change in $\varphi_{mn}^{(1)}$ and therefore, Lemmas 4 and 5 imply that
$\Delta\varphi_{mn}^{(1)}$ is given by eq. 5.32.

That $T(\theta)$ and the $S^{(i)}(\theta)$ generate an asymptotic
supertranslation and an asymptotic angle-dependent internal isometry between
the stationary eras follows from Equations 4.55 and 4.56 and that
$\mathcal{N}_{ab}=0$ in the stationary eras. ∎

We finally consider the spherical harmonic dependence of the change in the
metric coefficients $\Delta h_{AB}^{(1)},\Delta h_{Am}^{(1)}$ and
$\Delta\varphi_{mn}^{(1)}$. We first note that, by Lemma 5,
$\Delta\varphi_{mn}^{(1)}$ is clearly spanned only by $\ell=0$ spherical
harmonics. By Proposition 2, if $T(\theta)$ is spanned by $\ell=0,1$ spherical
harmonics then $\mathscr{D}_{A}\mathscr{D}_{B}T(\theta)=-q_{AB}T(\theta)$.
Therefore, it follows that the tracefree part of $\Delta h_{AB}^{(1)}$ on
$S^{2}$ is orthogonal to the $\ell=0,1$ spherical harmonics. Furthermore, by
the form of eq. 5.29, we have that $\Delta A_{Am}^{(1)}$ is orthogonal to the
$\ell=0$ spherical harmonics.

##  6 The Memory Effect in Compactified Spacetimes

###  6.1 Unification of memory effects

We now explore the geometric interpretation of Theorem 1 in terms of the
memory effect, which is an observable quantity. Physically, the memory effect
is the permanent relative displacement of a system of test particles,
initially at rest, caused by the passage of a burst of gravitational
radiation. The relative displacement of test particles is governed by the
geodesic deviation equation

$(v^{M}\nabla_{M})^{2}\xi^{N}=-R_{MPQ}{}^{N}v^{M}v^{Q}\xi^{P},$ (6.1)

where $v^{M}$ is the tangent vector of the worldline of the particle,
$\xi^{M}$ is the deviation vector and $R_{MPQ}{}^{N}$ is the Riemann tensor.
We are interested in the displacement of test particles located near future
null infinity and shall determine the leading order memory effects in a
$1\over r$ expansion in a neighborhood of null infinity.

We consider a spacetime where the metric near future null infinity is
stationary at leading order in $1\over r$, at asymptotically early and late
retarded times. In this subsection, we will simplify and integrate eq. 6.1 to
derive an explicit formula for the memory effect. This discussion is a
modification of a similar analysis found in [65]. There are subtle differences
when one considers compact internal manifolds, which makes the argument worth
revisiting.

Consider an array of initially stationary test particles in a neighborhood of
null infinity, which we model as a congruence of time-like geodesics whose
tangents $v^{A}$ initially point in the $(\partial/\partial u)^{M}$ direction.
In a neighborhood of null infinity, the spacetime metric deviates from the
Ricci-flat direct product metric (2.2) at order $1\over r$. Consequently, the
geodesic equation implies that $v^{M}$ differs from the corresponding integral
curve of $(\partial/\partial u)^{M}$ only at order $1\over r$ and therefore
$u$ will differ from an affine parameterization beginning at this order.

For an arbitrary internal manifold, the curvature is generically non-vanishing
at infinity. Nevertheless, these considerations imply that the quantity
$R_{MPQ}{}^{N}v^{M}v^{Q}$ in eq. 6.1 does vanish at infinity and is only non-
vanishing at order $1\over r$. Therefore, the deviation of $v^{M}$ from
$(\partial/\partial u)^{M}$ in eq. 6.1 can only affect $\xi^{N}$ at order
$1\over r^{2}$ and faster fall-off. Finally, by eq. 4.2, the Riemann tensor
differs from the Weyl tensor at $O\big{(}\frac{1}{r^{2}}\big{)}$ since the
stress energy falls off like $\frac{1}{r^{2}}$. Since we are only considering
the memory effect at leading order in $1\over r$, we can replace $v^{M}$ with
$(\partial/\partial u)^{M}$ and $R_{PML}{}^{N}v^{P}v^{L}$ with the electric
Weyl tensor $\mathcal{E}_{M}{}^{N}$ (as defined in eq. 4.11) in eq. 6.1 which
yields

$\frac{\partial^{2}}{\partial u^{2}}\xi^{M}=-\mathcal{E}^{M}{}_{N}\xi^{N}$
(6.2)

Indices on the right hand side of eq. 6.2 are raised and lowered with the
asymptotic metric $\hat{g}_{MN}$. Equation 6.2 implies that $\xi^{M}$ differs
from the integral curve of its initial value $\xi^{M}_{0}$ at order $1\over r$
and we may replace $\xi^{M}$ by its initial value in the right hand side of
eq. 6.2. Thus, at leading order in $1\over r$, we have

$\frac{\partial^{2}}{\partial
u^{2}}\xi^{(1)M}=-\mathcal{E}^{M}{}_{N}\xi^{N}_{(0)},$ (6.3)

where $\xi_{M}^{(1)}$ is the deviation vector at $O\big{(}\frac{1}{r}\big{)}$.
Integrating eq. 6.3 twice, we obtain

$\xi^{(1)M}\bigg{|}^{u=\infty}_{u=-\infty}=\Delta^{M}{}_{N}\xi^{N}_{(0)},$
(6.4)

where

$\Delta_{MN}\equiv-\int_{-\infty}^{\infty}du^{\prime}\int_{-\infty}^{u^{\prime}}du^{\prime\prime}\mathcal{E}_{MN}.$
(6.5)

We refer to $\Delta_{MN}$ as the memory tensor. This characterizes the memory
effect as a linear map on the initial displacement to the change in the
relative separation. Further, as noted in Lemma 3, the only non-vanishing
components of ${\mathcal{E}}_{MN}$ are ${\mathcal{E}}_{ab}=-{1\over
2}\partial_{u}\mathcal{N}_{ab}$ where $a,b$ are along
$S^{2}\times\mathcal{M}_{\textrm{int}}$. This gives a simpler manifestly
gauge-invariant relation for the memory,

$\Delta_{ab}(\theta,y)\equiv\frac{1}{2}\int_{-\infty}^{\infty}du\,\mathcal{N}_{ab}(u,\theta,y).$
(6.6)

From $(\ref{memoryandnews})$, it follows that

$\Delta_{ab}=\Delta_{ba},\quad\mathfrak{q}^{ab}\Delta_{ab}=q^{AB}\Delta_{AB}+\hat{g}^{mn}\Delta_{mn}=0,$
(6.7)

and clearly $\Delta_{ab}$ is time-independent. Additionally from eq. 4.19, we
see that

$\bm{D}^{2}\Delta_{AB}=0,\quad\bm{D}^{2}\Delta_{Am}=0,\quad\bm{D}^{2}\Delta_{mn}+2\mathcal{R}_{m}{}^{p}{}_{n}{}^{q}\Delta_{pq}=0.$
(6.8)

Using arguments identical to those in the proof of Lemma 3, we see that
$\Delta_{AB}$ is independent of internal coordinates $y^{m}$ and $\Delta_{Am}$
and $\Delta_{mn}$ can be uniquely decomposed in a basis of harmonic $1$-forms
$\overline{V}_{m}^{(i)}$ and Lichnerowicz zero modes
$\overline{t}_{mn}^{(i)}$, respectively,

$\Delta_{Am}=\sum_{i=1}^{b_{1}}\Delta^{(i)}_{A}(\theta)\otimes\overline{V}_{m}^{(i)}(y)\;\textrm{
and
}\;\Delta_{mn}=\sum_{i=1}^{{\hat{d}_{\textrm{L}}}}\Delta^{(i)}(\theta)\overline{t}_{mn}^{(i)}(y)+\frac{1}{D-4}\hat{g}_{mn}\hat{g}^{pq}\Delta_{pq}(\theta).$
(6.9)

The $\Delta_{A}^{(i)}$ are a collection of $b_{1}$ $1$-forms on $S^{2}$, and
the $\Delta^{(i)}$ are smooth functions on $S^{2}$.

We now provide a geometric interpretation of Theorem 1. In the gauge given in
Lemma 4, the news tensor can be expressed in terms of the leading order metric
eq. 4.46. This provides a direct relation between the change in the
$h_{AB}^{(1)},h_{Am}^{(1)}$ and $h_{mn}^{(1)}$ before and after the radiation
epochs:

$\Delta_{AB}=\frac{1}{2}\Delta h_{AB}^{(1)},\quad\Delta_{Am}=\frac{1}{2}\Delta
A_{Am}^{(1)}\quad\textrm{ and
}\quad\Delta_{mn}=\frac{1}{2}\Delta\varphi_{mn}^{(1)}\,.$ (6.10)

Using the results of Theorem 1 we can now relate the memory to the change in
the metric due to a burst of radiation. We first note that certain metric
components appearing in Theorem 1 can be directly related to definitions of
the Bondi mass aspect and electric charge aspect in $\mathbb{R}^{4}$.

$m_{B}\equiv-\frac{1}{2}\mathcal{E}_{rr}^{(3)}=\frac{1}{2}h_{uu}^{(1)}\text{
and }Q^{(i)}\equiv F_{ur}^{(2;i)}=A_{u}^{(1;i)}\text{ (in a stationary era)},$
(6.11)

where $F=dA$ using the exterior derivative on $\mathbb{R}^{4}$ and $A_{\mu m}$
is defined in eq. 3.1. Using the results of Theorem 1 and eq. 6.9 we see that

$\displaystyle\mathscr{D}^{A}\mathscr{D}^{B}\Delta_{AB}=2\Delta
m_{B}-\frac{1}{2}\Delta\phi-8\pi\mathcal{F}(\theta),\quad
q^{AB}\Delta_{AB}=-\frac{1}{2}\Delta\phi,$ (6.12)
$\displaystyle\mathscr{D}^{A}\Delta_{A}^{(i)}(\theta)=\frac{1}{2}\Delta
Q^{(i)}+8\pi\mathcal{J}^{(i)}(\theta),$ (6.13)
$\displaystyle\Delta^{(i)}=\frac{1}{2}\Delta\Phi^{(i)}\;\text{ and
}\;\hat{g}^{mn}\Delta_{mn}=\frac{1}{2}\Delta\phi.$ (6.14)

In analogy with the decomposition of the news in eqs. 4.20 and 4.21 we can
decompose the flux $\mathcal{F}(\theta)$ into gravitational, electromagnetic
and scalar contributions to the flux:

$\mathcal{F}(\theta)=\mathcal{F}_{\text{GR}}(\theta)+\mathcal{F}_{\text{EM}}(\theta)+\mathcal{F}_{S}(\theta),$
(6.15)

where

$\displaystyle\mathcal{F}_{\text{GR}}(\theta)=$
$\displaystyle-\int_{\mathbb{R}}du\,\bigg{(}\overline{T^{(2)}_{uu}}+\frac{1}{32\pi}N^{AB}N_{AB}\bigg{)},$
(6.16) $\displaystyle\mathcal{F}_{\text{EM}}(\theta)=$
$\displaystyle-\sum_{i=1}^{b_{1}}\int_{\mathbb{R}}du\,\mathcal{N}^{(i)A}\mathcal{N}^{(i)}_{A},$
(6.17) $\displaystyle\mathcal{F}_{\text{S}}(\theta)=$
$\displaystyle-2\int_{\mathbb{R}}du~{}N^{2}-\sum_{j=1}^{{\hat{d}_{\textrm{L}}}}\int_{\mathbb{R}}du~{}(\mathcal{N}^{(j)})^{2}.$
(6.18)

From the point of view of reduction, eq. 6.16 corresponds to the flux of four-
dimensional gravitational radiation energy as well as null stress energy.
Equation 6.17 corresponds to the flux of electromagnetic energy and eq. 6.18
is the flux of scalar energy where the first term is the contribution from the
volume mode and second term is the contribution from the volume-preserving
moduli.

We can give a physical interpretation to these relations, which express memory
in terms of fluxes. First consider eq. 6.12. The spherically symmetric part of
the left hand side vanishes. The right hand side defines a change in the
spherically symmetric part of the mass aspect. It is reasonable to view

$m=m_{B}-{1\over 4}\phi\qquad\text{ (in a stationary era) }\,,$ (6.19)

as the mass since the change in this quantity is determined by the energy flux
to $\mathscr{I}^{+}$ in analogy with the four-dimensional result
$(\ref{derivmemory-null})$. Similarly, $Q^{(i)}$ is the electric charge for
each asymptotic gauge-field $A^{(i;1)}_{\mu}$ since $\Delta Q^{(i)}$ is
determined by the charge flux to $\mathscr{I}^{+}$. Via $(\ref{scalarflux})$,
scalar memory is defined by the change in the scalar charge, given by the
coefficient of the ${1\over r}$ term in the expansion of the field near
$\mathscr{I}^{+}$, between early and late times. In this case, there is no
integrated flux term.

The memory effect $\Delta_{AB}$ corresponds to the permanent relative angular
displacement of a pair of freely falling test masses. $\Delta_{Am}$
corresponds to the displacement in the internal space directions (i.e. along
Killing directions) for a pair of test masses that are initially angularly
displaced. If the test masses had some initial displacement in the internal
space then, due to a change in scalar charge, the relative displacement in the
internal space will change by an amount $\Delta_{mn}$. Physically, the
internal space is small and therefore relative displacements of test masses
into the internal space are undetectable. Nevertheless, the four-dimensional
scalar and electromagnetic memory effects are usually described in terms of
velocity kicks [15, 14]. We should be able to recover this way of observing
memory from the higher-dimensional gravitational picture.

To see how this emerges, consider the geodesic motion of a test particle with
velocity $v^{M}$

$v^{M}\nabla_{M}v^{N}=0,$ (6.20)

which follows from varying the point-particle action

$\displaystyle S=-m\int\sqrt{-\hat{g}_{MN}(x)\,dx^{M}\,dx^{N}}\,.$ (6.21)

This equation of motion eq. 6.20 describes the motion of a point particle
following a timelike geodesic. We consider the case where the tangent $v^{M}$,
initially $v^{M}_{(0)}$, is of the form

$\displaystyle v^{M}_{(0)}\equiv c_{1}\bigg{(}\frac{\partial}{\partial
u}\bigg{)}^{M}+c_{2}\overline{V}^{m}(y)\bigg{(}\frac{\partial}{\partial
y^{m}}\bigg{)}^{M},$ (6.22) $\displaystyle
c_{1}^{2}=\frac{1+\sqrt{1+4q^{2}}}{2}\,,\qquad
c_{2}^{2}=\frac{-1+\sqrt{1+4q^{2}}}{2}\,,$ (6.23)

where $\overline{V}^{m}(y)$ is a unit normalized Killing vector, which is
automatically geodesic on $\mathcal{M}_{\textrm{int}}$:

$\overline{V}^{m}\bm{D}_{m}\overline{V}^{n}=0\text{ and
}\hat{g}^{mn}\overline{V}_{m}\overline{V}_{n}=1.$ (6.24)

This characterizes an initially stationary test particle with charge $q$
determined by the velocity in the internal direction at some early time
$u=u_{0}$. The vector field $\overline{V}^{m}$ must be Killing to ensure the
test particle is constructed from zero modes of the internal space. Since our
discussion is purely classical, we will not worry about quantization
conditions on the internal momentum, which force such momenta to be of order
the Kaluza-Klein scale.

We are interested in the velocity kick of this test particle relative to a
preferred class of asymptotic, stationary observers, which will define our lab
frame. To define a time-like vector field $v_{M}^{\text{lab}}$, we Lie-
transport the tangent vector $v_{M}^{(0)}$, so that $v_{M}^{\text{lab}}$ in
our coordinates agrees with the trivial extension $v_{M}^{(0)}$ for all
$u>u_{0}$. We note that this is an accelerated reference frame, which implies
that it differs from geodesic evolution of $v_{M}^{(0)}$ at order
$\frac{1}{r}$:

$v_{M}=v_{M}^{\text{lab}}+\frac{v_{M}^{(1)}(u,\theta,y)}{r}+O\bigg{(}\frac{1}{r^{2}}\bigg{)}.$
(6.25)

Expanding eq. 6.20 in powers of $\frac{1}{r}$ and integrating the geodesic
equation a straightforward computation yields in the gauge described by Lemma
4 that the non-vanishing components of the velocity kick are $\Delta v^{A(1)}$
and $\Delta v^{r(1)}$.

$\Delta
v^{(1)}_{A}(u,\theta)=c_{1}^{2}\int_{-\infty}^{u}du^{\prime}\partial_{u^{\prime}}h_{uA}^{(1)}+\frac{q}{2}\int_{-\infty}^{u}du^{\prime}\mathcal{N}_{Am}\overline{V}^{m}.$
(6.26)

The first term on the right hand side of $(\ref{kickone})$ is not proportional
to the charge! Rather it is finite as $q\rightarrow 0$ and corresponds to a
purely gravitational velocity kick. This effect actually has nothing to do
with the compact internal space and is present in just $\mathbb{R}^{4}$. It
would be very interesting to explore the potential observability of this
effect. The second term is the electromagnetic kick we expect. Note that
$\mathcal{N}_{Am}\overline{V}^{m}$ is independent of $y$ because of eq. 4.20.
Similarly, the radial velocity kick

$\Delta
v^{r(1)}(u,\theta)=\frac{c_{2}^{2}}{2}\int_{-\infty}^{u}du^{\prime}\mathcal{N}_{mn}\overline{V}^{m}\overline{V}^{n},$
(6.27)

is sensitive to radiation from the specific scalar zero modes associated to
the torus component in the decomposition theorem of [57].

The total velocity kicks in the angular and radial directions, respectively,
are given by

$\displaystyle\Delta v_{A}(\theta)$
$\displaystyle\equiv\lim_{u\to\infty}\Delta v_{A}^{(1)}(u,\theta)\,,$ (6.28)
$\displaystyle\Delta v^{r}(\theta)$
$\displaystyle\equiv\lim_{u\to\infty}\Delta v^{r(1)}(u,\theta)\,.$ (6.29)

Using eq. 5.35 we find that the integrand of the first term in eq. 6.26 can be
expressed in terms of an integral of the news:

$\partial_{u}h_{uA}^{(1)}=\frac{1}{2}\mathscr{D}^{B}N_{BA}+\frac{1}{4}\mathscr{D}_{A}N+\frac{1}{2}\mathscr{D}_{A}h_{ur}^{(1)}-\frac{1}{2}\partial_{u}^{2}\overline{h}_{rA}^{(2)}.$
(6.30)

Integrating eq. 6.30 and using eq. 5.36 implies that

$\Delta
h_{uA}^{(1)}(\theta)=\frac{1}{2}\int_{\mathbb{R}}du\mathscr{D}^{B}N_{BA}.$
(6.31)

Using eq. 6.6 yields the total velocity kick in terms of the memory

$\displaystyle\Delta v_{A}(\theta)=$ $\displaystyle
c_{1}^{2}\mathscr{D}^{B}\Delta_{BA}+q\Delta_{Am}\overline{V}^{m},$ (6.32)
$\displaystyle\Delta v^{r}(\theta)=$ $\displaystyle
c_{2}^{2}\Delta_{mn}\overline{V}^{m}\overline{V}^{n}.$ (6.33)

This leaves the question of how to detect radiation for moduli associated to
the simply-connected component of $\mathcal{M}_{\textrm{int}}$. It appears
that directly detecting such radiation requires a more sophisticated detector,
but we can make one comment on this issue. In principle, a detector can
measure $\mathcal{N}_{AB},\mathcal{N}_{Am}$ and the torus contribution to
$\mathcal{N}_{mn}$ by the motion of the arms of a LIGO-like detector and the
motion of a charged test particle. Squaring these contributions gives us all
of eq. 6.15 except any unknown null stress-energy, including contributions
from additional moduli. We can use the measured fluxes to compute what should
be the dominant contribution to the right hand side of eq. 6.12. Assuming the
size of the ordinary memory effect compared with the radiation contribution is
still small, and there is a sizeable discrepancy between the observed
gravitational memory and the flux computation, we can place upper bounds on
the possible contribution of any additional moduli.

###  6.2 The circle case

The original beauty of Kaluza-Klein theory was a unification of
electromagnetism, gravity and scalar field theory in a single $5$-dimensional
theory of gravity compactified on a circle. Let us revisit this beautiful and
simple example to unify the separately studied notions of memory for gravity
[18, 13], electromagnetism [15, 16, 17] and scalar theories [14] in the
framework of $5$-dimensional gravity using the discussion of section 6.1.

Let us take a spacetime metric with an exact $U(1)$ isometry,

$\hat{g}_{MN}dx^{M}dx^{N}=g_{\mu\nu}dx^{\mu}dx^{\nu}+e^{2\varphi(x)}(dy+A_{\mu}(x)dx^{\mu})^{2},$
(6.34)

where $y\sim y+2\pi L$ and $\varphi\rightarrow 0$ at infinity. Reducing the
$D=5$ Einstein-Hilbert action with zero cosmological constant on $y$ gives the
$4$-dimensional action,

$S={1\over 16\pi G}\int
d^{4}x\,e^{\varphi(x)}\sqrt{g}\left(R-\frac{1}{4}e^{2\varphi}F_{\mu\nu}F^{\mu\nu}+\partial_{\mu}\varphi\partial^{\mu}\varphi\right),$
(6.35)

where $F=dA$. This is a special case of $\mathcal{M}_{\textrm{int}}$ that we
studied earlier in the frame we have assumed in our discussion so far, which
is not Einstein frame! The $1\over r$ terms in the expansion of $A_{\mu}$ and
$e^{2\varphi(x)}$ can be identified with $A_{\mu y}^{(1)}$ and
$\varphi_{yy}^{(1)}$ defined in eq. 3.1 and discussed in the preceding
sections.

Specializing eq. 6.21 to the case of a $\mathbb{R}^{4}\times S^{1}$ gives the
geodesic equation,

$\displaystyle\frac{d^{2}x^{M}}{d\tau^{2}}+\tensor{\Gamma}{{}^{M}_{N}{}_{P}}\frac{dx^{N}}{d\tau}\frac{dx^{P}}{d\tau}=0\,,$
(6.36)

with the Christoffel symbols given to leading order in $\frac{1}{r}$ by

$\displaystyle\Gamma^{C}_{uu}=q^{CD}\partial_{u}h_{uD}^{(1)}\quad,\quad\Gamma^{C}_{uy}=\frac{1}{2}q^{CD}\partial_{u}A^{(1)}_{D}\quad,\quad\Gamma^{r}_{yy}=\frac{1}{2}\partial_{u}\phi^{(1)},$
(6.37)

where $\phi^{(1)}=2\varphi^{(1)}$. Assuming an initial $v^{M}_{(0)}$ of the
form eq. 6.22 gives the following leading order equations of motion,

$\partial_{u}v^{M(1)}=-c_{1}^{2}\Gamma_{uu}^{M}-2q\Gamma^{M}_{uy}-c_{2}^{2}K^{M}\Gamma^{r}_{yy},$
(6.38)

where $K^{M}\equiv\left(\frac{\partial}{\partial r}\right)^{M}$. In this case,
the time-dependent behavior of the angular and radial velocity kicks for a
particle with charge $q$, which might vanish, is determined using

$\partial_{u}v^{C;(1)}=-c_{1}^{2}q^{CD}\partial_{u}h_{uD}^{(1)}-qF_{uA}^{(1)}\,,\quad\partial_{u}v^{r;(1)}=-\frac{c_{2}^{2}}{2}\partial_{u}\phi^{(1)}.$
(6.39)

Using the analysis of section 6.1, the total velocity kick from the far past
($u\rightarrow-\infty)$ to the far future ($u\rightarrow+\infty)$ is given by

$\Delta v_{A}=c_{1}^{2}\mathscr{D}^{B}\Delta_{BA}+q\Delta_{Ay}\,,\quad\Delta
v^{r}=c_{2}^{2}\Delta_{yy},$ (6.40)

where $\Delta_{BA},\Delta_{By}$ and $\Delta_{yy}$ are found in eq. 6.10.

One final comment: in the context of subleading soft photon theorems, there
are proposals to permit gauge transformations in abelian gauge theory that
grow linearly with $r$ near $\mathscr{I}^{+}$ [66, 67]. This is an interesting
possibility, although the asymptotic behavior of the gauge parameter no longer
defines a $U(1)$ group element. In the Kaluza-Klein context, allowing such
gauge transformations becomes a statement about higher-dimensional gravity,
which would generalize the class of diffeomorphisms normally permitted,
assuming such a generalization is sensible. It would be interesting to explore
this embedding further.

###  6.3 Color memory

While most of the analysis in this paper assumes a Ricci-flat
$\mathcal{M}_{\textrm{int}}$, we cannot resist sketching how color memory
studied in [19, 55] should also emerge from Kaluza-Klein reduction. The
starting point is a higher-dimensional gravity theory which admits a space
with non-abelian isometries. We will assume a $D-4$ sphere for simplicity. Let
us take an action,

$S={1\over 2\kappa_{D}^{2}}\int
d^{D}x\sqrt{-g}\left(R-2\Lambda-|F_{D-4}|^{2}\right),$ (6.41)

where $F_{D-4}$ is a $D-4$-form field strength. Compactifying this theory on
$S^{D-4}$ with radius $L$ gives an effective four-dimensional potential for
the radius $L$ of the form:

$V_{\rm eff}={2\Lambda\over L^{D-4}}-{(D-4)(D-5)\over L^{D-2}}+{N^{2}\over
L^{3(D-4)}}.$ (6.42)

Here we assume the sphere metric is $L^{2}ds^{2}_{S^{D-4}}$, where
$ds^{2}_{S^{D-4}}$ is the metric for a sphere of unit volume. The parameter
$N$ is proportional to the amount of quantized $F_{D-4}$ flux through the
sphere. Since this is a classical gravity theory, we can chose $\Lambda$
conveniently to ensure the resulting spacetime is flat Minkowski. Under this
condition, the potential has a minimum with $L$ growing with $N$. This is all
we need. We have engineered Minkwoski spacetime from a compactification with
non-abelian isometries. In this case, the identity component of the isometry
group is $SO(D-3)$.

Let us return to the geodesic equation eq. 6.36 for a test particle with
velocity along the sphere. The novelty in this case, by comparison with the
Ricci-flat case, is that the internal velocity vector can rotate as higher-
dimensional gravitational radiation passes by. In the Ricci-flat case, the
Christoffel symbols along Killing directions vanish. For spaces with non-
abelian isometry groups, like the sphere, this is no longer true. From a four-
dimensional perspective, the color charge would therefore appear to change
because of a burst of radiation, in agreement with [19].

###  6.4 Frames

The final issue we need to address is the choice of frames. As illustrated in
the circle example of section 6.2, the natural four-dimensional frame that
corresponds to studying radiation in terms of the $D$-dimensional metric is
not Einstein frame. Let us parametrize the volume mode or breathing mode of
the internal metric in analogy with the circle case,

$ds^{2}_{\mathcal{M}_{\textrm{int}}}=e^{2\varphi(x)}\hat{g}_{mn}dy^{m}dy^{n},$
(6.43)

where $\varphi\rightarrow 0$ at infinity. To connect with our earlier
discussion, note that $\phi=2(D-4)\varphi^{(1)}$ where $\phi$ is defined in
Lemma 4. Reducing to four dimensions gives an effective action of the form,

$S={1\over 16\pi G}\int d^{4}x\,e^{(D-4)\varphi}\sqrt{-g}R+\ldots,$ (6.44)

where the omitted terms involve scalar and vector fields whose kinetic terms
typically depend on $\varphi$. Our analysis in terms of $\hat{g}$ gives
formulae for memory in this frame. To convert to Einstein frame with a
canonical Einstein-Hilbert action, we need to perform one conformal
transformation and use the relations described in section 1.1. The Einstein
frame metric is defined by

$\displaystyle g_{\mu\nu}^{(E)}$ $\displaystyle=e^{(D-4)\varphi}\,g_{\mu\nu},$
$\displaystyle=\left(1+(D-4)\frac{\varphi^{(1)}}{r}+\cdots\right)g_{\mu\nu}=\eta_{\mu\nu}+\frac{h_{\mu\nu}^{(1)}}{r}+(D-4)\frac{\varphi^{(1)}}{r}\eta_{\mu\nu}+\ldots,$
(6.45)
$\displaystyle=\eta_{\mu\nu}+\frac{h_{\mu\nu}^{(1)}}{r}+\frac{1}{2}\frac{\phi}{r}\eta_{\mu\nu}+\ldots.$
(6.46)

Therefore the leading order metric in Einstein frame is

$h_{\mu\nu}^{(1;E)}=h_{\mu\nu}^{(1)}+\frac{1}{2}\phi\eta_{\mu\nu}$ (6.47)

and so the Einstein news tensor is

$\mathcal{N}_{AB}^{(E)}=\mathcal{N}_{AB}-\frac{1}{2}Nq_{AB}=N_{AB}.$ (6.48)

Thus the Einstein news tensor is equivalent to the trace-free Bondi news
tensor – an Einstein frame observer is insensitive to the overall breathing
mode as we expect [24]. The components of electromagnetic and scalar radiative
degrees of freedom are unchanged:

$\mathcal{N}^{(E)}_{Am}=\mathcal{N}_{Am}\;\text{ and
}\;\mathcal{N}^{(E)}_{mn}=\mathcal{N}_{mn}.$ (6.49)

The memory effects as viewed by such an Einstein frame observer are then given
by

$\Delta^{(E)}_{AB}=\Delta_{AB}-\frac{1}{2}q_{AB}\left(q^{CD}\Delta_{CD}\right),\quad\Delta_{Am}^{(E)}=\Delta_{Am}\;\text{
and }\;\Delta_{mn}^{(E)}=\Delta_{mn}.$ (6.50)

## Acknowledgements

We would like to thank Mark Stern and Robert Wald for helpful discussions. C.
F. and S. S. are supported in part by NSF Grant No. PHY1720480 and PHY2014195.
C. F. is also supported by U.S. Department of Energy grant DE-SC0009999 and by
funds from the University of California. G. S. is supported in part by NSF
grants PHY 15-05124 and PHY18-04216 to the University of Chicago.

##  Appendix A Asymptotic Expansion of Einstein’s Equations

In this Appendix, we collect some technical results regarding the asymptotic
Einstein equations and the decay of certain components of the Ricci tensor
that will be used ubiquitously in this paper. To simplify our analysis we
assume that the metric is in the gauge described by Lemma 4.

###  A.1 Constraints on the asymptotic expansion

It is more convenient for our analysis to examine the trace-reversed Einstein
equations given by,

$R_{MN}=8\pi\mathcal{T}_{MN},$ (A.1)

where $\mathcal{T}_{MN}$ is the trace-reversed stress tensor:

$\mathcal{T}_{MN}=T_{MN}-\frac{1}{D-2}g_{MN}\left(g^{PQ}T_{PQ}\right).$ (A.2)

It is useful to split the Ricci tensor into a linear and nonlinear part using
the metric split $\hat{g}_{MN}+h_{MN}$ for some chosen $\hat{g}$. We define
the nonlinear part of the Ricci tensor as

$\mathcal{R}_{MN}\equiv R_{MN}-\widetilde{R}_{MN},$ (A.3)

where $R_{MN}$ is the Ricci tensor and $\widetilde{R}_{MN}$ is the linearized
Ricci tensor defined below:

$\widetilde{R}_{MN}\equiv-{1\over
2}\left(\Box_{\hat{g}}h_{MN}+2\hat{R}_{M}{}^{P}{}_{N}{}^{Q}h_{PQ}-2\hat{\nabla}_{(M}\hat{\nabla}^{P}h_{N)P}+\hat{\nabla}_{M}\hat{\nabla}_{N}h\right).$
(A.4)

On the right hand side, all differential operators along with Riemann are
defined with respect to $\hat{g}$. In the Appendices, we will denote the
linearized version of objects with a tilde, just as $\widetilde{R}_{MN}$ is
the linear part of $R_{MN}$.

In our analysis we defined $\hat{g}$ in $(\ref{etabondi})$ while $h_{MN}$ is
given by the collection of functions $(h_{\mu\nu},A_{\mu n},\varphi_{mn})$
appearing in $(\ref{ansatz})$. We will expand $(\ref{treversed})$ to find a
series of recursion relations of the form: (linearized Ricci) = (stress-
energy) - (non-linear Ricci). We find the following relations:

$\displaystyle[\mathscr{D}^{2}+(n-1)(n-2)]h_{uu}^{(n-1)}+2(n-1)\partial_{u}h_{uu}^{(n)}+\bm{D}^{2}h_{uu}^{(n+1)}+\partial_{u}^{2}(h^{(n+1)}+\phi^{(n+1)})$
(A.5)
$\displaystyle-2\partial_{u}\psi_{u}^{(n+1)}=-16\pi\mathcal{T}_{uu}^{(n+1)}+2\mathcal{R}_{uu}^{(n+1)},$
(A.6)
$\displaystyle[\mathscr{D}^{2}+n(n-3)]h_{ur}^{(n-1)}+2h_{uu}^{(n-1)}-2\mathscr{D}^{A}h_{Au}^{(n-1)}+2(n-1)\partial_{u}h_{ur}^{(n)}+\bm{D}^{2}h_{ur}^{(n+1)}+n\psi_{u}^{(n)}$
$\displaystyle-\partial_{u}\psi_{r}^{(n+1)}-n\partial_{u}(h^{(n)}+\phi^{(n)})=-16\pi\mathcal{T}_{ru}^{(n+1)}+2\mathcal{R}_{ur}^{(n+1)},$
(A.7)
$\displaystyle[\mathscr{D}^{2}+(n-1)(n-2)-1]h_{uA}^{(n-1)}-2\mathscr{D}_{A}(h_{uu}^{(n-1)}-h_{ur}^{(n-1)})+2(n-1)\partial_{u}h_{uA}^{(n)}+\bm{D}^{2}h_{uA}^{(n+1)}$
$\displaystyle-\mathscr{D}_{A}\psi_{u}^{(n)}-\partial_{u}\psi_{A}^{(n+1)}+\mathscr{D}_{A}\partial_{u}(h^{(n)}+\phi^{(n)})=-16\pi\mathcal{T}_{uA}^{(n+1)}+2\mathcal{R}_{uA}^{(n+1)},$
(A.8)
$\displaystyle[\mathscr{D}^{2}+(n-1)(n-2)-4]h_{rr}^{(n-1)}+4h_{ur}^{(n-1)}+2q^{AB}h_{AB}^{(n-1)}-4\mathscr{D}^{A}h_{Ar}^{(n-1)}+2(n-1)\partial_{u}h_{rr}^{(n)}$
$\displaystyle+\bm{D}^{2}h_{rr}^{(n+1)}+2n\psi_{r}^{(n)}+n(n-1)(h^{(n-1)}+\phi^{(n-1)})=-16\pi\mathcal{T}_{rr}^{(n+1)}+2\mathcal{R}_{rr}^{(n+1)},$
(A.9)
$\displaystyle[\mathscr{D}^{2}+(n-1)(n-2)-5]h_{rA}^{(n-1)}+4h_{uA}^{(n-1)}-2\mathscr{D}_{A}(h_{ur}^{(n-1)}-h_{rr}^{(n-1)})-2\mathscr{D}^{B}h_{BA}^{(n-1)}+\bm{D}^{2}h_{rA}^{(n+1)}$
$\displaystyle+2(n-1)\partial_{u}h_{rA}^{(n)}-\mathscr{D}_{A}\psi_{r}^{(n)}+n\psi_{A}^{(n)}-(n-1)\mathscr{D}_{A}(h^{(n-1)}+\phi^{(n-1)})$
(A.10)
$\displaystyle=-16\pi\mathcal{T}_{rA}^{(n+1)}+2\mathcal{R}_{rA}^{(n+1)},$
(A.11)
$\displaystyle[\mathscr{D}^{2}+(n-1)(n-2)-2]h_{AB}^{(n-1)}-4\mathscr{D}_{(A}h_{B)u}^{(n-1)}+4\mathscr{D}_{(A}h_{B)r}^{(n-1)}+2(n-1)\partial_{u}h_{AB}^{(n)}+\bm{D}^{2}h_{AB}^{(n+1)}$
$\displaystyle-2\mathscr{D}_{(A}\psi_{B)}^{(n)}-2(\psi_{r}^{(n)}-\psi_{u}^{(n)})q_{AB}+(\mathscr{D}_{A}\mathscr{D}_{B}-(n-1)q_{AB})(h^{(n-1)}+\phi^{(n-1)})$
$\displaystyle-
q_{AB}\partial_{u}(h^{(n)}+\phi^{(n)})+2(h_{rr}^{(n-1)}-2h_{ur}^{(n-1)}+h_{uu}^{(n-1)})q_{AB}=-16\pi\mathcal{T}_{AB}^{(n+1)}+2\mathcal{R}_{AB}^{(n+1)},$
(A.12)
$\displaystyle[\mathscr{D}^{2}+(n-1)(n-2)]A_{um}^{(n-1)}+2(n-1)\partial_{u}A_{um}^{(n)}+\bm{D}^{2}A_{um}^{(n+1)}-\bm{D}_{m}\psi_{u}^{(n+1)}-\partial_{u}\psi_{m}^{(n+1)}$
$\displaystyle+\bm{D}_{m}\partial_{u}(h^{(n+1)}+\phi^{(n+1)})=-16\pi\mathcal{T}_{um}^{(n+1)}+2\mathcal{R}_{um}^{(n+1)},$
(A.13)
$\displaystyle[\mathscr{D}^{2}+n(n-3)]A_{rm}^{(n-1)}+2A_{um}^{(n-1)}-2\mathscr{D}^{A}A_{Am}^{(n-1)}+2(n-1)\partial_{u}A_{rm}^{(n)}+\bm{D}^{2}A_{rm}^{(n+1)}+n\psi_{m}^{(n)}$
$\displaystyle-\bm{D}_{m}\psi_{r}^{(n+1)}-n\bm{D}_{m}(h^{(n)}+\phi^{(n)})=-16\pi\mathcal{T}_{rm}^{(n+1)}+2\mathcal{R}_{rm}^{(n+1)},$
(A.14)
$\displaystyle[\mathscr{D}^{2}+(n-1)(n-2)-1]A_{Am}^{(n-1)}-2\mathscr{D}_{A}(A_{um}^{(n-1)}-A_{rm}^{(n-1)})+2(n-1)\partial_{u}A_{Am}^{(n)}+\bm{D}^{2}A_{Am}^{(n+1)}$
$\displaystyle-\mathscr{D}_{A}\psi_{m}^{(n)}-\bm{D}_{m}\psi_{A}^{(n+1)}+\bm{D}_{m}\mathscr{D}_{A}(h^{(n)}+\phi^{(n)})=-16\pi\mathcal{T}_{Am}^{(n+1)}+2\mathcal{R}_{Am}^{(n+1)},$
(A.15)
$\displaystyle[\mathscr{D}^{2}+(n-1)(n-2)]\varphi_{mn}^{(n-1)}+2(n-1)\partial_{u}\varphi_{mn}^{(n)}+\bm{D}^{2}\varphi_{mn}^{(n+1)}+2\mathcal{R}_{m}{}^{p}{}_{n}{}^{q}\varphi_{pq}^{(n+1)}-2\bm{D}_{(m}\psi_{n)}^{(n+1)}$
$\displaystyle+\bm{D}_{m}\bm{D}_{n}(h^{(n+1)}+\phi^{(n+1)})=-16\pi\mathcal{T}_{mn}^{(n+1)}+2\mathcal{R}_{mn}^{(n+1)}.$
(A.16)

Here we have defined

$\psi_{M}\equiv\partial^{N}h_{NM},\qquad
h^{(n)}\equiv\eta^{\mu\nu}h_{\mu\nu}^{(n)},\qquad\phi^{(n)}\equiv\hat{g}^{mn}\varphi_{mn}^{(n)},$
(A.17)

so that

$\psi_{u}^{(n)}=\mathscr{D}^{A}h_{Au}^{(n-1)}+(3-n)(h_{ur}^{(n-1)}-h_{uu}^{(n-1)})-\partial_{u}h_{ur}^{(n)}+\bm{D}^{m}A_{um}^{(n)},$
(A.18)
$\psi_{r}^{(n)}=\mathscr{D}^{A}h_{Ar}^{(n-1)}+(3-n)(h_{rr}^{(n-1)}-h_{ur}^{(n-1)})-q^{AB}h_{AB}^{(n-1)}-\partial_{u}h_{rr}^{(n)}+\bm{D}^{m}A_{rm}^{(n)},$
(A.19)
$\psi_{A}^{(n)}=\mathscr{D}^{B}h_{BA}^{(n-1)}+(4-n)(h_{rA}^{(n-1)}-h_{uA}^{(n-1)})-\partial_{u}h_{rA}^{(n)}+\bm{D}^{m}A_{Am}^{(n)},$
(A.20)
$\psi_{m}^{(n)}=\mathscr{D}^{A}A_{Am}^{(n-1)}+(3-n)(A_{rm}^{(n-1)}-A_{um}^{(n-1)})-\partial_{u}A_{rm}^{(n)}+\bm{D}^{n}\varphi_{nm}^{(n)}\,.$
(A.21)

In the body of this work, we will need the expansion of Einstein’s equations
to order ${1\over r^{2}}$, and to order ${1\over r^{3}}$ for the special case
of a stationary era.

A direct calculation of $\mathcal{R}_{MN}^{(2)}$ in the gauge of Lemma 4 shows
that the non-vanishing components of $\mathcal{R}_{MN}^{(2)}$ can be written
entirely in terms of the news eq. 4.46. Explicitly the non-vanishing
components of $\mathcal{R}_{MN}^{(2)}$ are given by,

$\displaystyle\mathcal{R}_{uu}^{(2)}=$
$\displaystyle-\frac{1}{4}\mathcal{N}^{ab}\mathcal{N}_{ab}+\frac{1}{2}\partial_{u}\big{(}h^{(1)}_{ab}\mathcal{N}^{ab}\big{)},$
(A.22) $\displaystyle\mathcal{R}^{(2)}_{um}=$
$\displaystyle\frac{1}{4}(\bm{D}_{m}\Phi_{pq})\mathcal{N}^{pq}-\frac{1}{2}\Phi^{pn}\bm{D}_{p}\mathcal{N}_{mn}-\frac{1}{2(D-4)}\phi\bm{D}^{n}\mathcal{N}_{nm}+\frac{1}{2}\bm{D}_{m}(\Phi^{np}\mathcal{N}_{np}),$
(A.23) $\displaystyle\mathcal{R}_{mn}^{(2)}=$
$\displaystyle-\frac{1}{4}(\bm{D}_{(m}\Phi^{pq})(\bm{D}_{n)}\Phi_{pq})+(\bm{D}^{p}\Phi^{q}{}_{m})(\bm{D}_{[p}\Phi_{q]n})+\frac{1}{2}\Phi^{pq}\bm{D}_{p}\bm{D}_{q}\Phi_{mn}$
(A.24)
$\displaystyle+\frac{1}{2(D-4)}\phi\bm{D}^{2}\Phi_{mn}+\frac{1}{4}\bm{D}_{m}\bm{D}_{n}(\Phi^{pq}\Phi_{pq}),$

where the product in eq. A.22 is explicitly given by

$h_{ab}^{(1)}\mathcal{N}^{ab}=h^{(1)}_{AB}\mathcal{N}^{AB}+A_{Am}^{(1)}\mathcal{N}^{Am}+\Phi_{mn}\mathcal{N}^{mn}+\frac{1}{D-4}\,\phi\,\hat{g}^{mn}\mathcal{N}_{mn},$
(A.25)

and the scalars $\Phi_{mn}(u,\theta,y)$ and $\phi(u,\theta)$ are defined in
Lemma 4. The remaining components of $\mathcal{R}_{MN}^{(2)}$ vanish. In
Section 5.2, the zero modes of the nonlinear parts of the Ricci tensor appear
as “flux” terms for the change in metric. More precisley, we find that the
zero modes of $\mathcal{R}_{uu}^{(2)}$ and $\mathcal{R}_{um}^{(2)}$ determine
the change in the metric due to a burst of radiation. The zero mode of eq.
A.22 is manifestly non-vanishing unless $\mathcal{N}_{ab}=0$. To determine the
zero mode of $\mathcal{R}_{um}^{(2)}$ we contract with a Killing vector
$\overline{V}^{m}$ of $(\mathcal{M}_{\textrm{int}},\hat{g}_{mn})$ and
integrate over $\mathcal{M}_{\textrm{int}}$:

$\displaystyle\int_{\mathcal{M}_{\textrm{int}}}\mathcal{R}_{um}\overline{V}^{m}=$
$\displaystyle\frac{1}{4}\int_{\mathcal{M}_{\textrm{int}}}\bigg{[}\mathcal{N}^{pq}(\overline{V}^{m}\bm{D}_{m}\Phi_{pq})$
(A.26)
$\displaystyle-2\bm{D}_{p}\bigg{(}\Phi^{pn}\mathcal{N}_{mn}\overline{V}^{m}+\frac{\phi\mathcal{N}^{p}{}_{m}\overline{V}^{m}}{D-4}-\overline{V}^{p}\Phi^{mn}\mathcal{N}_{mn}\bigg{)}\bigg{]},$
(A.27) $\displaystyle=$
$\displaystyle\frac{1}{4}\int_{\mathcal{M}_{\textrm{int}}}\mathcal{N}^{mn}\pounds_{\bar{V}}\Phi_{mn},$
(A.28)

where in the first line we used the fact that $\Phi_{pq}$ is divergence free,
$\phi$ is constant on $\mathcal{M}_{\textrm{int}}$ and that $\bar{V}^{m}$ is
covariantly constant to write the last three terms in eq. A.23 as a total
derivative. In the second line we used the fact that $\bar{V}^{m}$ is
covariantly constant to write the directional derivative in terms of the Lie
derivative. However the decomposition theorem of [57] states that
$\mathcal{M}_{\textrm{int}}$ is a free quotient of a Riemannian product of a
torus and a connected Ricci-flat space with vanishing $b_{1}$. For such a
product, $\pounds_{\overline{V}}\Phi_{mn}=0$ since $\overline{V}$ is one of
the torus isometries.

At this stage, we want to check whether our ansatz $(\ref{ansatz})$ of an
expansion in powers of ${1\over r}$ makes sense as an asymptotic expansion.
This might seem fairly reasonable because in both pure gravity and Maxwell-
Einstein, there exists a large class of solutions which are smooth at
$\mathscr{I}^{+}$ in a particular gauge [68].121212Note that starting with
smooth initial data on a Cauchy surface and evolving that data does not
generically lead to a solution with an analytic expansion in ${1\over r}$ near
$\mathscr{I}^{+}$. Rather $\log(r)$ terms can be generated at subleading
orders in ${1\over r}$ even in pure gravity [69]. However, there exists a
class of initial data in pure gravity that guarantee $C^{k}$ differentiability
at $\mathscr{I}^{+}$ for any $k$ [68].  However, this is not the case for a
scalar field in four dimensions with null sources [65]. A scalar field $\phi$
in Minkowski spacetime satisfying

$\Box_{\eta}\phi=J,$ (A.29)

where $J$ is a source, does not admit a ${1\over r}$ expansion near
$\mathscr{I}^{+}$ when $J\sim{1\over r^{2}}$, which is a configuration with
finite flux through $\mathscr{I}^{+}$. Rather one must include
$\frac{\log(r)}{r^{n}}$ terms in the expansion. This is without dynamical
gravity.

In our case, there is a general obstruction to integrating in from
$\mathscr{I}^{+}$. Namely, if a specific scalar fluctuation of
$\mathcal{M}_{\textrm{int}}$ is obstructed, or equivalently gets a mass at
some order beyond the linearized approximation, then our ansatz is simply not
valid for that mode. The mode could never propagate to $\mathscr{I}^{+}$,
which we implicitly assume in our ansatz. We can see this obstruction emerge
in the ${1\over r}$ expansion. Consider the $mn$ component of the vacuum
Einstein’s equations at order ${1\over r^{2}}$, i.e., section A.1 for $n=1$
and $\mathcal{T}_{mn}^{(2)}=0$:

$\bm{D}^{2}\varphi_{mn}^{(2)}+2\mathcal{R}_{m}{}^{p}{}_{n}{}^{q}\varphi_{pq}^{(2)}-2\bm{D}_{(m}\psi^{(3)}_{n)}+\bm{D}_{m}\bm{D}_{n}(h^{(3)}+\phi^{(3)})=2\mathcal{R}_{mn}^{(2)}.$
(A.30)

After contracting both sides with a tensor field $t^{mn}(y)$ which is
annihilated by Lichnerowicz, it is straightforward to check that the right
hand side vanishes. We therefore get the following nonlinear obstruction to
our ansatz,

$\int_{\mathcal{M}_{\textrm{int}}}t^{mn}\mathcal{R}_{mn}^{(2)}=0.$ (A.31)

It is straightforward to check that the volume mode, as expected, is
unobstructed. Letting $t_{mn}=\hat{g}_{mn}(y)$ in eq. A.31 and using eq. A.24
gives

$\hat{g}^{mn}\mathcal{R}_{mn}^{(2)}=\bigg{(}\frac{1}{4}\bm{D}^{m}\Phi^{pq}\bm{D}_{m}\Phi_{pq}-\frac{1}{2}\bm{D}^{p}\Phi^{qm}\bm{D}_{q}\Phi_{pm}+\frac{1}{4}\bm{D}^{2}\Phi^{2}\bigg{)},$
(A.32)

where $\Phi^{2}=\Phi_{mn}\Phi^{mn}$. Integrating over
$\mathcal{M}_{\textrm{int}}$,

$\displaystyle\int_{\mathcal{M}_{\textrm{int}}}\hat{g}^{mn}\mathcal{R}_{mn}^{(2)}=$
$\displaystyle\frac{1}{2}\int_{\mathcal{M}_{\textrm{int}}}\bigg{(}\frac{1}{2}\bm{D}^{m}\Phi^{pq}\bm{D}_{m}\Phi_{pq}-\bm{D}^{p}\Phi^{qm}\bm{D}_{q}\Phi_{pm}\bigg{)},$
(A.33) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathcal{M}_{\textrm{int}}}\bigg{(}-\frac{1}{2}\Phi^{pq}\bm{D}^{2}\Phi_{pq}+\Phi^{qm}\bm{D}^{p}\bm{D}_{q}\Phi_{pm}\bigg{)},$
(A.34) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathcal{M}_{\textrm{int}}}\bigg{(}\mathcal{R}^{mpnq}\Phi_{mn}\Phi_{pq}-\mathcal{R}^{mpnq}\Phi_{mn}\Phi_{pq}+\Phi^{qm}\bm{D}_{q}\bm{D}^{p}\Phi_{pm}\bigg{)},$
(A.35) $\displaystyle=$ $\displaystyle~{}0,$ (A.36)

where we have used

$\bm{D}^{2}\Phi_{mn}+2\mathcal{R}_{m}{}^{p}{}_{n}{}^{q}\Phi_{pq}=0,$

and that $\Phi_{mn}$ is divergence-free. As we spelled out in section 2.2, the
space of exactly massless modes ${\hat{d}_{\textrm{L}}}\leq{d_{\textrm{L}}}$
is smaller than the kernel of Lichnerowicz. The exactly massless volume-
preserving moduli satisfy eq. A.31. Thus, as in Lemma 4, we truncate the
linearized massless moduli to exactly massless moduli and obtain a solution
consistent with our ansatz and Einstein’s equations at order ${1\over r^{2}}$.
As we will see in section A.3, this truncation also ensures that our ansatz is
consistent with Einstein’s equations at order ${1\over r^{3}}$. We fully
expect that restricting to exactly massless modes is necessary to obtain a
solution to Einstein’s equations to all orders in ${1\over r}$, however we
have not attempted to show this here. Note that this discussion motivates our
imposing a similar condition on $T_{mn}^{(2)}$; namely, that $T_{mn}^{(2)}$ be
orthogonal to the ${\hat{d}_{\textrm{L}}}+1$ exactly massless scalar modes.

###  A.2 Going to the stationary era gauge

We now want to show that a metric in the gauge of Lemma 4 can be further
restricted at order ${1\over r^{2}}$ in a stationary era. Specifically,

$\displaystyle h^{(2)}_{\mu\nu}=\overline{h}^{(2)}_{\mu\nu}(\theta),\qquad$
$\displaystyle A^{(2)}_{\mu
m}=\sum_{i=1}^{b_{1}}A_{\mu}^{(2;i)}(\theta)\otimes\overline{V}_{m}^{(i)}(y^{m})$
(A.37)

and

$\varphi_{mn}^{(2)}=\Phi^{(2)}_{mn}(\theta^{A},y^{m})+\bigg{(}\bm{D}_{m}\bm{D}_{n}-\frac{\hat{g}_{mn}}{d-4}\bm{D}^{2}\bigg{)}\Psi^{(2)}(\theta^{A},y^{m})+\frac{\hat{g}_{mn}}{d-4}\overline{\phi}^{(2)}(\theta).$
(A.38)

Note that $\varphi_{mn}^{(2)}$ is missing a vector term shown in Proposition
2, and $\overline{\phi}^{(2)}$ is constant on $\mathcal{M}_{\textrm{int}}$. To
achieve this gauge we first make a gauge transformation that is compatible
with our ansatz $(\ref{ansatz})$, stationarity conditions and Lemma 4. We
choose a gauge vector field of the form,

$\xi_{M}\sim\frac{\xi_{M}^{(2)}(\theta,y)}{r^{2}}+O\bigg{(}\frac{1}{r^{3}}\bigg{)},$
(A.39)

where $\xi_{M}$ is a $u$-independent gauge transformation. By an analysis
similar to the proof of Lemma 4 we see that $\bm{D}^{m}A^{(2)}_{\mu m}=0$ is
divergence free and $\varphi^{(2)}_{mn}$ admits the decomposition given in eq.
A.38. In a stationary era, $\mathcal{R}^{(2)}_{\mu\nu}=\mathcal{R}^{(2)}_{\mu
n}$ and $T_{MN}^{(2)}=0$. Therefore:

$(\mu\nu;2)$ $\displaystyle\quad\quad\quad\bm{D}^{2}h_{\mu\nu}^{(2)}=0,$
(A.40) $(\mu m;2)$ $\displaystyle\quad\quad\quad\bm{D}^{2}A_{\mu m}^{(2)}=0,$
(A.41) $(mn;2)$
$\displaystyle\quad\quad\quad\bm{D}^{2}\varphi_{mn}^{(2)}+2\mathcal{R}_{m}{}^{p}{}_{n}{}^{q}\varphi_{pq}^{(2)}+\bm{D}_{m}\bm{D}_{n}(-2h_{ur}^{(2)}+h_{rr}^{(2)}+q^{AB}h_{AB}^{(2)})$
(A.42)
$\displaystyle\quad\quad\quad-2\bm{D}_{(m}\bm{D}^{p}\varphi^{(2)}_{n)p}=2\mathcal{R}_{mn}^{(2)}.$

We conclude that

$h_{\mu\nu}^{(2)}=\overline{h}^{(2)}_{\mu\nu}(u,\theta)\quad\textrm{ and
}\quad A^{(2)}_{\mu
m}=\sum_{i=1}^{b_{1}}A_{\mu}^{(2)}(u,\theta)\otimes\overline{V}_{m}(y).$
(A.43)

Using these relations we now study $(mn;2)$. Taking the trace of $(mn;2)$
gives131313Just to remind the reader, $\Phi_{mn}$ without a superscript
denotes the leading order term as in $(\ref{radgauge})$.

$-2\bm{D}^{m}\bm{D}^{n}\varphi^{(2)}_{mn}=\frac{1}{2}\bm{D}^{m}\Phi^{pq}\bm{D}_{m}\Phi_{pq}-\bm{D}^{m}\Phi^{pq}\bm{D}_{p}\Phi_{mq}+\frac{1}{2}\bm{D}^{2}(\Phi^{pq}\Phi_{pq}),$
(A.44)

which yields the following equation for $\Psi^{(2)}$:

$\bigg{(}\frac{D-5}{D-4}\bigg{)}\bm{D}^{4}\Psi^{(2)}=-\frac{1}{4}\bm{D}^{m}\Phi^{pq}\bm{D}_{m}\Phi_{pq}+\frac{1}{2}\bm{D}^{m}\Phi^{pq}\bm{D}_{p}\Phi_{mq}-\frac{1}{4}\bm{D}^{2}(\Phi^{pq}\Phi_{pq}).$
(A.45)

We note that the above analysis implies that the right hand side has no zero
modes and therefore, we can solve for $\Psi^{(2)}$ in terms of $\Phi_{mn}$.
After solving for $\Psi^{(2)}$ we can then solve for $\Phi_{mn}^{(2)}$:

$\displaystyle
L[\Phi^{(2)}_{mn}]=-L[\mathfrak{D}_{mn}\Psi^{(2)}]+2\bigg{(}\frac{D-5}{D-4}\bigg{)}\bm{D}_{m}\bm{D}_{n}\bm{D}^{2}\Psi^{(2)}-\frac{1}{4}(\bm{D}_{(m}\Phi^{pq})(\bm{D}_{n)}\Phi_{pq})$
(A.46)
$\displaystyle+(\bm{D}^{p}\Phi^{q}{}_{m})(\bm{D}_{[p}\Phi_{q]n})+\frac{1}{2}\Phi^{pq}\bm{D}_{p}\bm{D}_{q}\Phi_{mn}+\frac{1}{4}\bm{D}_{m}\bm{D}_{n}(\Phi^{pq}\Phi_{pq})+\frac{\phi\bm{D}^{2}\Phi_{mn}}{2(D-4)}.$

Here $L[\cdot]$ is the Lichnerowicz operator and
$\mathfrak{D}_{mn}\equiv\big{(}\bm{D}_{m}\bm{D}_{n}-\frac{\hat{g}_{mn}}{D-4}\bm{D}^{2}\big{)}$.
As in our discussion of section A.1, we again truncate to exactly massless
scalar fluctuations for which the right hand side of eq. A.46 has no
Lichnerowicz zero modes. This guarantees solvability of eq. A.46. On a generic
Ricci flat manifold, $\psi^{(2)}$ will not be harmonic and $\Phi_{mn}^{(2)}$
does not satisfy the Lichnerowicz equation. In the special case of
$\mathcal{M}_{\textrm{int}}=\mathbb{T}^{k}$, we see that
$\bm{D}_{m}\Phi_{pq}=0$ and

$\bm{D}^{2}\Psi^{(2)}=0\quad\textrm{ $\implies$
}\quad\bm{D}^{2}\Phi^{(2)}_{mn}+2\mathcal{R}_{m}{}^{p}{}_{n}{}^{q}\Phi^{(2)}_{pq}=0\,\,\text{
for }\mathcal{M}_{\textrm{int}}=\mathbb{T}^{k}.$ (A.47)

###  A.3 Ricci in a stationary era

The last result we want to record is the behavior of the nonlinear part of the
Ricci tensor at order ${1\over r^{3}}$. By a lengthy but straightforward
calculation, the following components of the nonlinear part of the Ricci
tensor vanish in a stationary era and in our gauge at order ${1\over r^{3}}$ :

$\mathcal{R}_{\mu\nu}^{(3)}=0\quad\textrm{ and
}\quad\mathcal{R}_{um}^{(3)}=0\textrm{ in a stationary era},$ (A.48)

and the nonvanishing components are

$\displaystyle\mathcal{R}_{rm}^{(3)}=-\bm{D}_{m}(\Phi^{pq}\Phi_{pq})+\frac{1}{2}\bm{D}_{p}(\Phi^{pq}\Phi_{qm})\quad\textrm{
in a stationary era},$ (A.49)
$\displaystyle\mathcal{R}_{Am}^{(3)}=\frac{1}{4}\bm{D}_{m}\mathscr{D}_{A}(\Phi^{pq}\Phi_{pq})-\frac{1}{2}\bm{D}_{p}(\Phi^{pq}\mathscr{D}_{A}\Phi_{mq})\quad\textrm{
in a stationary era}.$ (A.50)

Finally, the $\mathcal{R}_{mn}^{(3)}$ component is given by

$\displaystyle\mathcal{R}_{mn}^{(3)}=$
$\displaystyle-\frac{1}{2}\bm{D}_{(m}\Phi^{pq}\bm{D}_{n)}\varphi^{(2)}_{pq}+(\bm{D}^{p}\Phi^{q}{}_{(m})(\bm{D}_{|p|}\varphi^{(2)}_{n)q})+\frac{1}{2}\bm{D}_{m}\bm{D}_{n}(\Phi_{pq}\varphi^{(2)pq})$
(A.51)
$\displaystyle-\bm{D}_{p}(\Phi^{pq}\bm{D}_{(m}\varphi^{(2)}_{n)q})+\frac{1}{2}\bm{D}_{p}(\Phi^{pq}\bm{D}_{q}\varphi_{mn}^{(2)})+\frac{1}{2}\bm{D}_{s}\big{[}\Phi^{s}{}_{P}\Phi^{Pq}\Xi_{mnq}\big{]}$
$\displaystyle-\frac{1}{2}\bm{D}_{m}\big{[}\Phi^{s}{}_{P}\Phi^{Pq}\Xi_{nsq}\big{]}-\frac{1}{2}\hat{g}^{kq}\Phi^{ls}\Xi_{msq}\Xi_{kls}+\frac{1}{2}\hat{g}^{kq}\Phi^{ls}\Xi_{lnq}\Xi_{kms}$
$\displaystyle+\textrm{non-zero modes},$

where $\Xi_{mrq}\equiv 2\bm{D}_{(m}\Phi_{r)q}-\bm{D}_{q}\Phi_{mr}$ and ‘non-
zero modes’ refers to modes orthogonal to the Lichnerowicz zero modes. Again
this obstruction to solving Einstein’s equations is generically non-trivial
for a Ricci-flat space, but
$\int_{\mathcal{M}_{\textrm{int}}}t^{mn}\mathcal{R}_{mn}^{(3)}=0$ if $t_{mn}$
is an exactly massless fluctuation, and hence the obstruction vanishes. Note
that for the special case of $\mathcal{M}_{\textrm{int}}=\mathbb{T}^{k}$,
$\mathcal{R}_{mn}^{(3)}=0$.

##  Appendix B A Gauge Invariant Derivation of Memory in Linearized Gravity
with Compact Extra Dimensions

In this section we will derive the memory effect in linearized gravity for
isolated systems with compact extra dimensions using the Bianchi identity. In
particular we shall assume, in any neighborhood of null infinity, there exists
a gauge in which the metric admits an asymptotic expansion of the form (3.5).
We now derive the memory effect in a manifestly gauge invariant way using the
Bianchi identity for the asymptotic Weyl tensor. Since we shall be working
with gauge invariant quantities, we shall only need that the expansion (3.5)
is valid in any local neighborhood of null infinity.

We denote the linearized Weyl tensor by $\widetilde{C}_{MNPQ}$. The linearized
Bianchi identity is

$\partial_{[M}\widetilde{C}_{NP]QR}=0.$ (B.1)

The linearized electric Weyl tensor is defined as

$\widetilde{E}_{PR}\equiv\widetilde{C}_{NPQR}n^{N}n^{Q},$ (B.2)

where $n^{N}\equiv(\partial/\partial u)^{N}$. Lemma 3 applies to the leading
order linearized electric Weyl tensor, which has non-vanishing components
$\widetilde{\mathcal{E}}_{AB}$ and $\widetilde{\mathcal{E}}_{Am}$ that are
harmonic on $\mathcal{M}_{\textrm{int}}$. The component
$\widetilde{\mathcal{E}}_{mn}$ satisfies the Lichnerowicz equation on
$\mathcal{M}_{\textrm{int}}$. Finally, we again have that
$q^{AB}\widetilde{\mathcal{E}}_{AB}=\hat{g}^{mn}\widetilde{\mathcal{E}}_{mn}$.

We now compute the memory effect from the Bianchi identity. We recall that

$\widetilde{\Delta}_{MN}=\int_{-\infty}^{\infty}du^{\prime}\int_{-\infty}^{u^{\prime}}du^{\prime\prime}\,\widetilde{\mathcal{E}}_{MN}.$
(B.3)

We start with the scalar memory effect. Since $\widetilde{\Delta}_{mn}$
satisfies the Lichnerowicz equation we can expand $\widetilde{\Delta}_{mn}$ as

$\widetilde{\Delta}_{mn}=\sum_{i=1}^{d_{L}}\widetilde{\Delta}^{(i)}T_{mn}^{(i)}+\frac{1}{D-4}\hat{g}_{mn}\hat{g}^{pq}\widetilde{\Delta}_{pq},$
(B.4)

in terms of $d_{L}$ trace-free, divergence-free symmetric tensors
$T_{mn}^{(i)}$ which satisfy the Lichnerowicz equation. Note that $\hat{g}$ is
defined in $(\ref{hat_metric})$. We note that $\Delta\Phi^{(i)}$ and
$\Delta\phi$ in Theorem 1 are actually gauge invariant quantities and
therefore, the derivation of scalar memory is exactly analogous to the
derivation in the nonlinear theory:

$\widetilde{\Delta}^{(i)}=\frac{1}{2}\Delta\Phi^{(i)}\,\textrm{ and
}\,\hat{g}^{mn}\widetilde{\Delta}_{mn}=\frac{1}{2}\Delta\phi.$ (B.5)

For the scalar case, working with gauge invariant variables does not buy us
much.

To derive the electromagnetic memory effect, we note that an explicit
computation using the linearized metric yields

$\overline{\widetilde{C}_{\mu\nu\rho
m}}=\sum_{i=1}^{b_{1}}\partial_{\rho}F_{\nu\mu}^{(i)}(x^{\mu})\otimes
V_{m}^{(i)}(y^{m}),$ (B.6)

where the bar on the left hand side denotes a projection to zero modes as
described in section 1.5. Viewing the left hand side as a $1$-form in the
internal space, this means projecting to harmonic $1$-forms on
$\mathcal{M}_{\textrm{int}}$ in agreement with the expression on the right
hand side. $F^{(i)}_{\mu\nu}$ is the field strength for the graviphoton
associated to $V_{m}^{(i)}$. This field strength is now gauge invariant and
$\partial_{\mu}$ is the derivative operator compatible with the flat metric
$\eta_{\mu\nu}$.

Since the Weyl tensor is trace-free and satisfies the first Bianchi identity,
it follows that $F_{\mu\nu}^{(i)}$ satisfies

$\partial^{\mu}F_{\mu\nu}^{(i)}=0\textrm{ and
}\partial_{[\mu}F_{\nu\sigma]}^{(i)}=0$ (B.7)

for all $i$. We then expand $F_{\mu\nu}^{(i)}$ in powers of $1\over r$ near
null infinity as given by eq. 3.5. Using Lemma 4, the the only non-vanishing
component of $F_{\mu\nu}^{(i)}$ at order $1\over r$ is $F_{uA}^{(i;1)}$ which,
by eq. B.6, is directly related to $\widetilde{\mathcal{E}}_{Am}$ in the
following way,

$\widetilde{\mathcal{E}}_{Am}=-\sum_{i=1}^{b_{1}}\partial_{u}F_{uA}^{(i;1)}(u,\theta)\otimes\overline{V}_{m}^{(i)}(y^{m}).$
(B.8)

The divergence equation for $F_{\mu\nu}^{(i)}$ at order $1\over r^{2}$
constrains the angular divergence of $F_{uA}^{(i;1)}$,

$\mathscr{D}^{A}F_{uA}^{(i;1)}=\partial_{u}F_{ur}^{(i;2)}.$ (B.9)

Similarly, applying $\epsilon^{AB}$ the Bianchi identity for
$F_{\mu\nu}^{(i)}$ at order $1\over r^{2}$ yields

$\epsilon^{AB}\mathscr{D}_{A}F_{uB}^{(i;1)}=\partial_{u}\epsilon^{AB}F_{AB}^{(2;i)}.$
(B.10)

Therefore, using eqs. B.8 and B.3 we find that

$\epsilon^{AB}\mathscr{D}_{A}\widetilde{\Delta}_{B}^{(i)}=\Delta\left(\epsilon^{AB}F_{AB}^{(2;i)}\right)\quad\textrm{
and
}\quad\mathscr{D}^{A}\widetilde{\Delta}_{A}^{(i)}=\Delta\left(F_{ur}^{(2;i)}\right).$
(B.11)

On the right hand side, $\Delta$ means the change in the quantity from
$u=-\infty$ to $u=+\infty$.

Finally we turn to the gravitational memory effect arising from asymptotic
dimensional reduction. Using the fact that the Weyl tensor is divergence-free
and satisfies the homogeneous wave equation one can show that the zero mode of
$\widetilde{C}_{\mu\nu\rho\sigma}$ satisfies

$\partial_{[\mu}\overline{\widetilde{C}_{\nu\rho]\sigma\kappa}}=0.$ (B.12)

We first focus on the relevant equations for
$\overline{\widetilde{E}_{\mu\nu}}$. By analogous manipulations that led to
eqs. 4.5 and 4.1 we find that

$\partial^{\mu}\overline{\widetilde{E}_{\mu\nu}}=0\quad\textrm{ and
}\quad\Box_{\eta}\overline{\widetilde{E}_{\mu\nu}}=0.$ (B.13)

Therefore, the $\mathbb{R}^{4}$ components of the linearized electric Weyl
tensor satisfy the same equations as the components of the linearized electric
Weyl tensor in flat spacetime. One major difference is that, when one has
compact extra dimensions, $\eta^{\mu\nu}\widetilde{E}_{\mu\nu}$ is non-
vanishing. In flat spacetime this quantity does vanish but, in the presence of
compact extra dimensions, the tracelessness of the Weyl tensor implies that
$\eta^{\mu\nu}\widetilde{E}_{\mu\nu}$ vanishes if only if
$\hat{g}^{mn}\widetilde{E}_{mn}$ vanishes. This is a crucial difference that
leads to contributions from the breathing mode of $\mathcal{M}_{\textrm{int}}$
to the observed gravitational memory in this frame. We will discuss the choice
of frame in section 6. Because of this subtlety we shall explicitly derive the
memory effects implied by the system of equations given in eq. B.13.

We now expand $\widetilde{E}_{\mu\nu}$ in powers of $1\over r$. The explicit
recursion relations relating Weyl tensor components order by order in $1\over
r$ can be found in [65]. By Lemma 4 the only non-vanishing component of
$\widetilde{\mathcal{E}}_{\mu\nu}$ is $\widetilde{\mathcal{E}}_{AB}$. Since
the trace $q^{AB}\widetilde{\mathcal{E}}_{AB}$ is equivalent to
$-\hat{g}^{mn}\widetilde{\mathcal{E}}_{mn}$ we shall focus on the trace-free
part of $\widetilde{\mathcal{E}}_{AB}$ on the $2$-sphere. Applying
$q^{CA}\mathscr{D}_{A}$ to the angle component of the divergence equation in
eq. B.13 at order $1\over r^{2}$ yields

$\mathscr{D}^{A}\mathscr{D}^{B}\textrm{TF}\big{[}\widetilde{\mathcal{E}}_{AB}\big{]}=-\frac{1}{2}\mathscr{D}^{2}q^{AB}\widetilde{\mathcal{E}}_{AB}+\partial_{u}\mathscr{D}^{A}\overline{\widetilde{E}^{(2)}_{Ar}},$
(B.14)

where $\textrm{TF}\big{[}\cdot\big{]}$ takes a symmetric $2$-tensor on $S^{2}$
and projects out the trace: $T_{AB}\rightarrow T_{AB}-{1\over
2}q_{AB}\left(q^{CD}T_{CD}\right)$.

The $r$-component of the divergence equation in eq. B.13 at order $1\over
r^{3}$ gives,

$\mathscr{D}^{A}\overline{\widetilde{E}_{Ar}^{(2)}}=\overline{q^{AB}\widetilde{E}_{AB}^{(2)}}+\partial_{u}\overline{\widetilde{E}_{rr}^{(3)}}.$
(B.15)

Finally applying $q^{AB}$ to the angle-angle components of the wave equation
in eq. B.13 at order $1\over r^{3}$ gives

$[\mathscr{D}^{2}-2]q^{AB}\overline{\widetilde{\mathcal{E}}_{AB}}+2\partial_{u}q^{AB}\overline{\widetilde{E}_{AB}^{(2)}}=0.$
(B.16)

Equations B.14, B.15 and B.16 imply that

$\mathscr{D}^{A}\mathscr{D}^{B}\textrm{TF}\big{[}\overline{\widetilde{\mathcal{E}}_{AB}}\big{]}=[\mathscr{D}^{2}-1]\overline{\hat{g}^{mn}\widetilde{\mathcal{E}}}_{mn}+\partial_{u}^{2}\overline{\widetilde{E}_{rr}^{(3)}},$
(B.17)

where we used that fact that
$q^{AB}\widetilde{E}_{AB}=-\hat{g}^{mn}\widetilde{E}_{mn}$.

Equation B.17 constrains the scalar part of
$\textrm{TF}[\widetilde{\mathcal{E}}_{AB}]$ on the $2$-sphere. We now consider
the vector part. The vector part of the angle-angle components of memory are
determined by the magnetic Weyl tensor on $\mathbb{R}^{4}$ given by,

$\widetilde{B}_{\mu\nu}\equiv\frac{1}{2}\epsilon^{\rho\sigma}{}_{\mu}\widetilde{C}_{\rho\sigma\nu
u},$ (B.18)

where $\epsilon_{\mu\nu\rho}$ is the spatial volume form on $\mathbb{R}^{4}$
which is related to the volume element on $\mathbb{R}^{4}$ by
$\epsilon_{\mu\nu\rho}=\epsilon_{u\mu\nu\rho}$; indices are raised with the
background flat metric $\eta_{\mu\nu}$. The magnetic Weyl tensor is symmetric,
has vanishing $u$-components and, by the first Bianchi identity, is traceless:

$\widetilde{B}_{u\nu}=0,\quad\widetilde{B}_{\mu\nu}=\widetilde{B}_{\nu\mu}\quad\textrm{
and }\quad\eta^{\mu\nu}\widetilde{B}_{\mu\nu}=0.$ (B.19)

Furthermore, the linearized Bianchi identity and the fact that all components
of the linearized tensor satisfies the wave equation implies that

$\partial^{\mu}\widetilde{B}_{\mu\nu}=0\quad\textrm{ and
}\quad\Box\widetilde{B}_{\mu\nu}=0.$ (B.20)

Therefore, the linearized magnetic Weyl tensor satisfies the same relations as
the linearized magnetic Weyl tensor in flat spacetime. In contrast to the
$\mathbb{R}^{4}$ components of the linearized electric Weyl tensor, the
magnetic Weyl tensor is traceless. The system of equations given by eq. B.20
are therefore identical to their analogous equations in flat spacetime. The
derivation of the vector part of memory for perturbations in flat spacetime
has been treated previously in [53]. Since these computations are identical to
the derivation of the vector part of $\widetilde{\Delta}_{AB}$, we will not
repeat this analysis here. Equation B.20 implies the following fall-off for
the magnetic Weyl tensor components:

$\widetilde{B}_{AB}\sim O\left({1\over r}\right),\quad\widetilde{B}_{r\mu}\sim
O\left({1\over r^{2}}\right),\quad\widetilde{B}_{rr}\sim O\left({1\over
r^{3}}\right).$ (B.21)

The final result from analyzing eq. B.20 together with eq. B.21 is

$\mathscr{D}^{A}\mathscr{D}^{B}\widetilde{B}_{AB}^{(1)}=\partial_{u}^{2}\widetilde{B}_{rr}^{(3)},$
(B.22)

where $\widetilde{B}_{AB}^{(1)}=-\left({1\over
2}\right)\epsilon_{A}{}^{C}\widetilde{\mathcal{E}}_{CB}$ and, explicitly,
$\widetilde{B}_{rr}^{(3)}=\left({1\over
2}\right)\epsilon^{AB}\widetilde{C}_{ABru}^{(3)}$.

After integrating eqs. B.22 and B.17 and using the fact that
$q^{AB}\widetilde{\Delta}_{AB}=-\hat{g}^{mn}\widetilde{\Delta}_{mn}$ we find
that

$\mathscr{D}^{A}\mathscr{D}^{B}\textrm{TF}[\widetilde{\Delta}_{AB}]=\frac{1}{2}[\mathscr{D}^{2}-1]\Delta\phi-\Delta\left(\overline{\widetilde{E}_{rr}^{(3)}}\right),$
(B.23)
$\epsilon^{CA}\mathscr{D}_{C}\mathscr{D}^{B}\widetilde{\Delta}_{AB}=-\Delta\left(\widetilde{B}_{rr}^{(3)}\right)\quad\textrm{
and }\quad q^{AB}\widetilde{\Delta}_{AB}=-\frac{1}{2}\Delta\phi.$ (B.24)

Equations B.23 and B.24 are consistent with the linearized form of eq. 6.12
since, by Lemma 5, $\Delta\left(\widetilde{B}_{rr}^{(3)}\right)$ vanishes and
$\Delta\phi$ is spherically symmetric under the strong stationarity conditions
we imposed.

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