In the equations for transforming between reference frames (Lorentz Transformations) one might discard faster than light (FTL) (v > c) as the result yields an imaginary number. However, I considered the possibility that complex numbers themselves, represent Spacetime coordinates. It is almost a mathematically tautology that they do so. I even re-derived the transformations, and they result in the same formulae, as the imaginary components cancel out in the solving of the system of equations.
Now, if you consider space coordinates the real component and temporal coordinates the imaginary component, speeds slower than light (STL) (v < c) result in the same transformations as with using only real numbers for both space and time coordinates. However, FTL speeds result in space and time coordinates inverting spots (due to imaginary numbers appearing and cancelling each other out for the time coordinate). Moreover, the transformation results in a temporal coordinate shift that implies events in the future for FTL reference frame appear to be in the past for STL reference frames, and vice reversa. The implication is that FTL speeds result in travelling backwards in time relative to STL reference frames.
Naturally, it seems impossible to accelerate a massive object FTL, as its inertia also increases with its speed, yielding an inertia approaching infinity as its speed approached that of light. However, I am currently in the process of deriving Einstein's field equations for General Relativity with complex spacetime coordinates. I will update with my results. My hypothesis is that a massive objected may be accelerated through a singularity to reverse its direction of travel through time.
I will update you all when I have finished.
Opinions, criticisms, and resources are welcome.