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.
