Acoustic Simultaneity

According to relativistic mechanics, two events occur simultaneously if the light from each of these two spatially separated events meet at the midpoint of the line adjoining them, at the same time.  Additionally if this simultaneity occurs in a reference frame that is considered to be stationary, then the events will not be generally regarded as simultaneous in a reference frame that is moving with a linear constant velocity v relative to the stationary frame.  This may be true for light waves, but it will not be true for sound waves, which rely for their propagation on a medium that passes easily through the porous conceptual walls of every inertial reference frame. The open still air will not be contained within the walls of both reference frames, in that the air molecules will be at rest according to the viewpoint of one reference frame, but at the same time in motion according to the viewpoint of the other reference frame.  This disengagement of the air molecules from the motion of any moving material object within a reference frame is the primary underlying proposition of this paper.

A thought experiment oft used to explicate simultaneity involves an archetypical Einstein train of length L travelling down a long level straight stretch of track, on a windless night, at the constant velocity, v.  The air/medium is at rest relative to the earth and track.  An observer, holding two mechanically identical clocks, is seated on the roof of the train at the midpoint between the engine and the caboose.   She is at rest in the train reference frame, but she feels the still air rushing past her face at the apparent velocity of w (v = w).  A storm threatens, and a number of lightning bolts have struck the ground around the rapidly moving train.  She prepares herself.

The engine and caboose are at the endpoints of the train, and they along with the midway point on the line joining them, have formed a tandem moving through space such that they maintain their distances of separation, whether the train is in motion or at rest.  After a few moments, two lightning bolts strike, one bolt at the engine end of the train, and the other bolt at the caboose end of the train. These two events occur simultaneously, so that the light generated by the strikes against the metal, at each end, should arrive at the midpoint observer at the same time, in the train reference frame, as is supposed by the Special Theory of Relativity. However, the sound wave that is generated by the lightning strike against the metal at each end of the train will not arrive at the midpoint observer at the same time due to the motion of the train reference frame through the still air.  Or conversely, so as to preserve mechanical symmetry for the train observer, an apparent wind must blow through the stationary train reference frame which causes the two travelling sound waves to arrive at the central location at different times.  So, the train observer determines to use these light signal to mark the departure events of the two sound waves within the train reference frame.  The light wave reaches her nearly instantaneously at this short distance, so she uses these flashes as the signals to start each of the clocks she holds so that they will now tick synchronously. 

The train meetting the wave.

Disregarding observer reaction times, the ticking clocks will essentially measure the time intervals t_x for each sound wave to reach the central point as the train is in motion. The sound waves travel at the same constant velocity c through the still air towards the middle location, but the moving train will shorten the distance of travel for the sound wave coming from the engine; and lengthen the distance of travel for the sound wave coming from the caboose. Thus, the two time intervals will not be equal, the arrival events of the two sound waves at her ears will occur at different times and positions within the train reference frame.  So, taking this into account, and that time equals distance divided by velocity, with the distance value from the endpoints to the midpoint mathematically being 0.5L:

♦t₁ = [0.5L – vt₁] / c = 0.5L / (c + v)

♦t₂ = [0.5L + vt₂] / c = 0.5L / (c – v)

-Since t₁ ≠ t₂, adding these two times gives,

♦T = t₁ + t₂ = 2[0.5Lc] / (c² – v²)

If the train were to be regarded as stationary while the earth and atmosphere are moving past it at the velocity w so that the air/medium remains at rest relative to the earth, then to maintain symmetry, an apparent wind must be summoned which will blow through the resting train reference frame.   This will cause the velocity of one sound wave to be decreased, and the velocity of the other sound wave to be increased:

♦T = t₃ + t₄ = [0.5 L / (c + w)] + [0.5 L / (c – w)] = 2[0.5Lc] / (c² – w²)           where t₃ ≠ t₄.

To restate this, each sound wave will travel the same distance from an endpoint to the midpoint.  However, the apparent wind will have a velocity w equal to the train’s velocity v which will slow down the sound wave coming from one direction and speed up the sound wave coming from the opposite direction, thusly the sound waves will not arrive at the midpoint between their departure points at the same time.  Since w = v, the result will be equivalent to considering the train to be in motion through the still air.

Both these sets of equations resemble the total time formula from the Michelson-Morley experiment to detect the aether wind.  However, neither equation takes the form of the total time that would be measured if the train, air, and earth were all at rest relative to one another:

♦T = t₅ + t₆ = 0.5L / c + 0.5L / c = 2[0.5L] / c           where t₅ = t₆.

Thus, adding these two measured time intervals, and then algebraically solving for v, the observer in the train reference frame should be able to find the train's velocity relative to the earth.  This value of v represents the direction and magnitude of the train’s velocity since the train should be moving in the direction of the time interval with the lower value. Additionally, this velocity value should be equal to the value found by the classical method of measuring the duration of time to travel between two landmarks, of a known distance apart.  But this new method, with slight alteration, can apply the Doppler Effect to the problem of the relative motion of material objects.   The Doppler frequency shift formula gives differing values depending on the whether the source is moving towards the receiver, or the receiver is moving towards the source.  This experiment can thusly be used to distinguish whether the earth and air is moving relative to a stationary train, or to preserve mechanical symmetry, the train is moving relative to a stationary earth and atmosphere.  By this experiment, the use of sound waves will allow an observer within the train reference frame to find the velocity of the train reference frame, in contradiction to the classical principle of relativity.  All the results of this thought experiment are based only on information available from within the train reference frame, without needing to utilize the Galilean or Lorentz transformation equations between reference frames.  The sound wave can discern relative motion between two reference frames, while the light wave cannot.