Speed of Light in Vacuum in the Case of Arbitrarily Non-uniform Motion of Reference Frames

Reference frames in arbitrary relative motion and with a coordinate transformation law are considered. It is shown that continuity of the time coordinate of one frame in the time coordinate of the other is a necessity for any physical coordinate transformation even if the velocity function for the relative motion of the reference frames is discontinuous in time. Then the differential Lorentz transform is used to infer an infinite speed of light in vacuum c, when the reference frames are in non-uniform relative motion. Based on the continuity of the time coordinate of one frame on the time coordinate of the other and on the integral representation of the differential Lorentz transform, a set of necessary and sufficient conditions is proved for the attainment of an infinite speed of light for reference frames in arbitrary relative motion. Using these conditions, it can be inferred that an initially finite c can be made infinite if and only if i) the reference frames are initially at rest and their motion, be it uniform rectilinear or arbitrarily non-uniform, starts with an initially discontinuous relative speed $v_{x}^{\prime}$ in time or ii) the frames are in arbitrary motion which at $t^{\prime}=0$ undergoes a jump discontinuity in $v_{x}^{\prime}$ with $v_{x}^{\prime}\neq 0$ at $t^{\prime}=0+$. After the infinite character of c is attained, c remains infinite regardless of whether the change in $v_{x}^{\prime}$ is smooth or not, until the motion comes to a stop not to start again with another initial discontinuous $v_{x}^{\prime}$. On the other hand, the differential Lorentz transform that holds in the immediate neighborhood of a fiducial observer, is only local and is therefore incapable of allowing drawing of global conclusions directly, such as the globally infinite c result of the paper. However, the proof that c is infinite locally can be generalized to hold for all space coordinates and for all future time, if one observes that unless the motion of the frames comes to a stop not to start again with a discontinuous $v_{x}^{\prime}$ the infinite character of c does not change when immediate neighborhoods of particular observers are jointed to constitute the full space and future time.