# Unifying Ordinary and Null Memory

Lavinia Heisenberg<EMAIL_ADDRESS>Institut für
Theoretische Physik, Philosophenweg 16, 69120 Heidelberg, Germany  Guangzi Xu
<EMAIL_ADDRESS>Institute for Theoretical Physics, ETH Zürich,
Wolfgang-Pauli-Strasse 27, 8093, Zürich, Switzerland  Jann Zosso
<EMAIL_ADDRESS>Institute for Theoretical Physics, ETH Zürich, Wolfgang-
Pauli-Strasse 27, 8093, Zürich, Switzerland

(September 3, 2024)

###### Abstract

Based on a recently proposed reinterpretation of gravitational wave memory
that builds up on the definition of gravitational waves pioneered by Isaacson,
we provide a unifying framework to derive both ordinary and null memory from a
single well-defined equation at leading order in the asymptotic expansion.
This allows us to formulate a memory equation that is valid for any unbound
asymptotic energy-flux that preserves local Lorentz invariance. Using
Horndeski gravity as a concrete example metric theory with an additional
potentially massive scalar degree of freedom in the gravitational sector, the
general memory formula is put into practice by presenting the first account of
the memory correction sourced by the emission of massive field waves.
Throughout the work, physical degrees of freedom are identified by
constructing manifestly gauge invariant perturbation variables within an SVT
decomposition on top of the asymptotic Minkowski background, which will in
particular prove useful in future studies of gravitational wave memory within
vector tensor theories.

##  I Introduction

The revolution in gravity physics at the beginning of the last century was
marked by a tremendous advance in the conception of the notions of space and
time. Using worldlines of free test particles as basic probes of spacetime,
the statement of the equivalence principle that fundamentally there exist no
test particles free of the gravitational interactions entails the far-reaching
consequence of viewing spacetime as a curved and dynamical concept. More
precisely, inertial observers of special relativity tracing out the flat and
rigid fabric of Minkowski spacetime are to be replaced by the notion of freely
falling frames, only locally recovering the Minkowski structure. Famously,
Einstein’s theory of general relativity (GR) emerged as the preeminent
framework for understanding gravity.

Yet, despite the remarkable success of GR, there remain several open
questions, in particular in the context of cosmology and the unification with
quantum physics, that drive the exploration of the theory space beyond GR.
Interestingly, the Einstein equivalence principle together with locality,
covariance, and a restriction to four space-time dimensions leaves room for a
broader theoretical framework beyond GR that might be worth exploring in this
context. While GR is identified through the Lovelock theorem [1] as the unique
local and covariant low energy gravity theory constructed out of a single
gravitational metric on a manifold, the larger theory space of metric theories
of gravity [2, 3, 4, 5, 6, 7, 8, 9] is characterized through the possibility
of describing additional degrees of freedom in the gravitational sector. In
doing so, it is however of imminent importance to maintaining the requirement
of the existence of a single physical metric that minimally and universally
couples to matter, in order to abide to the Einstein equivalence principle.

One popular alternative gravity theory is given by the Horndeski action [10,
11, 12, 13, 6, 14], representing the most general metric theory incorporating
an extra scalar field in the gravitational sector, with the additional
property of exhibiting equations of motion at most at second-order in
derivatives per field. This supplementary requirement ensures the absence of
any Ostrogradski instability [15, 16]. While such an extra scalar
gravitational degree of freedom naturally finds its application in cosmology,
its traces should equally be searched for within toady’s and tomorrow’s
gravitational wave data.

Indeed, the groundbreaking first direct discovery of a gravitational wave
signal [17, 18] opened a new window for testing the nature of the
gravitational interaction in entirely new regimes. In this context, the
framework of metric theories provides a viable interpretation of the
measurements on the basis of a well-defined geodesic deviation equation
governed by the electric part of the Riemann tensor associated to the physical
metric. In particular, metric theories of gravity therefore also lay out the
theoretical basis for discussing the gravitational wave memory effect beyond
general relativity [9].

Within GR, the memory effect, defined as a permanent geodesic displacement
after a gravitational wave burst, was first discovered within the linearized
theory, arising due to the presence of unbound matter components within an
otherwise isolated system that is emitting gravitational waves [19, 20, 21,
22]. It was subsequently understood that the gravitational waves themselves
represent an unbound energy component that results in a memory correction when
considering the full non-linear theory in asymptotically flat spacetime [23,
24, 25, 26, 27, 28] (see also [29, 30, 31, 32, 33, 34, 35, 36, 37, 38]). With
a precise notion of asymptotic null infinity at hand, it has proven useful to
let go of the historic classification between “non-linear” and “linear”
memory, in favor of a distinction between null and ordinary memory, denoting
memory components that are sourced by unbound energy-momentum fluxes that do,
respectively don’t reach the null boundary at asymptotic infinity [33]. More
precisely, the Bondi-Metzner-Sachs (BMS) flux-balance laws [39, 40, 41, 42]
naturally distinguish between massless memory source-fluxes traveling at the
speed of light and massive unbound components of the system [23, 26, 34, 43,
44, 45, 35, 46, 38].

Excitingly, the first detection of the gravitational wave memory effect is
soon to be expected [31, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
60], at the very least with next-generation ground based detectors such as the
Einstein Telescope [61, 62] or Cosmic Explorer [63, 64], as well as with the
Laser Interferometer Space Antenna (LISA) mission [65]. This in particular
also motivates the search for a deeper understanding of memory beyond GR to
harvest the full potential of gravitational memory in probing the theory of
gravity and its implications on the fabric of spacetime.

Here, we will build up on the novel perspective on displacement memory
provided in [9], which identified the Isaacson approach to gravitational waves
[66, 67] as the ideal framework to study gravitational wave memory within
generic metric theories. While [9] primarily focused on null memory by only
considering massless field perturbations, the present work will provide a
generalization to massive degrees of freedom in the gravitational sector in
the particular case of Horndeski gravity. This will not only provide the first
description of displacement memory that is sourced by the emission of a
massive scalar wave, but also demonstrate a unified derivation of ordinary and
null memory within the Isaacson picture.

After introducing the Horndeski metric theory, Sec. II will review the leading
order Isaacson equations of motion, with an emphasis on the importance of a
well-defined perturbation theory setup that might have been obscured in the
original work. Furthermore, we choose to resolve the associated gauge
ambiguity by explicitly identifying the gauge invariant variables within a
scalar-vector-tensor (SVT) decomposition of the field perturbations within an
assumed asymptotically flat region that preserves Lorentz invariance.
Subsequently, Sec. III introduces the geodesic deviation equation in generic
metric theories of gravity that is both at the root of an unambiguous
gravitational wave response, as well as the definition of gravitational wave
memory. Moreover, we seize the opportunity to offer a careful discussion of
gravitational wave polarizations of Horndeski theory within the SVT
decomposition that to our knowledge is hard to come by in the literature.
Finally, Sec. IV will dig into Isaacson’s low-frequency back-reaction
equation, providing a unified treatment of null and ordinary memory, including
a description of gravitational wave memory originating from the emission of a
massive Horndeski scalar degree of freedom in the gravitational sector.

In this work, we choose our metric signature to be $(-,+,+,+)$. Moreover, we
will use natural units in which $c=1$.

##  II Gravitational Waves in Horndeski Theories

###  II.1 Basics of Horndeski Gravity

Horndeski theory represents the most general scalar-tensor metric theory of
gravity with equations of motion that include only up to two derivative
operators per field, constructed out of a metric tensor $g_{\mu\nu}$ and a
real-valued scalar field $\phi$ [10, 11, 12, 13, 6, 14]. The action of
Horndeski gravity can be written in the following form [14]

$\displaystyle S=\frac{1}{2\kappa}\int
d^{4}x\sqrt{-g}\left(\sum_{i=2}^{5}\mathcal{L}_{i}[g,\phi]\right)+S_{\text{m}}[g,\Psi_{\text{m}}]\,,$
(1)

with $\kappa\equiv 8\pi G$ and where the Lagrangian’s $\mathcal{L}_{i}$ read

$\displaystyle\mathcal{L}_{2}=$ $\displaystyle\text{ }G_{2}(\phi,X)\,,$
$\displaystyle\mathcal{L}_{3}=$ $\displaystyle\text{
}-G_{3}(\phi,X)\Box\phi\,,$ $\displaystyle\mathcal{L}_{4}=$
$\displaystyle\text{
}G_{4}(\phi,X)R+G_{4,X}\left[(\Box\phi)^{2}-\nabla_{\rho}\nabla_{\sigma}\phi\nabla^{\rho}\nabla^{\sigma}\phi\right]\,,$
$\displaystyle\mathcal{L}_{5}=$ $\displaystyle\text{
}G_{5}(\phi,X)G_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\phi$
$\displaystyle-\frac{1}{6}G_{5,X}\left[(\Box\phi)^{3}+2(\nabla_{\rho}\nabla^{\sigma}\phi)(\nabla_{\gamma}\nabla^{\rho}\phi)(\nabla_{\sigma}\nabla^{\gamma}\phi)\right.$
$\displaystyle\left.-3(\Box\phi)(\nabla_{\rho}\nabla_{\sigma}\phi)(\nabla^{\sigma}\nabla^{\rho}\phi)\right]\,.$
(2)

The functions $G_{i}$’s are arbitrary scalar functions depending on the
scalars $\phi$ and $X\equiv-\nabla_{\mu}\phi\nabla^{\mu}\phi$, and
$G_{i,Z}\equiv\partial G_{i}/\partial Z$ denotes partial derivatives of these
scalar functions. Moreover, we further define
$\Box\phi\equiv\nabla_{\mu}\nabla^{\mu}\phi$ and
$S_{\text{m}}[g,\Psi_{\text{m}}]$ denotes the matter action in terms of
generic matter fields $\Psi_{\text{m}}$ that are minimally coupled to the
physical metric $g$ only.

As already mentioned, this last requirement is crucial to ensure the metric
theory character of Horndeski gravity that is for instance also crucial for
the existence of a locally conserved matter energy-momentum tensor

$T_{\mu\nu}[g,\Psi_{\text{m}}]\equiv\frac{-2}{\sqrt{-g}}\frac{\delta
S_{\text{m}}[g,\Psi_{\text{m}}]}{\delta g^{\mu\nu}}\,.$ (3)

Indeed, minimal and universal coupling together with the diffeomorphism
invariance of the action ensures its covariant conservation

$\nabla_{\mu}T^{\mu\nu}=0\,.$ (4)

This matter energy-momentum tensor in particular represents a source term for
the fields in the gravitational sector within the corresponding equations of
motion. For notational clarity, we will denote the metric and scalar field
equations as

$\displaystyle\mathcal{G}_{\mu\nu}[g,\phi]$
$\displaystyle=\kappa\,T_{\mu\nu}[g,\Psi_{m}]\,,$ (5a)
$\displaystyle\mathcal{J}[g,\phi]$ $\displaystyle=0\,.$ (5b)

The explicit form of the equations of motion can for instance be found in the
Appendix of [68].

As the most general scalar-tensor theory with second-order equations of
motion, Horndeski theory encompasses many specific gravity theories beyond GR.
Classic examples are Brans-Dicke (BD) theory [69, 70, 71, 3] that is recovered
through the specific choice of functionals

$G_{2}=\frac{\omega X}{\phi}\,,\;\;G_{4}=\phi\,,\;\;G_{3}=G_{5}=0\,,$ (6)

as well as $f(R)$ gravity [72, 73, 74, 75, 76] that is equivalent to choosing

$G_{2}=f(\phi)-\phi
f^{\prime}(\phi)\,,\;\;G_{4}=f^{\prime}(\phi)\,,\;\;G_{3}=G_{5}=0\,,$ (7)

and scalar Gauss-Bonnet (sGB) gravity [77, 78] arising from the combination
[68, 14]

$\displaystyle G_{2}$ $\displaystyle=X+8f^{(4)}(\phi)X^{2}(3-\ln X),$
$\displaystyle G_{3}$ $\displaystyle=4f^{(3)}(\phi)X(7-3\ln X),$
$\displaystyle G_{4}$ $\displaystyle=1+4f^{(2)}(\phi)X(2-\ln X),$
$\displaystyle G_{5}$ $\displaystyle=-f^{(1)}(\phi)\ln X,$ (8)

where $f^{(n)}\equiv\partial^{n}f/\partial\phi^{n}$ is the derivative of some
scalar function $f(\phi)$.

###  II.2 Isaacson’s Definition of Gravitational Waves

In practice, gravitational waves are usually defined within the framework of
linear perturbation theory (see e.g. [79, 80, 81, 82, 83]) that in the context
of Horndeski theory assumes an exact solution to the field equations
$\bar{g}_{\mu\nu}$ and $\bar{\phi}$ for some given energy-momentum tensor
written in some chart. Based on such an exact solution, more complex problems
can be studied by considering small deviations as approximate solutions of the
form

$g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}\,,\quad\phi=\bar{\phi}+\varphi\,,$ (9)

where

$|\bar{g}_{\mu\nu}|\,,\;|\bar{\phi}|\sim
1\,,\quad|h_{\mu\nu}|\,,\;|\varphi|\ll 1\,.$ (10)

These metric and scalar field perturbations are potentially but not
necessarily associated to perturbations in the matter distribution

$\Psi_{\text{m}}=\bar{\Psi}_{\text{m}}+\psi_{\text{m}}\,.$ (11)

The diffeomorphism invariance of metric theories defined on a manifold $M$
then translates into a gauge freedom of any perturbed field

$\Psi=\bar{\Psi}+\psi\,,$ (12)

on the manifold of the form

$\displaystyle\psi$
$\displaystyle\rightarrow\psi-\mathscr{L}_{\xi}\bar{\Psi}\,.$ (13)

with $\mathscr{L}_{\xi}$ the Lie derivative along a small vector field
$\xi^{\mu}$. This in particular includes the metric and scalar perturbations
$h_{\mu\nu}$ and $\varphi$, for which the gauge transformations become111Note
that the gauge freedom is larger than mere linearized coordinate
transformations, as is evident by the fact that also a scalar field
transforms.

$\displaystyle\mathscr{L}_{\xi}\bar{g}_{\mu\nu}$
$\displaystyle=\left(\xi^{\alpha}\partial_{\alpha}\bar{g}_{\mu\nu}+2\,\bar{g}_{\alpha(\nu}\partial_{\mu)}\xi^{\alpha}\right)\,,$
(14) $\displaystyle\mathscr{L}_{\xi}\bar{\phi}$
$\displaystyle=\xi^{\alpha}\partial_{\alpha}\bar{\phi}\,.$ (15)

The perturbative approach further allows a separation of the field equations
[Eq. (5)] into a background equation

$\displaystyle\mathcal{G}_{\mu\nu}[\bar{g},\bar{\phi}]$
$\displaystyle=\bar{\kappa}\,T_{\mu\nu}[\bar{g},\bar{\Psi}_{m}]\,,$ (16a)
$\displaystyle\mathcal{J}[\bar{g},\bar{\phi}]$ $\displaystyle=0\,,$ (16b)

and the equations of motion of the perturbations

$\displaystyle\sum_{i=1}\phantom{}{}_{\scriptscriptstyle{(i)}}\mathcal{G}_{\mu\nu}[h,\varphi]$
$\displaystyle=\kappa\,\sum_{i=1}\phantom{}{}_{\scriptscriptstyle{(i)}}T_{\mu\nu}[h,\psi_{\text{m}}]\,,$
(17a)
$\displaystyle\sum_{i=1}\phantom{}{}_{\scriptscriptstyle{(i)}}\mathcal{J}[h,\varphi]$
$\displaystyle=0\,,$ (17b)

where $\phantom{}{}_{\scriptscriptstyle{(i)}}O[A,B,\dots]$ denotes the $i$th
order expansion of the operator $O$ in the corresponding perturbation fields
$\\{A,B,\dots\\}$, and we omit the explicit dependence on the background.
Choosing the exact solution to be either flat Minkowski space far away from
any matter source or a homogeneous and isotropic cosmological spacetime, the
above equations allow the study of propagating waves defined as the dynamical
solutions of perturbations on top of the given exact solution.

However, strictly speaking a perturbative description of gravitational waves
is in general only well-defined if an additional assumption is made which is
often not stated explicitly [66, 67, 79, 84, 85, 9]: The definition of
gravitational waves propagating on top of a background spacetime requires a
clear separation of typical scales of variation, for instance in terms of a
low and high-frequency scale $f_{L}$ and $f_{H}$ respectively, where

$f_{L}\ll f_{H}\,.$ (18)

Only such a parametric separation of physical scales leads to an unambiguous
notion of dynamical waves propagating on a slowly varying or static background
with a meaningful association of a localized energy-momentum of the
propagating waves. Observe that for the simplest examples of a fixed Minkowski
or cosmological background, such an assumption is implicitly satisfied.

Yet, the explicit consideration of the so-called Isaacson assumption in Eq.
(18) turns out to be important regardless of the simplicity of the background.
This is foremost due to the intrinsic non-linearity of gravitation, implying
that any localized energy-momentum content of gravitational waves will
inevitably introduce corrections to the low-frequency background, regardless
of the exact solution about which one is perturbing. More precisely, a purely
linear treatment of gravitational waves fundamentally misses out on an
additional non-negligible contribution that in particular in the
asymptotically flat limit precisely corresponds to a gravitational wave memory
effect [23, 9]. In this work, we will show that a careful analysis of the
Isaacson approach to gravitational waves not only captures such a non-linear
memory effect of the gravitational null radiation, but is able to describe
both null and ordinary memory of any unbound energy-momentum flux in a unified
framework.

In the following, we will therefore assume that any field variable can be
split into a dependence on the low- and high-frequency scales, that will be
denoted by a corresponding super- or subscript $L$ and $H$. In most cases of
interest, the variables of the exact solution about which one is perturbing
are either static or slowly varying, such that
$\bar{g}_{\mu\nu}=\bar{g}_{\mu\nu}^{L}$ and $\bar{\phi}=\bar{\phi}^{L}$. On
the other hand, no assumption can be made on the perturbations, which
therefore admit a split

$\displaystyle h_{\mu\nu}$ $\displaystyle=h_{\mu\nu}^{L}+h_{\mu\nu}^{H}\,,$
(19a) $\displaystyle\varphi$ $\displaystyle=\varphi^{L}+\varphi^{H}\,,$ (19b)
$\displaystyle\psi_{\text{m}}$
$\displaystyle=\psi_{\text{m}}^{L}+\psi_{\text{m}}^{H}\,.$ (19c)

The potential presence of such low-frequency perturbations associated to the
slowly-varying background has important consequences. This is foremost due to
the fact that accounting for the typical scales of variation introduces the
additional small parameter $f_{L}/f_{H}\ll 1$, implying that Eqs. (17) can in
principle not simply be solved order by order in the label $(i)$. Moreover,
the perturbed equations of motion further admit a split into a low- and high-
frequency set of relations. In other words, starting from Eqs. (17) there are
two distinct sets of equations, whose leading order require a departure from
pure linear perturbation theory.

The general strategy for obtaining such a decomposition into low- and high-
frequency parts is via a (space-time) average that we will denote by
$\langle...\rangle$, whose details are not important as long as it satisfies a
list of criteria and in particular allows for integrations by parts of
derivative operators. Detailed requirements and properties of the averaging
operator can be found in [86]. The low-frequency contribution of any operator
$O$ is therefore given by

$\left[O\right]^{L}=\langle O\rangle\,,$ (20)

while the corresponding high-frequency part simply reads

$\left[O\right]^{H}=O-\langle O\rangle\,.$ (21)

###  II.3 The Leading Order Equations in the Asymptotically Flat Limit

In this work, we will refrain from a general treatment of the associated
equations and will directly analyze the simplest situation by considering the
asymptotically flat region only. For more details, we refer to the treatment
in [9].

In this case, we assume the existence of a source centered coordinate system
$\\{t,r,\theta,\phi\\}$, where we will often collectively denote the angles as
$\Omega=\\{\theta,\phi\\}$, in which the metric and the scalar field asymptote
towards a Minkowski background $\bar{g}_{\mu\nu}=\eta_{\mu\nu}$ and
$\bar{\phi}=\text{constant}$ with associated $\mathcal{O}(1/r)$ corrections
that naturally represent the perturbed quantities. This requires that at the
level of the exact solution the matter background vanishes in the asymptotic
limit $\bar{\Psi}_{\text{m}}=0$, with

$T_{\mu\nu}[\eta,\bar{\Psi}_{m}=0]=0\,,$ (22)

while asymptotic flatness further requires that any energy-momentum tensor is
localized enough such that
$\phantom{}{}_{\scriptscriptstyle{(1)}}T_{\mu\nu}=0$.

Thus, the assumption of an asymptotically flat spacetime naturally selects an
exact Minkowski solution $\\{\eta_{\mu\nu},\bar{\phi}\\}$ that preserves local
Lorentz invariance, about which one can consider perturbations associated to
the $\mathcal{O}(1/r)$ corrections. We then assume that the bulk motion of a
typical source of gravitational waves produces an asymptotic radiation of
typical characteristic frequency, setting the scale of $f_{H}$ of the problem.
While any localized source of course also entails static $\mathcal{O}(1/r)$
corrections to the asymptotic fields, these contributions remain non-dynamical
to a first approximation and are therefore uninteresting for any gravitational
wave experiment [79, 84]. In such a situation, any potential presence of
propagating low-frequency perturbations therefore only arise at the non-linear
level. We therefore assume the presence of so-called primary waves described
by the quantities $h^{H}_{\mu\nu}$ and $\varphi^{H}$ and solve Eqs. (16) and
(17).

In this case, the purely low-frequency equations for the exact Minkowski
background [Eqs. (16)] in Horndeski theory become

$\displaystyle\langle\mathcal{G}_{\mu\nu}[\eta,\bar{\phi}]\rangle$
$\displaystyle=\frac{1}{2}\eta_{\mu\nu}G_{2}(\bar{\phi},0)=0\,,$ (23a)
$\displaystyle\langle\mathcal{J}[\eta,\bar{\phi}]\rangle$
$\displaystyle=G_{2,\phi}(\bar{\phi},0)=0\,.$ (23b)

Given our assumptions on the background fields, these background equations
translate to two conditions on the functional form of the lowest-order free
functionals

$\bar{G}_{2}=0\,,\quad\bar{G}_{2,\phi}=0\,,$ (24)

where we defined the shorthand notation $\bar{G}_{i}\equiv
G_{i}(\bar{\phi},0)$.

At the next order, the leading order high-frequency equations within [Eqs.
(17)] provide a propagation equations for the primary waves that still remains
at the linear order in perturbations (see also [9]). In the case of Horndeski
theory, these are of the form

$\displaystyle\left[\phantom{}{}_{\scriptscriptstyle{(1)}}\mathcal{G}_{\mu\nu}\right]^{H}$
$\displaystyle=\phantom{}_{\scriptscriptstyle{(1)}}\mathcal{G}_{\mu\nu}[h^{H},\varphi^{H}]=0$
(25a) $\displaystyle=\bar{G}_{4,\phi}$
$\displaystyle\,(\partial_{\mu}\partial_{\nu}\varphi^{H}-\eta_{\mu\nu}\Box\varphi^{H})-\phantom{}_{\scriptscriptstyle{(1)}}G_{\mu\nu}[h^{H}]\,\bar{G}_{4}\,,$
$\displaystyle\left[\phantom{}{}_{\scriptscriptstyle{(1)}}\mathcal{J}\right]^{H}$
$\displaystyle=\phantom{}_{\scriptscriptstyle{(1)}}\mathcal{J}[h^{H},\varphi^{H}]=0$
(25b) $\displaystyle=\varphi^{H}\bar{G}$
${}_{2,\phi\phi}+\Box\varphi^{H}(\bar{G}_{2,X}-2\bar{G}_{3,\phi})+\phantom{}_{\scriptscriptstyle{(1)}}R[h^{H}]\bar{G}_{4,\phi}\,,$

where $R[g]$ denotes the Ricci scalar, while

$G_{\mu\nu}[g]\equiv R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\,,$ (26)

is the Einstein tensor with respect to the metric $g_{\mu\nu}$.

On the other hand, it is at the low-frequency level of the equations of motion
where the linear order in high-frequency perturbations needs to be left
behind. This is because any contribution involving only one instance of the
high-frequency fields averages out to zero on the low-frequency scales, such
that only a coarse grained contribution at second order survives that acts as
a source for the low frequency background (see also [79, 9]). The associated
leading order low-frequency perturbation equations therefore read

$\phantom{}{}_{\scriptscriptstyle{(1)}}\mathcal{G}_{\mu\nu}[h^{L},\varphi^{L}]$
$\displaystyle=-\langle\phantom{}_{\scriptscriptstyle{(2)}}\mathcal{G}_{\mu\nu}[h^{H},\varphi^{H}]\rangle+\kappa\langle\phantom{}_{\scriptscriptstyle{(2)}}T_{\mu\nu}\rangle\,,$
(27a) $\phantom{}{}_{\scriptscriptstyle{(1)}}\mathcal{J}[h^{L},\varphi^{L}]$
$\displaystyle=-\langle\phantom{}_{\scriptscriptstyle{(2)}}\mathcal{J}[h^{H},\varphi^{H}]\rangle\,,$
(27b)

In the reminder of this section we will however only be interested in
considering the high-frequency equations [Eqs. (25)] that describes the
propagation of the gravitational waves. The low-frequency equations above
giving rise to the memory corrections will then be considered in Sec. IV
below.

###  II.4 SVT Decomposition and the Propagation Equation of the Primary Waves

Let’s therefore analyze the leading order propagation equations [Eqs. (25)]
more closely. First of all, note that both these linear equations involve both
types of high-frequency perturbation fields, which we would like to decouple
in order to describe the propagation of individual dynamical degrees of
freedom. Moreover, as discussed above [see Eq. (13)], the field perturbations
only describe physical modes up to gauge transformations. We should therefore
both disentangle the field perturbations and identify the propagating physical
modes.

A universal way to accomplish both of these tasks is to split the linear
perturbation fields in a Helmholtz decomposition within a given asymptotic
chart. This is done by first splitting all field perturbations into
irreducible parts under the rotation group $SO(3)$, and further decompose the
individual pieces into scalar, vector and tensors of the $SO(2)$ subgroup
around a given direction. Given the spacial rotational invariance of the
background, the resulting scalar, vector and tensor modes in the linear
equations of motion will automatically be separated from each other. Moreover,
this setup allows the identification of manifestly gauge-invariant, and
therefore physical, quantities.

Very generally, the SVT decomposition of $h^{H}_{\mu\nu}$ as a two index
tensor field reads

$\displaystyle h^{H}_{00}$ $\displaystyle=2A\,,$ (28a) $\displaystyle
h^{H}_{0i}$ $\displaystyle=B^{\text{T}}_{i}+\partial_{i}B\,,$ (28b)
$\displaystyle h^{H}_{ij}$
$\displaystyle=\frac{1}{3}\delta_{ij}D+E^{\text{TT}}_{ij}+\partial_{(i}E^{\text{T}}_{j)}+(\partial_{i}\partial_{j}-\frac{1}{3}\delta_{ij}\partial^{2})E\,,$
(28c)

where $D=\delta^{ij}h^{H}_{ij}$ captures the trace of $h^{H}_{ij}$,
$B^{\text{T}}_{i}$ and $E^{\text{T}}_{i}$ are transverse vectors such that
$\partial^{i}B^{\text{T}}_{i}=0$ and $\partial^{i}E^{\text{T}}_{i}=0$ , and
$E_{ij}^{\text{TT}}$ is the transverse-traceless (TT) part of
$h^{H}_{\mu\nu}$, and therefore satisfies
$\partial^{i}E_{ij}^{\text{TT}}=\delta^{ij}E_{ij}^{\text{TT}}=0$.

In such a decomposition it is useful to introduce an explicit spacial
orthonormal basis set by a particular radial direction $\hat{n}$. At each
given location one can choose two transverse vectors $\hat{u}$ and $\hat{v}$
to $\hat{n}$, such that the basis $\left\\{\hat{n},\hat{u},\hat{v}\right\\}$
satisfies the completion relation (see e.g. [85, 3])

$\delta_{ij}=n_{i}n_{j}+u_{i}u_{j}+v_{i}v_{j}\,.$ (29)

With this basis at hand, it is possible to construct the transverse projection
operator

$P_{ij}(\hat{n})\equiv\delta_{ij}-n_{i}n_{j}=u_{i}u_{j}+v_{i}v_{j}\,,$ (30)

to extract the different components within the SVT decomposition introduced
above. For instance, we have

$B_{i}^{\text{T}}=P_{ij}h_{H}^{0j}\,.$ (31)

Similarly, we define the symmetric transverse-traceless projector

$\displaystyle\Lambda_{ij,kl}(\hat{n})\equiv
P_{ik}P_{jl}-\frac{1}{2}P_{ij}P_{kl}\,,$ (32)

that allows for the extraction of any TT component, for example

$\displaystyle E^{\text{TT}}_{ij}=\Lambda_{ij,kl}h_{H}^{kl}\,.$ (33)

While the above decomposition naturally disentangles the scalar, vector and
tensor sectors of the equations of motion, of course not all of the SVT
components are physical due to the gauge transformations in Eq. (13) generated
by an infinitesimal high-frequency vector field $\xi^{\mu}_{H}$. Indeed, while
for the given asymptotically flat background the scalar perturbation remains
invariant under the transformation in Eq. (15)

$\varphi^{H}\rightarrow\varphi^{H}\,,$ (34)

the metric perturbation transforms as [Eq. (14)]

$h^{H}_{\mu\nu}\rightarrow h^{H}_{\mu\nu}-2\partial_{(\mu}\xi^{H}_{\nu)}\,.$
(35)

Thus, only six of the ten modes within the metric decomposition in Eq. (28)
are physical.

To explicitly identify these physical modes we also Helmholtz split the gauge
transformation into

$\xi^{H}_{\mu}=(\xi_{0},\xi^{\text{T}}_{i}+\partial_{i}\xi)\,,$ (36)

where $\xi_{i}$ is transverse, such that the SVT components of
$h^{H}_{\mu\nu}$ transform as

$\displaystyle A$ $\displaystyle\rightarrow A-\dot{\xi_{0}}\,,$ (37a)
$\displaystyle B^{\text{T}}_{i}$ $\displaystyle\rightarrow
B^{\text{T}}_{i}-\dot{\xi}^{\text{T}}_{i}\,,$ (37b) $\displaystyle B$
$\displaystyle\rightarrow B-\xi_{0}-\dot{\xi}\,,$ (37c) $\displaystyle D$
$\displaystyle\rightarrow D-2\nabla^{2}\xi\,,$ (37d) $\displaystyle E$
$\displaystyle\rightarrow E-2\xi\,,$ (37e) $\displaystyle E^{\text{T}}_{i}$
$\displaystyle\rightarrow E^{\text{T}}_{i}-2\xi^{\text{T}}_{i}\,,$ (37f)
$\displaystyle E_{ij}^{\text{TT}}$ $\displaystyle\rightarrow
E_{ij}^{\text{TT}}\,.$ (37g)

While therefore $E_{ij}^{\text{TT}}$ is already gauge invariant, it is not
difficult to verify that the following two scalar and two vector combinations
represent the additional gauge invariant quantities we seek

$\displaystyle\Phi$ $\displaystyle\equiv-A+\dot{B}-\frac{1}{2}\ddot{E}\,,$
(38a) $\displaystyle\Theta$
$\displaystyle\equiv\frac{1}{3}\left(D-\nabla^{2}E\right)\,,$ (38b)
$\displaystyle\Xi_{i}$ $\displaystyle\equiv
B_{i}-\frac{1}{2}\dot{E}^{\text{T}}_{i}\,.$ (38c)

We have therefore identified the seven physical degrees of freedom of the
problem, given by the three gauge invariant scalars $\Phi$, $\Theta$ and
$\varphi^{H}\equiv\varphi$, the two vector modes within $\Xi_{i}$ together
with the two TT tensor modes $E_{ij}^{\text{TT}}$. It remains to plug this
gauge invariant decomposition into the equations of motion of the theory in
order to extract the subset of physical modes that are propagating. As we
explicitly show in Appendix A, the linear high-frequency propagation equations
within Horndeski theory dictate that, as expected, only the scalar field
perturbation $\varphi$ and the TT part of the metric $E_{ij}^{\text{TT}}$
satisfy dynamical equations of motion

$\displaystyle\Box E^{\text{TT}}_{ij}=0\,,\quad(\Box-m^{2})\varphi=0\,.$ (39)

with the effective mass of the scalar field perturbation defined as

$\displaystyle
m^{2}\equiv\frac{\bar{G}_{2,\phi\phi}}{\bar{G}_{2,X}-2\bar{G}_{3,\phi}+3(\bar{G}_{4,\phi})^{2}/\bar{G_{4}}}\,,$
(40)

while all other gauge invariant variables are constraints to take the
following values

$\displaystyle\Theta$ $\displaystyle=-\sigma\varphi\,,$ $\displaystyle\Phi$
$\displaystyle=\frac{1}{2}\sigma\varphi\,,$ $\displaystyle\Xi^{i}$
$\displaystyle=0\,,$ (41)

where

$\sigma\equiv\bar{G}_{4,\phi}/\bar{G}_{4}\,.$ (42)

Hence, the linear high-frequency EOMs [Eq. (39)] indicate that the Horndeski
theory has $2+1=3$ propagating degrees of freedom reflected in the three
independent modes of the high-frequency primary waves, while the remaining
four physical modes are constrained, non-dynamical variables.

The solutions to the homogeneous wave equations in Eqs. (39) in the large $r$
limit are given in Fourier space by a superposition of plane waves and because
of the fixed radial propagation $\hat{n}$ one can indeed without loss of
generality consider each plane wave mode individually222Note that interactions
between individual modes would become important when considering stochastic
solutions without fixed propagation directions [85]., hence

$\displaystyle E^{\text{TT}}_{ij}(x)$
$\displaystyle=\frac{1}{r}C^{\text{TT}}_{ij}(\Omega)e^{ik_{\mu}x^{\mu}}\,,$
(43a) $\displaystyle\varphi(x)$
$\displaystyle=\frac{1}{r}C(\Omega)e^{ip_{\mu}x^{\mu}}\,,$ (43b)

with

$k^{i}\,C^{\text{TT}}_{ij}=0\,,\quad\delta^{ij}C^{\text{TT}}_{ij}=0\,,$ (44)

and where the perturbations are characterized by their $\mathcal{O}(1/r)$
fall-off. Note that in terms of the Fourier momenta, the direction of
propagation is given by

$n_{i}=k_{i}/k=p_{i}/p=\partial_{i}r\,.$ (45)

Moreover, Eqs. (39) imply the following dispersion relations

$\displaystyle k^{0}(k)$ $\displaystyle=c\,k\,,$ (46a) $\displaystyle
p^{0}(p)$ $\displaystyle=c\,\sqrt{p^{2}+m^{2}}\,,$ (46b)

where we have defined $k\equiv\sqrt{k_{i}k^{i}}$ and
$p\equiv\sqrt{p_{i}p^{i}}$. This directly implies that the TT perturbations
propagate with the speed of light $c$, while the scalar field perturbation
propagates with a group velocity of

$\displaystyle v\equiv\frac{\partial p^{0}(p)}{\partial
p}=\,\sqrt{1-\frac{m^{2}c^{2}}{(p^{0})^{2}}}\,.$ (47)

This spacial velocity is to be understood as the velocity in the chosen
asymptotic source centered frame and is assumed to be non-zero.

Therefore, the solutions of the three propagating degrees of freedom in Eq.
(43) can be written as

$\displaystyle E^{\text{TT}}_{ij}(x)$
$\displaystyle=\frac{1}{r}C^{\text{TT}}_{ij}(\Omega)\,e^{-ik^{0}(ct-r)}\,,$
(48a) $\displaystyle\varphi(x)$
$\displaystyle=\frac{1}{r}C(\Omega)\,e^{-ip^{0}(ct-\beta r)}\,,$ (48b)

where we have defined $\beta\equiv v/c$. In other words, the asymptotic
solutions of the high-frequency wave modes take on the following general form

$\displaystyle E^{\text{TT}}_{ij}(x)$
$\displaystyle=\frac{1}{r}f^{\text{TT}}_{ij}(ct-r,\Omega)\,,$ (49a)
$\displaystyle\varphi(x)$ $\displaystyle=\frac{1}{r}f(ct-\beta r,\Omega)\,.$
(49b)

for some transverse-traceless and scalar functions $f^{\text{TT}}_{ij}$ and
$f$. Observe that these universal asymptotic forms imply that to the leading
order in $1/r$, the spatial derivatives and the temporal derivatives of
$E^{\text{TT}}_{ij}$ are related by

$\partial_{i}E_{\text{TT}}^{ij}=-\frac{1}{c}n_{i}\,\partial_{0}E_{\text{TT}}^{ij}\,,$
(50)

while similarly for the massive scalar mode we have

$\partial_{i}\varphi=-\frac{\beta}{c}\,n_{i}\,\partial_{0}\varphi\,.$ (51)

For simplicity, we set $c=1$ from now on.

##  III Gravitational Polarization Modes

As an interlude, we want to offer in this section a self-consistent account of
the gravitational polarization modes present in Horndeski theory. While
propagating degrees of freedom are defined as the linearly independent gauge
invariant solutions to the equations of motion of the leading-order perturbed
fields, we define here gravitational polarization modes to represent the
detectable modes of the physical metric that couples minimally to the matter.
When discussing memory beyond GR in the subsequent Section IV a clear
distinction between the concepts of propagating degrees of freedom of the
previous section and gravitational polarizations will be essential. Indeed,
the geodesic deviation equation at the basis of the definition of
gravitational polarization will set the stage for the very definition of
memory, as well as a classification of different types of displacement memory
that might be present in beyond GR theories. Furthermore, while the
polarization content of Horndeski theory was already discussed for instance in
[87] it is useful to rederive these results within the manifestly gauge-
invariant SVT decomposition approach introduced in Sec. II.4. To start, we
should therefore discuss the concept of additional polarizations in general
metric theories beyond GR.

###  III.1 Additional Polarizations Beyond GR

The foundation of any light-travel time based measurement of gravitational
waves is given by the geodesic deviation equation, which for two time-like
geodesics with spacial proper distance separation vector $s_{i}$ is to
leading-order governed by the electric part of the Riemann tensor (see e.g.
[79, 85, 83])

$\displaystyle\ddot{s}_{i}=-R_{i0j0}s_{j}\,,$ (52)

where $R_{\mu\nu\rho\sigma}$ represents the Riemann tensor of the physical
metric $g_{\mu\nu}$. Note that this equation solely follows from the minimal
and universal coupling requirement and is therefore valid in any metric theory
of gravity, regardless of the precise form of the equations of motion. Since
the electric part of the Riemann tensor possesses six independent components,
we can already conclude at this stage, that the physical response to
gravitational waves is in principle associated to six distinct gravitational
polarization modes. Naturally, these polarization modes are associated to the
six gauge invariant degrees of freedom within the perturbations of the
physical metric that we identified in Sec. II.4.

To explicitly derive this association within the asymptotically flat setup
employed in this work, we are interested in computing the leading order
contribution to the Riemann tensor of the physical metric

$\displaystyle
R_{\mu\nu\rho\sigma}=\phantom{}_{\scriptscriptstyle{\text{(}}1)}R_{\mu\nu\rho\sigma}+\mathcal{O}\left(\frac{1}{r^{2}}\right)\,,$
(53)

where the leading order term on the Minkowski background associated to a
metric perturbation $h_{\mu\nu}$ reads

$\phantom{}{}_{\scriptscriptstyle{\text{(1)}}}R_{i0j0}=-\frac{1}{2}\left(\partial_{0}\partial_{0}h_{ij}+\partial_{i}\partial_{j}h_{00}-\partial_{0}\partial_{i}h_{0j}-\partial_{0}\partial_{j}h_{0i}\right).$
(54)

At this point it should be noted that the metric perturbation $h_{\mu\nu}$ in
principle represents here both the low- and high-frequency dynamical
perturbations. However, the SVT decomposition together with the identification
of gauge invariant modes explicitly performed in Sec. II.4 for the high-
frequency modes can also be applied in exactly the same manner to the low-
frequency components. Thus, in this section, all perturbative quantities are
to be understood as the full perturbations that also include potential
dynamical low-frequency components. We will however refrain from introducing
new symbols for total perturbations in order to keep the notation simple.

With this comment out of the way, we can rewrite the general expression in Eq.
(54) in terms of the gauge invariant quantities within a general SVT
decomposition and obtain

$\displaystyle\phantom{}_{\scriptscriptstyle{\text{(1)}}}R_{0i0j}=\frac{-1}{2}\partial_{0}\partial_{0}E_{ij}^{\text{TT}}-\frac{\delta_{ij}}{2}\partial_{0}\partial_{0}\Theta+\partial_{0}\partial_{(i}\Xi_{j)}-\partial_{i}\partial_{j}\Phi.$
(55)

As expected, the gauge invariant linear perturbation of the Riemann tensor can
be written entirely in terms of gauge invariant variables of the metric. While
therefore all six gauge-invariant modes of the physical metric can in general
be involved in the physical response to a gravitational wave, not all of them
must be excited. Indeed, for GR for instance, only the two TT tensor modes are
present.

In general, only the components associated to a dynamical mode contribute to
the detector response, as otherwise their derivatives lead to higher-order
terms. However, in order to contribute, the different components of the
physical metric must not be dynamical themselves, but it suffices if the gauge
invariant variables are associated to a dynamical mode in the gravitational
sector. This is for instance the case for the two scalar components $\Theta$
and $\Phi$ that are excited through their non-trivial relation with the
dynamical degree of freedom described by $\varphi$ in Eq. (41). To continue
the analysis we have to make use of this fact and assume that any mode that
contributes must be associated to a dynamical behavior and therefore assume an
asymptotic form similar to Eq. (49) together with the asymptotic relation Eq.
(51) for some velocity $\beta$ of the associated plane-wave mode (see also
[3]).

This allows us to rewrite Eq. (55) as

$\phantom{}{}_{\scriptscriptstyle{\text{(1)}}}R_{0i0j}=-\frac{1}{2}\partial_{0}\partial_{0}A_{ij}$
(56)

where the response matrix is given by

$A_{ij}\equiv E_{ij}^{\text{TT}}+\delta_{ij}\Theta+2\beta
n_{(i}\Xi_{j)}+2\beta^{2}n_{i}n_{j}\Phi\,.$ (57)

With this equation at hand, the geodesic deviation equation [Eq. (52)] is
easily integrated to give to leading order

$\displaystyle\Delta s_{i}(\tau)=\frac{1}{2}\Delta
A_{ij}(\tau)s^{j}(\tau_{0})\,,$ (58)

where

$\Delta s_{i}(\tau)\equiv s_{i}(\tau)-s_{i}(\tau_{0})\,,$ (59)

for some initial reference proper time $\tau_{0}$ before the presence of any
gravitational wave.

The response matrix is therefore what ultimately governs the physical stretch
in proper distances within an idealized detector, and is thus also known as
the polarization matrix. The six independent components of the polarization
matrix can conveniently be expanded in a specific polarization basis as

$\displaystyle A_{ij}\equiv$ $\displaystyle
P_{+}\,e^{+}_{ij}+P_{\times}\,e^{\times}_{ij}+P_{u}\,n_{(i}u_{j)}+P_{v}\,n_{(i}v_{j)}$
$\displaystyle+P_{b}\,(u_{i}u_{j}+v_{i}v_{j})+P_{l}\,n_{i}n_{j}\,.$ (60)

where

$\displaystyle e^{+}_{ij}$ $\displaystyle\equiv u_{i}u_{j}-v_{i}v_{j}\,,\quad
e^{\times}_{ij}\equiv u_{i}v_{j}+v_{i}u_{j}\,,$ (61a) $\displaystyle
e^{u}_{ij}$ $\displaystyle\equiv n_{i}u_{j}+u_{i}n_{j}\,,\quad
e^{v}_{ij}\equiv n_{i}v_{j}+v_{i}n_{j}\,,$ (61b) $\displaystyle\quad
e^{b}_{ij}$ $\displaystyle\equiv u_{i}u_{j}+v_{i}v_{j}\,,\quad
e^{l}_{ij}\equiv n_{i}n_{j}\,,$ (61c)

For concreteness, it is useful to align the direction of travel $\hat{n}$ with
$z$-axis, in which case the polarization matrix becomes

$\displaystyle A_{ij}=\begin{bmatrix}P_{+}+P_{b}&P_{\times}&P_{u}\\\
P_{\times}&-P_{+}+P_{b}&P_{v}\\\ P_{u}&P_{v}&P_{l}\end{bmatrix},$ (62)

The spin-weighted scalars $P_{A}$ are ultimately what we define as the
gravitational polarizations. Of course $P_{+}$ and $P_{\times}$ are the two
familiar tensor polarization modes that show up in standard GR, while $P_{u}$
and $P_{v}$ represent two additional vector modes, $P_{l}$ is the scalar
longitudinal mode, and $P_{b}$ is known as the breathing polarization.

Comparing the above expressions in Eq. (60) with Eq. (57) while utilizing the
knowledge that $n_{i}$, $u_{i}$, $v_{i}$ form an orthonormal basis, we finally
obtain the following relations between the gravitational polarizations and the
gauge invariant components of the physical metric perturbations

$\displaystyle P_{+}$
$\displaystyle=\frac{1}{2}E^{\text{TT}}_{ij}e^{ij}_{+}\,,$ $\displaystyle
P_{u}$ $\displaystyle=\beta u^{i}\Xi_{i}\,,$ $\displaystyle P_{b}$
$\displaystyle=\Theta\,,$ (63a) $\displaystyle P_{\times}$
$\displaystyle=\frac{1}{2}E^{\text{TT}}_{ij}e^{ij}_{\times}\,,$ $\displaystyle
P_{v}$ $\displaystyle=\beta v^{i}\Xi_{i}\,,$ $\displaystyle P_{l}$
$\displaystyle=\Theta+2\beta^{2}\Phi\,.$ (63b)

###  III.2 Gravitational Polarizations in Horndeski Theory

Based on the general considerations above, it is now relatively
straightforward to assess the polarization content within Horndeski theory.
First, since $\Xi^{i}=0$ [Eq. (41)] Horndeski theory has no additional vector
polarization modes, hence $P_{u}=P_{v}=0$. However, on top of the two tensor
polarizations $P_{+}$ and $P_{\times}$, Horndeski theory offers the
possibility for the presence of additional scalar gravitational polarizations
that might be excited through the non-minimal coupling between the additional
scalar degree of freedom and the physical metric. Concretely, through Eqs.
(41) and (63) we have that

$\displaystyle P_{b}=-\sigma\varphi\,,\qquad
P_{l}=\sigma\varphi(-1+\beta^{2})\,.$ (64)

Therefore, depending on the parameters of the theory, up to four gravitational
polarizations may be present within Horndeski gravity. As can easily be seen,
the breathing polarization is always present for a nontrivial non-minimal
coupling parameter $\sigma$ defined back in Eq. (42), whereas the longitudinal
gravitational polarization has the additional requirement of $\beta\neq 1$,
and is therefore only present for massive modes, where the mass $m$ is given
by Eq. (40). Note, however, that the two scalar polarizations are not
independent of each other, as they are excited by the same dynamical
gravitational degree of freedom in the theory.

It is particularly illuminating to illustrate the different possibilities
within Horndeski theory by considering the concrete known theories beyond GR
that are captured within its framework, introduced at the end of Sec. II.1.
First of all, for Brans-Dicke theory [Eq. (6)] we have that
$\sigma=1/\bar{\phi}\neq 0$, and therefore BD gravity does excite an
additional scalar gravitational polarization. However, since the scalar
perturbations remains massless $m=0$, the speed of propagation of the wave is
always luminal, hence $\beta=0$, such that only the breathing mode is present.
On the other hand, for $f(R)$ gravity, the background equations require that
$\bar{\phi}=f(\bar{\phi})=0$, while however $f^{\prime}(\phi)$ and
$f^{\prime\prime}(\phi)$ are non-zero. This implies that the theory comes with
parameters $\sigma=f^{\prime\prime}(\bar{\phi})/f^{\prime}(\bar{\phi})\neq 0$
and $m^{2}=f^{\prime}(\bar{\phi})/(3f^{\prime\prime}(\bar{\phi}))\neq 0$ and
thus excites both $P_{b}$ and $P_{l}$. Finally, sGB gravity has $\sigma=0$ and
therefore does not come with any additional gravitational polarizations. These
examples nicely illustrate the difference between the concepts of propagating
degrees of freedom and gravitational polarizations. While all three theories
above possess three propagating DOFs in the gravity sector, the scalar mode
can either excite none, one or even two additional gravitational
polarizations.

##  IV Ordinary and Null Memory in Horndeski Theory

The gravitational wave memory effect is defined as a permanent displacement
between test masses in an idealized detector after the passage of a burst of
gravitational waves. More formally, within the solution to the geodesic
deviation equation in Eq. (58), gravitational memory is defined as a non-zero
value of the leading order spacial proper distance displacement between two
timelike geodesics in the limit of the absence of any gravitational wave
disturbance

$\lim_{\tau\rightarrow\infty}\Delta s(\tau)\neq 0\,.$ (65)

Therefore, the memory effect relies on a non-zero difference before and after
a gravitational wave within one of the components of the polarization matrix

$\lim_{\tau\rightarrow\infty}\Delta A_{ij}(\tau)\neq 0$ (66)

and can therefore through Eq. (60) naturally be associate to one of the
gravitational polarization types.

Since, as discussed in Sec. III.2, in Horndeski theory up to four
gravitational polarizations can be excited, gravitational memory can in
principle arise both within the tensor and the scalar sector. Concretely,
through Eqs. 63 memory in Horndeski theory can in principle be associated to a
non-zero permanent offset within either one of the following gauge invariant
variables

$\displaystyle\Delta A_{ij}=$ $\displaystyle\frac{1}{2}\Delta
E^{\text{TT}}_{kl}\left(e_{+}^{lk}\,e^{+}_{ij}+e_{\times}^{lk}\,e^{\times}_{ij}\right)$
$\displaystyle+\Delta\Theta(e^{b}_{ij}+e^{l}_{ij})+\Delta\Phi(2\beta^{2}e^{l}_{ij})\,.$
(67)

In this work, we will however entirely focus on the observationally relevant
tensor memory within the TT tensor polarizations.

At first sight, the presence of such a non-zero permanent offset is not
guaranteed. Indeed, in Sec. II.4 we have elaborated on the asymptotic solution
of the propagating high-frequency perturbations given by a superposition of
plane waves. Very generally, these solutions do not give rise to any memory
contribution, as the high frequency part of the gauge invariant variables are
subject to symmetric attenuation effects that die out any signal with zero
displacement

$\lim_{\tau\rightarrow\infty}[\Delta E^{\text{TT}}_{ij}]^{H}=0\,.$ (68)

However, as we will now show explicitly, the solution of the asymptotic
leading order the low-frequency equations of motion [Eq. (27a)] will give rise
to a propagating low-frequency correction to the background spacetime, which
precisely provides a memory effect in the sense defined above.

In the following, we will therefore first derive the precise form of the low-
frequency Isaacson equation in Horndeski theory within the SVT decomposition
introduced in Sec. II.4, and solve for the resulting memory evolution that is
sourced by the back-reaction of a massive or massless field perturbations in
the limit to null infinity. For simplicity, we will from now on denote the
low-frequency part of the metric perturbation as

$[h^{L}_{ij}]^{\text{TT}}=[E^{L}_{ij}]^{\text{TT}}=\delta
h^{\text{TT}}_{ij}\,,$ (69)

while the high-frequency waves are simply given

$[h^{H}_{ij}]^{\text{TT}}=[E^{H}_{ij}]^{\text{TT}}=h_{ij}^{\text{TT}}\,,\quad\varphi^{H}=\varphi\,.$
(70)

###  IV.1  Memory Evolution Equation in Horndeski Theory

We have obtained and solved the linear propagating equations in the previous
parts. Let us now consider the low-frequency back-reaction equation Eq. (27)
in the asymptotic limit, which provides us with the memory evolution equation
in Horndeski gravity. Focusing on the tensor memory, we will however only
consider the metric equations of motion and single out the TT part of the
equation by the use of the projection operator introduced in Eq. (32).

First, observe that the left-hand side of this equation takes the same form as
the high-frequency metric field equation. Hence, by performing the same 3+1
and SVT decomposition and grouping the perturbation fields into gauge-
invariant quantities as in Sec. II.4, the dynamical part of the left-hand side
of the back-reaction equation becomes

$\displaystyle\left(\phantom{}{}_{\scriptscriptstyle{(1)}}\mathcal{G}_{ij}[h^{L},\varphi^{L}]\right)^{\text{TT}}=\frac{-1}{2}\bar{G}_{4}\Box\delta
h^{\text{TT}}_{ij}\,.$ (71)

The dynamical TT part of the low frequency equation therefore becomes

$\displaystyle\frac{-1}{2}\bar{G}_{4}\Box\delta
h^{\text{TT}}_{ij}=\Lambda_{ij,kl}(-\langle\phantom{}_{\scriptscriptstyle{(2)}}\mathcal{G}^{kl}[h^{H},\varphi^{H}]\rangle+\kappa\langle\phantom{}_{\scriptscriptstyle{(2)}}T^{kl}\rangle)\,.$
(72)

Using the SVT decomposition, together with the properties of the spacetime
averaging [Sec. II.2] that allows for integrations by parts, as well as the
leading propagation equation of the high-frequency fields, the first term of
the right-hand side can then be evaluated to give

$\displaystyle\langle\phantom{}_{\scriptscriptstyle{(2)}}\mathcal{G}_{ij}\rangle=-\frac{1}{4}\bar{G}_{4}\left\langle\partial_{i}h^{\text{TT}}_{kl}\partial_{j}h_{\text{TT}}^{kl}\right\rangle-\frac{\bar{G}_{2,\phi\phi}}{2m^{2}}\left\langle\partial_{i}\varphi\partial_{j}\varphi\right\rangle\,.$
(73)

Thus, the TT part of the leading order low-frequency equation [Eq. (72)] in
Horndeski theory can be written as

$\displaystyle\Box\delta
h^{\text{TT}}_{ij}=-2\Lambda_{ij,kl}\left(\kappa^{\text{($h$)}}t^{kl}_{\text{($h$)}}+\kappa^{\text{($\varphi$)}}t^{kl}_{\text{($\varphi$)}}+\kappa^{\text{(m)}}t^{kl}_{\text{(m)}}\right)\,,$
(74)

where

$\displaystyle t^{kl}_{\text{($h$)}}$
$\displaystyle=\left\langle\partial^{k}h^{TT}_{kl}\partial^{l}h_{TT}^{kl}\right\rangle\,,$
(75a) $\displaystyle t^{kl}_{\text{($\varphi$)}}$
$\displaystyle=\left\langle\partial^{k}\varphi\partial^{l}\varphi\right\rangle\,,$
(75b) $\displaystyle t^{kl}_{\text{(m)}}$
$\displaystyle=\langle\phantom{}_{\scriptscriptstyle{(2)}}T^{kl}\rangle\,,$
(75c)

represent the effective energy-momentum tensors of the high-frequency waves as
well as the unbound matter components, and the coefficients read

$\displaystyle\kappa^{\text{($h$)}}$ $\displaystyle=\frac{1}{4}\,,$ (76a)
$\displaystyle\kappa^{\text{($\varphi$)}}$
$\displaystyle=\frac{\bar{G}_{2,X}-2\bar{G}_{3,\phi}}{2\bar{G}_{4}}+\frac{3}{2}\sigma^{2}=\frac{\bar{G}_{2,\phi\phi}}{2\bar{G}_{4}m^{2}}\,,$
(76b) $\displaystyle\kappa^{\text{(m)}}$
$\displaystyle=\frac{\kappa}{\bar{G}_{4}}\,.$ (76c)

The solution of the memory evolution will then be a superposition of the
solutions of inhomogeneous wave equations of the three different types of
asymptotic energy-momentum sources.

###  IV.2  Asymptotic Energy-Momentum Fluxes

Next we want to show that any asymptotic, radial energy-momentum tensor is
entirely characterized by its 00-component, hence the energy density. More
precisely, any energy-momentum flux with constant radial velocity $v$ in the
radiation zone takes on the general form in Eq. (83).

Before we give the general argument, we want to explicitly verify this
statement in the present specific case of Horndeski theory through direct
computation. For this, we first present also the 00 component of the second-
order metric equations of motion of Horndeski gravity within the SVT
decomposition

$\langle\phantom{}_{\scriptscriptstyle{(2)}}\mathcal{G}_{00}\rangle=-\frac{1}{4}\bar{G}_{4}\left\langle\partial_{0}h^{\text{TT}}_{kl}\partial_{0}h_{\text{TT}}^{kl}\right\rangle-\frac{\bar{G}_{2,\phi\phi}}{2m^{2}}\left\langle\partial_{0}\varphi\partial_{0}\varphi\right\rangle\,,$
(77)

and compare the result to Eq. (73), such that

$\displaystyle t_{00}^{\text{($h$)}}$
$\displaystyle=\left\langle\partial_{0}h^{TT}_{kl}\partial_{0}h_{TT}^{kl}\right\rangle\,,$
(78a) $\displaystyle t_{00}^{\text{($\varphi$)}}$
$\displaystyle=\left\langle\partial_{0}\varphi\partial_{0}\varphi\right\rangle\,.$
(78b)

First of all, as in GR, the massless high-frequency wave satisfies the
asymptotic relation given in Eq. (50), which implies that

$\displaystyle
t_{ij}^{(h)}=n_{i}n_{j}\left\langle\partial_{0}h^{\text{TT}}_{kl}\partial_{0}h_{\text{TT}}^{kl}\right\rangle=n_{i}n_{j}t_{00}^{(h)}\,.$
(79)

This precisely corresponds to Eq. (83) with the luminal speed of $v=1$.
Similarly, the potentially massive field wave $\varphi$ satisfies Eq. (51),
such that

$\displaystyle
t_{ij}^{\text{($\varphi$)}}=\beta^{2}n_{i}n_{j}\left\langle\partial_{0}\varphi\partial_{0}\varphi\right\rangle=\beta^{2}n_{i}n_{j}t_{00}^{\text{($\varphi$)}}\,.$
(80)

As an example for the asymptotic energy-momentum tensor of unbound matter,
that we will later use, consider also the case of a freely moving particle of
mass $M$ that is gravitationally unbound from other particles. In this case,
the asymptotic energy-momentum tensor of matter takes the form (see e.g. [71])

$\displaystyle T_{\mu\nu}^{\text{(m)}}(x)=M\gamma
V_{\mu}V_{\nu}\delta^{(3)}(\vec{x}-\vec{x}^{\prime}(t))\,,$ (81)

with velocity vector $V_{\mu}=(1,-\vec{\beta})$ and where $x^{\prime}(t)$
denotes the radial trajectory of the particle. Hence, component-wise we can
write $V_{i}=-\beta n_{i}V_{0}$, and then

$\displaystyle
T_{ij}^{\text{(m)}}(x)=\beta^{2}n_{i}n_{j}T_{00}^{\text{(m)}}(x)\,.$ (82)

A multi-particle system can be considered as the sum of all single-particle
systems.

Thus, indeed, any of the three asymptotic energy momentum tensors
$T^{\text{a}}_{\mu\nu}$ described above has the special form

$T^{\text{a}}_{ij}(u,r,\Omega)=T^{\text{a}}_{00}\,\beta^{2}\,n_{i}n_{j}\,,$
(83)

with $\beta=1$ for luminal fluxes. As mentioned, this is a very general result
that holds for any asymptotic energy-momentum tensor $T^{\text{a}}_{\mu\nu}$
that carries away energy from the source at some constant asymptotic radial
velocity $v$.

To derive this general result, note first that by definition the energy
density $T^{\text{a}}_{00}$ of an energy flux with constant asymptotic
velocity ratio $\beta$ defines the energy flux

$\beta
T^{\text{a}}_{00}(t,r,\Omega)=\frac{1}{r^{2}}\frac{dE(t-r/\beta,\Omega)}{dtd\Omega}\,.$
(84)

Hence, the energy flux only depends on the angles $\Omega$ and the asymptotic
retarded time

$u=t-\frac{r}{\beta}\,.$ (85)

As a consequence of the dependence on the specific combination of asymptotic
retarded time, to leading order in $1/r$, the energy density satisfies

$\partial_{i}T_{\text{a}}^{00}=-\frac{n_{i}}{\beta}\partial_{0}T_{\text{a}}^{00}\,,$
(86)

such that

$\partial_{0}T_{\text{a}}^{00}=-\beta\,n^{i}\partial_{i}T_{\text{a}}^{00}\,.$
(87)

On the other hand, the time component of the conservation of the energy-
momentum tensor in the radiation zone

$\partial_{\mu}T_{\text{a}}^{\mu\nu}=0\,,$ (88)

also implies that

$\partial_{i}T_{\text{a}}^{i0}=-\partial_{0}T_{\text{a}}^{00}=\beta^{i}\,\partial_{i}T_{\text{a}}^{00}\,,$
(89)

where in the last equality we have employed Eq. (87). Assuming that we are
only interested in the radial components of the energy momentum tensor, one
can integrate this equation over a spherical shell in the radiation zone of
volume element $dV=drr^{2}d\Omega$ and reduce the integral through Stocke’s
theorem to the outer surface with unit normal $n_{i}$ (see also [85]) to
obtain

$\int_{S^{2}}d\Omega\,n_{i}\,T_{\text{a}}^{i0}=\int_{S^{2}}d\Omega\,\beta\,T_{\text{a}}^{00}\,.$
(90)

Up to transverse terms that we are neglecting, we thus have the relation

$T_{\text{a}}^{i0}=\beta\,n^{i}\,T_{\text{a}}^{00}\,.$ (91)

Similarly, the spacial components of Eq. (88) require

$\partial_{i}T_{\text{a}}^{ij}=-\partial_{0}T_{\text{a}}^{0j}=-\beta
n^{j}\,\partial_{0}T_{\text{a}}^{00}=\beta^{2}\,n^{i}n^{j}\,\partial_{i}T_{\text{a}}^{00}\,,$
(92)

such that using similar arguments as above, up to transverse terms one indeed
recovers the relation

$T_{\text{a}}^{ij}=\beta^{2}\,n^{i}n^{j}\,T_{\text{a}}^{00}\,.$ (93)

###  IV.3 A Unified Treatment of Null and Ordinary Memory

Given the general considerations of asymptotic energy momentum tensors
discussed above, we will now be able to provide a very general form of the
solution for the low-frequency memory evolution equation that reads [Eq. (74)]

$\displaystyle\Box\delta
h^{\text{TT}}_{ij}=-2\kappa_{\scriptscriptstyle{\text{eff}}}\Lambda_{ij,kl}T_{\text{a}}^{kl}\,.$
(94)

As discussed, in the case of Horndeski theory, the energy-momentum tensor
$T_{\text{a}}^{kl}$ can either be the one associated to the tensor waves
$t_{\text{(h)}}^{kl}$, the gravitational scalar field perturbation
$t_{\text{($\varphi$)}}^{kl}$ or the unbound matter $T^{kl}_{\text{(m)}}$.
However, the memory evolution equation can be solved in the same way
regardless of the type of conserved energy-momentum source, as long as it
satisfies the asymptotic form in Eq. (83). The steps necessary to obtain the
general solution are very close to what was already carried out for instance
in [28, 37], although the underlying context within these works is rather
different. Moreover, our treatment offers a generalization to wave sources
associated to massive field perturbations with asymptotic speeds different
from $c$.

First, a general solution to Eq. (94) is of course provided by introducing a
standard Green’s function $G(x-x^{\prime})$ that satisfies

$\displaystyle\Box_{x}G(x-x^{\prime})=\delta^{(4)}(x-x^{\prime})\,,$ (95)

with $\Box_{x}$ being the d’Alembert operator with respect the variable $x$.
The corresponding solution to Eq. (94) is then given by

$\displaystyle\delta h^{\text{TT}}_{ij}(x)$
$\displaystyle=-2\kappa_{\scriptscriptstyle{\text{eff}}}\,\Lambda_{ij,kl}\int
d^{4}x^{\prime}\;G(x-x^{\prime})\,T_{\text{a}}^{kl}(x^{\prime})\,.$ (96)

Here the appropriate Green’s function is given by (see e.g. [85])

$\displaystyle G(x-x^{\prime})=-\frac{\delta(x^{0}_{\text{ret}}-x^{\prime
0})}{4\pi\left|\vec{x}-\vec{x}^{\prime}\right|}\,,$ (97)

where $x^{\prime 0}=t^{\prime}$, $x^{0}_{ret}=t_{\text{ret}}$, and
$t_{\text{ret}}$ is the retarded time for the propagating gravitational wave,

$\displaystyle t_{\text{ret}}=t-\left|\vec{x}-\vec{x}^{\prime}\right|\,.$ (98)

However, we want to evaluate the memory within the massless TT polarization
mode travelling at the speed of light in the limit to null infinity outside
its own source. This limit to null infinity is provided by the large $r$
limit, but at fixed asymptotic retarded time $u=t-r/c$. We should therefore
transform the coordinates to

$(t,x,y,z)\rightarrow(u,r,\theta,\phi)\,.$ (99)

Note, however, that as before we will only change coordinates in the arguments
of the fields while treating the indices still in source-centered Minkowski
coordinates.

Similarly, given the special form of the asymptotic energy flux in Eq. (83)
together with Eq. (84) of asymptotic radial speed $v$ equal or less than $c$,
it is convenient to perform the transformation

$(t^{\prime},x^{\prime},y^{\prime},z^{\prime})\rightarrow(u^{\prime},r^{\prime},\theta^{\prime},\phi^{\prime})\,,$
(100)

of the source coordinates, where in this case the asymptotic retarded time
reads $u^{\prime}=t^{\prime}-r^{\prime}/v$. Moreover, the volume form in these
new coordinates becomes

$\displaystyle d^{4}x^{\prime}\rightarrow r^{\prime
2}du^{\prime}dr^{\prime}d\Omega^{\prime}\,.$ (101)

Under these changes of coordinates, we can also of course also write
$\vec{x}=r\vec{n}$, $\vec{x}^{\prime}=r^{\prime}\vec{n}^{\prime}$, where
$\vec{n}$ provides the direction of the observer with respect to the
coordinate origin, while $\vec{n}^{\prime}$ represents the direction from the
same origin, at which the unbounded source creating the back-reaction is
evaluated. Note that as asymptotic radial directions, both $\vec{n}$ and
$\vec{n}^{\prime}$ correspond to the asymptotic longitudinal direction already
employed above.

It is important to realize, however, that while the source terms of the memory
equation in Eq. (94) are non-zero in the asymptotic region, there still exists
an effective hierarchy between $r$ and $r^{\prime}$ as the memory component
evaluated at $x$ only depends on the unbounded source terms for which
$r^{\prime}\ll r$ holds (see [37, 9]). This allows us to expand the source-to-
observer vector in the usual way as

$\displaystyle\left|\vec{x}-\vec{x}^{\prime}\right|$
$\displaystyle=\sqrt{(\vec{x}-\vec{x}^{\prime})^{2}}$
$\displaystyle=\sqrt{r^{2}+r^{\prime
2}-2rr^{\prime}\vec{n}\cdot\vec{n}^{\prime}}$ $\displaystyle\simeq
r-r^{\prime}\vec{n}^{\prime}\cdot\vec{n}\,.$ (102)

Moreover, under this approximation and by transforming to the asymptotic
retarded time coordinates the Green’s function can be simplified to the
following form

$\displaystyle G(x-x^{\prime})$
$\displaystyle=-\frac{\delta((u-u^{\prime})+r^{\prime}(\vec{n}^{\prime}\cdot\vec{n}-1/\beta))}{4\pi(r-r^{\prime}\vec{n}^{\prime}\cdot\vec{n})}$
$\displaystyle=-\frac{\beta\,\mathcal{V}\,\delta(r^{\prime}-v(u-u^{\prime})\mathcal{V})}{4\pi(r-r^{\prime}\vec{n}^{\prime}\cdot\vec{n})}\,.$
(103)

where we have defined

$\mathcal{V}\equiv\frac{1}{1-\beta\vec{n}^{\prime}\cdot\vec{n}}$ (104)

The last step was obtained by the formula

$\delta(g(x))={\textstyle\sum_{i}}\delta(x-x_{i})/\left|g^{\prime}(x_{i})\right|\,,$
(105)

where $x_{i}$’s are roots of $g(x)$.

Gathering all the ingredients above while killing the $r^{\prime}$ integral
with the $\delta$-function, the gravitational wave memory in Eq. (96) becomes

$\displaystyle\delta
h_{ij}^{\text{TT}}(x)=\frac{\kappa_{\scriptscriptstyle{\text{eff}}}}{2\pi}$
$\displaystyle\int_{-\infty}^{u}du^{\prime}\int_{S^{2}}d\Omega^{\prime}\,\frac{dE}{du^{\prime}d\Omega^{\prime}}\times$
$\displaystyle\times\frac{\beta^{2}\Lambda_{ij,kl}(\Omega)n^{\prime}_{k}n^{\prime}_{l}}{r(1-\beta\vec{n}^{\prime}\cdot\vec{n})-v(u-u^{\prime})\vec{n}^{\prime}\cdot\vec{n}}\,.$
(106)

We can now readily perform a limit to null infinity (defined as
$r\rightarrow\infty$ while $u$ remains constant) to obtain the leading order
expression for the gravitational wave memory:

$\displaystyle\delta h_{ij}^{\text{TT}}(x)$
$\displaystyle=\frac{\kappa_{\scriptscriptstyle{\text{eff}}}}{2\pi r}\int
du^{\prime}d\Omega^{\prime}\frac{dE}{du^{\prime}d\Omega^{\prime}}\,\frac{\beta^{2}\Lambda_{ij,kl}(\Omega)n^{\prime}_{k}n^{\prime}_{l}}{1-\beta\vec{n}^{\prime}\cdot\vec{n}(\Omega)}\,,$
$\displaystyle=\frac{\kappa_{\scriptscriptstyle{\text{eff}}}}{2\pi r}\int
d\Omega^{\prime}\;\frac{dE(u)}{d\Omega^{\prime}}\,\left[\frac{\beta^{\prime}_{i}\beta^{\prime}_{j}}{1-\vec{\beta}^{\prime}\cdot\vec{n}}\right]^{\text{TT}}\,,$
(107)

where in the last step we have defined

$\frac{dE(u)}{d\Omega^{\prime}}\equiv\int_{-\infty}^{u}du^{\prime}\,\frac{dE}{du^{\prime}d\Omega^{\prime}}\,,$
(108)

as well as

$\beta^{\prime}_{i}\equiv\beta n^{\prime}_{i}=\beta n_{i}(\Omega^{\prime})\,.$
(109)

We want to stress here, that due to the unique generic structure of the
asymptotic energy-momentum tensor in Eq. (83), the above limit to null
infinity in fact does not depend on the precise form of the unbound energy
flux, but is entirely governed by the Green’s function.

###  IV.4 The Memory Effect in Horndeski Theory

With the formula in Eq. (IV.3) for the displacement memory associated to an
arbitrary asymptotic energy momentum tensor, we can now assess the different
types of memory arising in Horndeski theory by simply inserting the
corresponding energy fluxes.

First of all, for the massless TT tensor modes traveling at the speed of light
$\beta=1$ the corresponding memory formula reads

$\displaystyle\delta h_{ij}^{(h)\text{TT}}(u,r,\Omega)$
$\displaystyle=\frac{1}{8\pi r}\int
d\Omega^{\prime}\;\frac{dE^{(h)}}{d\Omega^{\prime}}\,\left[\frac{n^{\prime}_{i}n^{\prime}_{j}}{1-\vec{n}^{\prime}\cdot\vec{n}}\right]^{\text{TT}}\,,$
(110)

with

$\displaystyle\frac{dE^{(h)}(u)}{d\Omega^{\prime}}=\int_{-\infty}^{u}du^{\prime}r^{2}\langle\partial_{0}h_{kl}^{\text{TT}}\partial_{0}h^{kl}_{\text{TT}}\rangle\,.$
(111)

This follows from Eqs. (79), (84) and (78a) of the unbound energy-momentum
content as well as the value of the coefficient $k^{(h)}$ in Eq. (76a). This
expression agrees with the gravitational wave memory formula in GR [23, 28,
32, 9].

Next, from Eq. (IV.3) it is also possible to recover the well known formula
for memory sourced by asymptotic matter fluxes, in particular of unbound
single particle contributions of mass M and radial velocity $\beta$. In this
case, the energy flux contribution is given by [Eq. (81)]

$\displaystyle\frac{dE^{\text{(m)}}}{du^{\prime}d\Omega^{\prime}}(x^{\prime})=\beta
r^{\prime
2}\langle\phantom{}_{\scriptscriptstyle{(2)}}T_{00}\rangle=M\gamma\beta
r^{\prime
2}\delta^{(3)}(\vec{x}^{\prime}-\vec{x}^{\prime\prime}(t^{\prime}))\,,$ (112)

with $x^{\prime\prime}(t^{\prime})$ denoting the trajectory of the particle.
Since our goal is to study the memory effect in the limit
$r\rightarrow\infty$, we only want to keep the radial component of the
trajectory of the particle. By performing a change to spherical coordinates in
which the delta function becomes

$\displaystyle\delta^{(3)}(\vec{x}^{\prime}-\vec{x}^{\prime\prime}(t^{\prime}))=\frac{\delta(r^{\prime}-r^{\prime\prime}(t^{\prime}))}{r^{\prime
2}}\delta^{(2)}(\Omega^{\prime}-\Omega^{\prime\prime})\,,$ (113)

the trajectory is thus given by the radial motion
$r^{\prime\prime}(t^{\prime})=r^{\prime\prime}_{0}+\beta t^{\prime}$ at a
fixed pair of angular coordinates $\Omega^{\prime\prime}=\text{constant}$. In
terms of the asymptotic retarded time
$u^{\prime}=t^{\prime}-r^{\prime}/\beta$, we thus have
$r^{\prime\prime}=r^{\prime\prime}_{0}+\beta u^{\prime}+r^{\prime}$. Without
loss of generality, we can set the constant $r^{\prime\prime}_{0}$ to be zero,
such that the energy flux reads

$\displaystyle\frac{dE^{\text{(m)}}}{du^{\prime}d\Omega^{\prime}}(x^{\prime})$
$\displaystyle=M\gamma\beta\delta(\beta
u^{\prime})\delta^{(2)}(\Omega^{\prime}-\Omega^{\prime\prime})$
$\displaystyle=M\gamma\delta(u^{\prime})\frac{\delta(\theta^{\prime}-\theta^{\prime\prime})}{\sin\theta^{\prime}}\delta(\phi^{\prime}-\phi^{\prime\prime})\,.$
(114)

Plugging this expression into our general form of memory Eq. (IV.3), we obtian

$\displaystyle\delta h^{\text{(m)TT}}_{ij}(x)=\frac{\kappa^{\text{(m)}}}{2\pi
r}\frac{M}{\sqrt{1-v^{2}}}\left[\frac{\beta^{\prime\prime}_{i}\beta^{\prime\prime}_{j}}{1-\vec{\beta}^{\prime\prime}\cdot\vec{n}}\right]^{\text{TT}}\,,$
(115)

where here the source direction is given by the fixed angle
$\Omega^{\prime\prime}$ and

$\beta^{\prime\prime}_{i}=\beta n_{i}(\Omega^{\prime\prime})\,.$ (116)

For a multi-particle system, the result is then just a sum of the memory
contribution from each particle,

$\displaystyle\delta h^{\text{(m)TT}}_{ij}(x)=\frac{\kappa^{\text{(m)}}}{2\pi
r}\sum_{A=1}^{N}\frac{M_{A}}{\sqrt{1-v_{A}^{2}}}\left[\frac{\beta^{\prime\prime}_{i}{}^{A}\beta^{\prime\prime}_{j}{}^{A}}{1-\vec{\beta}^{\prime\prime}_{A}\cdot\vec{n}}\right]^{\text{TT}}\,,$
(117)

thus recovering the known result from GR (see e.g. [22, 27]) with [Eq. (76c)]

$\kappa^{\text{(m)}}=\kappa=8\pi G\,,$ (118)

since in this case $\bar{G}_{4}=1$.

Finally, the general formula in Eq. (IV.3) also allows for the derivation of
new results. In the case of Horndeski theory, for instance, through the memory
formula associated to the massive scalar wave $\varphi$ emitted by the source.
This asymptotic wave can be described as a superposition of radially outward
plane waves of speed $\beta$ depending on the frequency content of the wave.
Thus, each plane wave contributes to the memory through a contribution of the
form

$\displaystyle\delta h_{ij}^{\text{($\varphi$)TT}}(x)$
$\displaystyle=\frac{\kappa^{\text{($\varphi$)}}}{2\pi r}\int
d\Omega^{\prime}\;\frac{dE^{\text{($\varphi$)}}}{d\Omega^{\prime}}\,\left[\frac{\beta^{\prime}_{i}\beta^{\prime}_{j}}{1-\vec{\beta}^{\prime}\cdot\vec{n}}\right]^{\text{TT}}\,,$
(119)

where $\kappa^{\text{($\varphi$)}}$ is given in Eq. (76b) and [Eq. (78b)]

$\displaystyle\frac{dE^{(\varphi)}(u)}{d\Omega^{\prime}}=\int_{-\infty}^{u}du^{\prime}r^{2}\langle\partial_{0}\varphi\partial_{0}\varphi\rangle\,.$
(120)

However, this only represents the contribution of the monochromatic plane wave
modes with a given frequency, leading to the asymptotic radial velocity
$\beta$ within the chosen source centered frame. Indeed, the group velocity of
a massive field wave defined in Eq. (47) depends on its frequency. Thus, just
as for the multi-particle systems discussed above, through linearity, the
total contribution is given by summing all the contributions from every plane
wave frequency and the memory formula for a realistic emission of massive
scalar waves is of the form

$\displaystyle\delta
h_{ij}^{\text{($\varphi$)TT}}=\frac{\kappa^{\text{($\varphi$)}}}{4\pi
r}\int_{0}^{1}d\beta\int
d\Omega^{\prime}\;\frac{dE^{\text{($\varphi$)}}(\beta)}{d\Omega^{\prime}}\,\left[\frac{\beta^{\prime}_{i}\beta^{\prime}_{j}}{1-\vec{\beta}^{\prime}\cdot\vec{n}}\right]^{\text{TT}}\,.$
(121)

The total displacement memory effect within Horndeski theory is given by the
superposition of all the three contributions in Eqs. (110), (117) and (121)

$\displaystyle\delta h_{ij}^{\text{TT}}(x)=\delta
h_{ij}^{\text{($h$)TT}}(x)+\delta h_{ij}^{\text{(m)TT}}(x)+\delta
h_{ij}^{\text{($\varphi$)TT}}(x)\,.$ (122)

##  V Discussion and Conclusion

Foremost, this paper provides a unifying framework to discuss ordinary and
null memory within metric theories of gravity. With Horndeski theory as an
explicit example, it is shown that the solution to the leading-order low-
frequency equation within the Isaacson approach to gravitational waves only
depends on a universal form of the coarse-grained source energy-momentum and
thus provides a very general form of the contributions to the tensor
displacement memory.

While the correspondence between the different types of memory were studied
before for the specific example of the point particle approximation [27, 44,
45], the existing individual derivations of the corresponding formulas for
ordinary and null memory in the general case were rather distinct in nature.
In contrast, the Isaacson approach not only unambiguously identifies memory
contributions as low-frequency corrections to a primary energy-momentum
emission, but provides the means of discussing all types of memory based on a
single memory evolution equation, and in particular a continuous transition
between null and ordinary memory.

The resulting general memory formula in Eq. (IV.3) not only recovers known
results but also includes the novel contribution given by the tensor memory
sourced by the emission of a massive scalar wave in Horndeski gravity offered
in Eq. (121). Massive scalar wave emissions represent a viable signature
beyond GR as studied for instance in [88, 89, 90, 91, 92, 93, 94], such an
extension of the memory formula to massive scalar fields represents an
important step towards a complete study of beyond GR effects that might be
captured by future memory measurements. However, while the memory signal from
a nearly monochromatic scalar emission is expected to be unobservable, a
direct application of the result in Eq. (121) for a burst of massive scalar
perturbations would first require a decomposition of the signal into a
superposition of plane waves, each characterized by their own propagation
speed. In this context, computing the size and shape of the memory
contribution for a specific promising example would represent an interesting
follow-up of this work.

Moreover, observe that a restriction to a constant asymptotic value of the
additional scalar field in the gravitational sector ensures that the
asymptotic Minkowski background preserves local Lorentz invariance. Another
interesting extension of the present considerations would therefore be to
analyze the consequences of giving up the requirement of asymptotic local
Lorentz invariance.

In particular also vector-tensor theories with a non-trivial asymptotic value
that (spontaneously) break local Lorentz invariance represent a natural
description of such a situation. Thus, an extension of the present work to
additional massive vector degrees of freedom within the gravitational sector
of metric theories would be desirable. In this case, the gauge-invariant SVT
decomposition introduced in Section II.4 might especially prove useful when
decoupling the propagating modes of the theory.

Generally, it is expected that in such Lorentz breaking scenarios the
luminality of the tensor waves is lost as well, therefore introducing an
alternative causal structure on the asymptotic spacetime. In this context, the
very broadly applicable Isaacson viewpoint of understanding asymptotic
radiation and memory seems essential, as other methods relying on a well-
defined boundary of the spacetime at null infinity would fail.

###### Acknowledgements.

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).

##  Appendix A Derivation of the Gauge Invariant Leading Order Propagation
Equations

In this appendix, we derive the gauge-invariant linear propagation equations
in Horndeski theory, as shown in Eqs. (39) and (41).

Consider first the leading order metric field equations at the high frequency
level in Eq. (25a). Within the 3+1 and SVT decomposition in the asymptotic
region presented in Sec. II.4, we can separately treat the different
components of the equations of motion and rewrite them purely in terms of
manifestly gauge invariant quantities. Firstly, the 00-component of the metric
equation converts to

$\displaystyle-\bar{G}_{4}\nabla^{2}\Theta=\bar{G}_{4,\phi}\nabla^{2}\varphi\,.$
(123)

On the other hand, the 0i-part of the metric equation can be further
decomposed into its transverse and longitudinal components

$\displaystyle-\frac{1}{2}\bar{G}_{4}\nabla^{2}\Xi^{i}$ $\displaystyle=0\,,$
(124a) $\displaystyle-\bar{G}_{4}\partial^{i}\dot{\Theta}$
$\displaystyle=\bar{G}_{4,\phi}\partial^{i}\dot{\varphi}\,.$ (124b)

Finally, the spatial ij-component can be separated into its transverse-
traceless, longitudinal, trace, and vector contributions:

$\displaystyle\frac{1}{2}\bar{G}_{4}\Box E_{ij}$ $\displaystyle=0\,,$ (125a)
$\displaystyle-\bar{G}_{4}\ddot{\Theta}$
$\displaystyle=\bar{G}_{4,\phi}\ddot{\varphi}\,,$ (125b)
$\displaystyle\bar{G}_{4}(-3\ddot{\Theta}+\nabla^{2}\Theta-2\nabla^{2}\Phi)$
$\displaystyle=\bar{G}_{4,\phi}(3\ddot{\varphi}-2\nabla^{2}\varphi)\,,$ (125c)
$\displaystyle\bar{G}_{4}(-\partial_{(i}\dot{\Xi}_{j)})$ $\displaystyle=0\,.$
(125d)

Here, the symbol $\nabla^{2}$ denotes the Laplacian operator.

Since the system is only physical when $\bar{G}_{4}\neq 0$, the above gauge
invariant EOMs can be simplified by dividing out $\bar{G}_{4}$. Employing the
notation $\sigma\equiv\bar{G}_{4,\phi}/\bar{G}_{4}$, the independent
components of the above field equations reduce to three constraint equations
and one dynamical equation

$\displaystyle\nabla^{2}(\Theta+\sigma\varphi)=0\,,$ (126a)
$\displaystyle\nabla^{2}\Xi^{i}=0\,,$ (126b)
$\displaystyle\nabla^{2}(\Phi-\frac{1}{2}\sigma\varphi)=0\,,$ (126c)
$\displaystyle\Box E^{\text{TT}}_{ij}=0\,.$ (126d)

The last equation indicates that as in GR only the TT part of the high-
frequency metric perturbation $h^{H}_{\mu\nu}\,$ is dynamical. The other three
Laplacian equations unveil how the scalar field imprints on the metric.
Because the scalar field and all gauge invariant quantities fulfill a well-
behaved physical boundary condition at $r\rightarrow\infty$, the solutions of
the Laplacian equations are simply given by

$\displaystyle\Theta$ $\displaystyle=-\sigma\varphi\,,$ (127a)
$\displaystyle\Phi$ $\displaystyle=\frac{1}{2}\sigma\varphi,$ (127b)
$\displaystyle\Xi^{i}$ $\displaystyle=0\,.$ (127c)

Using the above solutions, we can in turn drastically simplify the leading
order propagation equation of the scalar field perturbation in Eq. (25b) by
replacing the gauge invariants $\Theta$ and $\Phi$ with scalar field
perturbation $\varphi$. The scalar field propagating equation can then be
written in terms of the gauge invariant quantities as

$\displaystyle(\bar{G}_{2,X}-2\bar{G}_{3,\phi}+\frac{3(\bar{G}_{4,\phi})^{2}}{\bar{G_{4}}})(-\ddot{\varphi}+\nabla^{2}\varphi)+\varphi\bar{G}_{2,\phi\phi}=0\,.$
(128)

For the physically interesting solutions, we impose

$\bar{G}_{2,X}-2\bar{G}_{3,\phi}+3(\bar{G}_{4,\phi})^{2}/\bar{G_{4}}\neq 0\,,$
(129)

and define the effective mass as

$\displaystyle
m^{2}\equiv\frac{\bar{G}_{2,\phi\phi}}{\bar{G}_{2,X}-2\bar{G}_{3,\phi}+3(\bar{G}_{4,\phi})^{2}/\bar{G_{4}}}\,.$
(130)

The high-frequency scalar field equation then simplifies to the Klein-Gordon
equation given in Eq. (39).

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