  The gravitational memory effect and its electromagnetic (EM) analog are
potential probes in the strong gravity regime. In the literature, this effect
is derived for static observers at asymptotic infinity. While this is a
physically consistent approach, it restricts the space-time geometries for
which one can obtain the EM memory effect. To circumvent this, we evaluate the
EM memory effect for comoving observers (defined by the 4-velocity $u_{\mu}$)
in arbitrary curved space-times. Using the covariant approach, we split
Maxwell's equations into two parts -- projected parallel to the 4-velocity
$u_{\mu}$ and into the 3-space orthogonal to $u_{\mu}$. Further splitting the
equations into $1+1+2$-form, we obtain \emph{master equation} for the EM memory
in an arbitrary curved space-time. We provide a geometrical understanding of
the contributions to the memory effect. We then obtain EM memory for specific
space-time geometries and discuss the salient features.
