Retarded Fields of Null Particles and the Memory Effect
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 ${t}_{0}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ 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. As already noted, the second limit does not exist in the scalar case or for the Lorenz gauge vector potential and Lorenz gauge metric perturbation in the electromagnetic and linearized gravitational cases. However, in the electromagnetic case, we obtain a well-defined distributional limit for the electromagnetic field strength, and in the linearized gravitational case, we obtain a well-defined distributional limit for the linearized Riemann tensor. In the gravitational case, this limit agrees with the Aichelberg-Sexl solution. There is no memory effect associated with this limiting solution. This strongly suggests that the memory effect---including nonlinear memory effect of Christodoulou---should not be interpreted as arising simply from the passage of (effective) null stress energy to null infinity but rather as arising from a ``burst of radiation'' associated with the creation of the null stress energy [as in case (i) above] or, more generally, with radiation present in the spacetime that was not ``produced'' by the null stress energy.