The Lorentz factor in a reverse coordinate system

Abstract

In the present study, we have derived the Lorentz factor using a coordinate system with antiparallel X-axes. Using a thought experiment, common in relativistic literature, we have used the case of a pulse of light moving along the X-axis. Next, we have argued that, consequently with the isotropy of space, the result must be the same if the trajectory of the light pulse is any angle α > 0 • , thus obtaining an alternative transformation factor that generates the same results as the Lorenz factor at any angle of the trajectory of the light pulse, therefore confirming the relativistic isotropy of space in the context of the Minkowski spacetime. Nevertheless, a particularly important novelty is that the derived alternative transformation factor allows the reference frames involved to move at speeds greater than that of light.

A Transformation Factor for Superluminal Motion That Preserves Symmetrically the Spacetime Intervals

“…Specifically, the positive side of the X axis points to the right, while the positive side of the X' axis points to the left (Figure 1). We used this method in our prior work [23], from which we will take some of the derivations that we expanded upon, and especially those for which we provided a broader interpretation. As noted by Friedman and Scarr [22], using aligned coordinate systems breaks the symmetry in the relative motion of frames, since frame S moves in a negative direction with respect to frame S', leading to a difference in the structure of the transformation equations and their reverses [24], which differ by the sign of v. Although the choice of a particular type of coordinate does not affect the results, using reversed coordinates can help highlight certain features, such as those related to symmetry.…”

“…In this way, as explained by Friedman and Scarr [22], in Galilean transformations, the relationship between x and x within an aligned coordinate system is determined by x = (x − vt). To apply the same conversion in a reversed coordinate system, we must reverse the x axis (x −→ [−x ]) [23], resulting in:…”

“…Additionally, it is important to note that this factor applies to four-dimensional physics, and there is no relative movement in the Y and Z axis directions. The Lorentz transformations can be expressed in matrix form (2) and in their inverse form (23).…”