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.

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.

A Magick Carpet Ride

A young woman and her two wyrd sisters are practicing their mysterious magicks tonight.  They will exercise their telekinetic and mathematical skills with the levitation of a massive material object.   In their alchemical experiment they will test whether two events that would appear simultaneous in a reference frame that is at rest, would these events still appear simultaneous if the reference frame were in motion? Can she make the invisible, visible?

It is a windless night, during the witching hour (air / medium at rest relative to earth).  Since her youth, a thousand years in the past, she has known that three is a magick number, so the sisters can begin their session under a beneficial sign.  The young woman stands at the centerline of a soccer field with one sister at each endline.   This field has the length L meters.

Now, beforehand, the thrice wyrd sisters had planned their rite.   The young woman’s sisters have agreed to take certain actions in response to her initiations.  While in a mystical trance she conjures up two digital stopwatches which hover in the air before her, stacked, so that they are perpendicular to a line that runs from endline to endline.   She continues her dark rite by whispering the secret words.  Then, the entire green turf (and the reference frame attached to it) lifts itself from the dust, and rises above the ancient high treetops. It begins to fly away at the constant velocity c, straight and level, like a magick carpet, off into the starry night.   Then, it turns, and flies similarly back to the stadium.  A warlock soccer fan, with his supporters scarf, was observing the sisters rehearse their magicks from his stationary stadium seat in a reference frame attached to the earth (at rest).

The Blue Djinn on her Magick Carpet

As the carpet tandem passes once again through the stadium, within view of the warlock she will, with some incantations, make a ball of golden light appear above her head.   At this flash of light, her sisters, with their supernatural reflexes, will let out a banshee’s wail upon seeing the nearly instantaneous flash from this light.   In the same instant, two disembodied bony fingers waft as smoky wisps awaiting, for at the appearance of the light they will start the timing devices. The warlock sees this flash and begins each of his two hidden clocks.  Now their clocks are synchronized, so they will measure the same time intervals between these events in their separate reference frames.

As each of her sisters’ wail reaches her, the ball of golden light flashes to green when the first sound wave arrives at her central position, then to red when the second sound wave arrives at her central position, the warlock will witness each arrival flash.  The difference in arrival times of each sisters’ sound wave is due to the carpet’s forward motion. If the sound waves had arrived at the same time then that would mean the tandem was not moving through the still air and the light would instead flash to blue.

Thrice Wyrd Sisters ~ MacBeth

They have made a three-seated tandem (aligned parallel to the direction of motion) so that they maintain the same distances relative to each other, no matter how quickly, or slowly, the tandem moves through space; or possibly not even moving at all.  The distance L moves through space, neither increasing nor decreasing. So, each pair of clocks will measure two times, one from the forward sister and one from the rearward sister.  The following equation adds these two times for a total time T (both the witch at the center position and the warlock seated nearby do this addition):

♦T = t₁ + t₂ = [0.5L / (c + v )] + [0.5L / (c – v )] = Lc / (c² – v² ) = [L / c ] × (1 – [v² /c² ])^-1

If the carpet were at rest (v = 0) then this formula would reduce to:

♦ T_r = t₃ + t₄ = [0.5L / (c + 0)] + [0.5L / (c - 0)] = L / c

The zero velocity airy particles allow the porous conceptual walls of the reference frames to pass by without hindrance, so the air molecules velocity will neither increase nor decrease the velocity of the sound wave c travelling through the still air.  It will remain constant despite the motion, or lack of motion, of the wailing banshees.

The magick carpet will move straight, level, and true at the same time as the sound waves are in flight.   The length L of the tandem will neither increase nor decrease.  As the carpet moves forward, the distance from one sister to the center will appear to shorten, while the distance from the other sister to the center will appear to lengthen.  This equation will account for this by inserting the velocity v.

She puts some numbers into the calculator that she has been carrying beneath her pointed witchy hat; L = 100 meters; c = 343 meters / second; assume v = 30 meters /second; T_v = total time measured by each pair of clocks:

♦T_v = t₁ + t₂ = [Lc ] / (c² – v² ) = [100 m × 343 m/s] / [(343m)² – (30 m/s)² ] = 0.294 s

as compared to the total time measured when the carpet is at rest:

♦T_r = t₃ + t₄ = L / c = 100 m / 343 m/s = 0.292 s

thus, her velocity is:

♦v = √c² – [(Lc ) / T_v )] = √(343 m/s)² – [(100 m × 343 m/s) / 0.294 s] = 31.3 m/s

Thus, she has made her velocity form from the night’s shadows (detecting her absolute motion relative to the still air/medium) using sound waves.  She can make her magick carpet fly, through the night sky.  This will contradict Galileo, Newton, and Einstein and their precious principle of relativity, which says that what she has done is impossible.

The Wind

A young woman stands before a high flat concrete wall on a blustery day. She directly faces it, at a distance of L meters away. The wind sweeps down past the wall at the constant velocity v and blows directly perpendicular from the wall to her face. She feels compelled to shout at the wall in some way, but she takes the stopwatch from her pocket and decides upon the experiment that she shall perform (akin to the Michelson-Morley aether wind experiment).

The formula for a sound wave to echo back from a hard reflective surface fixed to the earth, when the air is still (medium at rest) is:

♦ 2L = cT

However, I speculate that this is not the formula when there is a wind blowing at the constant velocity v in the direction directly opposite to the sound wave emission source. The velocity of the air molecules (medium) will have a measurable impact on the velocity of the sound wave as it travels from the source to the wall, and then back.


Our home planet hurtles through interstellar space at a tremendous speed, 30 km/second, but the atmosphere does not get swept away, off into the cosmos. Fortunately for us, the molecular bonds of attraction and repulsion, and the force of gravity, hold a thin layer of atmosphere snugly against the earth’s surface. Though terrestrial winds may surpass 120 km/hour, most of the air molecules we depend upon to fill our lungs cannot attain enough velocity to escape the earth’s embrace. This balancing of hydrostatic pressure and gravity thus bestows upon us, the breath of life. So the earth makes its yearly orbital journey with a thin layer of atmosphere grasping tenuously to it; tornadoes and hurricanes may blow, but we shall breathe.

Girl playing by the wall

So, when the air appears still, it is actually moving at the velocity of the earth. It is supposed that we cannot detect this motion by any mechanical experiment in the reference frame of the earth; however, it is worth exploring the scenario when the air molecules are disengaged, such as by a wind, from the rapidly moving surface of the land and sea. She stands, with her mouth and ears at the ready, opposite the high and hard reflective surface, forming a moving tandem with it; that any shout she might make would come back to her some moments later. If she is standing at a reasonable distance from this reflective surface on a windless day, then the first formula applies; but if a wind is blowing as I have described before, then the maths are different.

The hydrostatic pressure casts a cloak of invisibility over the motion of the stationary earthbound tandem, and the stationary air molecules trapped near the surface of the earth. The earth, the tandem, the earthbound air molecules, are all traveling through the galaxy at the same velocity v locked together in their motion. That is, when the air is still, but a wind will cleave this triumvirate.

Despite the tandem being fixed to the earth’s surface during its daily gyre, there is no Doppler effect upon the sound wave traveling from the girl to the wall because the wave crests are squeezed together near the source (girl), but pulled apart near the receiver (wall) by an equal amount; and vice versa on the reflection’s trip, so she would observe no change in the wavelength or frequency of the wave. In the presence of a wind, the Doppler change in frequency vanishes, but the Doppler wind formula remains present and measurable:

♦ c` = c ± w, where c` is the Dopplerian speed of the sound wave in the presence of the wind.

Because of the wind’s speed and direction, the new wave speed is c` = c – w as the sound wave travels away from the emitter (her mouth); but it is c`= c + w when the wave is reflected back towards its original source (her ear). That is, when compared to speed of sound in still air, the wind slows down (subtract from) the sound wave speed as it travels in one direction; but speeds up (adds to) the sound wave speed when it travels in the opposite direction. Thusly, the total trip time interval for the sound wave is:

♦not, T = [2L/c]

♦but, T` = [L / (c + w)] + [L / (c – w)] = [2Lc] / (c² – w²) = [2L / c] [1 / (1 – [w²/c²])

♦time = distance / speed.

With the pen and pad from her other pocket, she begins to make her calculations. The given variable values for her experiment are: c = speed of sound in still air, 340 meters/second; w = speed of wind, 50 m/s; L = 100 meters.

So, in the first echo scenario (no wind):

♦T = [2L] / c; T = [2 × 100m] / [340m/s] = 0.588s

And, in the second echo scenario (wind):

♦T ` = [2Lc] / (c<sup>2</sup> – w<sup>2</sup>); T ` = [2 × 100m × 340m/s] / [(340m/s)²] – [(50m/s)²] = 0.601s

She makes note of these differing measured time values. This leads her to ponder her two scenarios of air motion: molecules at rest in a stationary reference frame, and molecules passing unencumbered through the porous walls of an apparently stationary reference frame. There is a measureable difference between an enclosed compartment and a reference frame. The “conceptual walls” of the reference frame do not compel the air molecules within it to go at the velocity of the reference frame. These free-spirit airy particles are not possessed by the earthbound reference frame. But it is difficult to say to which reference frame they belong; they belong to no reference frame, and are in all earthly reference frames. This alternative echo formula is only a close approximation.

The moving air/medium has been disengaged from the stationary earthbound girl-wall tandem in a mathematically revelatory way. This has profound implications for the motion of any material object, when that motion is investigated by sound waves. At the slow speeds of the wind, the measured time interval does not suffer the Special Relativistic effects of time dilation and length contraction; the gamma factor value is negligible at this speed. Thus, the passage of time is nearly absolute, on the scale of her everyday life behind the wall.