# Electromagnetic memory in arbitrary curved space-times

Susmita Jana<EMAIL_ADDRESS>Department of Physics, Indian Institute
of Technology Bombay, Mumbai 400076, India  S. Shankaranarayanan
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Bombay, Mumbai 400076, India

###### Abstract

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 _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.

##  I Introduction

LIGO-VIRGO-KAGRA has detected close to 100 gravitational wave (GW) sources. GW
signals emanating from a black hole or neutron star binaries have opened many
new research avenues in astronomy, cosmology, and fundamental physics
2009-Sathyaprakash.Schutz-LRR ; Arun:2013bp ; 2019-Barack.etal-CQG ;
2022-Shanki.Joseph-GRG . GWs provide a unique way to test gravity’s most
extreme, non-linear regime in novel ways. The planned third-generation ground-
based detector (Cosmic Explorer and the Einstein Telescope) will allow us to
peer far deeper, and LISA will open a new observational window at low
frequencies. With more sensitive detectors shortly, the focus has been to
understand the physical effects of GWs. _Gravitational wave memory_ is one
such effect Zeldovich:1974gvh ; PhysRevD.15.2069 ; kovacs1978generation ;
Braginsky:1986ia ; 1987-ThorneBraginskii-Nature ; PhysRevD.45.520 ;
PhysRevD.44.R2945 ; 2010-Favata-CQG ; 2014-Bieri.Garfinkle-PRD .

GW memory effects — physically observable phenomena that modify the state of
gravitational-wave detectors a little bit from their original undisturbed
state — are one of the key predictions of general relativity PhysRevD.15.2069
; kovacs1978generation ; 1987-ThorneBraginskii-Nature ; 1991-Christodoulou-PRL
. GW memory effects can be divided into two types 2010-Favata-CQG ;
2014-Bieri.Garfinkle-PRD : _null memory_ that occurs when radiation or
massless particles escape from a system to null infinity, and _ordinary
memory_ that occurs when the detector recoils relative to its initial center
of mass frame. The GW memory is characterized as a gravitational wave signal
approaching a nonzero finite value. This aspect of the GW signal is yet to be
observed, although LISA is predicted to observe it 2009-Favata-ApJL .

Recently, it has been realized that the memory effect can be thought of as a
vacuum transition between two different states related by an asymptotic
transformation 2014-Strominger-Zhibeodov-JHEP ; 2017-Zhang.etal-PRD . Since
such asymptotic transformations also occur for other gauge theories, there has
been an intense activity to obtain analogous memory effects in other gauge
theories Bieri:2013hqa ; Winicour:2014ska ; 2014-Strominger-JHEP ;
2017-Pate.etal-PRL ; 2019-Satishchandran.Wald-PRD . Since electromagnetic (EM)
theory is the simplest of all gauge theories and can be a potential probe,
_electromagnetic memory_ has received much attention 2012-Bieri-CQG ;
2015-Susskind-hep-th ; 2017-Hamada-JHEP ; 2018-Hamada.Sotaro-JHEP ; 2019-Mao-
PRD ; 2019-Jokela.Sarkkinen-PRD ; 2020-Mao.Tan-PRD ; 2021-Mao-PRD ;
2022-Taghiloo-hep-th ; 2022-AtulBhatkar-PRD ; 2022-Seraj-hep-th . Like in GW
memory, an EM wave generates a permanent change in the relative velocity of
test-charged particles attached to a detector in the 2-D surface perpendicular
to the direction of propagation of the wave while passing through the detector
[cf. (1)]. In other words, EM waves directly displace test particles by giving
them a momentum (kick), resulting in a relative velocity change. This is
different from GW memory as the GW does not displace test particles. Instead,
GW distorts the space-time geometry itself, which causes a change in
separation between two test particles.

Figure 1: Electromagnetic memory effect that lies in the 2-D surface
orthogonal to the direction of the coming wave.

Bieri and Garfinkle were the first to propose the memory effect due to
electromagnetic waves Bieri:2013hqa . Like in GW memory, they showed that EM
waves produce two types of momentum kicks. In Ref. Winicour:2014ska , Winicour
showed the absence of memory effect generated by the electromagnetic field
coming from distant sources for a bound charge distribution and the non-
existence of memory effect due to the magnetic field.

In the case of GW memory, gravitational radiation must reach the detector.
Likewise, EM radiation also has to reach null infinity to generate _null kick_
memory. Hence to calculate EM memory, one needs to know the properties of the
electric field and radiation at null infinity Bieri:2013hqa . More
specifically, the original approach by Bieri and Garfinkle requires prior
knowledge about the behavior of the fields in asymptotic limits. It can be
extended to conformally flat space-times 2022-AtulBhatkar-PRD ;
2022-EnriquezRojo-JHEP . Also, the analysis does not provide any physical
understanding of why the EM memory has such a form in flat and conformally
flat space-times.

This leads us to the following questions: Can we derive a master equation for
_EM memory_ in a generic curved space-time? What role does curved geometry
play in EM memory? Can we have a physical understanding of the various
contributions to EM memory? This work addresses these three questions using
$1+3$ covariant formalism 1955-Heckmann-zap ; 1955-A.K.Raychaudhuri-PRL ;
1996-vanElst.Ellis-CQG ; 1998-Ellis-NATOSci ; 2008-Tsagas.etal-PRep ;
2012-ellis_maartens_maccallum-Book .

There are two reasons why covariant formalism is better suited to studying EM
memory. First, as mentioned earlier, when the EM wave propagates in a given
spatial direction, the net momentum experienced by the particle lies in the
$2$-D surface orthogonal to the direction of propagation of the EM wave (for a
pictorial representation, see 1). In other words, the EM memory affects the
test particle lying on the 2-D surface. Hence, it is more natural to have a
formalism that identifies such a dynamical 2-D surface and evaluates EM
memory. Second, like in fluid mechanics, we can observe the flow of EM
radiation in two ways. First, as in Refs. Bieri:2013hqa ; Winicour:2014ska ,
an asymptotic stationary observer monitors changes in Electric and Magnetic
fields of the incoming EM radiation. Second, a comoving observer monitors
changes in Electric and Magnetic fields. In fluid mechanics, these are
referred to as the Lagrangian and Eulerian descriptions of flow, respectively.
It is well-known that the Eulerian description is better suited for fluids and
in cosmology 1996-vanElst.Ellis-CQG ; 1998-Ellis-NATOSci ;
2012-ellis_maartens_maccallum-Book .

In this work, we evaluate the memory effect using the $1+1+2$ covariant
formalism 1996-vanElst.Ellis-CQG ; 2003-Clarkson.Barrett-CQG ;
2004-Clarkson.P.K.S-APJ ; 2005-Tsagas-CQG ; 2021-Mavrogiannis.Tsagas-PRD . The
$1+1+2$ decomposition of space-time is a natural extension of the $1+3$
formalism in which the three-space is further decomposed to a given spatial
direction. This approach is also referred to as _semi-tetrad formalism_
2017-Sayuri.Sunil.Maharaj-PRD ; 2019-Sayuri.Sunil.Maharaj-JMP ;
2020-Hansraj.Sunil.Maharaj-GRG ; 2022-Singh-Sunil.Maharaj ;
2020-Khambule.SunilMaharaj-CQG . The principle advantage is that we can
evaluate the net momentum (kick) vector on the 2-D surface for arbitrary
space-time. Since this affects all the test particles on the 2-D surface, we
refer to this as _memory vector_. This can also be understood using the fact
that the electric and magnetic fields are transverse to the direction of
propagation of the EM wave. Using the $1+1+2$ covariant formalism, we obtain
the master equation for the EM memory in arbitrary space-time. We provide a
geometrical understanding of the various contributions to the memory effect.
We then obtain the EM memory for specific space-times.

The rest of this work is organized as follows: In Sec. II, we provide an
overview of the two — $1+3$ and $1+1+2$ — covariant formalisms and obtain the
key geometrical quantities. Then, in Sec. III, we rewrite Maxwell’s equation
in $1+3$ and $1+1+2$ covariant formalisms in arbitrary space-time. Next, in
Sec. IV, we obtain the master equation for the EM memory in arbitrary space-
time and discuss the key features. In Sec. V, we then obtain EM memory for
specific space-times and compare them with the known results in the
literature. Finally, in Sec. VI, we summarise our results and discuss possible
future directions.

In this work, we use $(-,+,+,+)$ metric signature and set
$c=1/(4\pi\epsilon_{0})=1$. A dot denotes a derivative with respect to the
proper time $\tau$. A prime denote derivative w.r.t the space-like vector
$n^{\mu}$. For easy comparison, we follow the notations of Ref.
2012-ellis_maartens_maccallum-Book .

##  II Overview of Covariant formalism

A covariant theory like general relativity does not favor any particular
coordinates. However, splitting tensors in time and space is typically
required for its physical meaning. Thus, the splitting achieves this by
rewriting Einstein’s equations as a set of constraint and evolution equations
in a three-dimensional framework. This allows for an intuitive evaluation of
the relevant physical system.

A choice of coordinates defines a threading of space-time into lines and a
slicing into hypersurfaces 1994-Boersma-Dray-GRG . Thus, the splitting
procedure can be carried out in two distinct ways: First, by employing the so-
called $(3+1)-$ formalism or slicing of space-time 2008-Alcubierre-Book .
Second, by employing $(1+3)-$ formalism, or threading of space-time
1996-vanElst.Ellis-CQG ; 1998-Ellis-NATOSci ; 2012-ellis_maartens_maccallum-
Book . In the $(3+1)-$ decomposition, the time is a label of space-like slices
$\Sigma_{t}$ with space coordinates $x_{i}$. In contrast, in the $(1+3)-$
splitting, the time-like world lines have coordinate $\tau$ and are labeled by
$x^{\mu}$. In the $(3+1)-$ formulation, the construction only requires space-
like hypersurfaces and does not demand causality of the time curves. However,
in the $(1+3)-$ approach, every tensor is split into the parallel and
orthogonal directions to a time-like vector (curves). Furthermore, it does not
provide any condition on the causality of the spatial distances. Though the
two approaches provide different points of view, it has been shown that they
are equivalent for space-times with symmetries 1994-Boersma-Dray-GRG . We use
the covariant $1+3$ formalism in this work to obtain EM memory. As mentioned
in the introduction, covariant formalism provides a physical understanding of
the origin of EM memory in arbitrary space-time.

###  II.1 Covariant 1+3 Formalism

Figure 2: Visualisation of $1+3$ formalism.

Heckmann, Schucking, and Raychaudhuri developed the covariant approach to
General relativity in the 1950s 1955-Heckmann-zap ; 1955-A.K.Raychaudhuri-PRL
and was later used in different gravitational and cosmological models
1996-vanElst.Ellis-CQG ; 1998-Ellis-NATOSci ; 2008-Tsagas.etal-PRep ;
2012-ellis_maartens_maccallum-Book . To decompose the 4-D space-time in
$(1+3)-$ formalism, we introduce a family of observers with worldlines tangent
to a timelike $4$-velocity vector $u^{\mu}$ satisfy the following:

$\displaystyle u^{\mu}=\frac{dx^{\mu}}{d\tau};\quad u^{\mu}u_{\mu}=-1\,,$ (1)

where $\tau$ is the proper time measured along the fundamental world line. See
2. Using the $4$-velocity ($u^{\mu}$) we can define the following projection
tensors 2012-ellis_maartens_maccallum-Book ; 1998-Ellis-NATOSci :

$\displaystyle U^{\mu}\,_{\nu}=-u^{\mu}u_{\nu};\quad
U^{\mu}\,_{\nu}\,U^{\nu}\,_{\gamma}=U^{\mu}\,_{\gamma};\quad
U^{\mu}\,_{\mu}=1$ (2a) $\displaystyle
h_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu};\quad
h^{\mu}\,_{\nu}\,h^{\nu}\,_{\gamma}=h^{\mu}\,_{\gamma};\quad
h^{\mu}\,_{\mu}=3;\quad h_{\mu\nu}\,u^{\nu}=0$ (2b)

$u^{\mu}$, and hence $U^{\mu}\,_{\nu}$, projects physical quantities parallel
to the $4$-velocity of the observer and $h_{\mu\nu}$ projects quantities into
the $3$-space orthogonal to $u^{\mu}$. The tensor $h_{\mu\nu}$ provides the
metric properties of the instantaneous $3$-space as well in the absence of
rotation or vorticity.

In this formalism, the projection of the vector ($V^{\nu}$) orthogonal to
$u^{\mu}$ is defined as $V_{<\mu>}$. Similarly, the trace-less part of a
rank-$2$ tensor ($S^{\alpha\beta}$) projected into space orthogonal to
$u^{\mu}$ is defined as $S_{<\mu\nu>}$. Mathematically, these are given by:

$\displaystyle V_{<\mu>}:=h_{\mu\nu}\,V^{\nu};\hskip
14.22636ptS_{<\mu\nu>}:=\left(h_{\mu\alpha}h_{\nu\beta}-\frac{1}{3}h_{\mu\nu}h_{\alpha\beta}\right)\,S^{\alpha\beta}$
(3)

The projection of the time derivative and orthogonal spatial derivative of any
vector ($V^{\nu}$) and tensor ($S^{\alpha\beta}$) are defined as:

$\displaystyle\dot{V}^{<\mu>}:=h^{\mu}\,_{\alpha}u^{\nu}\nabla_{\nu}\,V^{\alpha};\hskip
14.22636ptD_{\alpha}\,S^{\beta\gamma}:=h^{\mu}\,_{\alpha}\,h^{\beta}\,_{\nu}\,h^{\gamma}\,_{\rho}\,\nabla_{\mu}\,S^{\nu\rho}$
(4)

The covariant derivative of $u_{\mu}$ can be split into two parts: $1)$
directional derivative along the tangent to the world line, $2)$ spatial
derivative in the 3-space orthogonal to $u^{\nu}$. This can further be split
into trace, traceless symmetric and anti-symmetric tensor:

$\displaystyle\nabla_{\nu}u_{\mu}=\frac{\Theta}{3}h_{\mu\nu}+\sigma_{\mu\nu}+\omega_{\mu\nu}-\dot{u}_{\mu}u_{\nu}\,.$
(5)

In the above equation, $\sigma_{\mu\nu}$ is the symmetric expansion tensor
that describes the distortion in the matter flow, $\Theta$ corresponds to the
expansion rate of the matter w.r.t the observer, $\omega_{\mu\nu}$ is the
anti-symmetric vorticity tensor describing the rotation of the matter w.r.t a
non-rotating frame. The last term refers to the relativistic acceleration
vector (the directional derivative) $\dot{u}_{\mu}=u^{\nu}\nabla_{\nu}$ which
corresponds to the degree to which the matter moves under forces other than
gravity plus inertia. Further, using the vorticity tensor, we can define the
following quantity called the vorticity vector:

$\displaystyle\omega^{\nu}=-\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}\omega_{\alpha\beta}\,u_{\mu}$
(6)

where,
$\epsilon^{\mu\nu\alpha\beta}=\frac{1}{\sqrt{-g}}\eta^{\mu\nu\rho\sigma}$ is
fully antisymmetric tensor, $\eta^{\mu\nu\rho\sigma}$ is Levi-Civita symbol
whose values are $\pm 1$ and we set $\eta^{0123}=1=-\eta_{0123}$ 1971-Ellis-
Proc . The Levi-Civita $3$-tensor is defined as:

$\displaystyle\epsilon_{\mu\nu\alpha}\equiv\epsilon_{\mu\nu\alpha\beta}u^{\beta}\,,$
(7)

and satisfies the following relations: $\epsilon_{\mu\nu}u^{\nu}=0$ and
$\epsilon^{\mu\nu\alpha\beta}=2\left(\,u^{[\mu}\epsilon^{\nu]\alpha\beta}-\epsilon^{\mu\nu[\alpha}u^{\beta]}\,\right)$.
The square bracket w.r.t the indices refers to antisymmetrization.

###  II.2 1+1+2 covariant formalism

The $1+3$-covariant formalism is well-suited for relativistic cosmology
because, at the largest observable scales, the universe is homogeneous and
isotropic 1998-Ellis-NATOSci . These symmetries allow the slicing or threading
of the 4-D space-time manifold into a one-parameter family of spacelike
hypersurfaces corresponding to cosmic time. Interestingly, it is easy to show
that in the Friedmann-Lemaitre-Robertson-Walker (FLRW) background, all
physical quantities except for the volume expansion $\Theta$ and the energy
density vanish. Using the Stewart-Walker lemma, in this formalism, it was
possible to construct gauge invariant quantities up to second order in
cosmological perturbations 1989-Ellis.Bruni-PRD ; 1989-Ellis.Hwang.Bruni-PRD .
However, the $1+3$-formalism is not suited if the space-time is inhomogeneous,
like spherical symmetry or space-times with local rotational symmetry (LRS)
2003-Clarkson.Barrett-CQG . In such cases, splitting the $3$-space orthogonal
to the time-like congruence into one spacelike direction and a $2$-space is
apt 1996-vanElst.Ellis-CQG . Thus, the $1+1+2$ decomposition of space-time is
a natural extension of the $1+3$ formalism in which the three-space is further
decomposed to a given spatial direction. This approach is called semi-tetrad
formalism 2017-Sayuri.Sunil.Maharaj-PRD ; 2019-Sayuri.Sunil.Maharaj-JMP ;
2020-Hansraj.Sunil.Maharaj-GRG ; 2022-Singh-Sunil.Maharaj ;
2020-Khambule.SunilMaharaj-CQG .

As mentioned in the Introduction, our interest is to evaluate the net momentum
experienced by a test particle after the electromagnetic wave passes through
the space-time point. In the covariant $1+3$ formalism, the test particle is
the fundamental time-like observer. As depicted in (1), when the EM wave
propagates in a given spatial direction, the net momentum experienced by the
particle lies in the $2$-D surface orthogonal to the direction of propagation
of the EM wave. In other words, the net momentum (kick) vector lies in the
$2$-D surface. Thus, the net memory effect of the test particle will lie on
the 2-D surface; hence, we will refer to this as the _memory vector_. This can
also be understood using the fact that the electric and magnetic fields are
transverse to the direction of propagation of the EM wave. Thus, it is cogent
to further split the 3-space to $1+2$-space.

More specifically, choosing a generic space-like vector ($n^{\mu}$), we split
the $3$-space into 1 + 2-space 2003-Clarkson.Barrett-CQG ;
2004-Clarkson.P.K.S-APJ ; 2005-Tsagas-CQG ; 2021-Mavrogiannis.Tsagas-PRD . The
space-like vector ($n^{\mu}$) satisfies the following conditions:

$n^{\mu}n_{\mu}=1,\quad n^{\mu}u_{\mu}=0\,.$

Like in the $1+3$-formalism, we project the vectors and tensors defined in
$3$-space along the space-like direction ($n^{\mu}$) and into the $2$-space
that is orthogonal to $n^{\mu}$. Here again, the projection tensor
($\tilde{h}_{\mu\nu}$) need to be defined:

$\displaystyle\tilde{h}_{\mu\nu}=h_{\mu\nu}-n_{\mu}n_{\nu};\quad\tilde{h}^{\mu}\,_{\nu}\,\tilde{h}^{\nu}\,_{\gamma}=\tilde{h}^{\mu}\,_{\gamma};\quad\tilde{h}^{\mu}\,_{\mu}=2;\quad\tilde{h}_{\mu\nu}\,u^{\nu}=0;\quad\tilde{h}_{\mu\nu}\,n^{\nu}=0\,.$
(8)

All the vectors and tensors defined in the $3$-space in the $1+3$-formalism
can be split into $1+2$ form. For instance, an arbitrary space-like vector
$V^{\mu}$ (defined in the $3$-space) can be written as:

$\displaystyle V^{\mu}=\mathscr{V}n^{\mu}+\mathscr{V}^{\mu}$ (9)

where, $\mathscr{V}=V^{\mu}n_{\mu}$ and
$\mathscr{V}^{\mu}=\tilde{h}^{\mu}\,_{\nu}V^{\nu}$. Similarly an arbitrary
tensor $v_{\mu\nu}$ on the 3-space can be split as:

$\displaystyle
v_{\mu\nu}=V\left(n_{\mu}n_{\nu}-\frac{1}{2}\tilde{h}_{\mu\nu}\right)+2V_{(\mu}n_{\nu)}+V_{\mu\nu}\,,$
(10)

where $V_{(\mu}n_{\nu)}=(V_{\mu}n_{\nu}+n_{\nu}V_{\mu})/2$. Similarly, the
relative acceleration of the time-like observer and other geometrical
quantities defined in $3$-space can be written in $1+2$ space as:

$\displaystyle\dot{u}^{\mu}=\mathscr{A}n^{\mu}+\mathscr{A}^{\mu}$ (11)
$\displaystyle\dot{n}^{\mu}=\mathscr{A}u^{\mu}+\alpha^{\mu}$ (12)
$\displaystyle\omega^{\mu}=\Omega n^{\mu}+\Omega^{\mu}$ (13)
$\displaystyle\sigma_{\mu\nu}=\Sigma\left(n_{\mu}n_{\nu}-\frac{1}{2}\tilde{h}_{\mu\nu}\right)+2\Sigma_{(\mu}n_{\nu)}+\Sigma_{\mu\nu}$
(14)

where $\dot{n}^{\mu}:=u_{\nu}\nabla_{\nu}\ n^{\mu}$ is the relative
acceleration of the space-like vector along the time-like observer. Here,
$\mathscr{A}^{\mu},\alpha^{\mu},\Sigma_{\mu\nu},\,\Omega^{\mu}$ are orthogonal
to $n^{\mu}$ as well as $u^{\mu}$. Also,
$\mathscr{A}^{\mu},\Omega^{\mu}(\Sigma_{\mu\nu})$ are the vectors (tensor)
projected on the 2-space. In this formalism, we define the alternating Levi-
Civita $2$-tensor

$\displaystyle\epsilon_{\mu\nu}\equiv\epsilon_{\mu\nu\alpha}n^{\alpha}$ (15)

which is orthogonal to $n^{\mu}$ and has components only in the $2$-space.
Given an arbitrary vector $V^{\mu}$ in the $2$-space, we can construct another
vector $\epsilon_{\mu\nu}V^{\nu}$ that is orthogonal to $V^{\mu}$ which is in
the 2-space and has the same length.

The $1+2$ splitting of the 3-space leads to a new directional derivative along
the space-like vector $n^{\mu}$:

$\displaystyle v_{\mu\nu}^{\prime}\equiv n^{\alpha}D_{\alpha}v_{\mu\nu}$ (16)
$\displaystyle\tilde{D}_{\alpha}v_{\mu\nu}\equiv\tilde{h}_{\alpha}\,^{\beta}\tilde{h}_{\mu}\,^{\rho}\tilde{h}_{\nu}\,^{\sigma}D_{\beta}v_{\rho\sigma}\,.$
(17)

The derivative in Eq. (16) physically correspond to the variation of the
physical quantities on the 2-space along the space-like vector $n^{\mu}$. The
derivative ($\tilde{D}$) in Eq. (17) corresponds to the variation of the
physical quantities that lie in the $2$-space. These will contribute to the
memory vector.

As we split the covariant derivative of $u_{\mu}$ in Eq. (5), similarly we can
split the covariant derivative of $n_{\mu}$ as:

$\displaystyle
D_{\nu}n_{\mu}=\tilde{D}_{\nu}n_{\mu}+n_{\mu}n_{\nu}^{\prime}=\tilde{\sigma}_{\mu\nu}+\tilde{\omega}_{\mu\nu}+\frac{1}{2}\tilde{\Theta}\tilde{h}_{\mu\nu}+n_{\mu}n_{\nu}^{\prime}$
(18)

where, $\tilde{\sigma}_{\mu\nu}\equiv\tilde{D}_{<\nu}n_{\mu>}$,
$\tilde{\omega}_{\mu\nu}\equiv\tilde{D}_{(\nu}n_{\mu)}$ and
$\tilde{\Theta}=\tilde{D}^{\mu}n_{\mu}$ are shear, vorticity and the surface
expansion-contraction scalar respectively and $n_{\mu}^{{}^{\prime}}$ is the
spatial derivative along $n^{\mu}$. Thus, $\tilde{D}_{\nu}n_{\mu}$ describes
the kinematic properties or the relative motion of the space-like curves in
the $2$-surface orthogonal to $n^{\mu}$. We can obtain the relation between
the kinematic quantities derived from the motion of time-like vector $u_{\mu}$
and kinematic quantities in $2$-space derived from the space-like vector
$n^{\mu}$. See, for instance, Ref. 2021-Mavrogiannis.Tsagas-PRD .

##  III Electromagnetic theory in covariant formalism

The covariant formalism has been extensively employed in studying the
evolution of electromagnetic fields in curved space-time 2005-Tsagas-CQG . In
the covariant formulation, the dynamics and kinematics are constricted by the
Bianchi and Ricci identities. The $(1+3)-$ covariant formulation permits the
classification of cosmological models, a fluid description of the matter field
in FLRW universes. However, as mentioned earlier, the $1+3$-formalism is not
suited if the space-time is inhomogeneous, like spherical symmetry or space-
times with LRS 2003-Clarkson.Barrett-CQG . In such cases, the
$1+1+2$-covariant or semi-triad formalism are better suited.

Since we aim to derive EM memory for arbitrary space-times, we use
$1+1+2$-covariant formalism. We obtain a generic form of the EM memory effect
by evaluating the change in the velocity vector $\Delta u^{\mu}$ that lie in
the $2$-space. In order to do so, we fix the space-like direction to be the
direction of the propagation of the wave. In the case of spherically symmetric
space-time, this naturally translates to the radial direction. One key
advantage is that the electromagnetic theory in the $1+1+2$ formalism helps to
understand the evolution and dynamics of the EM fields along the space-like
direction and in the $2$-space normal to $n^{\mu}$ and $u^{\mu}$. Our approach
makes geometrical contributions to the memory effect more transparent.

In the next subsection, we rewrite Maxwell’s equations in $1+3$ formalism in
an arbitrary space-time. Later, we formulate the evolution equations of the EM
fields in the $2$-space and two constraint equations of the same along
$u^{\mu}$ and $n^{\mu}$ 2021-Mavrogiannis.Tsagas-PRD . The key advantage is
that we can obtain the memory vector from the projected acceleration vector
onto the $2$-space.

###  III.1 In 1+3 formalism

The fundamental objects are the Maxwell electromagnetic field tensor
$F^{\mu\nu}$. The $(1+3)$ covariant formalism of Maxwell’s electromagnetic
theory provides a way to study the interaction of EM fields with different
components of general space-time geometry 2005-Tsagas-CQG . With the $(1+3)$
decomposition, it is possible to split $F^{\mu\nu}$ into the electric and
magnetic fields. Note that the local coordinates are mathematical parameters
that label the points of the space-time manifold $M$; therefore, the electric
and magnetic fields may not have a direct physical meaning. In order to make
measurements, an observer brings in an additional structure on $M$ by
introducing the orthonormal coframe field. This gives rise to the split of
Maxwell’s tensor $F$ into the physical electric and magnetic fields.

Specifically, formalism allows us to split the equations of motion of the
fields and currents into two parts:

  1. 1.

projected parallel to the $4$-velocity $u^{\mu}$ of the fundamental observer

  2. 2.

projected into the $3$-space orthogonal to $u^{\mu}$.

To keep the calculations tractable, we perform all the calculations in source-
free and lossless regions. However, the EM memory analysis can be
straightforwardly extended to these regions. In the source-free regions,
Maxwell’s equations are:

$\displaystyle\nabla_{\nu}F^{\mu\nu}=0$ (19)
$\displaystyle\nabla_{[\gamma}F_{\mu\nu]}=0;\quad\text{or}\quad\nabla_{\nu}{F^{*}}^{\mu\nu}=0\,,$
(20)

where ${F^{*}}^{\mu\nu}$ is the dual to $F^{\mu\nu}$ and is defined as
${F^{*}}^{\mu\nu}=(1/2)\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$.

In the $1+3$ formalism, by projecting $F^{\mu\nu}$ and ${F^{*}}^{\mu\nu}$
along the time-like $4$-velocity vector, we can decompose them into electric
and magnetic parts. The electric ($E^{\mu}$) and magnetic ($B^{\mu}$)
4-vectors are defined as:

$\displaystyle E^{\mu}:=F^{\mu\nu}u_{\nu}$ (21) $\displaystyle
B^{\mu}:={F^{*}}^{\mu\nu}u_{\nu}$ (22)

From the above definitions, we infer:

$\displaystyle E^{\mu}u_{\mu}=0;\quad B^{\mu}u_{\mu}=0$ (23)

which implies $E^{\mu}$ and $B^{\mu}$ have only spatial components. Given
this, we can rewrite $F_{\mu\nu}$ and ${F^{*}}^{\mu\nu}$ as:

$\displaystyle F_{\mu\nu}$
$\displaystyle=u_{\mu}E_{\nu}-u_{\nu}E_{\mu}+\epsilon_{\mu\nu\alpha\beta}B^{\alpha}u^{\beta}$
(24) $\displaystyle\tilde{F}^{\alpha\beta}$
$\displaystyle=\epsilon^{\alpha\beta\mu\nu}u_{\mu}E_{\nu}+\left(\,u^{\alpha}B^{\beta}-u^{\beta}B^{\alpha}\,\right)\,.$
(25)

From the above expressions, we see that the simultaneous transformations
$E^{\mu}\rightarrow-B^{\mu}$, $B^{\mu}\rightarrow E^{\mu}$ leads to
${F^{*}}^{\mu\nu}\rightarrow F^{\mu\nu}$. This implies that we can obtain the
second Maxwell’s equation (20) from the first Maxwell’s equation (19) or vice-
versa. More specifically, if we obtain the time-like part and space-like part
of Maxwell’s equations (20), we can write the time-like part and space-like
part of the other Maxwell’s equations (19) by substituting
$E^{\mu}\rightarrow-B^{\mu}$, $B^{\mu}\rightarrow E^{\mu}$.

In the rest of this subsection, we obtain Maxwell’s equations by projecting
along $u_{\mu}$ (time-like part) and $h_{\mu\nu}$ (space-like part)
2009-Subramanian-AsNa . We first obtain the time-like part of Eq. (20) by
multiplying it with $u_{\mu}$:

$\displaystyle
u_{\alpha}\left(\,\nabla_{\beta}\tilde{F}^{\alpha\beta}\,\right)=0$ (26)

Using the decomposition in Eq. (25), the above expression becomes:

$\displaystyle\nabla_{\beta}B^{\beta}-B^{\beta}\dot{u}_{\beta}+\left(\nabla_{\beta}u_{\alpha}\right)\,\epsilon^{\alpha\beta\mu\nu}u_{\mu}E_{\nu}=0$
(27)

We simplify the above equation using the following steps: First, we combine
the first two terms in the LHS. From Eq. (26), we have
$B^{\beta}\dot{u}_{\beta}=-u_{\beta}\dot{B}^{\beta}=-u_{\beta}u^{\alpha}\nabla_{\alpha}{B}^{\beta}$.
Substituting in the second term of the above expression, we have
$\delta_{\beta}^{\alpha}\,\nabla_{\alpha}B^{\beta}+u_{\beta}u^{\alpha}\nabla_{\alpha}{B}^{\beta}=h^{\alpha}_{\beta}\left(\nabla_{\alpha}B^{\beta}\right)$.
Substituting $\nabla_{\beta}u_{\alpha}$ from Eq. (5) and using the definition
of vorticity vector in Eq. (6), the third term in the LHS of the above
expression simplifies to $-2\omega^{\beta}E_{\beta}$. Thus, the time-like part
of Eq. (20) reduces to:

$\displaystyle D_{\beta}B^{\beta}=2\omega^{\beta}E_{\beta}\,.$ (28)

The space-like part of Eq. (20) can be obtained by multiplying it with
$h_{\mu}\,^{\nu}$,

$\displaystyle
h_{\alpha}\,^{\rho}\left(\,\nabla_{\beta}\tilde{F}^{\alpha\beta}\,\right)=0$
(29)

Using a series of steps, the above expression can be rewritten as:

$\displaystyle\dot{B}^{<\rho>}=\left[\sigma^{\rho}\,_{\beta}+\omega^{\rho}\,_{\beta}-\frac{2\Theta}{3}\,h^{\rho}\,_{\beta}\right]B^{\beta}-\epsilon^{\rho\mu\nu}\,\dot{u}_{\mu}E_{\nu}-\epsilon^{\rho\mu\nu}\,\nabla_{\mu}E_{\nu}\,.$
(30)

where, $\epsilon^{\mu\nu\alpha}$ is defined in Eq. (7). The above equation
provides the dynamical evolution of the magnetic field, while Eq. (28) is the
constraint equation.

As mentioned above, performing simultaneous transformation
$E^{\mu}\rightarrow-B^{\mu}$ and $B^{\mu}\rightarrow E^{\mu}$ in Eqs. (31) and
(32), we obtain the time-like and space-like parts of the first Maxwell’s
equation (19):

$\displaystyle D_{\beta}E^{\beta}=-2\omega^{\nu}B_{\nu}$ (31)
$\displaystyle\dot{E}^{<\rho>}=\left[\sigma^{\rho}\,_{\beta}+\omega^{\rho}\,_{\beta}-\frac{2\Theta}{3}\,h^{\rho}\,_{\beta}\right]E^{\beta}+\epsilon^{\rho\mu\nu}\,\dot{u}_{\mu}B_{\nu}+\epsilon^{\rho\mu\nu}\,D_{\mu}B_{\nu}\,.$
(32)

Similarly, the above equation provides the dynamical evolution of the electric
field, while Eq. (31) is the constraint equation.

###  III.2 In 1+1+2 formalism

We aim to calculate the memory effect of EM fields. As the memory vector
resides in the $2$-surface orthogonal to the direction of propagation of the
in-coming wave, we need to decompose the 3-space to $1+2$-space w.r.t a given
spatial direction. In this subsection, we rewrite Maxwell’s equations (19, 20)
using the space-like vector $n^{\nu}$ and the projection tensor (8) in $1+1+2$
formalism.

To do this, we first express the EM fields and currents in $3$-space into
$1+2$ form:

$\displaystyle E^{\mu}=\mathscr{E}n^{\mu}+\mathscr{E}^{\mu}$ (33)
$\displaystyle B^{\mu}=\mathscr{B}n^{\mu}+\mathscr{B}^{\mu}\,.$ (34)

where, $\mathscr{E}\equiv E^{\mu}n_{\mu}$,
$\mathscr{E}^{\mu}\equiv\tilde{h}^{\mu}\,_{\nu}E^{\nu}$, $\mathscr{B}\equiv
B^{\mu}n_{\mu}$, and $\mathscr{B}^{\mu}\equiv\tilde{h}^{\mu}\,_{\nu}B^{\nu}$.
Following the discussion in Sec. (II.2), it follows that
$\epsilon_{\mu\nu}\mathscr{E}^{\nu}$ is orthogonal to $\mathscr{E}^{\mu}$ and,
similarly, $\epsilon_{\mu\nu}\mathscr{B}^{\nu}$ is orthogonal to
$\mathscr{B}^{\mu}$. If electric and magnetic fields are orthogonal to each
other in $2$ space, then we have

$\displaystyle\mathscr{E}^{\nu}=\epsilon_{\mu\nu}\mathscr{B}^{\nu}\quad\mathscr{B}^{\nu}=-\,\epsilon_{\mu\nu}\mathscr{E}^{\nu}\,.$
(35)

These relations will play an important role in Sec. (IV) to derive the memory
effect.

The second step is to split the evolution equations (30, 32) interms of
$\mathscr{E},\mathscr{E}^{\mu},\mathscr{B},\mathscr{B}^{\mu}$. To do that, we
project Eq. (32) along spacelike direction $n^{\mu}$ and multiply Eq. (32)
with projection tensor (8). After a long calculation, we obtain the following
evolution equations for $\mathscr{E}$ (along $n^{\mu}$) and
$\mathscr{E}^{\mu}$ (in the orthogonal $2$-space):

$\displaystyle\dot{\mathscr{E}}+\Theta\mathscr{E}=$
$\displaystyle~{}\alpha^{\mu}\mathscr{E}_{\mu}-2\tilde{\omega}\mathscr{B}+\epsilon_{\mu\rho}\tilde{D}^{\mu}\mathscr{B}^{\rho}$
(36)
$\displaystyle\dot{\mathscr{E}}_{\bar{\mu}}+\frac{\Theta}{2}\mathscr{E}_{\mu}=$
$\displaystyle-\left(\alpha_{\mu}+2\epsilon_{\mu\rho}\Omega^{\rho}\right)\mathscr{E}+\left(\Sigma_{\mu\rho}+\Omega\epsilon_{\mu\rho}\right)\mathscr{E}^{\rho}+\epsilon_{\mu\rho}\left(\mathscr{A}^{\rho}-n^{\prime\rho}+\tilde{D}^{\rho}\right)\mathscr{B}$
$\displaystyle-\epsilon_{\mu\rho}\left(\mathscr{A}\mathscr{B}^{\rho}+\mathscr{B}^{\prime\rho}-\left(\tilde{D}^{\rho}\mathscr{B}_{\nu}\right)n^{\nu}\right)\,,$
(37)

where, $\tilde{\omega}=\tilde{\omega}_{\mu\nu}\,\epsilon^{\mu\nu}$, $\Theta$
is the expansion factor defined in Eq. (5), $\mathscr{A}^{\mu}$ is the
relative acceleration vector in 2-space defined in Eq. (11), $\tilde{\omega}$
is the vorticity defined in Eq. (18). $\Omega^{\mu}$, $\Omega$ is defined in
Eq. (13) and $\Sigma_{\mu\nu}$ is in Eq. (14). The $2$-space component of
$\dot{n}^{\mu}$ is $\alpha^{\mu}$ which is defined in Eq. (12), whereas
$\mathscr{A}=n^{\mu}\dot{u}_{\mu}=-u^{\mu}\dot{n}_{\mu}$ mentioned in Eq.
(11), (12).

We want to highlight the following points regarding the above expressions:
First, the above equations generalize Ampere’s law for arbitrary space-time.
For example, in Eq. 36, the first term in the LHS corresponds to the time
derivative of the electric field along space-like direction $n^{\mu}$ and the
last term in RHS is the curl of the magnetic field in $2$-space. Similarly,
the LHS of Eq. (37) is the time derivative of the electric field in $2$-space,
and in the last term in the RHS is the curl of $\mathscr{B}^{\rho}$. Second,
in the flat space-time, the expansion factor ($\Theta$), the relative
acceleration vector ($\alpha^{\mu}$), and vorticity ($\tilde{\omega}$) vanish,
and the above expression lead to Ampere’s law in flat space-time. Thus,
background space-time introduces new couplings between the electric and
magnetic field components. Lastly, we showed that the simultaneous
transformation $E^{\mu}\rightarrow-B^{\mu}$, $B^{\mu}\rightarrow E^{\mu}$
leads to ${F^{*}}^{\mu\nu}\rightarrow F^{\mu\nu}$. Substituting
$\mathscr{E}\rightarrow\mathscr{B}$;
$\mathscr{E}^{\mu}\rightarrow\mathscr{B}^{\mu}$ and
$\mathscr{B}\rightarrow-\mathscr{E}$;
$\mathscr{B}^{\mu}\rightarrow-\mathscr{E}^{\mu}$ in Eqs. (36, 37), we have:

$\displaystyle\dot{\mathscr{B}}+\Theta\mathscr{B}=$
$\displaystyle\mathscr{B}^{\mu}\alpha_{\mu}+2\tilde{\omega}\mathscr{E}-\epsilon_{\mu\rho}\tilde{D}^{\mu}\mathscr{E}^{\rho}$
(38)
$\displaystyle\dot{\mathscr{B}}_{\bar{\mu}}+\frac{1}{2}\Theta\mathscr{B}_{\mu}=$
$\displaystyle-\left(\alpha_{\mu}+2\epsilon_{\mu\rho}\Omega^{\rho}\right)\mathscr{B}+\left(\Sigma_{\mu\rho}+\Omega\epsilon_{\mu\rho}\right)\mathscr{B}^{\rho}-\epsilon_{\mu\rho}\left(\mathscr{A}^{\rho}+\tilde{D}^{\rho}-n^{\prime\rho}\right)\mathscr{E}$
$\displaystyle+\epsilon_{\mu\rho}\left(\mathscr{A}\mathscr{E}^{\rho}+\epsilon_{\mu\rho}\mathscr{E}^{\prime\rho}-\left(\tilde{D}^{\rho}\mathscr{E}_{\nu}\right)n^{\nu}\right)$
(39)

Note that we obtain the above equations by projecting Eq. (30) along spacelike
direction $n^{\mu}$ and multiply Eq. (30) with projection tensor (8). Again,
the above equations generalize Faraday’s law for arbitrary space-time.

The last step is to split the constraint equations (31, 28) interms of
$\mathscr{E},\mathscr{E}^{\mu},\mathscr{B},\mathscr{B}^{\mu}$. Substituting
(33, 34) and the kinematic quantities (11-14), we get:

$\displaystyle\tilde{D}^{\mu}\mathscr{E}_{\mu}+n^{\mu}\mathscr{E}_{\mu}^{\prime}+\mathscr{E}^{\prime}+\tilde{\Theta}\mathscr{E}+2\left(\Omega\mathscr{B}+\Omega^{\mu}\mathscr{B}_{\mu}\right)=0$
(40)
$\displaystyle\tilde{D}^{\mu}\mathscr{B}_{\mu}-n^{\prime\mu}\mathscr{B}_{\mu}+\mathscr{B}^{\prime}+\tilde{\Theta}\mathscr{B}-2\left(\Omega\mathscr{E}+\Omega^{\mu}\mathscr{E}_{\mu}\right)=0$
(41)

where $\tilde{\Theta}$ is the expansion along the space-like vector defined in
Eq. (18). The above equations are generalizations of Gauss law. Here again, in
the flat space-time, the expansion factor ($\tilde{\Theta}$), the relative
acceleration vector ($\alpha^{\mu}$), vorticity (${\Omega}$) vanish, and the
above expressions lead to Gauss law in flat space-time.

###  III.3 Energy-momentum tensor of the electromagnetic field

As we will show in the next section, the electromagnetic stress tensor plays a
crucial role in understanding the memory effect. This subsection evaluates the
electromagnetic stress tensor in $1+1+2$ formalism for an arbitrary space-
time. The EM action in an arbitrary background is:

$S=-\frac{1}{4}\int
d^{4}x~{}\sqrt{-g}~{}F_{\mu\nu}F_{\rho\sigma}g^{\mu\rho}g^{\nu\sigma}\,.$ (42)

Varying the above action w.r.t the metric $(g^{\mu\nu})$ leads to the
following energy-momentum tensor:

$T_{\mu\nu}=\frac{1}{2}g^{\rho\sigma}F_{\mu\rho}F_{\nu\sigma}-\frac{1}{8}g_{\mu\nu}g^{\rho\sigma}g^{\alpha\beta}F_{\rho\alpha}F_{\sigma\beta}\,.$
(43)

In $1+3$-formalism, the stress-tensor of matter field ($T_{\mu\nu}$) can
written as:

$\displaystyle
T_{\mu\nu}=\rho\,u_{\mu}u_{\nu}+2\,{S}_{(\mu}\,u_{\nu)}+{W}_{\mu\nu}\,,$ (44)

where, the energy-density $\rho$, the energy flux $S^{\alpha}$ and stress-
tensor $W^{\alpha\beta}$ as measured in the observer’s worldline are given by
1982-Thorne.MacDonald-MNRAS :

$\rho=\mathrm{T}^{\mu\nu}{u}_{\mu}{u}_{\nu},\quad{S}^{\alpha}=-h_{\mu}^{\alpha}\,{T}^{\mu\nu}u_{\nu},\quad
W^{\alpha\beta}=h_{\mu}^{\alpha}\,{T}^{\mu\nu}h_{\nu}^{\beta}$ (45)

For the electromagnetic fields in $1+3$-formalism, $\rho,\,S_{\mu}$ and
${W}_{\mu\nu}$ are:

$\displaystyle\rho\equiv\frac{1}{2}\left(E^{\mu}E_{\mu}+B^{\mu}B_{\mu}\right);\quad
S_{\mu}\equiv\epsilon_{\mu\nu\rho}E^{\nu}B^{\rho}$ (46)
$\displaystyle{W}_{\mu\nu}\equiv\frac{1}{2}\left(E^{\mu}E_{\mu}+B^{\mu}B_{\mu}\right)h_{\mu\nu}-E_{\mu}E_{\nu}-B_{\mu}B_{\nu}$
(47)

Rewriting $\rho$ interms of the variables
($\mathscr{E},\mathscr{E}^{\mu},\mathscr{B},\mathscr{B}^{\mu})$ in $1+1+2$
formalism, we have:

$\displaystyle\rho$
$\displaystyle=\frac{1}{2}\left(\mathscr{E}^{2}+\mathscr{B}^{2}\right)+\frac{1}{2}\left(\mathscr{E}^{\mu}\mathscr{E}_{\mu}+\mathscr{B}^{\mu}\mathscr{B}_{\mu}\right)=\rho_{(n)}+\rho_{\rm
2-space}$ (48)

Thus, $\rho_{(n)}$ corresponds to the energy of the EM field along $n_{\mu}$
and $\rho_{\rm 2-space}$ corresponds to the energy of the EM field in the
2-space. The energy flux $S_{\mu}$ (a vector in $3$-space) can be rewritten in
$1+2$ space as:

$\displaystyle{S}_{\mu}$ $\displaystyle=\mathscr{S}n_{\mu}+\mathscr{S}_{\mu}$
(49)

where $\mathscr{S}$ is the Poynting vector of the EM field along the space-
like vector $n^{\mu}$ and $\mathscr{S}_{\mu}$ is the energy flux in the
2-space. These are given by:

$\displaystyle\mathscr{S}$
$\displaystyle={S}_{\mu}n^{\mu}=\epsilon_{\mu\nu}\mathscr{E}^{\mu}\mathscr{B}^{\nu}\,$
(50) $\displaystyle\mathscr{S}_{\mu}$
$\displaystyle=-\epsilon_{\mu\nu}\left(\mathscr{E}\mathscr{B}^{\nu}-\mathscr{B}\mathscr{E}^{\nu}\right)=-\left(\mathscr{E}\mathscr{E}^{\nu}+\mathscr{B}\mathscr{B}^{\nu}\right)$
(51)

In deriving the last expression, we have used the orthogonality condition
between the electric and magnetic fields in the $2$-space, i. e.,
$\mathscr{E}_{\nu}=\epsilon_{\nu\mu}\mathscr{B}^{\mu}$. As we will see in the
next section, the memory vector depends on the part of the electromagnetic
energy density $\rho$ and $\mathscr{S}_{\mu}$.

##  IV Memory effect in arbitrary space-time

Having written Maxwell’s equations in $1+1+2$ formalism for an arbitrary
space-time, we now evaluate the memory effect. Usually, in the literature, one
uses the Lorentz force equation to derive EM memory. The equation of motion of
a charged body (of mass $m$ and charge $e$) in both gravitational and
electromagnetic fields are:

$m\frac{du_{\alpha}}{d\tau}-\frac{m}{2}g_{\beta\gamma,\alpha}u^{\beta}u^{\gamma}=eF_{\alpha\beta}u^{\beta}$
(52)

However, the above expression does not consider the new couplings between the
electric and magnetic field components in Eqs. (36) - (39). Hence, we use the
complete Maxwell’s equations (36) - (41) and explicitly obtain the change in
velocity ($\Delta u^{\mu}$) of the time-like observer. More specifically,
using Eqs. (37, 39), we first calculate the acceleration vector
$\mathscr{A}^{\mu}$ in the $2$-space. We can then integrate the expression for
the acceleration vector ($\mathscr{A}^{\mu}$ in the $2$-space) with respect to
time $t$ or null time coordinate $u\equiv(t-r)$ leading to the memory vector.

In the rest of this section, we calculate $\mathscr{A}^{\mu}$ for observers
whose tangents are congruent to the space-like geodesics. This implies
$n^{\sigma}D_{\sigma}n^{\rho}=n^{\prime\rho}=0$, i. e., $n^{\mu}$ is tangent
to a congruence of space-like geodesics 2021-Mavrogiannis.Tsagas-PRD . Using
this condition and substituting
$\dot{\mathscr{E}}_{\bar{\mu}}=\tilde{h}_{\mu\nu}\dot{\mathscr{E}}^{\nu},\,\mathscr{B}^{\prime\,\rho}=n^{\nu}D_{\nu}\mathscr{B}^{\rho}$
in Eqs. (37, 39), we get:

$\displaystyle\tilde{h}_{\mu\nu}\dot{\mathscr{E}}^{\nu}+\epsilon_{\mu\rho}n^{\nu}D_{\nu}\mathscr{B}^{\rho}=$
$\displaystyle-\frac{1}{2}\Theta\mathscr{E}_{\mu}-\left(\alpha_{\mu}+2\epsilon_{\mu\rho}\Omega^{\rho}\right)\mathscr{E}+\left(\Sigma_{\mu\rho}+\Omega\epsilon_{\mu\rho}\right)\mathscr{E}^{\rho}$
$\displaystyle+\left(\epsilon_{\mu\rho}\mathscr{A}^{\rho}+\epsilon_{\mu\nu}\tilde{D}^{\nu}\right)\mathscr{B}-\epsilon_{\mu\nu}\left(\tilde{D}^{\nu}n^{\rho}\right)\mathscr{B}^{\rho}-\epsilon_{\mu\rho}\mathscr{A}\mathscr{B}^{\rho}$
(53)
$\displaystyle\left(\tilde{h}_{\mu\nu}\dot{\mathscr{B}}^{\nu}-\epsilon_{\mu\rho}n^{\nu}D_{\nu}\mathscr{E}^{\rho}\right)$
$\displaystyle=-\frac{1}{2}\Theta\mathscr{B}_{\mu}-\left(\alpha_{\mu}+2\epsilon_{\mu\rho}\Omega^{\rho}\right)\mathscr{B}+\left(\Sigma_{\mu\rho}+\Omega\epsilon_{\mu\rho}\right)\mathscr{B}^{\rho}$
$\displaystyle-\left(\epsilon_{\mu\rho}\mathscr{A}^{\rho}+\epsilon_{\mu\nu}\tilde{D}^{\nu}\right)\mathscr{E}+\epsilon_{\mu\nu}\left(\tilde{D}^{\nu}n^{\rho}\right)\mathscr{E}_{\rho}+\epsilon_{\mu\rho}\mathscr{A}\mathscr{E}^{\rho}$
(54)

Multiplying Eq. (53) with $\mathscr{B}$, multiplying Eq. (54) with
$\mathscr{E}$ and subtracting the resultant equations leads to:

$\displaystyle\epsilon_{\mu\nu}\mathscr{A}^{\nu}$
$\displaystyle=-\,\frac{\epsilon_{\mu\nu}}{2}\,\frac{D^{\nu}(\mathscr{E}^{2}+\mathscr{B}^{2})}{(\mathscr{E}^{2}+\mathscr{B}^{2})}+\left(\Sigma_{\mu\nu}+\Omega\epsilon_{\mu\nu}-\frac{\Theta}{2}\tilde{h}_{\mu\nu}\right)\,\frac{(\mathscr{E}\mathscr{B}^{\nu}-\mathscr{B}\mathscr{E}^{\nu})}{(\mathscr{E}^{2}+\mathscr{B}^{2})}$
$\displaystyle+\epsilon_{\mu\nu}\,\left(\tilde{\sigma}^{\rho\nu}+\tilde{\omega}^{\rho\nu}+\frac{\tilde{\Theta}}{2}\tilde{h}^{\rho\nu}\right)\,\frac{(\mathscr{B}\mathscr{B}_{\rho}+\mathscr{E}\mathscr{E}_{\rho})}{(\mathscr{E}^{2}+\mathscr{B}^{2})}+\frac{\epsilon_{\mu\rho}\mathscr{A}\left(\mathscr{E}\mathscr{E}^{\rho}+\mathscr{B}\mathscr{B}^{\rho}\right)}{(\mathscr{E}^{2}+\mathscr{B}^{2})}$
$\displaystyle+\frac{\mathscr{B}}{(\mathscr{E}^{2}+\mathscr{B}^{2})}\left(\tilde{h}_{\mu\nu}\dot{\mathscr{E}}^{\nu}+\epsilon_{\mu\rho}n^{\nu}D_{\nu}\mathscr{B}^{\rho}\right)-\frac{\mathscr{E}}{(\mathscr{E}^{2}+\mathscr{B}^{2})}\left(\tilde{h}_{\mu\nu}\dot{\mathscr{B}}^{\nu}-\epsilon_{\mu\rho}n^{\nu}D_{\nu}\mathscr{E}^{\rho}\right)$
(55)

To have a transparent understanding, we substitute the definitions (48) - (51)
in the expression above, resulting in:

$\displaystyle\epsilon_{\mu\nu}\mathscr{A}^{\nu}$
$\displaystyle=-\,\frac{\epsilon_{\mu\nu}}{2}\,\frac{D^{\nu}\rho_{(n)}}{\rho_{(n)}}-\frac{\epsilon^{\nu\alpha}}{2}\left(\Sigma_{\mu\nu}+\Omega\epsilon_{\mu\nu}-\frac{\Theta}{2}\tilde{h}_{\mu\nu}\right)\,\frac{\mathscr{S}_{\alpha}}{\rho_{(n)}}-\frac{\epsilon_{\mu\nu}}{2}\,\left(\tilde{\sigma}^{\rho\nu}+\tilde{\omega}^{\rho\nu}+\frac{\tilde{\Theta}}{2}\tilde{h}^{\rho\nu}\right)\,\frac{\mathscr{S}_{\rho}}{\rho_{(n)}}$
$\displaystyle\,-\frac{\epsilon_{\mu\rho}\mathscr{S}^{\rho}\,\mathscr{A}}{2\rho_{(n)}}+\frac{\mathscr{B}}{2\rho_{(n)}}\left(\tilde{h}_{\mu\nu}\dot{\mathscr{E}}^{\nu}+\epsilon_{\mu\rho}n^{\nu}D_{\nu}\mathscr{B}^{\rho}\right)-\frac{\mathscr{E}}{2\rho_{(n)}}\left(\tilde{h}_{\mu\nu}\dot{\mathscr{B}}^{\nu}-\epsilon_{\mu\rho}n^{\nu}D_{\nu}\mathscr{E}^{\rho}\right)\,.$
(56)

This is the master equation for the EM memory in arbitrary space-time
regarding which we would like to discuss the following points: First, to our
understanding, this is a first time the EM memory has been obtained for an
arbitrary space-time. In the previous calculations Bieri:2013hqa ;
Winicour:2014ska , the authors have restricted to asymptotic flat space-times.
Second, the last two terms in the RHS of the above expression vanishes in the
asymptotic limit. To see this, let us consider a spherically symmetric space-
time. Let $t$ refer to the time coordinate and $r$ to the radial coordinate
and the null coordinate is $u\equiv t-r$. In the asymptotic limit
$\partial_{u}\sim\partial_{t}$ and $\partial_{u}\sim-\partial_{r}$. Setting
$u^{\mu}\equiv(1,\,0,\,0,\,0)$ and $n^{\mu}\equiv(0,\,1,\,0,\,0)$, the
penultimate term in the RHS of the above equation simplifies to:

$\displaystyle\tilde{h}_{\mu\nu}\dot{\mathscr{E}}^{\nu}+\epsilon_{\mu\rho}n^{\nu}D_{\nu}\mathscr{B}^{\rho}$
$\displaystyle\simeq\tilde{h}_{\mu\nu}u^{0}\nabla_{0}\mathscr{E}^{\nu}+\epsilon_{\mu\rho}n^{1}\nabla_{1}\mathscr{B}^{\rho}\simeq\tilde{h}_{\mu\nu}\partial_{u}\mathscr{E}^{\nu}-\epsilon_{\mu\rho}\partial_{u}\mathscr{B}^{\rho}$
$\displaystyle=f(u)\partial_{u}\left(\bar{\tilde{h}}_{\mu\nu}\mathscr{E}^{\nu}-\bar{\mathscr{E}}_{\mu\nu}\mathscr{B}^{\nu}\right)$
(57)

where, $\tilde{h}_{\mu\nu}=f(u)\bar{\tilde{h}}_{\mu\nu}$ and
$\bar{\epsilon}_{\mu\nu}=f(u)\epsilon_{\mu\nu}$. The terms with bar represent
their time independent parts. The above expression vanishes if
$\mathscr{E}^{\nu}$ and $\mathscr{B}^{\nu}$ are orthogonal to each other in
the 2-space. As we mentioned earlier (35), in 2-space, the electric and
magnetic fields are always orthogonal to each other. Similarly, the last term
can also be shown to vanish in the asymptotic limit. Thus, the above
expression reduces to:

$\displaystyle\epsilon_{\mu\nu}\mathscr{A}^{\nu}=$
$\displaystyle-\,\frac{\epsilon_{\mu\nu}}{2}\frac{D^{\nu}\rho_{(n)}}{\rho_{(n)}}-\frac{\epsilon^{\nu\alpha}}{2}\left(\Sigma_{\mu\nu}+\Omega\epsilon_{\mu\nu}-\frac{\Theta}{2}\tilde{h}_{\mu\nu}\right)\frac{\mathscr{S}_{\alpha}}{\rho_{(n)}}$
$\displaystyle\,-\frac{\epsilon_{\mu\nu}}{2}\left(\tilde{\sigma}^{\rho\nu}+\tilde{\omega}^{\rho\nu}+\frac{\tilde{\Theta}}{2}\tilde{h}^{\rho\nu}\right)\frac{\mathscr{S}_{\rho}}{\rho_{(n)}}-\frac{\epsilon_{\mu\rho}}{2\rho_{(n)}}\,\mathscr{S}^{\rho}\,\mathscr{A}$
(58)

Third, the above expression provides a nice geometrical understanding of the
various contributions to memory effect. The first term in the RHS corresponds
to the change in the EM field energy ($\rho_{(n)}$) along $n_{\mu}$ in the
2-space. This does not contain any contribution from the kinematical
properties of the space-time. In other words, this term will vanish if the EM
field energy does not change in the 2-space, like a 2-D flat sheet. However,
as we show in the next section, this is non-zero in flat space-time expressed
in spherical coordinates. The next two terms in the RHS are proportional to
the energy flux ($\mathscr{S}_{\alpha}$) in the 2-space. However, both these
terms have different kinematical information of the space-time and vanish for
flat space-time. The second term in the RHS carries information about shear
$(\Sigma_{\mu\nu})$, vorticity scalar $(\Omega)$ related to $n^{\mu}$ and
expansion scalar $(\Theta)$ corresponding to time-like observer $u^{\mu}$. The
third term in the RHS carries information about shear
$(\tilde{\sigma}^{\mu\nu})$, vorticity tensor $(\tilde{\omega}^{\mu\nu})$ and
expansion scalar $(\tilde{\Theta})$ corresponding to the space-like vector
$n^{\mu}$.

Fourth, as mentioned earlier, we have not included external currents or
charges in our analysis. Hence, the acceleration vector does not have
contribution from the external sources. Hence, the memory vector we obtain is
equivalent to the null-kick derived in Refs. Bieri:2013hqa ; Winicour:2014ska
. It is also important to note that these authors did not obtain the
contributions due to the kinematical properties of the space-time. However, as
we will see in the next section, their contribution can be significant.

Lastly, to obtain the memory vector, we need to integrate the above expression
w.r.t the proper time of the observer — $\Delta u^{\mu}$ is the memory vector.
It is interesting to note that initially if the observer has non-zero velocity
_only_ along the time direction, at a later time, due to the memory effect,
there is a non-zero velocity in the 2-space.

##  V Application to specific space-times

In the previous section, we obtained a master equation for the EM vector for
an arbitrary 4-D space-time using $1+1+2$-formalism. As we discussed, the
memory vector has three distinct contributions. In order to illustrate this
fact, we consider specific examples and obtain the memory vector. In this
section we obtain memory vector for flat, FLRW, pp-wave and Kerr space-times.

###  V.1 Minkowski space-time

In order to compare the master equation with the existing results
Bieri:2013hqa , we first consider Minkowski space-time in spherical
coordinates:

$\displaystyle ds^{2}=-dt^{2}+dr^{2}+r^{2}\,\gamma_{AB}$ (59)

where,

$\displaystyle\gamma_{AB}=\left(\begin{array}[]{cc}1&0\\\
0&\sin^{2}\theta\end{array}\right)$ (62)

is the metric describing unit $2$-sphere. In Minkowski space-time, the
$4$-velocity of the time-like congruence observer is
$u^{\mu}\equiv(1,\,0,\,0,\,0)$ and the space-like vector is
$n^{\mu}\equiv(0,\,1,\,0,\,0)$. Since $\nabla_{\mu}u_{\nu}=0$ and
$\nabla_{\mu}n_{\nu}=0$, the kinematics quantities, defined in Sec. (II.1,
II.2) vanish for the Minkowski space-time. Hence only the first term in Eq.
(56) will be non-zero, i. e.,

$\displaystyle\mathscr{A}_{\rm Flat}^{\nu}$
$\displaystyle=-\,\frac{1}{2}\,\frac{D^{\nu}\rho_{\rm n}}{\rho_{\rm n}}\,.$
(63)

As mentioned earlier, the acceleration vector corresponds to acceleration in
the 2-Sphere. Hence, it is appropriate to switch to the 2-Sphere index:

$\mathscr{A}^{A}=u^{\mu}\nabla_{\mu}u^{A}=u^{0}\partial_{0}u^{A}\,+\,2u^{0}\Gamma^{A}_{0\,B}u^{B}\,.$

Since the 4-velocity $u^{\mu}$ is zero in the 2-Sphere, we have
$\mathscr{A}^{A}=u^{0}\partial_{0}u^{A}=\partial_{t}u^{A}$. In null
coordinate, this becomes $\mathscr{A}^{A}=\partial_{t}u^{A}$. Substituting the
above expression in Eq. (63) and integrating in the null coordinate, we have:

$\displaystyle\Delta
u^{A}\equiv\int\,du\,\mathscr{A}^{A}=-\frac{1}{2}\,\int\,du\,\frac{D^{A}\rho_{\rm
n}}{\rho_{\rm n}}\,.$ (64)

The above expression is velocity kick w.r.t the _Eulerian observers_. To
compare this with the net momentum (kick) vector as seen by the asymptotic
static observers (_Lagrangian observers_), we need to do a coordinate
transformation. Specifically, we need to transform from coordinate basis
$\left(\vec{e}^{\,\theta},\vec{e}^{\,\phi}\right)$ to orthogonal coordinate
basis $\left(\hat{\theta},\hat{\phi}\right)$. In terms of
$\left(\hat{\theta},\hat{\phi}\right)$, we have $\Delta\vec{u}\equiv\Delta
u^{\mu}\vec{e}_{\mu}$, where,
$\vec{e}^{\,\theta}=\hat{\theta}/r\,,\,\vec{e}^{\,\phi}=\hat{\phi}/(r\,\sin\theta)$.
Thus, the velocity kick w.r.t the asymptotic static observers is given by:

$\displaystyle\Delta\vec{u}_{\rm Flat}$ $\displaystyle=\frac{1}{r}\left(\Delta
u^{\theta}\hat{\theta}+\frac{\Delta u^{\phi}}{\sin\theta}\hat{\phi}\right)$
(65)

Interestingly, the EM memory vector in Minkowski space-time is inversely
proportional to ${r}$ and matches with Ref. Bieri:2013hqa . This passes the
first test that the master equation (56) indeed describes the EM memory
vector. In the rest of this section, we obtain the memory vector for non-flat
geometries and show the robustness of our approach.

###  V.2 FLRW space-time

The conformally flat FLRW metric in spherical coordinates is:

$\displaystyle
ds^{2}=a(\eta)^{2}\,\left(-d\eta^{2}+dr^{2}+r^{2}\,\gamma_{AB}\right)$ (66)

where, the conformal time ($\eta$) is related to the cosmic time ($t$) by
$dt=a(\eta)\,d\eta$. In $1+3$ formalism, the fundamental observer with time-
like $4-$velocity in FLRW metric is
$u^{\mu}=dx^{\mu}/dt=dx^{\mu}/(a(\eta))\,d\eta=\left(\,1,\,0,\,0,\,0\,\right)/a(\eta)$.
For this choice of observer, the $3-$space projection tensor $(h_{\mu\nu})$
orthogonal to $u^{\mu}$ is:

$\displaystyle h_{\mu\nu}=\left(\begin{array}[]{cc}a^{2}(\eta)&0\\\
0&a^{2}(\eta)\,r^{2}\,\gamma_{AB}\end{array}\right)\,.$ (69)

Since the FLRW line-element is homogeneous and isotropic, only the expansion
scalar ($\Theta$) is non-zero:

$\Theta=3\frac{{\cal H}(\eta)}{a(\eta)}\quad\mbox{where}\quad{\cal
H}=\frac{a^{\prime}(\eta)}{a(\eta)}$

where ′ refers to derivative w.r.t. $\eta$. Other kinematic quantities vanish,
i. e., $\sigma_{\mu\nu}=\omega_{\mu\nu}=0$.

We now spilt the 3-space into $1+2$ by choosing the following space-like
vector $n^{\mu}=(0,\,{1},\,0,\,0)/{a(\eta)}$. This satisfies the conditions:
$n^{\mu}n_{\mu}=1$ and $u^{\mu}n_{\mu}=0$. Repeating the steps discussed in
Sec. (II.2) for the line-element (66), we get:

$\tilde{\Theta}=\frac{2}{a(\eta)}\frac{1}{r}\,,\tilde{\sigma}_{\mu\nu}=\tilde{\omega}_{\mu\nu}=0.$

It is important to note that while $\Theta$ is a function of $\eta$ only,
$\tilde{\Theta}$ depends on both $\eta$ and $r$. Also, $\Theta$ depends on the
Hubble parameter ${\cal H}$, while $\tilde{\Theta}$ is inversely proportional
of $r$. Hence, at large distances, $\tilde{\Theta}$ decays faster compared to
$\Theta$.

Substituting the above expressions in Eq. (58), we have:

$\displaystyle\mathscr{A}^{\nu}_{\rm FLRW}$
$\displaystyle=-\,\frac{1}{2}\,\frac{D^{\nu}\rho_{\rm n}}{\rho_{\rm
n}}+\frac{1}{4\,\rho_{\rm n}}\,\mathscr{S}^{\nu}(\Theta-\tilde{\Theta})\,.$
(70)

Like Minkowski space-time, $\mathscr{A}^{\nu}$ will have components only in
the $2$-Sphere. Using the fact that the fundamental observers have zero
velocity in the 2-Sphere and repeating the earlier analysis, we have

$\mathscr{A}^{A}=u^{0}\partial_{0}u^{A}=\frac{1}{a(\eta)}\,\frac{\partial
u^{A}}{\partial\eta}\,.$

In terms of the null coordinate $u(\equiv\eta-r)$, we have:

$\mathscr{A}^{A}=\frac{1}{a(u)}\,\frac{\partial u^{A}}{\partial u}\,.$

Substituting the above expression in Eq. (70), we have:

$\displaystyle\frac{\partial u^{A}}{\partial
u}=-\,\frac{a(u)}{2}\,\frac{D^{A}\rho_{\rm n}}{\rho_{\rm
n}}+\frac{a(u)}{4\,\rho_{\rm n}}\,\mathscr{S}^{A}(\Theta-\tilde{\Theta})\,.$
(71)

Integrating the above expression w.r.t $u$, leads to the following memory
vector:

$\displaystyle\Delta u_{\rm FLRW}^{A}=-\frac{1}{2}\,\int
du\,\frac{a(u)}{\rho_{\rm n}}\,D^{A}\rho_{\rm n}+\frac{1}{4}\,\int
du\,\frac{a(u)}{\rho_{\rm n}}\,\mathscr{S}^{A}(\Theta-\tilde{\Theta})$ (72)

This is the expression for the memory vector in FLRW space-time regarding
which we want to highlight the following points: First, unlike Minkowski
space-time, here the fundamental observers are Eulerian, and hence, we do not
have to transform the above expression to Lagrangian observers. Second, our
results differ from the results of Ref. 2022-EnriquezRojo-JHEP . In Ref.
2022-EnriquezRojo-JHEP , the authors show that the EM memory effect in FLRW
differs from the Minkowski only by the conformal factor $a(\eta)$ or $a(u)$.
In other words, their analysis did not account for the geometric contribution
to the memory effect. As mentioned earlier, the geometric contribution leads
to a non-zero energy flux ($\mathscr{S}^{A}$) contribution. Also note that the
ordinary memory derived in Ref. 2022-EnriquezRojo-JHEP  is not present in Eq.
(72) as we have assumed any external charge or current to be zero. Third, we
find that $\rho_{(n)}$ and the energy flux ($\mathscr{S}^{A}$) contribute
oppositely. It will be interesting to see whether the two contributions
nullify the EM memory.

###  V.3 pp-wave space-times

In this subsection, we derive the EM memory for a special kind of plane-
fronted wave with parallel rays (pp-waves) called plane-wave metric 2003-Blau-
CQG :

$\displaystyle ds^{2}=-2dudv-\mathscr{F}(u,\,x,\,y)\,du^{2}+dx^{2}+dy^{2}$
(73)

where, $\mathscr{F}(u,\,x,\,y)=A(u)(x^{2}-y^{2})+2B(u)xy$ describes the plane
wave and $A(u),B(u)$ are arbitrary functions such that $\mathscr{F}>0$. Note
that $u,v$ are not light-cone coordinates. $u$ is time-like coordinate and $v$
is a null coordinate.

We split the above 4-D space-time into $1+3$ form and later into $1+1+2$-form
by considering the following time-like velocity vector ($u^{\mu}$) and space-
like vector $(n^{\mu})$:

$u^{\mu}\equiv\left(\mathscr{F}(u,\,x,\,y)^{(-1/2)},\,0,\,0,\,0\right),\quad
n^{\mu}\equiv\left(\mathscr{F}(u,\,x,\,y)^{(-1/2)},\,-\,\mathscr{F}(u,\,x,\,y)^{(1/2)},\,0,\,0\right)\,.$

For the above choice of time-like vector, the 3-space projection tensor
$(h_{\mu\nu})$ is:

$\displaystyle h_{\mu\nu}$ $\displaystyle=\begin{bmatrix}\,0&0&0&0\,\\\
\,0&\frac{1}{\mathscr{F}(u,\,x,\,y)}&0&0\,\\\ \,0&0&1&0\,\\\
\,0&0&0&1\,\end{bmatrix}$ (74)

Substituting these in the definitions in Sec. (II), only non-zero quantity is
the expansion scalar ($\Theta$):

$\displaystyle\Theta=-\frac{\left(x^{2}-y^{2}\right)\,A^{\prime}(u)\,+\,2xy\,B^{\prime}(u)}{2\,\left(2B(u)\,xy\,+\,A(u)\,\left(x^{2}-y^{2}\right)\,\right)^{3/2}}\,.$
(75)

The non-zero projection tensor $\tilde{h}_{\mu\nu}$ components in the 2-space
are $\tilde{h}_{xx}=1,\,\tilde{h}_{yy}=1$. Thus, the memory vector for the
special kind of pp-wave space-times is:

$\displaystyle\mathscr{A}_{\rm PP}^{\nu}$
$\displaystyle=-\,\frac{1}{2}\,\frac{D^{\nu}\rho_{\rm n}}{\rho_{\rm
n}}+\frac{\Theta}{4\,\rho_{\rm n}}\,\mathscr{S}^{\nu}\,.$ (76)

Here, the acceleration of the time-like observer is confined to the $x-y$
plane, i. e.,

$\displaystyle\mathscr{A}_{\rm PP}^{A}$
$\displaystyle=-\,\frac{1}{2}\,\frac{D^{A}\rho_{\rm n}}{\rho_{\rm
n}}+\frac{\Theta}{4\,\rho_{\rm n}}\,\mathscr{S}^{A}\,,$ (77)

where, the index $A,\,B$ corresponds to $(x,\,y)$. Evaluating the acceleration
vector along $x$ and $y$, we have:

$\displaystyle\mathscr{A}^{\rm(PP)}_{x(y)}$
$\displaystyle=-\frac{1}{2\,\rho_{\rm n}}\,\partial_{x(y)}\left(\rho_{\rm
n}\right)+\frac{\Theta}{4\,\rho_{\rm n}}\,\mathscr{S}_{x(y)}\,.$ (78)

Integrating the above equation w.r.t $u$, we have:

$\displaystyle\Delta u^{\rm PP}_{x(y)}=-\frac{1}{2}\int
du\,\,\frac{\partial_{x(y)}\left(\rho_{\rm n}\right)}{\rho_{\rm
n}}\,+\,\frac{\Theta}{4}\,\int du\frac{\mathscr{S}_{x(y)}}{\rho_{\rm n}}\,.$
(79)

The above expression for the velocity kick is for a generic plane-wave metric.
To gain some physical intuition, we consider two specific forms — Penrose
limit of the Schwarzschild and FLRW space-times 2003-Blau-CQG . For
Schwarzschild space-time, we have

$A(u)=\frac{6}{25u^{2}};\quad B(u)=0$

Substituting these in Eq. (75), we have:

$\Theta_{\rm PP,Sch}=\frac{5}{\sqrt{6(x^{2}-y^{2})}}\,.$

It is interesting to note that although the space-time metric does not
differentiate between the two spatial coordinates $(x,y)$, in order for
$\Theta$ to be real, the above expression demands that $x>y$. Thus, velocity
kick due to EM wave in PP-wave limit of Schwarzschild space-time can only
occur if $x>y$ and is given by:

$\displaystyle\Delta u^{\rm PP\,Sch}_{x(y)}=-\frac{1}{2}\int
du\,\,\frac{\partial_{x(y)}\left(\rho_{\rm n}\right)}{\rho_{\rm
n}}\,+\,\frac{5}{4\sqrt{6(x^{2}-y^{2})}}\,\int
du\frac{\mathscr{S}_{x(y)}}{\rho_{\rm n}}\,.$ (80)

In the case of Penrose limit of FLRW space-time with power-law scale factor
$a(t)\sim t^{h}$, we have:

$A(u)=-\frac{h}{(1+h)u^{2}},\quad B(u)=0\,.$

Substituting these in Eq. (75), we have:

$\displaystyle\Theta_{\rm
PP,FLRW}=\sqrt{\frac{(1+h)}{h(y^{2}-x^{2})}};\quad\Delta u^{\rm
PP\,FLRW}_{x(y)}=-\frac{1}{2}\int du\,\frac{\partial_{x(y)}\left(\rho_{\rm
n}\right)}{\rho_{\rm n}}+\frac{\sqrt{(1+h)}}{4\sqrt{h(y^{2}-x^{2})}}\int
du\frac{\mathscr{S}_{x(y)}}{\rho_{\rm n}}\,.$ (81)

Here again, we see that in-order for $\Theta$ to be real, the above expression
demands that $y>x$. Thus, velocity kick due to EM wave in PP-wave limit of
FLRW space-time occurs in a different region of the 2-space compared to
Schwarzschild. Thus, EM memory has a distinct signature for different space-
times and can potentially be used as a probe.

###  V.4 Kerr space-time

In this section, we derive the memory effect in Kerr space-time. In Boyer-
Lindquist coordinates $\left(t,\,r,\,\chi,\,\phi\right)$, the Kerr space-time
is:

$\displaystyle ds^{2}=$
$\displaystyle\left(\frac{2mr}{r^{2}+a^{2}\chi^{2}}-1\right)dt^{2}+\left(\frac{r^{2}+a^{2}\chi^{2}}{r^{2}-2mr+a^{2}}\right)dr^{2}+\left(\frac{r^{2}+a^{2}\chi^{2}}{1-\chi^{2}}\right)d\chi^{2}$
$\displaystyle-\left[\frac{4mar\left(1-\chi^{2}\right)}{r^{2}+a^{2}\chi^{2}}\right]dtd\varphi+\left(1-\chi^{2}\right)\left[r^{2}+a^{2}+\frac{2ma^{2}r\left(1-\chi^{2}\right)}{r^{2}+a^{2}\chi^{2}}\right]d\varphi^{2}\,.$
(82)

where $\chi\equiv\cos\theta$. In this case, the time-like observer 4-velocity
$(u^{\mu})$ and the space-like vector $(n^{\mu})$ are
2021-Hansraj.SunilMaharaj-gr-qc :

$u^{\mu}=\left[\sqrt{\frac{r^{2}-2mr+a^{2}}{r^{2}+a^{2}\chi^{2}}},0,0,0\right],n^{\mu}=\left[0,\sqrt{\frac{r^{2}-2mr+a^{2}}{r^{2}+a^{2}\chi^{2}}},0,0\right]\,.$

We give below the kinematical quantities (discussed in Sec. (II.2)) for Kerr
space-time in $1+1+2$ formalism obtained in Ref. 2021-Hansraj.SunilMaharaj-gr-
qc :

$\displaystyle\Theta=0;$ $\displaystyle\quad\Sigma_{\mu\nu}=0\,;$ (83)
$\displaystyle\Omega=-\frac{2mar\chi\sqrt{\mathscr{L}}}{J\sqrt{\mathscr{K}^{3}}};$
$\displaystyle\quad\tilde{\Theta}=\frac{\mathscr{W}}{\mathscr{J}\sqrt{\mathscr{K}^{3}\mathscr{L}}}\,;$
(84) $\displaystyle\tilde{\omega}_{\mu\nu}=\tilde{\omega}\epsilon_{\mu\nu}=0;$
$\displaystyle\quad\mathscr{A}=-\frac{m\mathscr{D}\sqrt{\mathscr{L}}}{J\sqrt{\mathscr{K}^{3}}}\,;$
(85)
$\displaystyle\tilde{\sigma}_{\mu\nu}=\left[\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\
0&0&-\frac{1}{2}\frac{a^{2}(m-r)\sqrt{\mathscr{K}}}{\mathscr{J}\sqrt{\mathscr{L}}}&0\\\
0&0&0&\frac{1}{2}\frac{a^{2}(m-r)\mathscr{M}^{2}\sqrt{\mathscr{L}\mathscr{K}}}{\mathscr{J}^{2}}\end{array}\right]$
(90)

where,

$\displaystyle\mathscr{M}=\chi^{2}-1;$
$\displaystyle\quad\mathscr{D}=-r^{2}+a^{2}\chi^{2};\quad\mathscr{L}=r^{2}-2mr+a^{2}$
(91) $\displaystyle\mathscr{J}=r^{2}-2mr+a^{2}\chi^{2};$
$\displaystyle\quad\mathscr{K}=r^{2}+a^{2}\chi^{2}$ (92)
$\displaystyle\mathscr{W}=2r^{3}(r-2m)^{2}+a^{4}\chi^{2}\left(m+r-m\chi^{2}+r\chi^{2}\right)+a^{2}r^{2}\left(-3m+r+\chi^{2}(3r-5m)\right)$
(93)

Substituting these expressions in Eq. (58), and noting that the memory vector
lies in the 2-D surface, we get:

$\displaystyle\mathscr{A}^{A}=$
$\displaystyle-\,\frac{1}{2}\frac{D^{A}\rho_{(n)}}{\rho_{(n)}}-\frac{\Omega}{2}\frac{\epsilon^{AB}\,\mathscr{S}_{B}}{\rho_{(n)}}\,-\frac{1}{2}\left(\tilde{\sigma}^{AB}+\frac{\tilde{\Theta}}{2}\tilde{h}^{AB}\right)\frac{\mathscr{S}_{B}}{\rho_{(n)}}-\frac{\mathscr{A}}{2\rho_{(n)}}\,\mathscr{S}^{A}\,$
(94)

This is the EM memory vector for an Eulerian observer in Kerr space-time. Note
that this is a generic result for any value of angular momentum. For a better
physical insight, we consider $a\rightarrow 0$ limit. Substituting $a\to 0$ in
Eqs. (83 \- 93), we have

$\displaystyle\mathscr{M}_{0}=\chi^{2}-1;$
$\displaystyle\quad\mathscr{D}_{0}=-r^{2};\quad\mathscr{L}_{0}=r^{2}-2mr$ (95)
$\displaystyle\mathscr{J}_{0}=r^{2}-2mr;$
$\displaystyle\quad\mathscr{K}_{0}=r^{2};\quad\mathscr{W}_{0}=2r^{3}(r-2m)^{2}$
(96) $\displaystyle\Omega_{0}=\tilde{\sigma_{0}}^{\mu\nu}=0;$
$\displaystyle\quad\tilde{\Theta}_{0}=2\sqrt{\frac{(r-2m)}{r^{3}}};\quad\mathscr{A}=\frac{m}{\sqrt{r^{3}(r-2m)}}$
(97)

Substituting the above quantities in Eq. (94), we have:

$\displaystyle\mathscr{A}^{A}=-\,\frac{1}{2}\frac{D^{A}\rho_{(n)}}{\rho_{(n)}}-\frac{1}{2}\sqrt{\frac{r-2m}{r^{3}}}\frac{\mathscr{S}^{A}}{\rho_{(n)}}-\frac{1}{2\rho_{(n)}}\frac{m}{\sqrt{r^{3}(r-2m)}}\,\mathscr{S}^{A}\,.$
(98)

This is the EM memory vector for an Eulerian observer in Schwarzschild space-
time, regarding which we want to mention the following points: First, in the
limit, $r\rightarrow\infty$, reduces to Minkowski space-time expression (63).
Second, in the limit $r\rightarrow\infty$, the subleading term is proportional
to $r^{-1}$. Third, to derive the memory vector $\Delta u^{A}$, we have to
switch to the null time coordinate $u=t-r$ and integrate Eq. (98) with respect
to $u$ at the asymptotic limit. Lastly, to evaluate the memory effect
experienced by the static asymptotic (Lagrangian) observer, we need to do the
transformation from $\left(\vec{e}^{\,\theta},\vec{e}^{\,\phi}\right)$ to the
orthogonal coordinate basis $\left(\hat{\theta},\hat{\phi}\right)$ like in
Sec. (V.1).

##  VI Conclusions

In this work, we have derived a master equation for electromagnetic memory in
an arbitrary space-time. We used the covariant formalism to obtain the same.
More specifically, we used the $1+1+2$ covariant formalism. The $1+1+2$
decomposition of space-time is a natural extension of the $1+3$ formalism in
which the three-space is further decomposed using a given spatial direction.
This choice of covariant formalism is because the net momentum (kick) vector
lies on the 2-D surface for arbitrary space-time. Also, the electric and
magnetic fields are transverse to the direction of propagation of the passing
EM wave.

The EM memory (58) has three distinct contributions: First contribution is due
to the change in the EM field energy ($\rho_{(n)}$) along $n^{\mu}$ in the
2-space. This is non-zero for Minkowski space-time. The second contribution is
proportional to the energy flux ($S^{\alpha}$) in the 2-space. This has
kinematical information of the space-time and vanishes for the flat space-
time. The third contribution is proportional to the acceleration ${\cal A}$
along the time-like vector $u^{\mu}$. To our understanding, the earlier
approaches could not isolate the different contributions to the EM memory as
done in this work.

We then obtained the EM memory for different space-times. In the case of FLRW
space-time, we showed that the earlier analysis did not account for the
geometric contribution to the memory effect 2022-EnriquezRojo-JHEP .
Specifically, their analysis did not account for the geometric contribution
leading to a non-zero energy flux ($\mathscr{S}^{A}$) contribution. We have
also obtained the EM memory for Kerr space-time. We also showed that the EM
memory has a distinct signature for different pp wave space-times and can
potentially be used as a probe.

It would be interesting to extend our analysis for black holes with multiple
horizons and those that are not asymptotically flat. These may be particularly
relevant for using EM memory as a probe to PBH. Finally, our analysis points
to the possibility of using $1+1+2$ covariant formalism to understand
gravitational memory. These are currently under investigation.

###### Acknowledgements.

The authors thank A. Chowdhury, S. Mahesh Chandran and A. Kushwaha for
comments on the earlier version of the manuscript. The work is supported by
the SERB-Core Research grant.

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