Derivation of the Lorentz Transformation Equations for Determination of their Matrix Form

Abstract

Abstract: This article introduces a modified version of the Lorentz transformation equations that transform spacetime coordinates between two inertial frames when the relative motion between them occurs along the X-, Y-, and Z-directions, and represents an extension of the one-dimensional Lorentz transformation equations to three spatial dimensions. Making use of the invariance of the spacetime interval, the paper demonstrates that an event in the spacetime continuum can be represented by six coordinates, of which the first three represent the spatial coordinates, and the remaining three represent the time coordinates. By employing the notion of a position six-vector, the correct matrix form of the Lorentz transformation equations of order 6 × 6 has been thoroughly developed. In addition, the D’Alembert operator, the basic ingredient of the wave equation, is shown to be form-invariant under the modified Lorentz transformation equations. Furthermore, the relativistic velocity addition formulas, as well as the Lorentz transformations of linear momentum and energy, have been theoretically analyzed on the basis of the extended Lorentz transformations. Finally, the particular purpose of this work is to present equal and opposite relativistic spacetime coordinate transformation equations between inertial frames, which properly allow for the formulation of the correct matrix form of the Lorentz transformation equations in terms of the position six-vector.