Fig. 2.

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Space-time depictions of the past light cone of the observation point, P, located at xP (the dashed blue lines) and the possible forms of the trajectory, R(t) = |x(t)−xP|, of a super-luminally rotating source point, S, with the position vector x (the red curves). In (a), where |dR/dt|< c, the trajectory of S intersects the past light cone of any P only once, so the field observed at P at time tP is determined by the single wave front that was emitted by S at the retarded time, t. In (c), where the component of the velocity of S along the radiation direction is greater than c, the trajectory of S intersects the past light cone of certain observers, P, at three points, so the field at P receives contributions from three distinct values of the retarded time simultaneously. In (b), where both dR/dt = −c and d2R/dt2 = 0 at the point where the trajectory of S intersects the past light cone of P tangentially, the field is determined by the coalescence of three contributions originating from a retarded time at which the source point approaches the observer along the radiation direction not only at the speed of light but also with zero acceleration.
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