# AN APPLICATION OF SOUND WAVES TO THE SPECIAL RELATIVISTIC DEFINITION OF SIMULTANEITY

Einstein’s Special Theory of Relativity defines simultaneity as: if two spatially separated events occur such that the light waves generated by these two events arrive at the midpoint of the line adjoining them, at a same time *t*, then these two events are considered simultaneous. However, if these two events occur in open still air -- which is disengaged from the motion of a material object through space -- then any sound waves that might also be generated at the light flash events may not arrive at this midpoint, at the same time. The events occur at the endpoints of their adjoining line and form a tandem, of length *L*, where all the discrete points on the line tandem (e.g., a high-speed train) are moving at a constant velocity *v* along a line parallel to the line adjoining the collection of points. The time and distance intervals measured in the tandem reference frame relative to the still air/earth reference frame may then be mathematically determined using a modified formula from the Michelson-Morley experiment in which the value of *c* is switched from the speed of light to the speed of sound. This switch is made plausible by the concept of the velocity constancy of wave phenomena. This methodology of using sound waves to investigate the motion of a material object through air thus calls into question the classical principle of relativity by dispensing with the need for a Galilean or Lorentz transformation between relatively moving reference frames. All needed physical information is available from within a single reference frame whether that frame is stationary or in motion.

According to Special Relativistic (STR) mechanics, two events occur simultaneously if the light from each of those two spatially separated events meet at the midpoint of the line adjoining them, at the same time *t*. 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 on a medium for their propagation, sound does not propagate in a vacuum. The velocity of the medium has a measurable effect on the velocity *c* of propagating sound waves which follows the formulas experimentally observed by Doppler. The medium’s velocity may be zero or have any other value relative to the source and receiver and as a result the arrival times of the sound waves at the midpoint will be staggered due to the motion of the line tandem reference frame through the still air.

An important but generally disregarded characteristic of this air/medium is that the air molecules pass easily through the porous conceptual walls of any inertial reference frame whose motion is disengaged from the open air. The still air will not be delimited by the walls of any stationary or moving reference frame in the same way as any air molecules contained within an enclosed compartment. A material object in flight within a reference frame follows a trajectory that is essentially the same as the object’s trajectory within an enclosed compartment; the object’s velocity will only be minimally impacted by any air resistance or wind. For sound waves however not every reference frame is an enclosed compartment. In the reference frame attached to the train the air molecules will have the velocity of the train only if they are in an enclosed compartment or sealed train car. This is because the solid walls of the compartment have imparted a mechanically invisible component of velocity upon the air molecules/medium contained within it. The non-zero velocity of the air then would increase or decrease the velocity of the sound wave and thus mechanically cloak the compartment’s motion during any experiment conducted within the enclosed compartment. On the other hand, the open still air outside any train compartment will be at rest relative to the moving train. This zero air velocity will result in the sound wave propagating at a constant velocity *c* relative to the train. Each scenario will consequently manifest a different velocity for any sound waves propagating through a medium within a reference frame based on the velocity of the medium relative to the sound wave.

An objective of any test of simultaneity would be to determine if two events occur at the same time or if one event occurs before or after some other event. This would require some type of time measurement that could make a temporal distinction between what is earlier and what is later in observable mechanical terms. A possible means of distinguishing whether the abovementioned events are simultaneous involves utilizing sound waves to mechanically measure time intervals and distance intervals. So sending a sound wave along the length *L* parallel to its extension in space and then applying mathematical formulas that will allow the measurement and comparison of time and distance intervals in a way which is not constrained by any single reference frame could be a means to mechanically reflect the physics of simultaneity.

A thought experiment oft used to explicate simultaneity involves an archetypical Einstein train of length *L* (distance between engine and caboose) travelling down a long level straight stretch of track, on a windless day, at the constant velocity, *v*. The air/medium is at rest relative to the earth and track. Suppose additionally that there is an observer seated on the roof at the midpoint of the train situated so as to see both the engine and the caboose and enjoying the view of the landscape. At some point in time two lightning bolts strike the cast iron hulk of the train, simultaneously, one at the engine end and one at the caboose end. At the occurrence of these two light flash events there are also two sound wavefronts generated. The departure events of the two sound waves are consequently also simultaneous.

The arrival events of the light waves at the midpoint of the train will be simultaneous according to the STR. However, the arrival events of the sound waves will not be simultaneous due to the forward motion of the train through the stationary air. The relativistic formulas from STR require the acceptance of the mathematical pretense that if the observer is working from within the train reference frame then that frame is to be considered as being at rest. As a result, the propagating light waves will traverse a particular distance in a particular duration of time without taking into consideration the velocity of the train. However, the formulas for the propagating sound waves will be different as a consequence of the porous conceptual walls of the train reference frame which will allow the train reference frame to pass easily through the air, or the air to pass easily through the train reference frame. In the moving train reference frame the still air molecules outside the solid walls of any particular train compartment must be philosophically assigned to either the train reference frame or the earth reference frame or maybe both. The free passage of the external air molecules through the train reference frame will require a more complicated mathematical approach which takes into account the train and sound wave velocities relative to the still air. So the propagating sound wave will manifest a behavior mathematically different from the light wave in the train reference frame though they are occupying the same region of space.

The train observer does not necessarily have to actually perform an actual mechanical experiment. She needs only to do some algebra to determine the mathematical solution that will state the simultaneous or non-simultaneous nature of events in the train reference frame. If she held two mechanically identical clocks at a single location she could find the flight time (Newtonian universal time) for each sound wave to reach the midpoint of the train. She would use the light waves as nearly instantaneous signals to indicate that she should start her clocks; at the lengths and speeds of a typical train this approximation should be valid. In addition, the effects of the gamma factor from the STR is very minimal at the speeds of a typical train in motion. Thusly, disregarding her reaction times, she could start the clocks simultaneously and the identical clocks would proceed to tick synchronously in an identical manner. Then by marking the clock readings for the arrival events of each sound wave at the midpoint she could make a decision as to the simultaneity of the sound waves arriving at her ears. If the light wave arrival events are apparently simultaneous but the sound wave arrivals are not, she might conclude that this may be due to the motion of the train. Another observer on the nearby platform could do the same if he had two clocks and he would come to a similar conclusion. In addition, there is not any type of direct communication between the two observers mechanical or otherwise.

The train tandem of cars moves through space with each discrete point at a fixed distance of separation from any other point on the tandem. Working completely from within the train reference frame and using only information available to her from that reference frame then there are only two reasonable mathematical options to pursue. For the propagating sound wave she must take into mathematical consideration the state of motion or state of rest of the medium and apply the Doppler wind formulas for the flight time of the sound waves from the endpoints to the midpoint through the still air. As a prelude, each light wave, one from the engine and one from the caboose, will traverse the distance 0.5L at the constant velocity *c*. So, according to the STR the formula that best reflects the flight time (relativistic proper time) of the light wavefront coming from either one of two spatially opposite directions in a stationary reference frame is:

[0.5L] / c = t = [0.5L] / c

In a reference frame that is considered as being at rest then the sound wave will propagate in a mathematically similar way according to the classical kinematics formula *time* = *distance* / *velocity*. However, if the reference frame is regarded as being in motion at the train’s constant velocity *v* through the still air/medium, then each sound wave one from the engine and one from the caboose will consequently traverse unequal distances. One distance will be less than 0.5L and the directly opposite distance will be greater than 0.5L due to the motion of the train. The sound wave will travel these altered distances at the one constant velocity *c*. Since the symbol *c* is commonly used to represent both the speed of sound and the speed of light in many scientific reference texts then the formulas that best reflect a sound wave coming from a direction parallel to the motion of a reference frame moving with the constant velocity *v* is:

t_{1} = [0.5L + vt_{1}] / c = [0.5L] / (c – v)

and from the opposite direction,

t_{2} = [0.5L – vt_{2}] / c = [0.5L] / (c + v)

These two time intervals are self-evidently different, t_{1} ≠ t_{2}. Both the train observer and the platform observer will determine the same value for the length interval *L* and the constant velocity of the train *v* by classical methods though they are in motion relative to one another. A particular classical method might be one in which a material object passes certain landmarks a known distance apart in a certain duration of time. This second pair of formulas will achieve nearly identical time results when used by either observer in his or her own reference frame. So this time difference could be used to determine simultaneity or not simultaneity due to the motion of a particular reference frame relative to some other reference frame. Also these two mathematical expressions bear a remarkable resemblance to the formulas that arose from the considerations of the Michelson-Morley experiment to detect the aether wind. That is, the time formulas that were applied to the light traveling along the interferometer arm that was aligned parallel to the direction of the earth’s orbital motion around the sun as an effort to investigate the earth’s motion through space. The goals of the Michelson-Morley experiment are very similar to the objectives of the thought experiment presented here.

The first pair of formulas imply that the train is at rest or the reference frame attached to the train acts as an enclosed compartment. This would follow the Galilean and Lorentz reasoning of considering the reference frame attached to a material object to be at rest, although that object is in motion. Meanwhile, the second set of formulas include the velocity of the train relative to the earth in a mathematical way that recognizes the conceptual porosity of the walls of a moving reference frame following the reasoning of the Michelson-Morley experiment. The sound waves are in essence either meeting or overtaking the observer at the central location depending on the direction of motion of the sound waves relative to this central observer. Deriving the formulas recognizes that the distance between events increases for one direction such that the flight time between events also increases by some factor that includes the train velocity *v*. In the directly opposite direction the distance the wave travels decreases such that the time of flight for the wave decreases by a similar factor. The train reference frame will then appear to not be in motion at least according to any mechanical measurements of sound wave velocity made within an enclosed compartment on the train. While a sound wave travelling through the external still air can to a great approximation detect the train’s motion from within the train reference frame.

Thus by mechanical hypothesis the time and distance interval values are invariant across the relatively moving reference frames. As a result, the variables can be assumed to be equal in both the train reference frame and the platform reference frame. Consequently, being able to mathematically determine the relative velocity then permits the finding of the simultaneity of events across reference frames which contradicts the STR since the train reference frame and the platform reference frame can use the same formula to investigate simultaneity. The STR states that the train observer and the platform observer must use different formulas which include the variable for the speed of light waves. However, the train observer can compare the differing times of sound waves arriving at her ears such that she can come to a decision about the approximate simultaneity of the lightning strikes by factoring in the motion and velocity of the train. She might conclude that what has caused the staggered times of the sound wave arrival events is the motion of the train. She may wonder why this is not true for light.

If the train were regarded as being at rest, for the reference frames to preserve mechanical equivalence between the scenarios of a moving or stationary train then an apparent Dopplerian wind of velocity *w* must be summoned. The relative velocity *v* represents either the train moving past a stationary earth and atmosphere or the entire earth and sky are moving past a stationary train. The air/medium must retain the value of zero relative to the earth in both scenarios and the air must observably move past the stationary train or the train must move past the stationary air at either *w* or *v*. So this Doppler wind would appear to slow down the sound wave coming from one direction and speed up the sound wave coming from the opposite direction. Each sound wave would nonetheless travel along the same full length 0.5*L* between the endpoints and the midpoint on the train but at apparently different velocities:

t_{3} = [0.5L] / (c + w)

and from the directly opposite direction,

t_{4} = [0.5L] / (c – w)

where t_{3} ≠ t_{4}. Since w = v, then the pair t_{3} and t_{4} is mathematically identical to the pair t_{1} and t_{2}. This consequently means that the train observer and the platform observer could use the same formulas for measuring the time intervals between the sound wave arrival events. That is, each reference frame can use the one and the same set of formulas to find the invariant time intervals as viewed from each reference frame.

Neither set of formulas specifically refers to measurements that are available only from the platform observer nor does the train observer need any especial information from the platform reference frame to find an algebraic solution for simultaneity. This algebraic solution will establish a mathematical relationship between relatively moving reference frames that dispenses with the need for any type of transformation equation. An observer at rest on a nearby platform would also see the sound wave from the engine end of the train arrive at the central location before the sound wave from the caboose. He could also use the abovementioned formulas with the identical variables to determine the time interval values for the departure and arrival events for each sound wave. Additionally, both observers would see the sound wave flight durations from each direction as measurably different by the same amounts. The single constant velocity *c* for the propagating sound waves will manifest in both reference frames though this velocity will have the appearance of having differing values as viewed by each relatively moving observer. In these two apparently mechanically different scenarios the reference frame from which the velocity of the train is viewed does not matter. All the variables are readily accessible from within the train reference frame, she simply has to do the algebra.

CONCLUSION

Typical mechanical experiments involving material objects or sound waves which are conducted in an enclosed compartment will usually not reveal the motion of the compartment relative to that which is outside the compartment. However, an experiment involving sound waves which is conducted outside of an enclosed compartment would expose the sound waves to the open motionless external air. This would present a description of a type of relative motion between reference frames which does not require either a Galilean or Lorentz transformations. It would establish a mathematical relationship between reference frames that are in motion relative to one another which allows the observers in each reference frame to use identical formulas for making invariant time, distance, and velocity measurements. The pretense of a stationary system from the Galilean principle of relativity and the Einstein STR can then be discarded and there would be no need for Galilean or Lorentz addition of velocities with respect to sound waves propagating in open still air. In addition, there is a lack of formulaic influence from the Lorentz gamma factor because *v* is so much less than *c*, where that *c* represents the speed of light

The abovementioned formulas thusly displace the Galilean and Lorentz transformation equations to become a new form for expressing the mathematical relationship between relatively moving reference frames and in doing so challenge the validity of the classical principle of relativity. Since the observers have used identical formulas though they are in relatively moving reference frames then they will be in agreement as to the time measurements that would distinguish between the simultaneous and the non-simultaneous event scenarios. This would contest the validity of the STR which states that simultaneity can only be a relative concept; in other words, events are only simultaneous in a reference frame that is at rest, but are not necessarily simultaneous in a relatively moving reference frame. By these sets of formulas events will appear to be simultaneous when viewed by an observer located in a reference frame that is stationary and at the same time when viewed by an observer located in a reference frame that is in motion. This does not align with the formal definition of simultaneity as stated in the Special Theory of Relativity which is more strictly associated with propagating light waves.

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