A person walking on a moving conveyor belt represents a classic example of compound relative motion in mechanics. The passenger is at the airport, 900 from the gate and walking at a speed of 1.5 , with only 5 minutes remaining before the plane departs. However, by stepping onto a moving conveyor belt, the passenger succeeds in catching the flight (see Fig. 1). How can this outcome be explained?

The scenario will be analysed using distinct approaches: through the Lorentz transformation (LT), the currently accepted Galilean transformation (GT), the transformed Galilean transformation (TGT) [1, 2], and through an artificial intelligence-generated solution to this relativity problem. Finally, the results from all approaches will be compared and discussed. Artificial intelligence (AI) systems are built with two primary functions relevant to scientific reasoning. On one hand, they are trained to align with established scientific theories, including the foundational principles of modern physics. On the other hand, they are optimized to solve problems efficiently – often favouring the most direct and practical solution, particularly in the domain of basic mechanics. This dual design can sometimes produce outcomes that are consistent with practical experience but formally at odds with established theoretical predictions. The emergence of a distinct solution from a logically consistent, non-biased AI system when addressing the same physical problem is presented here as an additional argument in favour of adopting a new mathematical formalism in relativity mechanics: the transformation of the Galilean transformation. While this alone is another justification that further supports the scientific soundness of the transformed Galilean transformation (TGT), which has already been introduced with analytical rigor, coherent geometric interpretation, strong philosophical foundations, and full agreement with observable physical results [1, 2].

where and represent the coordinates of a moving body in their respective frames of reference, and and are the corresponding times, which differ from one another depending on the relative velocity between the frames. The constant denotes the speed of light, which – according to modern physics – is considered part of all forms of motion, even in cases where no physical connection to light exists. The parameter is a scaling factor known as the Lorentz factor.

This formulation – precisely the second equation, which is arbitrary – was constructed to justify the LT, specifically to support the often-repeated distinction: in classical physics, time is absolute, whereas in modern physics, time is relative. It is supposed that the quantities in equations (2) play the same role as those in equations (1).

A passenger running to catch the plane gets on the conveyor belt (red passenger in Fig. 1). The traveller walks at speed , and the conveyor belt has speed . This motion is a compound relative motion, commonly used in the study of relativity. Why does the traveller get on the conveyor belt? If the equation for time , as claimed by today’s GT (2), were valid for this motion, then the traveller would have no reason to use the conveyor belt. Also, if the different time intervals would be only of nanometric scale, as claimed by LT (1), then again, the traveller would have no reason to use the conveyor belt. The fact that a body can travel the same distance in different amounts of time depending on its speed has been known and used by humans since ancient times. This knowledge is applied in the use of conveyor belts, which are widely implemented in practical settings. During the era of industrialization, they were introduced into factories for a single purpose: to maximize time efficiency – essentially, to take advantage of the relativity of the time required to complete a task. Similarly, conveyor belts have been installed in areas where passengers move, not only to assist them physically but also to help them save time, to use the relativity of time. In the case of the traveller trying to catch a plane, the time gained – while maintaining the same personal walking speed – depends on the speed of the conveyor belt and can be calculated precisely using the TGT (3).

Given Parameters:

Materials Needed for the Experiment:

Steps of the Experiment:

the traveller reaches the gate on time.

Example with Numbers:

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.

Even here, the agreement among the three transformations and the system of equations (4) holds only algebraically, since the interpretation of this system is not the same across the three theories. For , the differences between the transformations become more substantial, clearly exposing the underlying theoretical positions of each framework. These differences are not merely formal, mathematical, or limited to numerical results — they also reflect deep physical and conceptual divergences, reaching the level of a paradigm shift. Therefore, the comparative analysis of these transformations in a concrete problem – such as the case of the passenger on the moving conveyor belt – serves as an experimental test to evaluate the stability and practical usefulness of each, as well as for drawing theoretical conclusions.

whereas for the GT and the TGT, it would be taken from this equation:

For the LT, Eq. (5) faces two main obstacles that prevent it from solving the problem. The first obstacle is: how do we determine the length ? According to LT (Special theory of relativity, STR), as summarized in the concept of length contraction, and represent the same path, with one being contracted depending on the velocity . That is, LT (STR) does not account for the possibility that represents the total distance, composed of the part walked by the passenger on their own () and the part covered with the help of the conveyor belt (). Therefore, conceptually, under LT (STR), and in our problem () differ so little that the difference can effectively be considered zero. The second obstacle arises when we substitute the eventual solution of Eq. (5) back into the second equation of the LT (1). It becomes even clearer that the difference between the times and is extremely small – negligible – and thus cannot resolve the passenger’s problem. Why does this happen? Because just like with length, according to LT (STR), the time intervals and represent the same event, measured by “two different clocks”. In other words, LT (STR) does not consider the possibility that and are time intervals of different developments, measured by “the same clock”. Algebraically speaking, by inserting the speed of light in place of the actual relative speed (composed of and ), LT (STR) makes it impossible for and to be meaningfully distinguished from and [1, 2]. Therefore, the ticking of a clock moving with velocity does not result in a time difference large enough to benefit our passenger. Hence, we conclude that the LT does not solve the problem posed here.

For the current GT, Eq. (6) faces only one obstacle in solving the problem. This is because “the relativity of space” is accounted for in the current GT [9, 10] – specifically, it shares the second limitation found in the LT. Using the current GT, the required conveyor belt speed at which the passenger would catch the plane can be easily and accurately calculated. However, the second equation of the current GT (2) does not justify the gained time, because by definition in this transformation, .

For the TGT, Eq. (6) provides the desired solution and presents no obstacle in fully explaining the resolution of the problem, using the time transformation equation. This is the case because, according to TGT (3), represents the total distance covered by the passenger with the help of the conveyor belt, is the distance covered by the belt itself, and is the distance the passenger walks on the moving belt by his own effort. What, then, is the relativity of distance in this context? For a given distance and time interval, the difference between and depends solely on the speed . When , the entire distance must be covered by the passenger alone, moving at their own speed , and thus . If , then is shorter than , because the conveyor belt assists the passenger in covering the total distance , contributing an additional path segment . In this case, distance is not contracted, but rather expressed as the sum of two components: . The same reasoning applies to time. For a given distance and time interval, the difference between and also depends only on the speed . When , the entire distance is covered by the passenger at their own walking speed , and therefore (see Eq. 3), meaning the passenger gains no time. More precisely, in this case:

and time represents a time interval of a single event. If , then in addition to the distance , the combined velocity also comes into play, and now, for the event time , we have:

Equation (8) shows that is shorter than because the conveyor belt assists the passenger in covering the distance , resulting in the additional time interval . Thus, in this case, time does not undergo contraction or dilation, but simply represents the sum of two components: .

First conclusion: Just as the total distance , covered jointly by the conveyor belt and the passenger at a combined (relative) speed during the event time , is the sum of two distances; and just as the relative speed is the sum of the two individual speeds; likewise, the total time interval of the event is the sum of two time intervals. In other words, corresponding to the equations for and , there is a matching equation for :

Second conclusion: Historically, discussions in relativity have focused only on the time intervals and . However, the status of the time intervals , , and is the same, since we are dealing with three observers – three reference frames – that, according to the principle of relativity, are not privileged with respect to one another.

Third conclusion: The appearance of time relativity (as well as the relativity of distance and speed) can be neglected in cases where the difference between the component speeds of the relative velocity is large, but it becomes unavoidable when these two speeds are close in value (see cases 1 and 4 in Tables 1 and 2). According to modern physics, since the speed of light appears in the LT equations for every type of compound motion, the time interval is always considered negligible – zero – even for a speed such as that in case 4 (Table 1). In contrast, under TGT, the time interval is zero only when . If becomes very large, then the time interval loses its significance (see case 4, Table 2).

Fourth conclusion: Relativity of time gives meaningful content to the term “gained time”. From the example involving the conveyor belt and from the calculations, we understand why the relative time can also be interpreted as gained time in the case of everyday events involving compound relative motion. The second equation of the TGT (3) is consistent with the law of conservation of energy, as it expresses the contributions (in terms of time) made by the conveyor belt and the passenger to cover the distance during the total time of the event :

This equation can and should be referred to as “the longitudinal time Doppler effect” [1, 2]. Now let us pose the following question: How much time did the passenger gain by stepping onto the conveyor belt moving at speed (Fig. 1)? This is determined by the difference between two time intervals: the event time , when the distance is covered by the passenger alone, and the event time , when the same distance is covered with the assistance of the conveyor belt (10).

Fifth conclusion: The relativity of time and time dilation, as scientific terms, are not the same thing. The relativity of time refers to the shortening of a traveller's time interval when covering a distance , as his speed increases; while time dilation refers to the idea that, in such a case, the traveller’s clock runs more slowly [11]. Time relativity is easy to understand and explain, has a clear physical meaning, is easily measurable, widely applicable, and practically unavoidable. On the other hand, time dilation is complex and not easily explained, lacks a clear physical interpretation, is difficult to measure, and is used mainly in science for theoretical or “exotic practical” cases – and in most situations, it can easily be ignored. Relative time has been known and used by humans since antiquity, and it was also defined by Newton in the Principia as time measured by means of motion [12].

Author has no conflicts of interest to disclose.

There is no funding to declare.

I am a single author! The idea of manuscript is mine, and I wrote the manuscript text and prepared figure.