THEORY OF RELATIVITY - IT IS POSSIBLE THAT A PARTICLE CAN TRAVEL FASTER THAN THE SPEED OF LIGHT

PRESENTATION

Contrary to what is said in the book by A. Einstein entitled: "On the theory of special and general relativity" (Ediciones Altaya S.A. 1999), where it is stated that the speed of light (c) cannot be exceed, we make a mathematical approach to show that a particle can exceed this speed.

When we transcribe some content of the book that we take as a reference, we will write it in quotation marks and italics.

 

1.-WHERE DOES IT APPEAR THAT THE SPEED OF LIGHT CANNOT BE EXCEEDED?

On page 36 of the book that we take as a reference, after making an application of the formula of the space of Lorentz Transformations, the following sentence appears:

“For speed v = c it would be

and for even higher speeds the root would become imaginary. From this we infer that in the theory of relativity speed (c) plays the role of a limiting speed that no real body can reach or exceed. We add that this role of velocity (c) as limiting velocity follows from the Lorentz Transformation equations themselves because they lose all meaning when (v) is chosen greater than (c). “

The fact that an equation "loses all meaning" for some values of its variables, perhaps this sentence already indicates that it is not choosing the right path to assess what it intends. If this equation were adequate, we believe that, referring to the variables that compose it, it would have to define for which values it loses its field of validity. It should not be the equation that "loses its meaning" but we must define when its variables lose their field of validity.

Of course, these equations lose all meaning when (v) is greater than (c) since they do not precisely serve the purpose they want to give them… In this case, the observer has no field of vision of the Event to be observed. But he has nothing to do with the fact that the event does not exist and moves at any speed.

We must choose the appropriate mathematical approach so that between the comparison of the speed' (v) of a moving body and the speed of light (c) this incompatibility does not occur. This is what we are going to study.

 

2.- GEOMETRIC APPROACH TO THE OBSERVATION OF AN EVENT THAT OCCURS IN SIDERAL SPACE

In the study of the vision of the appearance of an Event that occurs in outer space, observed from a Mobile Reference System, we will proceed as follows:

We locate three points in outer space. A point (E) will be identified as the place where the Event appeared or was born. Another dot (F) will represent the end of the Event duration. We will identify it as the end of your EXTENSION. A third point (PO) will be the observation point of the Event. To carry out the calculations, these three points will be distributed in such a way that they form a right triangle.

The figure represents that the situation where the event is born is fixed (SRF). The point of observation (PO) is mobile (SRM). It can be moved, of course, always considering the figure of the right triangle.

With this geometric figure we can represent all the variables and parameters that intervene in the observation process carried out from a (SRF), of an Event that happens in a (SRM). With the help of this geometric figure we can make the approach that will allow us to obtain the mathematical expression:  

known as the Lorentz Factor.

The interpretation of the previous figure is the following:

We want to observe from a moving point (PO), the Extension of the Event. This Extension is represented in the drawing by: c.(tp ), where (c) is the speed of light and (tp) which we will call the Event's Own Time. Because the point (PO) is mobile, we include it within a Mobile Reference System (SRM).

The variable (td) represents the Displacement Time of an observer to locate himself at the point of observation (PO), having started from point (F). We identify this point (F) to mark the end of: c.(tp). We consider this point (F) as the starting point of the displacement, since we have to ensure that when the observation point (PO) is reached, the entire Extension of the Event has already been developed. (Note the reader that it is a geometric condition that we are imposing). The variable (tr) means the travel time. Therefore: c.(tr) is the space that exists between the point (E) of appearance of the Event and its observation point. It is the path of the image. It is necessary that the information on the appearance of the Event has arrived from the point (PO).

These will be the "rules of the game" that govern the observation of an Event that occurs in a certain place in outer space and that a mobile observer (relative movements) observes its appearance and duration.

 

3.-VISION CONDITIONS OF AN EVENT AND MATHEMATICAL APPROACHES TO MEET THIS CONDITION

In order to observe an Event, which occurs at a certain point (E) in outer space, from a mobile observation point (PO), two VISION CONDITIONS OF THE EVENT will be:

That the observer is already located at the point of observation (PO) and that the image of the Event has also reached this point.

We can choose two different mathematical approaches to meet the Viewing Conditions of the Event.

SYNCHRONIZATION CONDITION.

This condition requires that the travel time (tr) of the Event image, from point (E) to point (PO), be equal to the observer's displacement time (td), from point (F) to point (PO). That is (tr) = (td)

APPROACH TO COMPENSATION IN TRAVEL S

We will call another mathematical approach that we can give to fulfill the Viewing Conditions of the Event: “Compensation Approach in the Routes.

We will assume that, for the vision of the Event and the arrival of the observer to arrive together at the same point of observation (PO), if the speed (v) of the observer is greater than the speed of light (c) in the path performed by the light (the Event image), then we will also assume that its travel time (td) is less than the travel time (tr) of the Event image.

That is: If (v) > (c) implies that (td) < (tr)

When considering carrying out the necessary mathematical calculation to obtain the formula that allows us to assess the Own Time (tp) of an Event, we will forget about the Synchronization condition and we will consider the possibility that a higher velocity (v) can be compensated with a lower Travel Time (td)

This approach will make the final formula that allows us to value (tp) have a different structure from the one that would be obtained by applying the Synchronization condition and with this "do not lose meaning" as A. Einstein says in his book.

 

4.- TWO INCOMPATIBLE EVENTS IN MATHEMATICS

We can choose two paths (two mathematical models) to visualize and be able to quantify the value of (tp). But, although both allow the value of (tp) to be obtained, one of them is limited and produces a mathematical incompatibility depending on which values are assigned to its variable (v).

Starting from the right triangle that serves as a mathematical pattern, we say the following:

If we impose: (tr) = (td) (a condition)

Y

we assume (v) > (c) (another condition)

this assumption produces an inconsistent event in mathematics

since then implies that: (td). (v) > (tr). (c)

Y

In a right triangle, it is inconsistent that one leg is greater than its hypotenuse.

This mathematically incompatible path is the one chosen by the author of the aforementioned book to justify the invalidity of the factor

of Lorentz when (v) is equal to (c)

However, in a later essay we will see that the path of considering the Synchronization condition leads us to give us a result that will be useful to start another analysis. It is for these that in the following paragraph we explain its mathematical development.

 

5.- MATHEMATICAL DEVELOPMENT TO OBTAIN THE OWN TIME (TP) OF AN EVENT, APPLYING THE SYNCHRONIZATION CONDITION

In order to do the calculations to obtain the display and the value of (tp) from the point of observation of the Event (PO), we are going to impose the SYNCHRONIZATION condition. so it must start from the point (F) in which the full extent of the Event can already be seen. In addition, another condition is that it arrives at the point (PO) precisely when the Event image has arrived. Therefore, when the calculation process begins, we will impede the condition:

                                          (td) = (tr)

and the value (tr) is replaced by its equivalent (td).

To start the mathematical study we will observe the figure that we have drawn in paragraph 2, and we will proceed to develop the calculations, applying the Pythagorean Theorem.

                   
Demanding the fulfillment of the Synchronization Condition:

                                  (td) = (tr)

allows us to substitute (tr) for (td) with what is obtained:

 

Grouping terms we have:

We can transform the denominator as follows:

                

And from here we get:    

In the previous formula, the expression: 

is known as the Lorentz Factor. 

Let's remember that this approach is the one that produces an incompatible ucess in mathematics, but it is precisely the one that A accepted. Einstein and his followers.

6.- MATHEMATICAL DEVELOPMENT TO OBTAIN THE (Tp) USING THE COMPENSATION APPROACH IN THE ROUTES

We can give another approach to assess the Own Time (tp) of an Event (E), which occurs in a Fixed Reference System (SRF), from another Mobile Reference System (SRM).

In this other approach, we will enforce the VISION CONDITIONS OF THE EVENT in a different way than what we have explained in the previous paragraph. We will use the Compensation Approach in the Walkthroughs.

Remember that this approach implies that a higher value of the variable (v) implies a lower value of the variable (td), that is, we will justify the following:

                           If v > c implies that (td) < (tr)

With this justification and NOT taking into account the equality

(tr) = (td) that we had commented on in the previous case, we proceed to develop the path to obtain the value of the Own Time (tp) of the Event and with it its Extension (c. (tp)).

We will use the figure we have drawn in paragraph 2 as an observation guideline. The steps to follow are the following:

 

With what we have obtained the proper time of the event.

We ask ourselves: is this mathematical expression obtained valid? If it is valid, we can accept that a particle traveling in the direction of the axis (X) of the aforementioned drawing, can reach a speed: (v) > (c).

We must study the field of validity of the previous formula.

 

7.- FIELD OF VALIDITY OF THE RESPONSE OBTAINED USING THE COMPENSATION APPROACH IN THE ROUTES

We will analyze the field of validity of the formula:

            

that we have obtained in the previous paragraph.

We check if the condition is met:   

    

It would be zero when:

     

  = 

and this would imply c×tr=v×td

Considering the geometric pattern that has served as the basis for the mathematical development, we see that in this right triangle it is impossible for the hypotenuse to be the same as one of the legs. Consequently, the value of the radicand cannot be zero. So, using geometry, we can state mathematically that a particle can travel at speeds greater than the speed of light.

The formula would also be valid when (v) was much smaller than (c), that is: v<<<c. Then it would happen that, very approximately, tp = tr

Which would indicate that we were at the starting point (F) in which the path of the image of the light is equal to the Extension of the Event.

(NOTE: Delving a little deeper into the issue of the validity of the formula we have obtained, we could ask ourselves what would happen if we considered speeds (v) much higher than that of light. That is, if (v)>>>(c) This would vary the values of the right triangle on which the formulation of the formula is based, obtaining a very small displacement on the base leg and a very large TOwn time (tp), so that the Extension: c.(tp) of the event would be very large. Realize that now we are not evaluating exclusively the value: v^2/c^2 but its routes through its time. Perhaps this is the behavior of the particles?... We leave this question for to be judged by microparticle experts)

(NOTE: In the booklet "Theory of relativity.- Critique of nonsense analyzed in seven installments" you will find seven topics for debate on this theory.