Understand
We consider the retarded solution to the scalar, electromagnetic, and linearized gravitational field equations in Minkowski spacetime, with source given by a particle moving on a null geodesic.
- In the scalar case and in the Lorenz gauge in the electromagnetic and gravitational cases, the retarded integral over the infinite past of the source does not converge as a distribution, so we cut off the null source suitably at a finite time $t_0$ and then consider two different limits: (i) the limit as the observation point goes to null infinity at fixed $t_0$, from which the ``$1/r$'' part of the fields can be extracted and (ii) the limit $t_0 \to - \infty$ at fixed ``observation point.'' The limit (i) gives rise to a ``velocity kick'' on distant test particles in the scalar and electromagnetic cases, and it gives rise to a ``memory effect'' (i.e., a permanent change in relative separation of two test particles) in the linearized gravitational case, in agreement with previous analyses.
- Although the second limit does not exist for the Lorenz gauge potentials in the electromagnetic and linearized gravitational cases, we obtain a well defined distributional limit for the electromagnetic field strength and for the linearized Riemann tensor.
- In the gravitational case, this limit agrees with the Aichelberg-Sexl solution, but there is no ``memory effect'' associated with this limiting solution.