The Concept of Reference Frames in Relativity

The notion of relativity, born with Galileo and formalized by Newton, remained unchanged until Einstein. When he published his work in 1905, what would later be called the theory of special relativity revolutionized what was previously understood about relativity. Distance, time, and simultaneity, once considered invariant or absolute, became relative concepts. Where only velocity was considered relative, it was now the speed of light « c » that became the absolute reference for judging the relationship between reference frames, or the required clock for measuring events between reference frames. As there is extensive literature regarding the works on relativity by these three key figures, we will not delve further into their research here. It should be understood right away that this is not a training or additional information aimed at teaching the different theories of relativity.

Although the theory of special relativity is universally accepted, its origins are based on a strange paradox or curious misunderstanding. In fact, this misunderstanding is the inspiration of this publication arose.

Origins

Galilean relativity postulates the existence of inertial reference frames, allowing the observation and quantification of movements. This implies that time and space are absolute references. A beach next to a body of water or a park next to a railway track are fixed and by definition inertial reference frames. The same applies to a boat, an aircraft, or a train moving in uniform straight-line motion, as long as no other forces or accelerations alter the reference frame. Based on this postulate, Galileo asserted that « motion is like nothing » after observing that the fall of an object was identical whether observed from a fixed reference frame or from one in uniform straight-line motion. Comparative observations made from these different reference frames proved this point. Thus, comparing comparable situations allowed Galileo to support his postulate. The notion of relativity now begins to emerge when considering at this point the position of the observer in a distinct obviously inertial reference frame to measure and quantify any motion. From this came the law of composition of velocities, with the so-called Galilean transformations, and the notion of relativity. It is on this foundation that Newton will formulate his laws of motion while confirming Galileo's relativistic logic.

The Paradox of Light

At the beginning of the 20th century, among the unsolved mysteries of science was the constancy of the speed of light. As established by Maxwell's equations and confirmed by the Michelson-Morley experiment, the speed of light did not conform to what the laws of relativity then prescribed. In addition to carrying around an ether that could not be properly identified or defined to justify the propagation of electromagnetic waves, including light, these waves did not adhere to the law of velocity composition deduced from Galilean transformations. By substituting Lorentz transformations for Galilean transformations, Einstein resolved this paradox, thereby discarding the concept of ether, which was not leading to any concrete outcomes. But was this paradox really exist?

According to some interpretations, special relativity was born in response to the rules of Galilean relativity, which could not explain the speed of light constant nature. Einstein, relying on Maxwell's equations, based his theory on two postulates: the constancy of the speed of light and the non-existence of ether. With this new theory of relativity, any conjecture induced by this paradox could now be forgotten.

However, there is an aspect here that seems to have been naively overlooked by the scientific world of that time like it's still today. Like light, sound and waves on water do not glue to the law of velocity composition. If sound waves or water waves manage to conform to the law of velocity composition, it is primarily due to the observation criteria or mechanisms but not the relativity itself. The paradox in question, therefore, does not only concern light. If the theory of special relativity led us to believe this paradox were now solved, the confusion surrounding what is actually a simple misunderstanding remains unresolved.

Definition of Inertial Reference Frame

Let us first clarify that the terms « Galilean reference frame » or « Galilean relativity » will often be used. Since Galileo was the first to introduce this concept and initiated a scientific movement upon which future works and discoveries would be based, it will be referenced in these terms. Most of the rules and principles mentioned later were however clarified by other scientists than Galileo himself.

Now, concerning the rules of relativity and their formulations, it is important to remember that all inertial reference frames, to be considered such, must meet the following conditions:

• They must not undergo any acceleration, change of direction, or external forces, or those forces must cancel out. They are either at rest or in uniform straight-line motion.
• The experiments conducted and their results must be the same.
• All laws of physics must apply.

These are the essential conditions for defining one or more reference frames from which observations can be made and compared. Note that these same conditions apply within the framework of special relativity. However, within the framework of Galilean relativity, the following rules apply to ensure the principles are respected:

• For any change of reference frame, Galilean transformations apply.
• Velocities are relative and follow the law of velocity composition according to the aforementioned transformations, v = v' + U.
• Distances and durations are invariant.
• The simultaneity of events is absolute.

Let us clarify which types of observations can be made with waves of all kinds, including light. Without initially worrying about the nature of reference frames and the rules of relativity, it is possible to observe and measure the following phenomena:

• There are movements whose velocities are measurable based on an observation point, whether it is in motion or not.
• A measurable Doppler effect, whether the observation point or the reference frame itself is moving.
• A difference in time can be observed depending on the viewpoint and the chronology of events considered, seemingly simultaneous.

Additionally, the conditions of the observations themselves must be considered. A distinction must be made, as the observation conditions prevailing in the observed reference frame and that of the observer must be the same. It remains to ensure that these observations and the experiments conducted comply with the laws of relativity. It then becomes pertinent to ask the following question:

• Do the reference frames in question share something in common with each other, and if so, does this interfere with the interpretation of the results?

Reference Frame Test

In the first exercise, let’s consider a wave on water. Better yet, a stylus that, through a regular motion, generates a series of waves. One experiment takes place in a basin near a railway track, and the second inside a basin located in a train car moving along the said track. An observer is positioned in each reference frame, and each of these reference frames meets the validity criteria mentioned earlier. Since both styluses generate the same type of wave motion on the water, each observer will measure a difference in the speed of wave propagation, regardless of the observation point, as well as a measurable Doppler effect depending on the motion. This effect will be measurable based on the chosen observation point. From the observer's own reference frame, no observable Doppler effect can be measured, while such an effect is observed and measured when observing the other reference frame. When considering only the propagation speeds and applying Galileo’s transformations, we find that the waves generated on both sides are exactly the same, and the observed speed difference will simply be the result of the composition of speeds. Up to this point, everything is fine; there is no misunderstanding or any questions casting doubt on the laws of relativity, whatever they may be.

Now, let’s repeat the same experiment, but this time with just one basin for both reference frames. To avoid confusing the measurements, it can be carried out in two stages.

First stage: The stylus on the ground will generate the same waves as those observed in the first exercise. The observer on the ground will therefore obtain exactly the same results as before — the same speed and no Doppler effect. Similarly, the observer in the train will reach the same results while observing the basin on the ground. We still have the same conditions, and the Doppler effect that they measured previously remains the same.

Second stage: When the stylus aboard the train agitates the basin located on the ground, we notice that the speed of each wave crest is the same as those generated by the stylus on the ground. The difference is that the crests are much closer together in the direction of the train’s movement, while they move farther apart in the opposite direction. A Doppler effect, similar to what the observer on the ground could measure from the experience aboard the train in the first exercise with two basins, is now observed. In this final experiment, the measured Doppler effects remained roughly the same, but the most troubling aspect for relativity is that no mathematical transformation or speed composition could be applied.

Although it is too early to draw an exact comparison between what we observed and what we know about the behavior of light, a correlation can already be established. By sharing the same propagation medium, the observed propagation speed turned out to be the same regardless of the observation point. This is the same observation that was made with light during the 19th and 20th centuries. Therefore, it is possible to suggest that, in the second exercise, one or both reference frames could no longer be considered valid since both shared the same propagation medium. In fact, it was the train itself — or the wagon, to be more precise — that no longer met the validity criteria, as the chosen propagation medium did not fully match with it. It is also noteworthy that, in the first exercise, it was not the stylus moving with the train that imparted an additional speed to the generated waves, but the basin itself and its contents. As mentioned, the entire setup was consistent with the reference frame. No need has been required to quantify the observations or put them into an equation; logic alone has been enough to explain why, under these conditions, the speed composition law could not be applied.

Regarding Sound

To support the previous points, let’s turn to sound. It was mentioned earlier, but if the Doppler effect is a phenomenon that requires explanation, unless one is hard of hearing, everyone has experienced this effect. We’ve all had the chance to note the difference in sound when comparing the noise of a car approaching us to that of one moving away. This is a difference in the frequency of the sound waves perceived by our hearing. This is the same effect that was discussed earlier with the waves, but in a visual and measurable way in space. The distinction was observed between wave crests approaching each other and those moving in the opposite direction, moving farther apart.

That said, the speed of sound propagation directly depends on the medium in which it propagates. In air, the speed is approximately 340 meters per second or 1224 km/h. A loudspeaker, whose sound can be heard from the outside, inside a train moving at full speed versus another in a railway station, regardless of the tests and observations carried out, the speed of sound will remain unchanged. Thus, an observer inside the train, whether at the front or rear, will not notice any difference in sound based on their position, because they are traveling at the same speed as the train. On the other hand, an observer on the platform will notice the difference in sound when the train passes by. Only the Doppler effect will make a distinction since the speed of sound waves remains the same for both observers. Proof of this is the sonic boom produced by jets when they break the sound barrier, showing that movement does not alter the speed of sound itself. Again, here, the law of composition of speeds cannot be applied. If, therefore, both a train and a station can be considered valid reference frames for judging the launch of a projectile in each of them, this is not the case for sound or any other phenomenon that behaves like a wave if we cannot confine them to the observed reference frame.

Let’s not conclude too hastily, because if we limit ourselves to considering only the sound emitted by a loudspeaker inside one of the train’s cars, it then becomes possible to apply the principles of relativity. Since the propagation medium is isolated from the outside world, we can apply the law of composition of speeds. Although sound waves cannot be seen, there are ways to measure their impact on any mobile, yet fixed, object inside the car, allowing us to visualize the waves and perform the appropriate calculations. In such a context, we will find that the reference frame constituted by the car is valid. Moreover, if the train in question were to break the sound barrier, while for people outside only a sonic boom coming from the train would be perceived, the listeners inside the carriage would simply continue to have their ears lulled as if nothing had happened.

The simple logical exercise just demonstrated shows that the fact that two reference frames share the same propagation medium automatically discredits them. When dealing with wave phenomena like sound or light, we generally compare reference frames that share the same propagation medium, where the phenomenon observed depends on this same medium. Consequently, when two reference frames share an element intrinsically linked to the phenomenon being observed, they are no longer distinct, and relativity, as traditionally understood, no longer applies. The law of composition of speeds can no longer be applied. Thus, we see that a crucial rule is missing from the definition of a reference frame. The fact that a reference frame is considered inertial is not sufficient on its own to judge and measure the relativity of what is observed. The observable and measurable relationship between the events under consideration must also take into account the complete distinction between the reference frames involved.

Simultaneity of Events

To avoid making things too complex regarding the rules governing the principles of relativity, the simultaneity of events has not been addressed until now. Let’s now consider two projectiles launched at the same time with exactly the same velocity criteria. Both will hit the ground at the same time. Whether on the ground or inside any vehicle moving at a constant velocity, the result will be the same. This will also be the case when comparing the two reference frames, as long as the projectile launches are synchronized; all projectiles will reach the ground simultaneously. Up to this point, everything seems fine, and the integration of this rule into the definition of a reference frame cannot be questioned.

However, as in the cases mentioned earlier, if we instead consider events involving wave phenomena, the results will no longer be the same. With the waves on water in the two basins considered at the start, there is indeed simultaneity, but in all other cases evaluated, no simultaneity can be observed. For example, from the train, if we observe the movement of waves in a basin on the ground and limit the observation to the ends of the train car in which we are located, we will notice that the first waves will reach the back of the train before a single wave reaches the front. There is, therefore, no simultaneity. There’s no need to carry out new experiments with other wave phenomena—the results will be the same as here. In fact, this is one of the aspects of relativity that the theory of special relativity itself has demonstrated, even though no distinction about the propagation medium is made.

Given the demonstrations just performed on reference frames and their relationships, i.e., relativity itself, can we truly reject the simultaneity of events as one of the rules defining an inertial reference frame? The question can be raised, and the answer is simply relative. Nothing that has been presented so far tells us that simultaneity cannot be verified. It has only been demonstrated that, from incorrect reference frames, the simultaneity of events could not be verified. By establishing the correct calculation benchmarks to determine the occurrence of the considered events, it is then possible to determine whether they are simultaneous or not. Before rejecting this rule, it would need to reject itself in a context where the reference frames are truly valid without any doubt. The invalidity of the reference frames considered so far does not allow us to reject this rule within the framework of Galilean relativity. This remains to be taken lightly; the different situations and observation conditions will always require judging the relativity of events with respect to each other in order to determine whether they are indeed simultaneous. Like in special relativity, we must conclude that the simultaneity of events is not absolute, but relative.

Additional Clarifications on Reference Frames

To truly understand the nature of a reference frame and judge its interaction with another, as well as its validity, we must distinguish whether the phenomenon observed depends on the propagation medium, or how the comparisons made depend on it. We must also determine whether the propagation medium itself is in motion or not according to the reference frames.

The motion of a ball launched in two distinct reference frames does not depend on the medium in which the ball is projected. Its motion is only affected by it. Whether in air or in water, the conditions must simply be the same in each reference frame in order to compare the same things. Since the ball’s motion is not dependent on the propagation medium, but rather on what generates its motion, we need only distinguish whether this medium is in motion or not, and consequently, linked to the concerned reference frame.

It should be clarified that what propels the ball cannot depend on the medium in which it propagates, because its movement is generated by its source of impulse, the element that projects it, and its mechanical link to the reference frame in question. Its motion can only be altered by the nature of the medium, unlike a wave in water or sound waves in air, which depend directly on the medium itself to propagate, as this medium is their propagation medium. As for light, based on what we know today, it does not depend on any medium and has no propagation medium. Although light has a defined speed in a vacuum, it varies depending on the medium in which it propagates, but not depending on the vehicle from which it is emitted, nor on whether this vehicle is moving.

In the context of special relativity, it is common to use drawings, graphs, and animations of all kinds to demonstrate how to understand distances and time based on a beam of light or a laser emitted from a moving vehicle compared to the same device located in a completely inert environment. While such demonstrations are quite eloquent¹ , in most cases (if not all), the emitted beam is presented as if it must necessarily take part in the movement of the vehicle, which is not accurate, even though it’s not entirely false. Despite the admirable mathematical efforts, where distances are contracted on one side and time dilated on the other to reconcile events, even though the final result seems convincing, it is, in fact, a waste of effort and energy. We are naively attempting to compare the incomparable. Since the propagation medium of light is the same for both reference frames, they cannot be compared this way to explain the composition of speeds in special relativity, as in Galilean relativity.

However, note this distinction. Although it is essential that the propagation medium is exactly the same for the validity of the experiment, it must be independent of any other reference frame, which is not the case here. The two reference frames exist in the same medium. Additionally, even if the emission source follows the movement of the vehicle, the light itself is not affected in its course. Only a Doppler effect could be measured based on the vehicle's speed. By slowing down a photon to the point that its speed would only be a fraction of the vehicle's speed (something modern science is capable of doing), one would then observe its projection not in front of its emission source, but behind it. It would then be found that its propagation is exactly the same as that of the photon generated in the completely inert reference frame, of course, if the propagation medium remains the same for both reference frames.

Redefining What an Inertial Reference Frame Is

Throughout the discussions so far, we have encountered an issue with the very definition of an inertial reference frame. To address this shortcoming, it seems essential to introduce one or more corrections. What was initially postulated cannot be questioned and must remain as it is. However, new conditions must be added to clearly define the true nature of an inertial reference frame within the framework of any principle of relativity. Here are the conditions that must be respected:

• The reference frame must not experience any acceleration, change of direction, or external forces, or any external forces must cancel each other out. Therefore, they must be at rest or in uniform rectilinear motion.
• The experiments conducted and their results must be the same in all reference frames.
• All the laws of physics must apply.
• The conditions of the environment and the environment itself in which the experiments and observations are carried out must be absolutely identical. In other words, the surrounding environment in each reference frame must have exactly the same physical properties.
• The surrounding environment or medium in which the experiments and observations take place must be completely distinct from one reference frame to another.

These two new conditions now allow for the validation of any experiment or observation made to determine the actual relationship between the observed events, to be measured and quantified with accuracy. It should be noted that these new conditions do not specify a particular medium, such as those required for waves, sound waves, or other electromagnetic waves. They apply to all types of environments, including those that have not yet been defined or remain to be discovered.

Consequences

What conclusion can we draw from this thought exercise? Initially, it seems strange that no one has stopped to consider this fact—that the rules of relativity have always been interpreted incorrectly. With Einstein's theory of special relativity, any questioning or challenge to these rules seems to have been relegated to obscurity. Since it is a theory and not a law of relativity, it seems that all efforts have been directed towards validating or invalidating this theory, forgetting to revisit its foundations—the rules of relativity themselves. Talking about laws or rules essentially amounts to the same thing, as we must agree on the definitions that help us appropriately quantify any event we aim to measure. Once convinced of this, we can then speak of laws. These laws then become the ultimate reference.

Since all scientific approaches or experiments must be based on solid postulates, tools, and knowledge, we are justified in questioning several theories, including those that primarily rely on a definition of reference frames that was believed to be valid up until now. Like special relativity, new deficiencies have been demonstrated in the study of the systems involved—reference frames—which could undermine several theories and research known to date.

Without dismissing everything outright, we note that the proper use of reference frames defined appropriately paves the way for new hypotheses, leading to new theories and possibly new laws of physics. We are justified in believing that where scientific progress seems to be hindered by unexplained phenomena or theories that we cannot validate or invalidate, these can be re-evaluated with new insights.