦ D = ct + vt (t = t)

♦ v = [D / t] - c

Is she correct in thinking that she can find the velocity, v, of the train based on this total time, t; which she measures on her single clock? She imagines that this thought experiment assails the wall of separation between reference frames; between the reference frame attached to the moving train, and the other reference frame attached to the platform which is considered at rest. She has found a method for determining her speed relative to the earth; based only on information available from within the reference frame attached the train. She is furthermore sharing in the motion of train (she is at rest relative to the train).

The Newtonian laws of motion do not take their simplest form in either reference frame. Instead, each takes on a form related to the Galilean transformation equations (similar to the Michelson-Morley formulas: L = ct + vt; t = L / [c + v]). That is, in both the moving frame, and the frame at rest, the distance, L, between the two starting points stays the same. Despite the Galilean transformation, the distance, L, will be preserved across reference frames (L = (x₂ - vt) - (x₁ - vt) = x₂ - x₁ = L; the vt‘s have cancelled out). The observer at rest in the rest area will note that the caboose is in motion, as well as that the sound wave is in motion. But the times measured in each reference frame will be the same, thus allowing the observer on the caboose to solve for v, the velocity of the train. This contradicts the principle of relativity once again.