This work we consider that explains in a forceful way to all our work, because it identifies the vector space of the quad vector in the Special and General Relativity of Einstein, as a mathematical situation that when needing the measurement of these vectors and working with them, we can feel a vector space that has an additional structure provided with the scalar product, especially if mass is conceived as a vector. So in this article space is treated as such. Einstein does not even think about it because he does not really imagine mass as a vector, even though in addition to the invariant mass Einstein already fully identified an additional inertial and relativistic mass, which even increased as a mass to the beat of speed. It was missing was to identify the apparent gravitational mass also tied in some way to the speed, to complete the vectorial relationship of the mass.
1. Introduction
This introduction without resorting to the mathematical language of matrices or tensors, makes the deduction of the recognized equation of Special Relativity to identify, the essential point where we understand Einstein was confused. In addition, this work aims to finish justifying the previous works of the apparent gravitational mass and the new equations of kinetic energy and momentum.
A quad vector is a mathematical representation in the form of a vector of four dimensions of a vector magnitude in the theory of relativity. The works of Lorentz, Poincaré, Einstein, and Minkowski on classical electromagnetism led to the idea that it is not possible to define an absolute time that takes place in an identical way for all observers regardless of their state of movement. The non-existence of an absolute time required that there be a measure of time for each observer. Thus the set of events (space-time points) naturally led to define vectors of four dimensions:
Where E is the vectorial space and the four previous components representing the three spatial coordinates of the place in which something occurs and the moment in which it happens. For c is simply the speed of light that appears multiplied by the event's own time to translate the relative time of an observer.
Special relativity uses tensors and quadrivectors to represent a pseudo-Euclidean space. This space, however, is similar to three-dimensional Euclidean space in many ways and it is relatively easy to work on it. The metric tensor that gives the elementary distance (ds) in a Euclidean space is defined as:
Where dx, dy, dz are differentials of the three spatial Cartesian coordinates and ds is the resulting differential.
In the geometry of special relativity, to show the pseudoeuclidean character of space-time geometry, the fourth dimension of contracted light given in the product jcdt is added, where t is time, c the speed of light and j the unit of contraction. Being also consistent with that fourth dimension that is added in the approach of this article, it should always be considered in an orthogonal sense to the direction resulting from the three spatial Cartesian coordinates. The resulting quad vector is the differential of the light space and the relativistic interval remains, in differential form, in the following way:
Where dc is the differential of the light space or quadrivector, dx, dy, dz are the differentials of the three spatial Cartesian coordinates and jcdt is the fourth vector added.
In the same way that the velocity in Newtonian mechanics is the temporal derivative of the position with respect to time, in the special theory of relativity the quadrivelocity is the temporal derivative of the quadrivector position with respect to the own time of the particle. The quadrivelocity is a vectorial magnitude associated with the movement of a particle, used in the context of the theory of relativity, which is also tangent to the trajectory of said particle through four-dimensional space-time. Therefore, starting from the above equation number three (3) and transferring equivalent terms we obtain the quadrivelocity in the following way:
In the above equation number five (5) of the quadrivelocity, we can still observe the product jc that still only lasts from the fourth dimension initially added. This unit j of contraction or coefficient of contraction is precisely the mathematical element that provides the fixed substrate of general relativity, which appears in a relational way between two events of space-time since the vacuum is dependent on the trajectory of the observer in space-time . Exactly, j is equal to the quotient of the relation between masses, we assume the apparent gravitational mass and the own and invariant mass of a particle that moves with respect to an observer beside, that same unit of contraction j, also is equal to the Lorentz contraction as described in the following relationship:
Where j is the coefficient of contraction, mo is in our calculations the apparent gravitational mass and m is the invariant and own mass of the particle.
We can take any of the two equivalent values of j expressed in the previous relation six (6), to replace it in equation number five (5) of this work, whether we use the relationship between the masses that in our case always we make between the apparent and invariant gravitational mass or, let's take the Lorentz contraction as it seems was the choice and path that Einstein's calculations followed as expressed in the following relationships:
By replacing and mathematically transferring the Lorentz contraction in the whole equation, we have the previous relation number seven in the following way:
Here is the time when Einstein involves the mass as a simple scalar through using the same definition of Newton's momentum, since the whole relation and quadrivector above is multiplied by the same invariant scalar mass, leaving the number six relation as follows:
Then we distinguish the concept of apparent inertial mass or also called relativistic mass as follows:
Where mi is the apparent inertial mass or relativistic mass, m is the invariant mass and the recognized Lorentz contraction
This previous relationship leads us finally to the next, recognized and famous equation of special relativity:
Where mi is the apparent inertial mass or relativistic mass, m is the invariant mass and p is the amount of motion, v is the velocity of the particle and c is the speed of light.