Generalised transformation relations for superluminal particles in (3 + 1) dimensional system

Here, we have proposed new transformation relations between two inertial frames with relative velocity greater than speed of light and discussed the required conditions to detect superluminal particles in the physical realm. A novel theoretical formulation is conceptualized to uniquely depict the frame of reference, valid for all ranges of velocities, in a complex plane. In the tachyonic regime, the super-relativistic gamma factor is obtained by parameterizing the generalised gamma factor using the complementary frames of reference of superluminal and subluminal spaces. The formulations of transformation relations have been analysed in four different classes, i.e., two inter and two intra classes of frames, for subluminal and superluminal particles. The intra classes of frames encompass real gamma factors, whilst, inter classes of frames involve imaginary gamma factors. These transformation relations in all four different classes are expressed in 4-vector form with their respective transformation matrices \((\Lambda )\) and are found to be invariant under the transformations. This invariancy has led to the establishment of relationships among the transformation factors in stationary and moving frames. The velocity 4-vector relations for subluminal and superluminal particles, both in moving and stationary frames, have been evaluated. The 4-velocity transformation relations between particle velocity in stationary and moving frames are also introduced by using the transformation matrices. Based on these 4-velocity transformations, relative velocity addition relations have been deduced and discussed corresponding to each of the transformation relations. The generalised gamma factor has been concluded and validated for both frames of reference.