Electromagnetic memory in an expanding universe.

Abstract

The memory effect is the total change in fields over infinite time and has been developed in detail for gravitation. For electromagnetism the memory effect is a very recent study. In this paper we analyze and calculate the memory as proposed by Susskind (https://arxiv.org/abs/1507.02584). This memory is embedded in a spherical shell of superconducting nodes with penetration depth $\lambda$. Between these nodes a current will flow that measures the memory. This current is proportional to the gradient of the superconductor phase $\phi$. We also extend this memory to an expanding universe, since this connection has not yet been made for the electromagnetic memory. As a charge source we take a single charged particle moving in the $\hat{z}$ direction. In Minkowski space we find an explicit expression for the final phase which causes the memory current. For the expanding universe calculation we assumed the universe consists of a single component and undergoes decelerated expansion. We also look at times $t\gg \lambda$ only. Under these assumptions we have also found an explicit expression for the memory effect in the expanding universe. Both solutions oscillate in time, so the current we measure depends on the time of measurement.