(pseudo)tensor near null infinity can be decomposed into two terms: a standard
GR term and a term that depends on the perturbations of the additional
gravitational fields. In order to establish this property, we must first
massage the second-order effective action near null infinity. In the limit to
null infinity,

$\displaystyle
0=\phantom{}_{\scriptscriptstyle{(1)}}\mathcal{G}_{\mu\nu}[\eta_{0},\\{h^{H},\Psi^{H}\\}]$
$\displaystyle\propto\left[\frac{\delta\phantom{}_{\scriptscriptstyle{(}2)}S^{\scriptscriptstyle{flat}}_{\text{eff}}}{\delta
h^{H\mu\nu}}\right]^{H}\,,$ (132) $\displaystyle
0=\phantom{}_{\scriptscriptstyle{(1)}}\mathcal{J}[\eta_{0},\\{h^{H},\Psi^{H}\\}]$
$\displaystyle\propto\left[\frac{\delta\phantom{}_{\scriptscriptstyle{(}2)}S^{\scriptscriptstyle{flat}}_{\text{eff}}}{\delta\Psi^{H}}\right]^{H}\,,$
(133)

and hence, the leading-order propagation equations [Eqs. (123) and (124)] only
depend on the
$\phantom{}{}_{\scriptscriptstyle{(}2)}S^{\scriptscriptstyle{flat}}_{\text{eff}}$
of Eq. (116). Assumption (iii) of the theorem then implies that there exists a
set of redefined perturbation variables
$\\{\hat{h}_{\mu\nu}^{H},\hat{\Psi}^{H}\\}$ for which the effective action
$\phantom{}{}_{\scriptscriptstyle{(}2)}S^{\scriptscriptstyle{flat}}_{\text{eff}}$
can be written as

$\displaystyle\phantom{}_{\scriptscriptstyle{(}2)}S^{\scriptscriptstyle{flat}}_{\text{eff}}\propto$
$\displaystyle\int\differential^{4}x\sqrt{-\eta}\,\bigg{[}\hat{h}^{H\mu\nu}\mathcal{E}^{\alpha\beta}_{\mu\nu}\hat{h}^{H}_{\alpha\beta}+\sum_{\Psi}\left(\partial\hat{\Psi}^{H}\right)^{2}\bigg{]}\,.$
(134)

The first term is the usual Fierz-Pauli operator with indices contracted with
the independent background metric $\eta_{\mu\nu}$. The second term corresponds
schematically to the sum of kinetic terms of the high-frequency perturbations
of the additional gravitational fields. Up to integrations by parts, this is
the unique action that will lead to a set of decoupled massless wave equations
for the leading-order, high-frequency perturbations upon imposing the adequate
gauge choices of Eq. (127).

With this in hand, we can now establish property (b). Lemma 1 and the
perturbed generalized field equations imply that, in the limit to null
infinity,

$\Big{\langle}\phantom{}_{\scriptscriptstyle{(2)}}\mathcal{G}_{\mu\nu}[\eta_{0},\\{\hat{h}^{H},\hat{\Psi}^{H}\\}]\Big{\rangle}\propto\Bigg{\langle}\frac{\delta\phantom{}_{\scriptscriptstyle{(2)}}S^{\scriptscriptstyle{flat}}_{\text{eff}}}{\delta\eta^{\mu\nu}}\Bigg{\rangle}\propto\phantom{}_{\scriptscriptstyle{(2)}}t_{\mu\nu}\,.$
(135)

In other words, in this limit, the effective stress-energy (pseudo)tensor of
radiation, and therefore, also the right-hand side of the memory equation [Eq.
(122)], only requires knowledge of the flat, second-order effective action
$\phantom{}{}_{\scriptscriptstyle{(}2)}S^{\scriptscriptstyle{flat}}_{\text{eff}}$
as well. Since
$\phantom{}{}_{\scriptscriptstyle{(}2)}S^{\scriptscriptstyle{flat}}_{\text{eff}}$
can be written in the form of Eq. (134), it can be split into a term that only
depends on the metric perturbation $h^{H}_{\mu\nu}$ and another set of terms,
each of which only depends on the field perturbation $\Psi^{H}$ of every
additional field $\Psi$ and involves only two derivative operators. Each of
these terms can be treated separately when computing the radiative energy-
momentum (pseudo)tensor, and hence

$\phantom{}_{\scriptscriptstyle{(2)}}t_{\mu\nu}=\phantom{}_{\scriptscriptstyle{(2)}}t^{\scriptscriptstyle{\text{GR}}}_{\mu\nu}[\eta_{0},\hat{h}^{H}]+\sum_{\Psi}\phantom{}{}_{\scriptscriptstyle{(2)}}t^{\hat{\Psi}}_{\mu\nu}[\eta_{0},\hat{\Psi}^{H}]\,,$
(136)

where each term only involves two derivative operators. Moreover, since the
metric perturbation piece in the flat, effective second-order action in Eq.
(134) admits the same form as in GR, the first term is proportional to the
radiative energy momentum tensor of GR. After imposing the gauge of Eq. (127),
the latter reads

$\phantom{}{}_{\scriptscriptstyle{(2)}}t^{\scriptscriptstyle{\text{GR}}}_{\mu\nu}\propto\Big{\langle}\partial_{\mu}\hat{h}^{H}_{\alpha\beta}\partial_{\nu}\hat{h}^{H\alpha\beta}\Big{\rangle}\,,$
(137)

thus establishing property (b).

Finally, let us consider property (c), which states that the energy-momentum
(pseudo)tensor is invariant under infinitesimal coordinate transformations.
The split in Eq. (136) and the assumption that $\hat{\Psi}^{H}$ does not
depend on $h^{H}_{\mu\nu}$, directly implies that
$\phantom{}{}_{\scriptscriptstyle{(2)}}t_{\mu\nu}$ is invariant under
infinitesimal diffeomorphisms $x^{\mu}\rightarrow x^{\mu}-\xi_{H}^{\mu}$. This
is because such diffeomorphisms only affect the term
$\phantom{}{}_{\scriptscriptstyle{(2)}}t^{\scriptscriptstyle{\text{GR}}}_{\mu\nu}$,
which is invariant upon integration by parts and after imposing the Lorenz
gauge Eq. (127).

Since we have established Eq. (128) together with all the required properties
(a)–(c) of the radiative energy momentum tensor
$\phantom{}{}_{\scriptscriptstyle{(2)}}t_{\mu\nu}$, we have therefore proved
the theorem. ∎

####  V.2.3 Technical Remarks on the Theorem

Let us conclude the discussion of the theorem by making several technical
remarks. First, let us stress that in assumption (iii) we only require the
first-order propagation equations to reduce to a set of decoupled second-order
wave equations. The two main results of the theorem are therefore that 1) the
decoupling between fields remains intact even at $\mathcal{O}(\alpha^{2})$ at
the low-frequency level; 2) The low-frequency $\mathcal{O}(\alpha^{2})$ term
also only involves two derivative operators. In particular, the latter
directly implies that in any theory satisfying the assumptions of the theorem,
the memory equation will only directly depend on terms in the action that
involve two derivative operators. This is nicely exemplified in the SVT theory
result of the radiative energy momentum tensor in Eq. (87).

Furthermore, observe that the assumption (iii) is quite generic. Indeed, in
any theory satisfying second-order equations of motion, the first-order
propagation equation will only involve two derivative operators. But as we
will explore in Sec. V.4, this is even true for a large class of theories,
whose field equations are higher-order in derivatives. Local Lorentz
invariance as well as masslessness then assure that the leading-order
propagation equation generically takes the form of a massless wave equation.

Moreover, also a decoupling of the equations at first order in perturbations
is quite generic. First, note that a Minkowski background ensures that the
tensor, vector and scalar sectors313131The terms ”tensor”, ”vector” and
”scalar” refer here to the polarization type of each mode. can always be
decoupled at leading order in perturbations and each of these sectors can
therefore be treated individually. Therefore, potential couplings between
perturbations at the level of the leading-order perturbation equations could
only arise within each of these sectors. We are, however, not aware of any
concrete massless theory that admits such a coupling of first order
perturbations. Indeed, explicit examples of such coupled equations typically
only arise in theories that include an explicit mass term, such as for
instance in massive bigravity models (see e.g. [132, 133]) or massive multi-
Proca theories [134]. It would however be interesting to explore null memory
for such theories with coupled leading-order perturbation equations, a task we
leave for future work.

Also, any typical massless theory involving multiple interacting vector or
scalar fields at the level of the full action, such as non-Abelian vector
fields or typical scalar multi-field models (see e.g. [135]), naturally
decouple to leading order in perturbations on a Minkowski background, and
thus, still abide by the decoupling assumption in (iii).

###  V.3 Relation of the Memory Evolution Theorem to the Tensor Null Memory

The form of the memory-evolution equation [Eq. (128)] for dynamical metric
theories (see Def. 1) established through Theorem 1, precisely corresponds to
the memory equation [Eq. (80)] obtained for the specific example of SVT
gravity, considered in Sec. IV. The only difference is that the energy-
momentum (pseudo)tensor of the leading-order radiation
$t^{\scriptscriptstyle{\text{SVT}}}_{\mu\nu}$, is replaced by a more general
(pseudo)tensor $\phantom{}{}_{\scriptscriptstyle{(}2)}t_{\mu\nu}$, which is
still conserved and gauge invariant. The precise form, of course, depends on
the nature of the additional gravitational fields $\Psi$.

To make a statement about the resulting tensor null memory, we will now
further assume, that in the asymptotic region far from the source, the
radiative energy momentum tensor can be written in the form (cf. Eq. (82))

$\phantom{}_{\scriptscriptstyle{(}2)}t_{\mu\nu}=\phantom{}_{\scriptscriptstyle{(}2)}t_{00}\,\ell_{\mu}\ell_{\nu}\,.$
(138)

where $\ell_{\mu}=-\delta^{0}_{\mu}+\delta^{i}_{\mu}n_{i}$ and the scaling
with $r$ of $\phantom{}{}_{\scriptscriptstyle{(}2)}t_{00}$ is $\sim r^{-2}$
323232Recall that in the SVT example, this structure followed from the general
asymptotic fall-off behavior in Eq. (62) of the solutions to the wave equation
and the explicit structure of the energy-momentum tensor.. This is all that is
needed to apply the same steps presented in Secs. IV.2 and IV.3, in order to
write the resulting asymptotic tensor null memory as (see Eq. (96))

$\delta
h_{L}^{lm}=\frac{2\kappa_{\text{eff}}}{r}\sqrt{\frac{(l-2)!}{(l+2)!}}\int\differential^{2}\Omega^{\prime}\,\bar{Y}^{lm}\int\differential
u^{\prime}\,r^{2}\phantom{}_{\scriptscriptstyle{(}2)}t_{00}\,,$ (139)

where

$\phantom{}{}_{\scriptscriptstyle{(}2)}t_{00}=\phantom{}_{\scriptscriptstyle{(2)}}t^{\scriptscriptstyle{\text{GR}}}_{00}+\sum_{\Psi}\phantom{}{}_{\scriptscriptstyle{(2)}}t^{\hat{\Psi}}_{00}\,.$
(140)

Here, $\delta h_{L}^{lm}$ are the coefficients of a SWSH expansion of the
quantity $\delta h^{L}=\delta h^{L\,TT}_{ij}m_{i}m_{j}$, with $\delta
h^{L\,TT}_{ij}=\delta\hat{h}^{L\,TT}_{ij}$. This result follows immediately,
because, in the derivation in Secs. IV.2 and IV.3, the details of the
radiative energy momentum tensor are irrelevant, since the radiated energy
flux $r^{2}\phantom{}_{\scriptscriptstyle{(}2)}t_{00}$ remains untouched after
Eq. (138) is established.

The question is now the following: how general is the assumed structure of the
radiative energy-momentum in Eq. (138)? We will argue that this assumption is
not very restrictive, in the sense that it is expected to be generically
satisfied by dynamical metric theories of Def. 1, and thus by most of the
viable extensions to GR considered in the literature, whose fields are gauge
invariant and admit a Lorenz gauge.

To analyze this question carefully, we will first assume that any of the
modes, and hence any of the gauge-invariant solutions to the leading-order
wave-equations $h$ and $\psi$, scale as (cf. Eq. (62))

$\displaystyle\left(\hat{h}\,,\;\hat{\psi}\right)\sim\frac{1}{r}\left[f_{h}(u,\theta,\phi),f_{\psi}(u,\theta,\phi)\right]\,,$
(141)

near null infinity. This implies that the GR piece
$\phantom{}{}_{\scriptscriptstyle{(2)}}t^{\scriptscriptstyle{\text{GR}}}_{\mu\nu}\propto\big{\langle}\partial_{\mu}\hat{h}^{H}_{\alpha\beta}\partial_{\nu}\hat{h}_{H}^{\alpha\beta}\big{\rangle}$
satisfies Eq. (138) (cf. Sec. IV.2). Using the considerations in Sec. II.2,
the additional terms in
$\phantom{}{}_{\scriptscriptstyle{(}2)}t^{\hat{\Psi}}_{\mu\nu}$ can be
schematically written as

$\phantom{}{}_{\scriptscriptstyle{(}2)}t^{\hat{\Psi}}_{\mu\nu}\sim\,\eta_{0}^{n}\Big{\langle}\partial\hat{\Psi}^{H}\partial\hat{\Psi}^{H}\Big{\rangle}\,.$
(142)

where we have omitted the index structure to keep the expression general, and
where $\eta_{0}^{n}$ denotes $n$ background-field contributions with indices
contracted onto the derivatives or the tensor structure of the gravitational
fields themselves. The additional terms in
$\phantom{}{}_{\scriptscriptstyle{(}2)}t^{\hat{\Psi}}_{\mu\nu}$ take this form
because the leading-order term in
$\langle\phantom{}{\scriptscriptstyle{(2)}}\mathcal{G}_{\mu\nu}\rangle$ has
two derivatives acting on the two leading-order wave perturbations, which, at
most, can be multiplied by background fields $\eta_{0}$ in the limit to null
infinity. Moreover, due to Theorem 1, the assumption of decoupled high-
frequency propagation equations ensures that there are no mixed terms between
the metric perturbation $\hat{h}^{H}$ and the other gravitational fields
$\hat{\Psi}^{H}$ (e.g. of the form
$\langle\partial\hat{\Psi}^{H}\;\partial\hat{h}\rangle$) or between different
types of gravitational fields (e.g. of the form
$\langle\partial\hat{V}^{H}\;\partial\hat{\phi}^{H}\rangle$ if $\Psi$ contains
a vector $V$ and scalar $\phi$).

We will now partially restore the tensor structure. Requiring that the action
remain gauge invariant, we can rewrite Eq. (142) as

$\phantom{}{}_{\scriptscriptstyle{(}2)}t^{\hat{\Psi}}_{\mu\nu}\sim\,\Big{\langle}\partial_{\mu}\hat{\Psi}^{H}\partial_{\nu}\hat{\Psi}^{H}\Big{\rangle}\,,$
(143)

where we have explicitly included the indices on the partial derivatives and
assumed any tensor structure in the first $\hat{\Psi}^{H}$ is contracted onto
the tensor structure of the second $\hat{\Psi}^{H}$, is a trace or is a
scalar. The free indices $\mu$ and $\nu$ need to be placed on the two
derivative operators because otherwise the resulting term vanishes either due
to the first-order equations of motion or the Lorenz gauge, upon integration
by parts. We have here removed the n copies of the $\eta_{0}$ background
fields because they cannot be present if one wishes to preserve gauge
invariance. This is because, if the action is gauge invariant, it must be
constructed from gauge strength tensors, which are trivial at null infinity,
and must, therefore, not appear in Eq. (143). Together with the fall-off Eq.
(141) this implies that Eq. (138) holds.

To continue, we want to be more specific and consider a concrete type of
possible additional gravitational fields, which covers a large class of
theories considered in the literature. Namely, from now on, we will focus on
dynamical metric theories whose additional gravitational fields $\Psi$ are
$k$-form field potentials with an associated Abelian $U(1)$ gauge
symmetry333333Recall that a differential $k$-form field $\Psi$ is a totally
antisymmetric tensor field, which in a coordinate-induced basis can be written
as $\Psi=(1/k!)\,\Psi_{\mu_{1}...\mu_{k}}\,dx^{\mu_{1}}\wedge...\wedge
dx^{\mu_{k}}$, with $k<d$ and where $"\wedge"$ denotes the exterior product.
Such $k$-form fields naturally generalize $U(1)$ vector field potentials
because their field strength $\mathscr{F}\equiv d\Psi$ is invariant under
Abelian gauge transformations $\Psi\rightarrow\Psi+d\Lambda$, where $\Lambda$
is an arbitrary $(k-1)$-form and $d$ is the exterior derivative. See for
instance [136, 137] for a review of the topic.. The collection of these
additional dynamical k-form fields are assumed to describe $N$ additional
propagating gravitational degrees of freedom. Thus, the theory admits $N$
independent and propagating solutions to the wave equations, characterized
through $N$ modes in the canonically-normalized, second-order action, which we
will denote as $\hat{\psi}_{\lambda}$, where $\lambda=1,..,N$. Note that in
four spacetime dimensions, we only consider $k$-forms for $k<4$. A $0$-form
field simply corresponds to a scalar field, while a 1-form field naturally
describes an Abelian vector field, and, therefore, it carries two propagating
degrees of freedom. A 2-form field, on the other hand, again only describes
one dynamical mode equivalent to a scalar degree of freedom (see e.g. [138]).
A 3-form field will not contain any propagating modes in four dimensions
simply because the components of the associate 4-form field strength are
constant (see e.g. [139])343434However, a non-trivial coupling to the metric
of such fields can for example lead to a dynamical contribution to the
cosmological constant [140] and may thus still have physical implications..

In Appendix D we offer the explicit derivation of the associated tensor null
memory formula for this class of theories. Provided that the assumptions of
Theorem 1 hold, the end result is

$\begin{split}&\delta
h_{H}^{lm}(u,r)=\frac{1}{r}\sqrt{\frac{(l-2)!}{(l+2)!}}\int_{S^{2}}\differential^{2}\Omega^{\prime}\,\bar{Y}^{lm}(\Omega^{\prime})\\\
&\quad\times\,\int_{-\infty}^{u}\differential
u^{\prime}\,r^{2}\bigg{\langle}|\dot{\hat{h}}_{+}|^{2}+|\dot{\hat{h}}_{\times}|^{2}+\sum_{\lambda=1}^{N}|\dot{\hat{\psi}}_{\lambda}|^{2}\bigg{\rangle}\,,\end{split}$
(144)

where $\hat{h}_{+,\times}=h_{+,\times}$. Recall that for certain theories (see
Sec. III) it is necessary to redefine the tensor perturbation variable
$\hat{h}_{\mu\nu}^{H}=W(h^{H},\Psi^{H})$ to obtain a perturbation variable
that satisfies a first-order wave equation in the appropriate gauge. However,
the TT component of this redefined variable, and thus, also the polarization
modes $h_{+,\times}$ always correspond to the TT component of the physical
metric present in the detector response
$\hat{h}^{\text{TT}H}_{\mu\nu}=h^{\text{TT}H}_{\mu\nu}$. The same is true for
the memory component. The need of such a change of variables to decouple the
leading order equations is, typically, a sign of the presence of additional
gravitational polarizations in the theory.

Let us end this subsection by stressing that the tensor null memory result in
Eq. (144) was obtained without any knowledge of the precise form of the
Lagrangian of the theory, and it simply follows from Theorem 1 and the
resulting solution of the memory evolution equation. The coupling constants of
a specific theory would then enter through a transformation from the
canonically normalized modes $\hat{\psi}_{\lambda}$ to the physical modes of
the theory (recall the discussion around Eq. (87)). The expression in Eq.
(144) should be compared to the result in Eqs. (96) and (86) and it represents
a generalization of the explicit SVT theory example we considered in Sec. IV.

###  V.4 Scope of the Memory Evolution Theorem and its link to the Tensor Null
Memory

We will now explore the scope of the above statements and, in particular,
investigate which theories admit a tensor null memory of the form Eq. (144).
Such theories must, of course, satisfy the assumptions of Theorem 1. Let us
then restrict ourselves to dynamical metric theories of gravity that admit an
arbitrary number of additional $k$-form fields in the gravitational sector
(see App. D). In particular, this restriction implies that we focus on
theories with Abelian gauge groups, but similar conclusions also hold in the
non-Abelian case for at least some simple examples. A restriction to $k$-form
fields also implies a limitation to bosonic fields. Fermionic fields do not
usually play the role of a massless force carrier in known theories.
Nonetheless, in principle, it could still be interesting to enrich metric
theories with additional fermionic fields in the gravitational sector, an
investigation we leave for future work.

Given these restrictions, there is a large class of theories for which the
tensor null memory takes the form Eq. (144). The theorem clearly encompasses
any covariantized version of massless $k$-form Galileon theories [141, 142],
restricting the full equations of motion to second order. Such theories
represent a natural generalization of the SVT class of theories with second-
order equations of motion discussed in Sec. IV. As shown explicitly in Sec.
IV.4, these SVT theories contain Horndeski-type theories, including BD theory
[39, 40] and other similar scalar-tensor theories, sGB gravity [61, 63], and
double-dual Riemann gravity [88, 89].

The memory formula in Eq. (144), however, is not restricted to theories that
satisfy second-order equations of motion. A first interesting concrete example
of a theory that does not fall under the class of covariantized $k$-form
Galileon theories is dCS gravity [51] (see also [52]). This theory is defined
through the action

$\begin{split}S^{\scriptscriptstyle{\text{dCS}}}=\int\differential^{4}x\sqrt{-g}\bigg{(}&\frac{1}{16\pi
G}R+\frac{\epsilon^{2}}{4}\,\Theta\,R_{\mu\nu\rho\sigma}\tilde{R}^{\nu\mu\rho\sigma}\\\
&-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\Theta\nabla_{\nu}\Theta\bigg{)}\,,\end{split}$
(145)

where $\epsilon$ is a dimension-full coupling constant and $\Theta$ is a
(pseudo)scalar field353535A vanishing of the scalar field potential implies
that $\Theta$ is massless, and the theory is shift symmetric in $\Theta$. This
modified theory reduces to GR smoothly when $\Theta$ is constant or when
$\epsilon$ vanishes.. This theory is an example of the effective field
theories with higher-than-second-order field equations, discussed in Sec.
II.2. In recent years, dCS gravity has received some attention, mainly because
its rotating black holes have a non-trivial (pseudo)scalar profile or “hair”
[143]. This implies that when such hairy black holes accelerate, a scalar away
is sourced, which carries energy-momentum to null infinity.

This fact is what makes dCS interesting from the point of view of the memory
because the scalar wave should modify the tensor null memory. Just as in sGB
gravity (see Sec. IV.4.2), the dCS action expanded to second-order in
perturbations simply reduces to the GR one with a canonical scalar field near
null infinity, because the Pontryagin density is of higher order in this
limit. Therefore, the linear-order, high-frequency propagation equations are
just given by two decoupled, second-order wave equations for the metric and
the scalar field perturbations. In turn, this fact implies that dCS gravity
naturally falls within the scope of Theorem 1 and admits the tensor null
memory formula of Eq. (144), with $N=1$, $\Psi\to\Theta$ and corresponding
leading-order wave-mode $\hat{\psi}_{\lambda}\to\hat{\vartheta}=\vartheta$.
Note, however, that, just as in the sGB case, the coupling $\epsilon$ does not
enter explicitly into the memory equation, but rather it enters implicitly
through the dependence of the metric and scalar perturbations on the coupling.
For dCS gravity, this result is indeed confirmed by the explicit computation
of the associated BMS balance laws in [53], from which one can extract the
null memory following the same steps outlined in Appendix C for BD gravity.

The above result can be generalized to a wide class of theories with higher-
derivative field equations. Indeed, any GR correction involving higher powers
of curvature invariants, including non-minimally-coupled, massless scalar
fields, satisfies decoupled second-order wave equations at the linear level
[96]. In particular, assuming Eq. (36), this is even true when the GR
corrections involve terms that are quadratic in the curvature and do not form
topological invariants (as in the dCS cases). Unlike other GR corrections,
such theories do in fact lead to higher-order equations of motion at the
linear level in perturbations, when taken at face value. However, in the small
coupling approximation, more precisely, if we assume the strong condition Eq.
(36), these contributions are suppressed by factors of $\mathcal{O}(\epsilon
f_{H})$, where $\epsilon f_{H}\ll 1$, and, therefore, they do not appear at
leading order. Indeed, it was explicitly shown in [96] that iteratively
solving these higher-derivative equations through order reduction in fact
leads again to second-order wave-equations for the metric perturbations. All
of this then immediately implies that, even for these theories, Theorem 1
holds and the tensor null memory takes the form of Eq. (144).

Note that the arguments and conclusions above naturally also encompass metric
theories with higher-derivative terms in the action that do not introduce new
gravitational fields, as for example systematically explored in [106].
Moreover, the statements can be generalized to theories with non-minimal
couplings to other gauge invariant, Abelian, $k$-form fields. This is because
for $k>0$, gauge invariance forces any non-minimal coupling to occur through
the field strength, whose background value vanishes when evaluated at null
infinity, such that the derivation in [106] is essentially unaltered. The
massless and covariant (including local Lorentz invariant) conditions on the
other hand ensure that the leading-order propagation equations reduce to
massless wave equations at null infinity. As discussed in Sec. V.2.3, even if
multiple vector or scalar modes are present in the theory, these leading-order
propagation equations typically decouple and the assumptions of Theorem 1 are
satisfied. Furthermore, just as for the higher-derivative curvature terms, a
small-coupling approximation can in principle also be applied to any higher-
derivative terms of the additional $k$-form fields that are not tuned to lead
to second order equations of motion. Therefore, it is expected that very
generally, Theorem 1 can be applied to dynamical metric theory with additional
k-form fields, which further implies that the tensor null memory is given by
Eq. (144).

In summary, Theorem 1 suggests that, very generically, for dynamical metric
theories of gravity defined in Def. 1 in the small coupling approximation
(ensuring a viable EFT expansion), the tensor null memory is of the form of
Eq. (144). In other words, null memory is modified in comparison to GR in two
ways: (I) through contributions of the energy fluxes to null infinity of
additional, massless, dynamical degrees of freedom in the theory; and (II)
through modifications in the generation and propagation of the leading-order
tensor perturbations. This simple result could have interesting implications,
as we will explore in the discussions below. Clearly then, the theorem does
not cover metric theories (a) with additional massive degrees of freedom or a
massive physical metric, such as in massive gravity [144], (b) with non-local
effects, (c) with explicit Lorentz breaking, such as in Einstein-Æther
theories [145], and (d) with non-vanishing torsion or non-metricity, such as
in torsion bigravity [146, 147].

##  VI Discussion and Conclusion

Planned space-based detectors, such as LISA [148], or neXt Generation ground-
based detectors, such as the Einstein Telescope [149] or Cosmic Explorer [150]
are expected to provide the first detections of the tensor memory effect in
the near future. These observations of gravitational wave memory may play an
important role in establishing a better understanding of gravity and
constraining modifications of GR. This is because tensor memory is a very
special, non-linear correction to the gravitational wave response. Indeed, the
memory’s dominant null component can be understood as being sourced by the
leading-order wavefront itself, allowing for potential consistency checks
between the standard oscillatory part of the response and the memory itself.
The results of the work we have presented, including the computation of the
null memory in the most general, massless and gauge invariant SVT theory with
second-order equations of motion, together with its subsequent
generalizations, is but one of the first steps towards the possible future
exploitation of gravitational wave memory as a test of GR.

One of the main discoveries presented in this paper is a simple but very
generic conclusion (see Eq. (144)): the functional form of the tensor null
memory of massless dynamical metric theories of gravity is merely modified
from the GR expectation through the presence of additional null fluxes due to
other gravitational fields present in the beyond-GR extension. This result
could potentially be exploited to develop a largely model-insensitive test of
GR of perhaps the most straightforward manifestation of new physics: the
existence of additional scalar, vectorial or tensorial propagating degrees of
freedom. Since memory is sensitive to any kind of energy-momentum emitted from
the source, such a test would not only complement ongoing searches for
additional gravitational polarizations [151], but also extend the sensitivity
of such tests to scenarios in which additional gravitational fields do not
excite any other polarization modes of the physical metric.

Much work remains to be done to explicitly construct such a model-insensitive
null-memory test of GR, but let us suggest some general directions that could
be explored. One direction for future research is to study whether the
oscillatory part of the gravitational wave signal can be separated from the
non-oscillatory part. This could be accomplished by exploiting the fact that,
by definition of the memory evolution equation in the Isaacson picture, the
characteristic frequency of the oscillatory component of the signal is
parametrically separated from the frequency of the null memory part.
Therefore, Morlet wavelets [152] in a time-frequency analysis may be able to
separate these two components. Based on such a model-independent analysis, one
could imagine developing a consistency check between the extracted oscillatory
part of the gravitational wave signal and the non-oscillatory memory
contribution. Such a test would require knowledge of the inclination angle and
the extraction of both polarizations, which could, for example, be obtained
through an electromagnetic counterpart and the detection of the signal with
multiple interferometers. On a very general basis, a potential missmatch
between the two signals could then be interpreted as the presence of
additional propagating (scalar, vector or tensor) degrees of freedom.

Another possible approach would be to target as well the merger sensitivity of
the memory signal directly, by developing a parameterization for the beyond-GR
memory effect, akin to the parameterized post-Einsteinian framework used for
the oscillatory signal [153]. Such an approach would require the construction
of a parametric post-Einsteinian model for the GR deformation to the null
memory through simple transition functions, such as those studied in [154].
Based on a set of examples of the null memory in specific beyond-GR theories
computed through Eq. (144), such a parametric model could then be chosen to
ensure that the various specific beyond-GR modifications to the memory can be
recovered by the model. This would in turn allow for a Bayesian parameter
estimation study on the full signal to attempt to extract the parameters of
the post-Einsteinian tensor memory from the signal.

On the theoretical side, a very interesting future task will be to generalize
our study to ordinary memory [14, 155, 22], defined colloquially as the part
of the memory that is sourced by fields that do not reach null infinity. The
typical example of such a source are unbound components of the matter stress-
energy of the system itself, which, due to compact support, are typically
assumed to not reach null infinity. In theories beyond GR, however, the
possibility of additional massive gravitational fields in the theory
represents an additional source of ordinary memory. In particular, considering
massive fields would allow for the study of a much richer structure of
Ostrogradski-stable vector field theories [113], as well as a higher-
derivative massive 2-form interaction [138], which can be related to the
topological mass generation of BF theories [156].

Moreover, we would like to mention, that a definition of the memory evolution
equation as the leading-order low-frequency equation of motion in the Isaacson
picture, potentially allows for an even broader study of memory beyond GR.
Indeed, in principle, the Isaacson approach could also be applied to the field
equations of theories which we explicitly disregarded in this work, such as
theories with non-dynamical field-content or explicit Lorentz breaking. In
such a general setting, one could also think about solving the resulting
memory evolution equation not in the vicinity of null infinity, but in a
different appropriate limit.

This last point could potentially be explored for the study of the tensor
memory in metric theories with broken diffeomorphism invariance, such as
massive gravity theories [144]. Such an analysis may however turn out to be
less straightforward, as the current understanding of memory is heavily based
on the structure of the theory around null infinity. In the case of massive
gravity, other calculation techniques, such as point-particle source
approximations [157], might be more appropriate for the computation of its
memory contribution. For example, [158] applied such approximations to massive
Fierz-Pauli theory, but the analysis in this work may need to be generalized
to avoid an unphysical discrepancy with GR in the small mass limit.

Another avenue for future work would be to investigate a possible
generalization of the BMS balance laws to a wide class of theories, based on
the results obtained in this paper. In the light of the close connection
between BMS balance laws and tensor null memory established in Appendix C for
BD theory, our results strongly suggest that the approach of [45, 46, 47, 48]
for BD gravity may be readily generalized to the asymptotic structure of any
dynamical metric theories. It would be interesting to explore this conjecture
in detail, especially with regard to the ordinary memory, which we excluded in
our analysis. As concerns scalar or vector null memory, the connection to BMS
balance laws is less clear. In our work, we did not find any non-zero scalar
or vector memory, which parallels the observation that no analogous BMS
constraint can be formulated for scalar memory within BD theory [45, 46, 47,
48]. Recently, however, [159] suggested that scalar memory in BD theory could
be associated to asymptotic symmetries of a dual description of the scalar
sector instead, the implications of which still remain unexplored.

###### Acknowledgements.

We would like to express our gratitude to Luc Blanchet for pointing out the
relation in Eq. (91) and sharing his personal notes of the proof thereof.
Moreover, we thank Lydia Bieri, David Garfinkle and Henri Inchauspé for useful
discussions on the memory effect, Justin Ripley for helpful comments on
gravitational radiation in sGB theories and recent progress in numerical
relativity, as well as Fabio D’Ambrosio, Shaun Fell, Francesco Gozzini, David
Maibach and Stefan Zentarra for valuable conversations on asymptotic
symmetries and BMS flux blanace laws. LH is supported by funding from the
European Research Council (ERC) under the European Unions Horizon 2020
research and innovation programme grant agreement No 801781 and by the Swiss
National Science Foundation grant 179740. LH further acknowledges support from
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg
STRUCTURES Excellence Cluster). NY acknowledges support from the Simmons
Foundation through Award No. 896696. JZ is supported by an ETH Zürich
Doc.Mobility Fellowship.

## Glossary

gravitational field

    A tensor field (other than the physical metric) that is non-minimally coupled to the physical metric. In a metric theory of gravity, a gravitational field does not couple directly to matter fields, as only the physical metric does.
gravitational polarizations

    Polarization modes of the physical metric perturbation far away from the source.
gravitational radiation

    Radiation of the physical metric perturbation.
gravitational waves

    Waves of the physical metric perturbation.
limit to null infinity

    Leading-order behavior in the limit $r\rightarrow\infty$ at fixed $u=t-r$, in source centered coordinates $\\{t,x,y,z\\}$, with $r=\sqrt{x^{2}+y^{2}+z^{2}}$.
memory

    A component of a gravitational polarization that induces a permanent, non-oscillatory offset between two test-masses in the idealized response of a gravitational wave detector.
null memory

    Memory sourced by the energy-momentum of radiation.
ordinary memory

    Memory sourced by massive bodies that are unbound in their initial or final states (or both)
physical metric

    The metric that minimally couples to matter. In a metric theory of gravity, the physical metric is the only gravitational field that couples directly to matter.
polarization

    Property of a radiative mode that specifies the geometrical orientation of the oscillation, distinguishing it from other radiative modes.
radiation

    Field-perturbation that escape to null infinity.
radiative degrees of freedom

    Gauge-invariant and propagating solutions to the wave equation in the limit to null infinity.
radiative modes

    c.f.radiative degrees of freedom.
tensor memory

    (tensor/vector/scalar) Memory associated to a tensor/vector/scalar gravitational polarization.
tensor polarizations

    (tensor/vector/scalar) Modes that transform like a tensor, a vector or a scalar under rotations about the direction of propagation.
waves

    Field-perturbations with characteristic high frequency $f_{H}$.

##  Appendix A Transverse Traceless Multipole Expansion

For convenience, we gather in this appendix a collection of definitions and
formulas from the unifying review by Thorne [97] for different multipole
expansions used in this work. In particular, we will touch upon spin-weighted
spherical harmonics, pure spin transverse-traceless tensor harmonics, as well
as STF tensors, all tied to the $SO(3)$ rotation group and the irreducible
representations thereof. The notation for STF tensors will however be adapted
to a more contemporary custom (see for instance [41, 38, 94]). For more
details and derivations, we refer the reader to [97], while part of the
treatment can also be found in [94] and in the appendix of [160].

###  A.1 Spin-Weighted Spherical Harmonics

Spin-weighted spherical harmonics can be constructed from ordinary spherical
harmonics through applications of angular derivative operators

$\phantom{}{}_{\scriptscriptstyle{s}}Y^{lm}=\begin{cases}\sqrt{\frac{(l-s)!}{(l+s)!}}\eth^{s}Y^{lm}\,,&l\geq
s\geq 0\\\
(-1)^{s}\sqrt{\frac{(l+s)!}{(l-s)!}}\bar{\eth}^{-s}Y^{lm}\,,&0>s\geq-l\\\
0\,,&|s|>l\\\ \end{cases}$ (146)

where the angular derivative operator $\eth$ and its complex conjugate
$\bar{\eth}$ are defined through their action on functions $f_{s}$ of spin-
weight $s$. The application of the operator $\eth$ on a function $f_{s}$
defines a quantity with spin-weight $s+1$, while $\bar{\eth}$ lowers the spin-
weight by one unit.

$\displaystyle\eth f_{s}$
$\displaystyle\equiv-\sin^{s}\theta\left(\partial_{\theta}+i\,\csc\theta\right)(f_{s}\sin^{-s}\theta)\,,$
(147) $\displaystyle\bar{\eth}f_{s}$
$\displaystyle\equiv-\sin^{-s}\theta(\partial_{\theta}-i\,\csc\theta)\left(f_{s}\sin^{s}\theta\right)\,.$
(148)

They satisfy

$\displaystyle(-1)^{s+m}\,\phantom{}_{\scriptscriptstyle{-s}}\bar{Y}^{l-m}(\theta,\phi)$
$\displaystyle=\phantom{}_{\scriptscriptstyle{s}}Y^{lm}(\theta,\phi)\,,$ (149)
$\displaystyle\int_{S^{2}}\differential\Omega\,_{\scriptscriptstyle{s}}Y^{lm}(\theta,\phi)_{\scriptscriptstyle{s}}\bar{Y}^{l^{\prime}m^{\prime}}(\theta,\phi)$
$\displaystyle=\delta_{ll^{\prime}}\delta_{mm^{\prime}}\,,$ (150)
$\displaystyle\sqrt{(l-s)(l+s+1)}\,_{\scriptscriptstyle{s+1}}Y^{lm}$
$\displaystyle=\eth\,_{\scriptscriptstyle{s}}Y^{lm}\,,$ (151)
$\displaystyle-\sqrt{(l+s)(l-s+1)}\,_{\scriptscriptstyle{s-1}}Y^{lm}$
$\displaystyle=\bar{\eth}\,_{\scriptscriptstyle{s}}Y^{lm}\,.$ (152)
$\begin{split}&\int_{S^{2}}\differential^{2}\Omega\,_{\scriptscriptstyle{s_{1}}}Y^{l_{1}m_{1}}\,_{\scriptscriptstyle{s_{2}}}Y^{l_{2}m_{2}}\,_{\scriptscriptstyle{s_{3}}}Y^{l_{3}m_{3}}\\\
&\,=\,\sqrt{\frac{\prod_{i=1}^{3}(2l_{i}+1)}{4\pi}}\footnotesize{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\\
-s_{1}&-s_{2}&-s_{3}\end{pmatrix}}\,,\end{split}$ (153)

if $s_{1}+s_{2}+s_{3}=0$.

###  A.2 Pure Spin Transverse-Traceless Tensor Harmonics

The electric and magnetic parity pure-spin transverse-traceless (TT) harmonics
can also be constructed from ordinary spherical harmonics through

$\displaystyle T^{\scriptscriptstyle{E2}\,\scriptstyle{lm}}_{ij}$
$\displaystyle\equiv
r^{2}\sqrt{\frac{2(l-2)!}{(l+2)!}}\,\perp_{ijab}\partial_{a}\partial_{b}\,Y^{lm}\,,$
(154) $\displaystyle T^{\scriptscriptstyle{B2}\,\scriptstyle{lm}}_{ij}$
$\displaystyle\equiv
r\sqrt{\frac{2(l-2)!}{(l+2)!}}\,\perp_{ijab}\epsilon_{cd(a}\partial_{b)}r\,n_{d}\partial_{c}\,Y^{lm}\,.$
(155)

where $n_{i}$ is a unit radial vector and the TT projector $\perp_{ijab}$ is
defined in Eq. (60). They satisfy

$\displaystyle(-1)^{m}\bar{T}^{\scriptscriptstyle{P}\,\scriptstyle{l-m}}_{ij}(\theta,\phi)$
$\displaystyle=T^{\scriptscriptstyle{P}\,\scriptstyle{lm}}_{ij}(\theta,\phi)\,,$
(156)
$\displaystyle\int_{S^{2}}\differential\Omega\,T^{\scriptscriptstyle{P}\,\scriptstyle{lm}}_{ij}(\theta,\phi)\,T^{\scriptscriptstyle{P^{\prime}}\,\scriptstyle{l^{\prime}m^{\prime}}}_{ij}(\theta,\phi)$
$\displaystyle=\delta_{PP^{\prime}}\delta_{ll^{\prime}}\delta_{mm^{\prime}}\,.$
(157)

###  A.3 STF Tensors

Denoting a multi-index of $l$ spacial indices collectively as $L\equiv
i_{1}i_{2}...i_{l}$, the projection of a tensor component $A_{L}$ onto its STF
part will be referred to as $A_{\langle L\rangle}$. Note that $B_{L}A_{\langle
L\rangle}=B_{\langle L\rangle}A_{\langle L\rangle}$, where a sum over $l$ is
implicit. Moreover, $n_{L}$ stands for the tensor product of $l$ radial
vectors $n_{i}$. An expansion in terms of STF tensors $n_{\langle
L\rangle}=n_{\langle L\rangle}(\theta,\phi)$ is equivalent to an expansion in
spherical harmonics

$\displaystyle n_{\langle L\rangle}$
$\displaystyle=\frac{4\pi\,l!}{(2l+1)!!}\sum_{m=-l}^{l}\,\mathcal{Y}_{\langle
L\rangle}^{lm}\,Y^{lm}\,,$ (158) $\displaystyle Y^{lm}$
$\displaystyle=\mathcal{Y}_{L}^{lm}\,n_{L}=\mathcal{Y}_{L}^{lm}\,n_{\langle
L\rangle}\,,$ (159)

where $\mathcal{Y}_{L}^{lm}$ are angle-independent, STF tensors connecting the
two basis. They satisfy

$\displaystyle(-1)^{m}\bar{\mathcal{Y}}^{l-m}_{L}$
$\displaystyle=\mathcal{Y}^{lm}_{L}\,$ (160)
$\displaystyle\bar{\mathcal{Y}}_{L}^{lm}\mathcal{Y}_{L}^{lm^{\prime}}$
$\displaystyle=\frac{(2l+1)!!}{4\pi\,l!}\,\delta_{mm^{\prime}}\,.$ (161)

###  A.4 Relations Between Harmonics

The pure-spin TT harmonic tensors are related to spin weight $s=\pm 2$ spin-
weighted spherical harmonics through

$\displaystyle T^{\scriptscriptstyle{E2}\,\scriptscriptstyle{lm}}_{ij}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\phantom{}{}_{\scriptscriptstyle{-2}}Y^{lm}\,m_{i}m_{j}+\phantom{}_{\scriptscriptstyle{2}}Y^{lm}\,\bar{m}_{i}\bar{m}_{j}\right)\,,$
(162) $\displaystyle T^{\scriptscriptstyle{B2}\,\scriptscriptstyle{lm}}_{ij}$
$\displaystyle=-\frac{i}{\sqrt{2}}\left(\phantom{}{}_{\scriptscriptstyle{-2}}Y^{lm}\,m_{i}m_{j}-\phantom{}_{\scriptscriptstyle{2}}Y^{lm}\,\bar{m}_{i}\bar{m}_{j}\right)\,,$
(163)

while the relation to the STF basis is given by

$\displaystyle T^{\scriptscriptstyle{E2}\,\scriptscriptstyle{lm}}_{ij}$
$\displaystyle=\perp_{ijab}\sqrt{\frac{2(l-1)l}{(l+1)(l+2)}}\,\mathcal{Y}_{abL-2}^{lm}\,n_{L-2}\,,$
(164) $\displaystyle T^{\scriptscriptstyle{B2}\,\scriptscriptstyle{lm}}_{ij}$
$\displaystyle=\perp_{ijab}\sqrt{\frac{2(l-1)l}{(l+1)(l+2)}}\,\epsilon_{cd(a}\,\mathcal{Y}_{b)dL-2}^{lm}\,n_{cL-2}\,.$
(165)

###  A.5 Expansion of a Rank-2, TT Tensor

Using the definitions above, we will now relate the expansions of an
arbitrary, rank-2, TT tensor $H_{ij}^{\text{TT}}(u,r,\theta,\phi)$, in terms
of spin-weighted spherical harmonics, pure spin TT tensor harmonics, STF
tensors.

####  A.5.1 Spin-Weighted Spherical Harmonic Expansion

Using the transverse null vector of spinweight $s=-1$

$\bar{m}^{i}\equiv\frac{1}{\sqrt{2}}(u^{i}-iv^{i})\,,$ (166)

we can define the spinweight $s=-2$ scalar quantity

$H\equiv
H_{ij}^{\text{TT}}\,\bar{m}^{i}\,\bar{m}^{j}=H_{ij}\,\bar{m}^{i}\,\bar{m}^{j}\,,$
(167)

which we expand in terms of spin-weighted spherical harmonics as

$H=\sum_{l=2}^{\infty}\sum_{m=-l}^{l}\,H^{lm}\,\phantom{}_{\scriptscriptstyle-2}Y^{lm}\,,$
(168)

where

$\displaystyle
H^{lm}(u,r)=\int_{S^{2}}\differential^{2}\Omega\,\phantom{}_{\scriptscriptstyle-2}\bar{Y}^{lm}\,H\,.$
(169)

####  A.5.2 TT Tensor Expansion

The general expansion in terms of TT tensor harmonics reads

$\displaystyle
H_{ij}^{\text{TT}}=\sum_{l=2}^{\infty}\sum_{m=-l}^{l}\left[U^{lm}\,T^{\scriptscriptstyle{E2}\,\scriptstyle{lm}}_{ij}+V^{lm}\,T^{\scriptscriptstyle{B2}\,\scriptstyle{lm}}_{ij}\right]\,,$
(170)

where

$\displaystyle U^{lm}(u,r)$
$\displaystyle=\int_{S^{2}}\differential^{2}\Omega\,T^{\scriptscriptstyle{E2}\,\scriptstyle{lm}}_{ij}\,H^{\text{TT}}_{ij}\,,$
(171) $\displaystyle V^{lm}(u,r)$
$\displaystyle=\int_{S^{2}}\differential^{2}\Omega\,T^{\scriptscriptstyle{B2}\,\scriptscriptstyle{lm}}_{ij}\,H^{\text{TT}}_{ij}\,.$
(172)

The expansion coefficients $U^{lm}$ and $V^{lm}$ are the electric and magnetic
parity multipole moments respectively, also commonly known as mass and current
multipole moments. In the literature, it is common to explicitly factor out
the $r$ dependence of mass and current multipole moments, which we will not do
here. Since $H^{\text{TT}}_{ij}$ is real, they satisfy

$\displaystyle\bar{U}^{lm}=(-1)^{m}U^{l-m}\,,\quad\bar{V}^{lm}=(-1)^{m}V^{l-m}\,.$
(174)

####  A.5.3 STF Expansion

The corresponding STF multipole expansion is given by

$\displaystyle\begin{split}H_{ij}^{\text{TT}}=4\perp_{ijab}\sum_{l=2}^{\infty}&\frac{1}{l!}\Big{[}\,U_{klL-2}\,n_{L-2}\\\
&+\frac{2l}{l+1}\epsilon_{cd(a}\,V_{b)cL-2}\,n_{dL-2}\Big{]}\,,\end{split}$
(175)

where

$\displaystyle U_{ijL-2}$
$\displaystyle=\frac{1}{4\pi}\frac{l(l-1)(2l+1)!!}{2(l+1)(l+2)}\int\differential^{2}\Omega\,n_{L-2}\,H_{ij}^{\text{TT}}\,,$
(176) $\displaystyle V_{ijL-2}$
$\displaystyle=\frac{1}{4\pi}\frac{(l-1)(2l+1)!!}{4(l+2)}\int\differential^{2}\Omega\,n_{aL-2}\,\epsilon_{iab}H_{bj}^{\text{TT}}\,.$
(177)

####  A.5.4 Relations Between Expansion Coefficients

The multipole moments in a spin-weighted basis are related to the electric-
and magnetic-parity moments as

$H^{lm}=\frac{1}{\sqrt{2}}\left[U^{lm}-iV^{lm}\right]\,,$ (178)

which can be inverted to give

$\displaystyle U^{lm}$
$\displaystyle=\frac{1}{\sqrt{2}}\left[H^{lm}+(-1)^{m}\,\bar{H}^{l-m}\right]\,,$
(179) $\displaystyle V^{lm}$
$\displaystyle=\frac{i}{\sqrt{2}}\left[H^{lm}-(-1)^{m}\,\bar{H}^{l-m}\right]\,.$
(180)

On the other hand, the relation between a STF and a TT tensor basis is given
by

$\displaystyle U^{lm}$
$\displaystyle=\frac{16\pi}{(2l+1)!!}\sqrt{\frac{(l+1)(l+2)}{2(l-1)l}}\,\bar{\mathcal{Y}}_{L}^{lm}\,U_{L}\,,$
(181) $\displaystyle V^{lm}$ $\displaystyle=\frac{-32\pi
l}{(l+1)(2l+1)!!}\sqrt{\frac{(l+1)(l+2)}{2(l-1)l}}\,\bar{\mathcal{Y}}_{L}^{lm}\,V_{L}\,,$
(182) $\displaystyle U_{L}$
$\displaystyle=\frac{l!}{4}\sqrt{\frac{2(l-1)l}{(l+1)(l+2)}}\sum_{m=-l}^{l}\,\mathcal{Y}_{L}^{lm}\,U^{lm}\,,$
(183) $\displaystyle V_{L}$
$\displaystyle=\frac{-(l+1)!}{8l}\sqrt{\frac{2(l-1)l}{(l+1)(l+2)}}\sum_{m=-l}^{l}\,\mathcal{Y}_{L}^{lm}\,V^{lm}\,.$
(184)

##  Appendix B Proof of Equation (91)

In this appendix, we provide a derivation of the relation

$\left[\frac{n^{\prime}_{i}n^{\prime}_{j}}{1-\vec{n}^{\prime}\cdot\vec{n}}\right]^{\text{TT}}=\sum_{l=2}^{\infty}\frac{2(2l+1)!!}{(l+2)!}\left[n_{L-2}n^{\prime}_{\langle
ijL-2\rangle}\right]^{\text{TT}}\,,$ (185)

where a superscript TT implies a contraction of free indices with the TT
projector with respect to $\vec{n}$, which we defined in Eq. (60). The proof
below is based on private communications with Luc Blanchet.

###### Proof.

First, note that due to the TT projection, the left-hand side of Eq. (185)
vanishes for $\vec{n}^{\prime}=\vec{n}$, while this is also true for the
right-hand side. Thus, in the following, we consider the case
$\vec{n}^{\prime}\neq\vec{n}$, which implies
$|\vec{n}^{\prime}\cdot\vec{n}|<1$ such that we can expand the left-hand side
in terms of a geometric series

$\displaystyle\frac{n^{\prime}_{i}n^{\prime}_{j}}{1-\vec{n}^{\prime}\cdot\vec{n}}$
$\displaystyle=n^{\prime}_{i}n^{\prime}_{j}\sum_{l=2}^{\infty}(\vec{n}^{\prime}\cdot\vec{n})^{l-2}=\sum_{l=2}^{\infty}n_{L-2}n^{\prime}_{ijL-2}\,,$
(186)

where we use the notation introduced in Sec. A.3. The remaining task is to
rewrite $n^{\prime}_{ijL-2}$ in terms of its STF part $n^{\prime}_{\langle
ijL-2\rangle}$, by using the formula (see e.g. Eq. (A21a) in [161])

$n_{L}=\sum_{k=0}^{[l/2]}a_{k}^{l}\delta_{\\{2K}n^{\prime}_{\langle
L-2K\rangle\\}}\,,$ (187)

where

$a_{k}^{l}\equiv\frac{(2l-4k+1)!!}{(2l-2k+1)!!}\,,$ (188)

and where $[l/2]$ selects the integer part of $l/2$. Moreover, the operator
$\\{...\\}$ on tensor indices $A_{\\{L\\}}$ denotes the sum $\sum_{\sigma\in
S}A_{i_{\sigma(1)}...i_{\sigma(l)}}$, where $S$ is the smallest set of
permutations of $(1...l)$, which makes $A_{\\{L\\}}$ fully symmetric in $L$.
For instance,

$\begin{split}\delta_{\\{ab}n^{\prime}_{ij\\}}=&\,\delta_{ab}n^{\prime}_{ij}+\delta_{ai}n^{\prime}_{bj}+\delta_{aj}n^{\prime}_{bi}+\delta_{bi}n^{\prime}_{aj}\\\
&+\delta_{bj}n^{\prime}_{ai}+\delta_{ij}n^{\prime}_{ab}\,.\end{split}$ (189)

With this in hand,

$\displaystyle\frac{n^{\prime}_{i}n^{\prime}_{j}}{1-\vec{n}^{\prime}\cdot\vec{n}}=\sum_{l=2}^{\infty}n_{L-2}\sum_{k=0}^{[\frac{l-2}{2}]}a_{k}^{l}\delta_{\\{2K}n^{\prime}_{\langle
ijL-2-2K\rangle\\}}\,.$ (190)

Notice, however, that due to the TT projection with respect to $\vec{n}$ on
the free indices $ij$ and the contraction of the remaining indices with
$n_{L-2}$, any term which involves one of the indices $i$ or $j$ within the
Kronecker delta, hence terms containing $\delta_{ij}$, $\delta_{ia}$ or
$\delta_{ja}$ for any index $a$, will vanish. For example, for the term $l=4$
and $k=1$, written out in Eq. (189), only the first term will survive. Thus

$\displaystyle\left[\frac{n^{\prime}_{i}n^{\prime}_{j}}{1-\vec{n}^{\prime}\cdot\vec{n}}\right]^{\text{TT}}=\sum_{l=2}^{\infty}n_{L-2}\sum_{k=0}^{[\frac{l-2}{2}]}\left[\delta_{2K}n^{\prime}_{\langle
ijL-2-2K\rangle}\right]^{\text{TT}}$
$\displaystyle\quad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\times
a_{k}^{l}\,b_{k}^{l}\,,$
$\displaystyle=\sum_{l=2}^{\infty}\sum_{k=0}^{[\frac{l-2}{2}]}a_{k}^{l}\,b_{k}^{l}\left[n_{L-2-2K}n^{\prime}_{\langle
ijL-2-2K\rangle}\right]^{\text{TT}}\,,$ (191)

where

$b_{k}^{l}\equiv\frac{(l-2)!}{2^{k}k!(l-2-2k)!}\,,$ (192)

is the number of terms within the sum $\delta_{\\{2K}n^{\prime}_{L-2-2K\\}}$.

We can now re-arrange the sum over positive integers $l$ and $k$ as

$\displaystyle\sum_{l=2}^{\infty}\sum_{k=0}^{[\frac{l-2}{2}]}$
$\displaystyle=\sum_{l,k}[2\leq l\leq\infty][0\leq k\leq\frac{l-2}{2}]$
$\displaystyle=\sum_{l,k}[0\leq k\leq\frac{l-2}{2}\leq\infty]$
$\displaystyle=\sum_{p,k}[0\leq k\leq\frac{p+2k-2}{2}\leq\infty]$
$\displaystyle=\sum_{p,k}[2\leq p\leq\infty][0\leq k\leq\infty]\,,$ (193)

where we defined $p\equiv l-2k$ and, in this context, the angular brackets
$[B]$ of a boolean expression denote Inverson brackets363636Inverson Brackets
are defined as $[B]=\begin{cases}1\,&\text{if $B$ is true}\\\
0\,,&\text{otherwise}\end{cases}$., such that

$\displaystyle\left[\frac{n^{\prime}_{i}n^{\prime}_{j}}{1-\vec{n}^{\prime}\cdot\vec{n}}\right]^{\text{TT}}$
$\displaystyle=\sum_{p=2}^{\infty}\sum_{k=0}^{\infty}a_{k}^{p+2k}\,b_{k}^{p+2k}\left[n_{P-2}n^{\prime}_{\langle
ijP-2\rangle}\right]^{\text{TT}}$
$\displaystyle=\sum_{l=2}^{\infty}\underset{\equiv
S_{l}}{\underbrace{\sum_{k=0}^{\infty}a_{k}^{l+2k}\,b_{k}^{l+2k}}}\left[n_{L-2}n^{\prime}_{\langle
ijL-2\rangle}\right]^{\text{TT}}\,,$ (194)

where in the last step we simply relabeled $p\rightarrow l$. The sum over $k$
can indeed be resummed explicitly to give

$\displaystyle S_{l}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{(2l+1)!!}{(2l+2k+1)!!}\,\frac{(l+2k-2)!}{2^{k}k!(l-2)!}$
$\displaystyle=\frac{2^{2+l}\Gamma(l+\frac{3}{2})}{\sqrt{\pi}\Gamma(l+3)}$
$\displaystyle=\frac{2(2l+1)!!}{(l+2)!}\,,$ (195)

where in the last step we used the identities of gamma functions
$\Gamma(l+3)=(l+2)!$ and
$\Gamma(l+\frac{3}{2})=\frac{\sqrt{\pi}(2l+1)!!}{2^{1+l}}$. Comparing to Eq.
(185), this concludes the proof. ∎

##  Appendix C Displacement Memory from BMS Balance-Laws in Brans-Dicke
Gravity

Starting from the action in Eq. (98), the authors in [45, 46, 47, 48] arrive
at the BMS supermomentum flux balance law in spherical coordinates
$\\{u,r,x^{A}\\}$, $x^{A}=\\{\theta,\phi\\}$ (see for instance Eqs. (10)–(12)
in [48], from which we also adopt the notation)

$\begin{split}0=\frac{\varphi_{0}}{4\pi
G}&\int_{S^{2}}\differential^{2}\Omega\,\alpha\Bigg{\\{}\Delta
M+\int_{-\infty}^{\infty}\differential
u^{\prime}\Bigg{[}\frac{1}{2}N_{AB}N^{AB}\\\
&+\mathcal{D}_{A}\mathcal{D}_{B}N^{AB}+(2\omega+3)\left(\frac{N}{\varphi_{0}}\right)^{2}\Bigg{]}\Bigg{\\}}\,,\end{split}$
(196)

where $\mathcal{D}_{A}$ is the covariant derivative on $S^{2}$,
$\alpha=\alpha(\theta,\phi)$ is an arbitrary function on $S^{2}$ parametrizing
supertranslations, and $M$ denotes the Bondi mass aspect, while $\Delta
M\equiv M(u\rightarrow\infty)-M(u\rightarrow-\infty)$. Moreover, as in [48] we
write

$N_{AB}\equiv-\dot{\hat{c}}_{AB}\,,\quad N\equiv\dot{\varphi}_{1}\,,$ (197)

where $\hat{c}_{AB}$ is the symmetric, traceless and transverse shear tensor
and $\varphi_{1}$ the component of the scalar that falls off as $\sim 1/r$. We
additionally expand the shear as

$\hat{c}_{AB}=\bar{c}\,\bar{m}_{A}\bar{m}_{B}+c\,m_{A}m_{B}\,,$ (198)

where in spherical coordinates

$m=\frac{1}{\sqrt{2}}\left(\partial_{\theta}+i\sin\theta\partial_{\phi}\right)\,.$
(199)

Using Eqs. (197) and (198), as well as the definition of the angular
derivative operator [Eq. (147)], which implies that we have
$\mathcal{D}_{A}\mathcal{D}_{B}\hat{c}^{AB}=\frac{1}{2}\left(\eth^{2}c+\bar{\eth}^{2}\bar{c}\right)$,
the flux-balance law in Eq. (196) can be rewritten as

$\displaystyle\begin{split}\int_{S^{2}}\differential^{2}\Omega\,\alpha\,\Delta
M=&\frac{1}{4}\int_{-\infty}^{\infty}\differential
u^{\prime}\int_{S^{2}}\differential^{2}\Omega\,\alpha\Bigg{[}\,|\dot{c}|^{2}\\\
&-\real\eth^{2}\dot{c}+(2\omega+3)\left(\frac{\dot{\varphi}_{1}}{\varphi_{0}}\right)^{2}\Bigg{]}\,.\end{split}$
(200)

To single out the tensor null memory from the above relation, we can first set
the subdominant, left-hand side to zero. This gives rise to the so-called
linear or ordinary memory. We then rewrite the BMS supermomentum balance law
in Eq. (200) by moving the second to last term to the left, while carrying out
the $u^{\prime}$ integral to obtain

$\displaystyle\begin{split}\int_{S^{2}}\differential^{2}\Omega\,\alpha\real\eth^{2}\Delta{c}=\int_{-\infty}^{\infty}&\differential
u^{\prime}\int_{S^{2}}\differential^{2}\Omega\,\alpha\Bigg{[}\,|\dot{c}|^{2}\\\
&+(2\omega+3)\left(\frac{\dot{\varphi}_{1}}{\varphi_{0}}\right)^{2}\Bigg{]}\,.\end{split}$
(201)

Furthermore, expanding $c$ with spinweight $s=-2$ on the left-hand side as

$c(u,\theta,\phi)=\sum_{l=2}^{\infty}\sum_{m=-l}^{l}c_{lm}(u)\,_{\scriptscriptstyle-2}Y^{lm}(\theta,\phi)\,,$
(202)

and using the relation [Eq. (151)] (or equivalently integrating $\eth^{2}$ by
parts),

$\eth^{2}\phantom{}_{\scriptscriptstyle-2}Y^{lm}=\sqrt{\frac{(l+2)!}{(l-2)!}}Y^{lm}$
(203)

as well as choosing $\alpha(\theta,\phi)=\bar{Y}^{lm}(\theta,\phi)$, we obtain

$\displaystyle\begin{split}\frac{1}{2}\big{(}\Delta
c^{lm}+(-1)^{m}&\Delta\bar{c}^{l-m}\big{)}=\sqrt{\frac{(l-2)!}{(l+2)!}}\int_{S^{2}}\differential^{2}\Omega\,\bar{Y}^{lm}\\\
&\times\int_{-\infty}^{\infty}\differential
u^{\prime}\left[\,|\dot{c}|^{2}+(2\omega+3)\left(\frac{\dot{\varphi}_{1}}{\varphi_{0}}\right)^{2}\right]\,.\end{split}$
(204)

Note that by separating $c^{lm}$ into its electric and magnetic parity moments
of a spin-2, TT tensor harmonic expansion of the rank-2 TT tensor
$c_{ij}=e_{i}^{A}e_{j}^{B}\hat{c}_{AB}$ (178)

$c^{lm}=\frac{1}{\sqrt{2}}\left[U_{c}^{lm}-iV_{c}^{lm}\right]\,,$ (205)

the left-hand side in Eq. (204) precisely corresponds to the electric parity
part [Eq. (179)]. Therefore, finally we arrive at

$\displaystyle\begin{split}\Delta
U_{c}^{lm}=&\sqrt{\frac{2(l-2)!}{(l+2)!}}\int_{S^{2}}\differential^{2}\Omega\,\bar{Y}^{lm}\\\
&\times\int_{-\infty}^{\infty}\differential
u^{\prime}\left[\,|\dot{c}|^{2}+(2\omega+3)\left(\frac{\dot{\varphi}_{1}}{\varphi_{0}}\right)^{2}\right]\,.\end{split}$
(206)

From the balance laws, we can therefore single out the total tensor
displacement memory, and hence, the lasting non-zero component within the
electric parity multipole of the shear $c$, which ultimately induces a lasting
offset in the detector strain. Note, however, that the shear and the scalar
field which enter the balance laws are the total shear and scalar field at
$\mathcal{O}(r^{-1})$ within the full non-linear theory, and therefore, in
particular, they already contain all possible memory
contributions373737However, the generalized Bondi News, which enters the
right-hand side of the equation, vanishes as $u\rightarrow\pm\infty$, where by
assumption no gravitational waves reach null infinity.. In order to connect
this result with the computation in Eq. (99) within the setup of the present
work, we should therefore also distinguish between a high, and low frequency
part of the shear and the scalar

$c=c^{L}+c^{H}\,,\quad\varphi_{1}=\varphi_{1}^{L}+\varphi_{1}^{H}\,,$ (207)

and slightly reinterpret the flux balance law in order to use it as a tool to
compute the low-frequency displacement memory, characterized by the measurable
monotonically-increasing and non-oscillatory, time-dependent raise of the
memory, which is what gravitational wave detectors are sensitive to. We
therefore only gradually integrate over retarded time, while extracting the
low-frequency part of the expression by averaging out the small scales. Note
that in order to compute the full memory, such an averaging is irrelevant.

After averaging, any cross-terms of the form “$c^{L}c^{H}$” or
“$\varphi_{1}^{L}\varphi_{1}^{H}$” on the right-hand side in Eq. (206) will
vanish. Moreover, we assume that we can neglect any contribution of low-
frequency components “$c^{L}c^{L}$” or “$\varphi_{1}^{L}\varphi_{1}^{L}$”
which can be interpreted as the “memory of the memory”. In other words, we
assume that the source modes for the memory themselves have a negligible
memory component, which is indeed a reasonable assumption [126]. Furthermore
imposing $c(u\rightarrow-\infty)=0$, we therefore have

$\displaystyle c_{L}^{lm}(u)=$
$\displaystyle\sqrt{\frac{(l-2)!}{(l+2)!}}\int_{S^{2}}\differential^{2}\Omega\,\bar{Y}^{lm}$
(208) $\displaystyle\times\int_{-\infty}^{u}\differential
u^{\prime}\Bigg{\langle}\,|\dot{c}^{H}|^{2}+(2\omega+3)\left(\frac{\dot{\varphi}^{H}_{1}}{\varphi_{0}}\right)^{2}\Bigg{\rangle}\,,$

where $c_{L}^{lm}(u)$ is the resulting low-frequency correction to the shear,
given high-frequency radiation modes $c^{H}$ and $\varphi_{1}^{H}$. We have
here used Eq. (205) with $\delta V_{c}^{lm}=0$ to rewrite Eq. (207) in terms
of the shear.

The reason why such a reinterpretation of the balance law is useful in
practice, is because typical numerical relativity waveforms do not capture any
memory component due to various technical reasons (see e.g. [19]). Only
recently were people able to compute memory in numerical relativity based on
Cauchy-characteristic extraction [162].

As a last step before finally being able to compare results, we need to
connect the perturbative shear and scalar field defined here to the
perturbations used in the main text and ensure that these are indeed the same
quantities. In the case of BD theory, the easiest way to establish this
correspondence is to compare the corresponding leading
$\mathcal{O}\left(r^{-1}\right)$ terms of the electric part of the Riemann
tensor. In [45, 46], these terms were computed and found to be (see e.g. Eq.
(2.44) in [45])383838Note, however, that the authors in [45, 46] report the
result in an orthonormal tetrad basis, instead of the spherical coordinates
employed here.

$R_{uAuB}=-\frac{1}{2r}\left(\ddot{\hat{c}}_{AB}-\gamma_{AB}\frac{\ddot{\varphi}_{1}}{\varphi_{0}}\right)+\mathcal{O}\left(\frac{1}{r^{2}}\right)\,.$
(209)

By using the embedding of the unit $S^{2}$ basis $e^{i}_{A}=\frac{\partial
n^{i}}{\partial x^{A}}$, such that
$\gamma_{AB}=\delta^{ij}e^{i}_{A}e^{j}_{B}=2m_{(A}\bar{m}_{B)}$, we can
convert the leading-order expression to a $\\{t,x,y,z\\}$ Minkowski basis,
which yields

$\displaystyle R_{0i0j}$
$\displaystyle=e_{i}^{A}e_{j}^{B}R_{uAuB}=-\frac{e_{i}^{A}e_{j}^{B}}{2r}\left(\ddot{\hat{c}}_{AB}-\gamma_{AB}\frac{\ddot{\varphi}_{1}}{\varphi_{0}}\right)$
$\displaystyle=-\frac{1}{2r}\bigg{(}\underset{=\frac{1}{2}e^{+}_{ij}\,(\ddot{c}+\ddot{\bar{c}})+\frac{i}{2}e^{+}_{ij}\,(\ddot{c}-\ddot{\bar{c}})}{\underbrace{m_{i}m_{j}\,\ddot{c}+\bar{m}_{i}\bar{m}_{j}\,\ddot{\bar{c}}}}-(\delta_{ij}-n_{i}n_{j})\frac{\ddot{\varphi}_{1}}{\varphi_{0}}\bigg{)}$
$\displaystyle=-\frac{1}{2r}\left(e^{+}_{ij}\,\ddot{c}_{+}+e^{\times}_{ij}\,\ddot{c}_{\times}-(\delta_{ij}-n_{i}n_{j})\frac{\ddot{\varphi}_{1}}{\varphi_{0}}\right)\,,$

where we used that
$e_{i}^{A}e_{j}^{B}\,\gamma_{AB}=2m_{(i}\bar{m}_{j)}=\delta_{ij}-n_{i}n_{j}$,
$e_{i}^{A}m_{A}=m_{i}$ and we defined $c_{+}\equiv\real c$ and
$c_{\times}\equiv-\imaginary c$. Hence, comparing to Eq. (69) with
$\sigma=1/\varphi_{0}$ we obtain the correspondence

$\displaystyle c^{H}_{+}(u,\Omega)$
$\displaystyle=\lim_{r\rightarrow\infty}rh_{+}(u,r,\Omega)\,,$ (210)
$\displaystyle c^{H}_{\times}(u,\Omega)$
$\displaystyle=\lim_{r\rightarrow\infty}rh_{\times}(u,r,\Omega)\,,$ (211)
$\displaystyle\varphi^{H}_{1}(u,\Omega)$
$\displaystyle=\lim_{r\rightarrow\infty}r\varphi(u,r,\Omega)\,,$ (212)

while therefore as well

$c_{L}^{lm}=\lim_{r\rightarrow\infty}r\delta h^{lm}\,,$ (213)

such that Eq. (208) indeed corresponds to the result in Eq. (100).

##  Appendix D Tensor Null Memory in a Gauge-Invariant $k$-Form Metric Theory

In order to exemplify how generic the assumption given by Eq. (138) is, we
present in this appendix the general form of the tensor null memory of a
particular, but fairly general class of additional gravitational fields.
Namely, we consider an arbitrary number of additional dynamical $k$-form
connection fields that we schematically denote as $\Psi$, with an associated,
Abelian, gauge symmetry for $k>1$ (for $k=0$, the field simply corresponds to
a scalar field). The action in Eq. (117) is thus constructed out of curvature
invariants of the metric, as well as field-strengths $\mathscr{F}\equiv d\Psi$
that are invariant under Abelian gauge transformations

$\Psi\rightarrow\Psi+d\Lambda\,,$ (214)

where $\Lambda$ are arbitrary $(k-1)$-forms and $d$ is the exterior
derivative. These dynamical k-form fields are assumed to describe $N$
additional propagating gravitational degrees of freedom. Note that in a local
chart-induced basis, a $k$-form field reads

$\begin{split}\Psi&=\frac{1}{k!}\Psi_{\mu_{1}\mu_{2}...\mu_{k}}\,dx^{\mu_{1}}\wedge
dx^{\mu_{2}}\wedge...\wedge dx^{\mu_{k}}\,,\\\
&\equiv\frac{1}{k!}\Psi_{K}\,\differential X^{K}\,.\end{split}$ (215)

The assumptions of Theorem 1 are now natural. In particular, the leading-order
propagation equations [Eq. (124) of the theory reduce to a set of decoupled
wave-equations for the leading-order waves $\Psi^{H}_{K}$, upon imposing the
analog of the Lorentz gauge

$\partial^{\mu_{i}}\hat{\Psi}_{K}^{H}=0\,,$ (216)

on each of the field-perturbations. This is because, by assumption, the theory
is covariant, locally Lorentz-invariant and massless. Moreover, any term in
the action that would lead to higher-order derivatives in the equations of
motion of the leading-order waves does not contribute to leading-order, in the
small-coupling approximation [Eq. (24)]. An additional assumption here is
that, in the limit to null infinity, there is no coupling between different
fields at the level of the leading-order, high-frequency equation.

Following the proof of Theorem 1, the flat, effective, second-order action of
Eq. (134) can thus be written as

$\displaystyle\begin{split}\phantom{}{}_{\scriptscriptstyle{(}2)}S^{\scriptscriptstyle{flat}}_{\text{eff}}=&\frac{-1}{2\kappa_{\text{eff}}}\int\differential^{4}x\,\sqrt{-\eta}\bigg{[}\hat{h}^{\mu\nu}_{H}\mathcal{E}^{\alpha\beta}_{\mu\nu}\hat{h}^{H}_{\alpha\beta}\\\
&+\sum_{\Psi}\frac{1}{2q}\eta^{\mu_{1}\nu_{1}}...\eta^{\mu_{q}\nu_{q}}\hat{\mathscr{F}}^{H}_{\mu_{1}...\mu_{q}}\hat{\mathscr{F}}^{H}_{\nu_{1}...\nu_{q}}\bigg{]}\,,\end{split}$
(217)

where $q\equiv k+1$, $\kappa_{\text{eff}}=\kappa_{0}A(\eta_{0})$ and
$\hat{\mathscr{F}}^{H}=d\hat{\Psi}^{H}$ are the field strength of the
canonically normalized $k$-form perturbations $\hat{\Psi}^{H}$. The first term
in Eq. (217) is again the usual Fierz-Pauli operator with indices contracted
with the independent background metric $\eta_{\mu\nu}$, while the second term
corresponds to the sum of kinetic terms of all the additional gravitational,
gauge-invariant, $k$-form field perturbations.

With this explicit form of the flat effective action at hand, we can now
explicitly compute the resulting effective stress-energy tensor for the
additional leading-order waves (see also [163] for a study of the properties
of the energy momentum tensor of k-forms)

$\displaystyle\begin{split}\phantom{}{}_{\scriptscriptstyle{(}2)}t_{\mu\nu}^{\hat{\Psi}}=\frac{1}{2\kappa_{\text{eff}}}\sum_{\Psi}\bigg{\langle}&\hat{\mathscr{F}}^{H}_{\mu
K}\hat{\mathscr{F}}_{\nu}^{HK}-\frac{1}{2q}\eta_{\mu\nu}\hat{\mathscr{F}}^{H}_{Q}\hat{\mathscr{F}}^{HQ}\bigg{\rangle}\,.\end{split}$
(218)

Imposing the equations of motion [Eq. (126)], as well as the Lorenz gauge [Eq.
(216)], while recalling that the averaging allows for integrations by parts,
the only surviving terms are

$\displaystyle\phantom{}_{\scriptscriptstyle{(}2)}t_{\mu\nu}^{\hat{\Psi}}=\frac{1}{2\kappa_{\text{eff}}}\sum_{\Psi}\bigg{\langle}\partial_{\mu}\hat{\Psi}^{H}_{K}\partial_{\nu}\hat{\Psi}^{KK}\bigg{\rangle}\,.$
(219)

Including the Fierz-Pauli result, the total energy-momentum tensor therefore
reads

$\displaystyle\phantom{}_{\scriptscriptstyle{(}2)}t_{\mu\nu}=\frac{1}{2\kappa_{\text{eff}}}\bigg{\langle}$
$\displaystyle\frac{1}{2}\partial_{\mu}\hat{h}^{H}_{\alpha\beta}\partial_{\nu}\hat{h}^{H\alpha\beta}+\sum_{\Psi}\partial_{\mu}\hat{\Psi}^{H}_{K}\partial_{\nu}\hat{\Psi}^{HK}\bigg{\rangle}\,.$
(220)

This (pseudo)energy-momentum tensor is indeed conserved (and traceless), as
well as gauge invariant (both under linearized diffeomorphisms and gauge
transformations of the $k$-form fields, upon imposing a Lorenz gauge).

The above also means that the (pseudo)energy-momentum tensor can be written in
terms of the radiative modes of each field, which are the solutions to the
wave equations that we can expand in terms of a polarization basis. For the
metric perturbation, we have the usual TT tensor modes of the physical metric

$\hat{h}^{H\text{TT}}_{\mu\nu}=h^{H\text{TT}}_{\mu\nu}=\epsilon^{+}_{\mu\nu}\,\hat{h}_{+}+\epsilon^{\times}_{\mu\nu}\,\hat{h}_{\times}\,,$
(221)

where

$\epsilon^{+}_{\mu\nu}\epsilon^{+\mu\nu}=\epsilon^{\times}_{\mu\nu}\epsilon^{\times\mu\nu}=2\,,\quad\epsilon^{+}_{\mu\nu}\epsilon^{\times\mu\nu}=0\,.$
(222)

For the additional gravitational fields, $\Psi$, by assumption they describe
$N$ additional radiative modes $\psi_{\lambda}$, where $\lambda=1,..,N$. These
can be written in some polarization basis as393939A practical way of
completely fixing the gauge is the use of spacetime light-cone coordinates, as
exemplified for instance in [138] for 2-forms. These polarizations should,
however, not be confused with the gravitational polarizations of the physical
metric defined in Sec. III.2.2.

$\hat{\Psi}^{p}_{K}=\sum_{\text{P}}\epsilon^{\text{P}}_{K}\,\hat{\psi}_{\text{P}}\,,\quad\text{where}\quad\epsilon^{\text{P}}_{K}\epsilon^{\text{P}^{\prime}K}=\delta^{\text{PP}^{\prime}}\,.$
(223)

These solutions to the wave equation in the limit to null infinity take the
form (cf. Eq. (62))

$\displaystyle\left(\hat{h}_{+}\,,\;\hat{h}_{\times}\,,\;\hat{\psi}_{\lambda}\right)\sim\frac{1}{r}\left[f_{+}(u,\theta,\phi),f_{\times}(u,\theta,\phi),f_{\lambda}(u,\theta,\phi)\right]\,.$
(224)

for some real functions $f_{+,\times,\lambda}$. Together with the form of the
energy momentum tensor Eq. (220) this behavior near null infinity implies that
the radiative energy momentum tensor can indeed be written as (see Sec. IV.2)

$\phantom{}_{\scriptscriptstyle{(}2)}t_{\mu\nu}=\frac{1}{2\kappa_{\text{eff}}}\bigg{\langle}|\dot{\hat{h}}_{+}|^{2}+|\dot{\hat{h}}_{\times}|^{2}+\sum_{\lambda=1}^{N}|\dot{\hat{\psi}}_{\lambda}|^{2}\bigg{\rangle}\,\ell_{\mu}\ell_{\nu}\,,$
(225)

where $\ell_{\mu}=-\nabla_{\mu}t+\nabla_{\mu}r$ and the falloffs of Eq. (224)
impose the scaling $t_{\mu\nu}\sim r^{-2}$.

Following Sec. V.3, the tensor null memory formula therefore takes the form

$\begin{split}&\delta
h_{H}^{lm}(u,r)=\frac{1}{r}\sqrt{\frac{(l-2)!}{(l+2)!}}\int_{S^{2}}\differential^{2}\Omega^{\prime}\,\bar{Y}^{lm}(\Omega^{\prime})\\\
&\quad\times\,\int_{-\infty}^{u}\differential
u^{\prime}\,r^{2}\bigg{\langle}|\dot{\hat{h}}_{+}|^{2}+|\dot{\hat{h}}_{\times}|^{2}+\sum_{\lambda=1}^{N}|\dot{\hat{\psi}}_{\lambda}|^{2}\bigg{\rangle}\,.\end{split}$
(226)

Note that the $\hat{\psi}$ here are the canonically normalized modes. The
coupling constants of a specific theory will then enter the memory formula by
transforming back to the physical modes.

The example presented here could be readily generalized by the inclusion of
non-Abelian, 1-form gauge fields with a simple and compact but otherwise
arbitrary gauge group and field strength $d\mathrm{F}=d\Psi+\Psi\wedge\Psi$.
Such theories are a natural generalization of the SVT theory considered in
this work, but would not change the result significantly, as the background
solution requires a vanishing vector field in this case. Moreover, just as for
massless Abelian fields, no higher-order self-interaction terms exist in
$d=4$, which would still lead to second-order equations of motion [141].
Similarly, in the same way as Abelian 1-forms can be generalized to non-
Abelian gauge groups, $k$-forms have non-Abelian generalizations, which
typically require the use of gerbes (see e.g. [164, 165]). We conjecture, that
also in such a general framework an adapted version of the memory equation
[Eq. (144)] should hold.

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