On El Naschie’s complex time, Hawking’s imaginary time and special relativity

The idea of complex time, as first proposed by El Naschie in 1995, not only provided a very important mathematical utility in clarifying the nature of nowness, but also opened a definite possibility for the instantaneous transmission of information through the theoretical prediction of massless particles travelling at velocities larger than the speed of light. Based on a very simple thought experiment, here we show that the complex nature of time arises when two independent inertial observers, in relative uniform motion, communicate via a light signal in order to compare their own proper time measurements for the same event. The observation that the time employed by the signal to go from one observer to the other is calculable, but not measurable, permits to build up a general expression for the complex time, which not only complies with the possibility of time decomposition into two dimensions, but also conciliates with the idea of a complex space. In particular, we find that El Naschie’s complex time can be interpreted as an asymptotic limit when the velocity of the moving observer equals that of light. Within this new formulation, the inverse Lorentz transformations of special relativity follow as a direct consequence of the complex time.