Subleading non linear gravitational memory effect

Zahra Mirzaiyan

Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut),
Mühlenberg 1, D-14476 Potsdam, Germany.

ABSTRACT

We apply the new method based on null geodesics for detecting gravitational
memory and find the bulk memory in Newman-Unti gauge around the boundary of
the conformally compactified space time. We show how we use the newly found
conserved charges in the subleading orders of large-$r$ expansion of the BMS
charges to define the gravitational memory at each order in the non-linearised
gravitational theory. We also find the gravitational shift in the $r$
direction. It is shown that the longitudinal displacement at order $1/r$ is
the relative radius change between two detectors derived by Strominger and
Zhiboedov.

<EMAIL_ADDRESS>
##  1 Introduction

The gravitational memory effect discovered by Zeldovich and Polnarev [1] is
based on the asymptotic behaviour of the gravitational fields in the
asymptotic region of space time and is defined as the permanent change in the
relative spatial position of a pair of inertial observers after a finite
gravitational burst of energy. The passage of gravitational waves produces an
observable effect which LIGO [2, 3] and the next generation of gravitational
waves detectors like LISA [4, 5] are based on.
The phenomenon has been studied in the linearised theory of gravity [6, 7] and
generalised by Christodoulou and others [8, 9] to the non-linearised theory.
Christodoulou showed that every gravitational burst of energy has a non linear
memory and can not be neglected only because the gravitational wave sources
are at large distances from our detectors on the Earth. As we are extremely
far away from the sources of gravitational waves, the amplitude of the waves
are so small as it is assumed that linearised theory of gravity is sufficient
for the study of the gravitational waves. However, depending on the source of
the burst, the amplitude of the non linear memory111Also referred as
Christodoulou memory. can be of the same order as the dynamical part of the
burst of energy. Therefore, the study of gravitational memory effect in the
full non-linearised theory of gravity is important and plays a crucial role in
the future observation of the gravitational waves. The original formulation of
the gravitational memory by Christodoulou is based on the definition of test
particles which are initially at rest moving on timelike geodesics at the
boundary of the conformally compactified space time. The memory at null
infinity is then obtained by the geodesic equation which is not the case for
the gravitational displacement in the bulk of the space time due to the
gravitationally inward nature of timelike geodesics near a strongly
gravitational system, i.e a black hole. Therefore, recently another method for
finding the memory effect is provided based on null geodesics instead of
timelike geodesics [10].
The memory effect is indeed the interesting corner of the so called Infrared
(IR) triangle [11] discovered recently by Strominger [12] which connects the
memory with asymptotic symmetries [13] and the soft theorems [14]. Due to the
non-trivial connection of the corners at the IR limit, the observation of the
memory effect gives us information about the symmetries in the asymptotic
region of the space time. However, for the detection of the memory in the weak
field approximation, one needs to wait for the IR gravitational wave detection
[15]. Strominger and Zhiboedov revealed this connection and showed that the
relative position of the inertial observers differ by a BMS
supertranslation222It is also shown that the the nearby detectors experience a
relative time delay in addition to the standard gravitational spatial memory
effect.. Therefore, the displacement between a pair of inertial observers
using the supertranslated metric is precisely the standard formula for the
memory effect and can be derived from geodesic equation at the leading order
at null infinity.
Our aim in the paper is to find the the large-$r$ expansion of the
gravitational displacement close to the null infinity in the bulk of the four
dimensional asymptotically flat space time in the full non-linearised theory
of gravity using null geodesics. Although, all we have detected till now
showed a displacement memory predicted by the linearised gravity, it still
worth investigating the phenomenon in the non-linearised theory as well for
the reasons explained earlier333See [16] for a detailed review on linearised
and non-linearised gravitational wave memory effect.. Our motivation to study
the subleading memory effect around the null infinity is based on the new
interesting physics hidden in the subleading orders, especially the existence
of a new set of conserved integrable charges at order $r^{-3}$ of large-$r$
expansion of the BMS charges [17] which are directly related to the the non-
linearly conserved Newman-Penrose (NP) charges [18]. Since the connection of
the BMS symmetries and the memory effect is established in the bulk of the
space time444Also see section (2.3) of [10] for more details. [19, 20], one
needs to consider the subleading BMS diffeomorphisms and therefore the newly
found subleading BMS charges at the subleading orders. Our idea is simple and
the expectation is reasonable. At the leading order of the gravitational
memory, if the leading BMS charge, namely the Bondi mass aspect is conserved,
then there is no flux at null infinity and therefore no displacement memory
which we show it is not the case in the subleading order where we have a set
of conserved charges. If we let the mass to vary, we have memory which is
directly related to the change of the gravitational wave tensor, $\Delta
C_{IJ}$ at null infinity. At the subleading order $r^{-3}$, we have a set of
integrable and finite BMS charges which are related to some physical conserved
charges. We are especially interested in the subleading gravitational memory
if one considers newly found conserved charges at subleading orders. We will
also use the method introduced in [10] to find the longitudinal memory in the
$r$ direction.
The paper is as follows: We begin by the asymptotically flat metric in four
dimensions in section 2. Then we review the new method for the gravitational
memory calculation based on null geodesics in section 3. The towers of
gravitational memories is found in section 4, where, we show the memory is not
vanishing although having a set of conserved charges in some certain
subleading orders. In section 5, the longitudinal displacement is found and it
is shown that this shift only shows itself in the subleading orders, mainly in
order $1/r$. We conclude our work with some discussions in section 6.

##  2 Asymtotically 4.d flat metric in Bondi gauge

###  2.1 Metric definition in the Bondi gauge

The most general form of the metric in 4.d which is asymptotic to the flat
metric in retarded Bondi gauge with coordinates $(u,r,x^{I})$,
$x^{I}=\\{\theta,\phi\\}$ takes the form

$\displaystyle ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=-Fe^{2\beta}\
du^{2}-2e^{2\beta}\ dudr+g_{IJ}(dx^{I}-C^{I}du)(dx^{J}-C^{J}du),$ (2.1)

where $F$, $\beta$ and $C^{I}$ are some metric functions with the following
fall-off conditions

$\displaystyle F(u,r,x^{I})=1+\sum_{n=0}^{n}\frac{F_{n}(u,x^{I})}{r^{n+1}},\ \
\beta(u,r,x^{I})=\sum_{n=0}^{n}\frac{\beta_{n}(u,x^{I})}{r^{n+2}},\ \
C^{J}(u,r,x^{I})=\sum_{n=0}^{n}\frac{C_{n}^{J}(u,x^{I})}{r^{n+2}}.$

The inverse of the metric (2.1) reads as

$\displaystyle g^{\mu\nu}=\begin{pmatrix}0&-e^{-2\beta}&0\\\
-e^{-2\beta}&Fe^{-2\beta}&-e^{-2\beta}C^{J}\\\
0&-e^{-2\beta}C^{I}&g^{IJ}\end{pmatrix}.$ (2.3)

The spatial part of the metric has the following fall-off at large-$r$
expansion defined in [17] as

$\displaystyle
g_{IJ}(u,r,x^{I})=r^{2}h_{IJ}=r^{2}\gamma_{IJ}+rC_{IJ}(u,x^{I})+\frac{C^{2}\gamma_{IJ}}{4}+\frac{D_{IJ}(u,x^{I})}{r}+\frac{E_{IJ}(u,x^{I})}{r^{2}}+\mathcal{O}(r^{-2}).$

where, $\gamma_{IJ}$ is the metric on the 2-sphere, $C_{IJ}$ describes the
gravitational waves and the square of the so called “Bondi news tensor”,
$N_{IJ}=\partial_{u}C_{IJ}$ is proportional to the flux of energy at the null
boundary of the space time and $C^{2}=C_{IJ}C^{IJ}$. Fixing the gauge
condition as follows

$\displaystyle\frac{1}{r^{2}}\text{det}(g_{IJ})=\text{det}(\gamma_{IJ})=\sin^{2}\theta,$
(2.5)

keeps the spatial part of the metric topologically spherical and gives us the
following constraints about the functions $C_{IJ}$, $D_{IJ}$ and $E_{IJ}$,

$\displaystyle\frac{1}{r^{2}}\text{det}(g_{IJ})=\text{det}(\gamma_{IJ}+\frac{C_{IJ}}{r}+\frac{C^{2}\gamma_{IJ}}{4r^{2}}+\frac{D_{IJ}}{r^{3}}+\frac{E_{IJ}}{r^{4}}+{\mathcal{O}(r^{-4})})$
$\displaystyle=\text{det}(\gamma_{IJ})\
[1+\frac{C_{I}^{I}}{r}+\frac{D_{I}^{I}}{r^{3}}+\frac{E_{I}^{I}}{r^{4}}-\frac{C^{2}C_{I}^{I}}{4r^{3}}-\frac{C_{KL}D^{KL}}{r^{4}}{+}\frac{C^{4}}{16r^{4}}+..],$

which implies

$\displaystyle\text{tr}C=0,\ \ \text{tr}D=0,\ \
\text{tr}E=C_{IJ}D^{IJ}-\frac{C^{4}}{16}.$ (2.7)

Assuming appropriate fall-off condition for energy-momentum tensor at each
order, some of the metric functions in metric (2.1) read as555See section
(2.2) of [17] for the complete derivation of the metric functions in metric
(2.1).

$\displaystyle\partial_{u}F_{0}=-\frac{1}{2}D_{I}D_{J}\partial_{u}C^{IJ}+\frac{1}{4}\partial_{u}C^{IJ}\partial_{u}C_{IJ},$
(2.8) $\displaystyle\beta_{0}=-\frac{1}{32}\,C^{2},\ \
\beta_{2}=\frac{1}{128}\,(C^{2})^{2}-\frac{3}{32}D_{IJ}C^{IJ},$ (2.9)
$\displaystyle C_{0}^{I}=-{\textstyle{\frac{1}{2}}}D_{J}C^{IJ}.$ (2.10)

One can also re-parametrise $h_{IJ}$, the spatial part of the metric (2.1).
The parametrisation of the metric is indeed very useful and crucial for
calculating the coefficients in the metric functions described in (2.1) and in
all the calculations in the current paper. See Appendix A for the
parametrisation of the metric.

##  3 Gravitational memory in the bulk

The gravitational memory effect is defined as the permanent shift in the
relative distance of a pair of inertial observers666See [12] for a detailed
discussion on different types of observers. stationed near $\mathcal{I}^{+}$
due to the burst of energy. The main purpose of this paper is to find the
corrections to the gravitational memory away from null infinity which we call
“subleading gravitational memory”. In fact, we aim to calculate the
displacement memory in the bulk, close to the boundary of conformally
compactified space time. One can use null geodesics instead of timelike
geodesics and find the gravitational displacement [10]. It is shown that this
method can be easily used for finding the gravitational memory effect in the
bulk of the space time. We first briefly review this new method for the
formulation of the gravitational memory in the bulk using ingoing null
geodesics. One can measure the deviation between a pair of null geodesics
$\eta^{\mu}|_{v}$, when one shoots light rays at time $v$. At some later time
after the burst of the gravitational energy, another pair of light rays are
considered and the deviation between the second pair of null geodesics is
measured as $\eta^{\mu}|_{v^{\prime}}$. The new method for the formulation of
the gravitational memory is based on comparing the geodesic deviation between
two assumed pairs of null geodesics as follows

$\displaystyle\Delta\eta^{\mu}:=\eta^{\mu}|_{v^{\prime}}-\eta^{\mu}|_{v},$
(3.1)

which is well-defined in any radius. $\eta^{\mu}$ is the deviation between a
pair of null geodesics given by

$\displaystyle\eta^{\mu}(r)=x^{\mu}(r)-x_{0}^{\mu}(r).$ (3.2)

$x_{0}^{\mu}(r)$ is a geodesic generated by the vector
$n_{\nu}=-\partial_{\nu}v$ and the neighbourhood geodesic $x^{\mu}(r)$
generated by small deformations $f$ given by

$\displaystyle
x^{\mu}(r)=\int^{r}g^{\mu\nu}(n_{\nu}-\partial_{\nu}f)+z^{\mu}.$ (3.3)

$f$ is a function of the 2-sphere and $z^{\mu}$ indicates the ingoing location
of the geodesic. In Eq. (3.2), $r$ is an affine parameter. If we choose
Newman-Unti (NU) coordinate, at a sufficiently large radius $r=r_{L}$, the
worldlines $(v,r_{L},x^{I})$ are approximately inertial observers and the
gravitational displacement due to the burst of energy between $v$ and
$v^{\prime}$ can be computed easily by comparing the quantity $\eta^{\mu}$
before and after the flux at infinity. It is shown that this method is also
applicable to find the gravitational memory in the bulk of the space time. We
are in fact interested in the large-$r$ expansion of the gravitational memory
which is near null infinity and in the bulk of the space time.

##  4 Subleading memory effect in Newman-Unti gauge

The asymptotically flat space time metric can be written in NU coordinate
$(v,r,x^{I})$. In this gauge null foliations of space time are describe with
parameter $v$ while $r$ is affine for generators of geodesics in the
hypersurfaces $\Sigma_{v}$ of constant $v$. Starting from the form of the
metric (2.1) in Bondi gauge with radial coordinate $r$, one can find the
asymptotically flat metric in NU gauge by the following radial coordinate
change according to Barnich and Lambert [21] as

$\displaystyle r_{\text{NU}}=r-\int_{r}^{\infty}dr^{\prime}\
(e^{2\beta}-1)=r-\frac{2\beta_{0}}{r}-\frac{2(\beta_{0}^{2}+\beta_{2})}{3r^{3}}+{\mathcal{O}(r^{-3})}.$
(4.1)

As we are interested in the spatial gravitational displacement
$\Delta\eta^{I}$ at some fixed $r$, we need the spatial part of the metric in
NU gauge. We will address the displacement in the longitudinal direction,
$\Delta\eta^{r}$ as well later in this the paper. The spatial part of the
asymptotically flat metric in NU gauge can be found as

$\displaystyle
g^{IJ{(\text{NU})}}=\frac{\gamma^{IJ}}{r_{\text{NU}}^{2}}+\frac{\mathcal{A}^{IJ}}{r_{\text{NU}}^{3}}+\frac{\mathcal{B}^{IJ}}{r_{\text{NU}}^{4}}+\frac{\mathcal{G}^{IJ}}{r_{\text{NU}}^{5}}+\frac{\mathcal{H}^{IJ}}{r_{\text{NU}}^{6}}+\mathcal{O}(r_{{\text{NU}}}^{-7}),$
(4.2)

with

$\displaystyle\mathcal{A}^{IJ}=-C^{IJ},\ \
\mathcal{B}^{IJ}=\frac{3}{8}C^{2}\gamma^{IJ},\ \
\mathcal{G}^{IJ}=-(D^{IJ}+\frac{3}{16}C^{2}C^{IJ}),$
$\displaystyle\mathcal{H}^{IJ}=-E^{IJ}+\frac{9}{8}(D^{KL}C_{KL}-\frac{1}{16}C^{4})\
\gamma^{IJ}+\frac{5C^{4}}{64}\gamma^{IJ}.$ (4.3)

We assume the following large-$r$ expansion for the spatial gravitational
displacement777The spatial memory in Eq. (4.13) starts from $n=2$ but that is
just because of the factor $1/r^{2}$ in $g^{IJ}$.  as

$\displaystyle\mathscr{D}^{I}=\sum_{n=2}^{n}\frac{\mathscr{D}^{I(n-2)}}{r^{n}}.$
(4.4)

Using Eqs. (3.1) and (4.2), one can find the the gravitational memory in each
order. One should also consider the newly found BMS chrages at subleading
orders which are actually related to the conserved Newman-Penrose charges.
Therefore, the change in the geodesic deviation with leading order deviation
$d^{I}=\frac{1}{r}\gamma^{IJ}\partial_{J}f$, gives us a shift which by
considering the conserved charges at each order corresponds to the
gravitational displacement at each order.

###  4.1 Memory effect and the BMS charge at $\mathcal{O}(r^{0})$

According to [21], the variation of the leading BMS charge reads as

$\displaystyle\delta\mathcal{I}_{0}=\delta(-2sF_{0})+\frac{s}{2}\partial_{u}C_{IJ}\delta
C^{IJ},$ (4.5)

where $s$ is referred to the supertranslations and

$\displaystyle\delta C^{IJ}=s\partial_{u}C^{IJ}-2D^{I}D^{J}s,$ (4.6)

is the variation of $C^{IJ}$ under the action of supertranslation and $D^{I}$
is the covariant derivative with respect to $\gamma^{IJ}$. The first term in
charge (4.5) is the integrable part of the charge and $F_{0}$ is proportional
to the Bondi mass aspect $m_{B}$. The second term is the non-integrable part
of the BMS charge and corresponds to the radiation at null infinity. If one
sets the non-integrable part of charge to zero, then $\mathcal{I}_{0}$ is
integrable and $F_{0}$ is conserved which means there is no radiation at null
infinity and therefore $C^{IJ}$ is a free data at null infinity. But if we
allow $F_{0}$ to vary, there will be a flux of gravitational radiation at null
infinity and $\partial_{u}C^{IJ}$ is no longer zero. The gravitational memory
at leading order and right at null infinity reads as

$\displaystyle\Delta\eta^{I(0)}=-\frac{\partial^{J}f}{2r_{NU}^{2}}\Delta
C_{J}^{I}=-\frac{\partial^{J}f}{2r^{2}}\Delta C_{J}^{I}.$ (4.7)

Therefore,

$\displaystyle\mathscr{D}^{I(0)}=-\frac{d^{J}}{2r}\Delta C_{J}^{I}.$ (4.8)

At this order one can also find the gravitational shift between a pair of
timelike observers using the geodesic equation [12, 11]. See Appendix B for
the derivation of the leading memory at null infinity in NP formalism using
timelike geodesics.

###  4.2 Memory effect and BMS charge at $\mathcal{O}(r^{-1})$

At this order the BMS charge is identically zero due to the Einstein equations
and strong enough choice for the fall off of the energy-momentum tensor, thus
$\mathcal{I}_{1}=0$. Therefore, there is no conserved quantity in this order
and the gravitational shift at this order is

$\displaystyle\Delta\eta^{I(1)}=-\frac{\partial^{I}f}{8r_{NU}^{3}}\Delta(C^{2})=-\frac{\partial^{I}f}{8r^{3}}\Delta(C^{2}).$
(4.9)

Therefore,

$\displaystyle\mathscr{D}^{I(1)}=-\frac{d^{I}}{8r^{2}}\Delta(C^{2}).$ (4.10)

###  4.3 Memory effect and BMS charge at $\mathcal{O}(r^{-2})$

At this order the BMS charge is integrable as the non-integrable part of the
charge is zero $\delta\mathcal{I}_{2}^{\text{(non-int)}}=0$, for $l=0$ or
$l=1$ spherical harmonics with the constraint
$D_{I}D_{J}s=\frac{1}{2}\gamma_{IJ}\Box s$ on the supertranslation function
$s$. The integrable part of the charge is given by integrating

$\displaystyle\delta\mathcal{I}_{2}=D_{I}D_{J}s\
(-D^{IJ}+\frac{1}{16}C^{2}C^{IJ}).$ (4.11)

Due to the trace free property of $C_{IJ}$ and $D_{IJ}$ defined in (2.7), the
BMS charge at this order is zero and therefore the memory reads as

$\displaystyle\Delta\eta^{I(2)}=-\frac{\partial^{J}f}{4r_{NU}^{4}}(\Delta(D_{J}^{I}-\frac{1}{16}C^{2}C_{J}^{I})+\frac{1}{4}\Delta(C^{2}C_{J}^{I}))=-\frac{\partial^{J}f}{4r^{4}}\Delta(D_{J}^{I}-\frac{1}{16}C^{2}C_{J}^{I}).$
(4.12)

Thus,

$\displaystyle\mathscr{D}^{I(2)}=-\frac{d^{J}}{4r^{3}}\Delta(D_{J}^{I}-\frac{1}{16}C^{2}C_{J}^{I}).$
(4.13)

###  4.4 Memory effect and BMS charge at $\mathcal{O}(r^{-3})$

At this order the non-integrable part of the BMS charge is zero and if one
considers the supertranslation $s$ to be an $l=2$ spherical harmonic, there is
a new set of integrable charges which are directly related to the conserved NP
charges as

$\displaystyle\mathcal{I}_{3}=sD_{I}D_{J}(-E^{IJ}+\frac{1}{2}(D^{KL}C_{KL}-\frac{1}{16}C^{4})\gamma^{IJ}).$
(4.14)

The above expression is directly related to the component $\Psi_{0}^{1}$ of
the Weyl scalars888See Eq. (4.30) of [17].. Using (3.1), the gravitational
shift is given by

$\displaystyle\Delta\eta^{I(3)}=-\frac{\partial_{J}f}{4r_{NU}^{5}}\Delta(E^{IJ}+\frac{9}{5}\Big{(}-E^{IJ}+\frac{1}{2}(D^{KL}C_{KL}-\frac{1}{16}C^{4})\gamma^{IJ}\Big{)}+\frac{1}{16}C^{4}\gamma^{IJ})$
$\displaystyle=-\frac{\partial_{J}f}{4r^{5}}\Delta(E^{IJ}+\frac{9}{5}\Big{(}-E^{IJ}+\frac{1}{2}(D^{KL}C_{KL}-\frac{1}{16}C^{4})\gamma^{IJ}\Big{)}-\frac{1}{32}C^{4}\gamma^{IJ}).$
(4.15)

But due to the existence of set of integrable conserved charges at this order,
the gravitational shift is obtained by

$\displaystyle\mathscr{D}^{I(3)}=-\frac{d^{J}}{4r^{4}}\Delta(E_{J}^{I}-\frac{\delta_{J}^{I}}{32}C^{4}).$
(4.16)

The above equation shows that even if the charge remains conserved in this
order, still there is a non-trivial displacement. However, we just considered
the subleading BMS charges which are related to the real part of the weyl
scalars.

##  5 Longitudinal displacement

In addition to $\Delta\eta^{I}$, one can also consider the gravitational shift
in the longitudinal direction as $\Delta\eta^{r}$. Using Eq. (3.1), the shift
on the $r$ direction can be read as

$\displaystyle\Delta\eta^{r}=-\int^{r}g^{rI}\partial_{I}f.$ (5.1)

Using Eq. (2.10), the above equation can be written as999The large-$r$
expansion of $g^{rI}$ component of the metric reads as  $\displaystyle
g^{rI}=-\frac{C_{0}^{I}}{r^{2}}-\frac{C_{1}^{I}}{r^{3}}-\frac{2\beta_{0}C_{0}^{I}+C_{2}^{I}}{r^{4}}+\mathcal{O}(r^{-5}).$

$\displaystyle\Delta\eta^{r}=\frac{D^{J}\Delta
C_{IJ}}{2r}\partial^{I}f+\mathcal{O}(r^{-2}).$ (5.2)

But vanishing of the curvature in the asymptotic region implies

$\displaystyle\Delta C_{IJ}=2D_{I}D_{J}s,$ (5.3)

Thus, Eq. (5.2) shows the relative radius change between two null geodesics
which is the same shift in $r$ direction derived in[12]101010See Eq. (4.13) in
the reference. as $\delta r$ when one considers the action of the
supertranslations $s$ on the initial separation vector between the detectors
$\zeta$ as $\mathcal{L}_{s}\zeta$. Therefore, the shift in the $r$ direction
between two observers can be neatly recast using the new method for finding
the gravitational memory based on null geodesics.

##  6 Discussion

In this paper, we studied the non linear memory in the bulk of the space time
around null infinity. Considering the large-$r$ expansion of the BMS conserved
charges at each order, we showed that despite the existence of a set of
integrable charges at order $r^{-3}$, the memory is not zero contrary to the
case in the leading order. That may be related to the fact that the physical
meaning of the NP charges is not very clear especially in the non-linearised
theory of gravity while the interpretation of the charge in the leading order
is the Bondi mass aspect. However, we only considered the supertranslation
charges at subleading orders. Large-$r$ expansion of the superrotation charges
which may be related to some conserved physical charges and the connection to
the subleading memory remains to be explored in future works. However, the
observable for the spin memory is a time delay 111111See [22, 23] for the
connection of the spin memory and superrotations. while in this paper the
displacement memory is considered. Moreover, the gravitational waves and
therefore the infinite towers of memories are characterized by the asymptotic
shear $\sigma^{0}$ and according to [24], the subleading memories are defined
by different choice of the asymptotic shear. However as it is obvious from
Eqs. (4.13) and (4.16), the displacement memories are not characterized only
by the shear121212The tower of memories should be only dependent on the
functions $f_{0}$ and $g_{0}$ (See Appendix A). However, the subleading memory
at order $r^{-2}$ and $r^{-3}$ also depends on the functions $f_{2}$, $g_{2}$,
$f_{3}$ and $g_{3}$. Suppose in a physical system we let the Bondi mass be
conserved. Therefore, there will be no gravitational displacement memory at
null infinity and also not around the null boundary. which is related to the
flux at infinity. We may solve this in a future work by considering the tower
of dual BMS charges [25]. Also despite the leading order in which the memory
is produced by the non-integrable part of the leading charge, the non-
integrable parts of the subleading supertranslation charges do not play any
role in the subleading memory effect. Finally, in addition to the standard
memory, a displacement in the radial direction has been calculated. It is
shown that the shift in the $r$ direction shows itself only away from null
infinity and it is exactly the change in the relative radius between two
detectors when the supertranslations act on the initial separation vector
between two observers before passage of the gravitational waves. This confirms
the method we used in the body of the paper led to consistent results with
previous works.

## Acknowledgements

We are grateful of Maryam Aghaei, Henk Bart, Ajit Mehta and especially Hadi
Godazgar for useful discussions. We thank Axel Kleinschmidt and Maarten van de
Meent for the comments on the draft.

##  Appendix A Parametrisation of the spatial part of the asymptotically flat
metric

Following [13, 17], one can do a parametrization of the spatial part of the
metric (2.1) as

$\displaystyle
h_{IJ}dx^{I}dx^{J}=\frac{1}{2}(e^{2f}+e^{2g})d\theta^{2}+(e^{f-g}+e^{-f+g})\sin\theta\
d\theta d\phi+\frac{1}{2}\sin^{2}\theta\ (e^{-2f}+e^{-2g})d\phi^{2}.$

with $f(u,r,\theta,\phi)$ and $g(u,r,\theta,\phi)$ defined as

$\displaystyle f(u,r,\theta,\phi)=\sum_{n=0}^{\infty}\frac{f_{n}}{r^{n+1}},\ \
\ \ g(u,r,\theta,\phi)=\sum_{n=0}^{\infty}\frac{g_{n}}{r^{n+1}}.$ (A.2)

with the constraint $f_{1}=g_{1}=0$, that comes from the Sommerfeld radiation
condition131313The condition is also called the regularity condition on the
metric and it is the consistent behaviour of the metric functions at far
distances. See section 4 of [13] and especially Eq. (4.10) of [26]. which
describes the behaviour of the field in asymptotic region as
$r\rightarrow\infty$. One can expand the spatial metric (A) in terms of
$f_{n}$ and $g_{n}$ and find the tensors $C_{IJ}$, $D_{IJ}$ and $E_{IJ}$ in
terms of $f$ and $g$ as

$\displaystyle C_{00}=-\frac{1}{\sin^{2}\theta}C_{11}=f_{0}+g_{0},\ \
C_{01}=C_{10}=(f_{0}-g_{0})\sin\theta,$ (A.3) $\displaystyle
D_{00}=-\frac{1}{\sin^{2}\theta}D_{11}=f_{2}+g_{2}+\frac{2}{3}(f_{0}^{3}+g_{0}^{3}),$
$\displaystyle
D_{01}=D_{10}=(f_{2}-g_{2}+\frac{1}{6}(f_{0}-g_{0})^{3})\sin\theta,$ (A.4)
$\displaystyle
E_{00}=f_{3}+g_{3}+\frac{1}{3}f_{0}^{4}+2f_{0}f_{2}+2g_{0}g_{2}+\frac{1}{3}g_{0}^{4},$
$\displaystyle
E_{11}=-(f_{3}+g_{3}-\frac{1}{3}(f_{0}^{4}+6f_{0}f_{2}+6g_{0}g_{2}+g_{0}^{4}))\sin^{2}\theta,$
$\displaystyle E_{01}=E_{10}=(f_{3}-g_{3})\sin\theta.$ (A.5)

The the above parametrisation is indeed essential for finding the metric
functions defined in [17].

##  Appendix B Memory effect at null infinity in Newman-Penrose formalism

The gravitational memory can be also written in NP formalism141414For the NP
formalisation of the infinite towers of gravitational memory see [24].. The
Newman-Penrose formalism begins with a choice of complex null frame
$\\{l,n,m,\bar{m}\\}$. We begin with the asymptotic form of the metric as

$\displaystyle ds^{2}=-Fe^{2\beta}\ du^{2}-2e^{2\beta}\
dudr+r^{2}h_{IJ}(dx^{I}-C^{I}du)(dx^{J}-C^{J}du)$
$\displaystyle=-e^{2\beta}du\
(Fdu+2dr)+2r^{2}(\hat{m}_{(I}\bar{\hat{m}}_{J)})(dx^{I}-C^{I}du)(dx^{J}-C^{J}du).$
(B.1)

Using $g_{ab}=E_{a}^{\mu}E_{b}^{\nu}\eta_{\mu\nu}$, the orthonormal tetrad
$\\{E_{\alpha}\\}$ reads as

$\displaystyle E_{0}^{\perp}=e^{2\beta}du,\ \
E_{1}^{\perp}=\frac{1}{2}(Fdu+2dr),\ \
E_{2}^{\perp}=r\hat{m}_{I}(dx^{I}-C^{I}du),\ \
E_{3}^{\perp}=r\bar{\hat{m}}_{I}(dx^{I}-C^{I}du).$

Accordingly, a complex null frame can be introduced as

$\displaystyle l=\partial_{r},\ \
n=e^{-2\beta}[\partial_{u}-\frac{1}{2}F\partial_{r}+C^{I}\partial_{I}],\ \
m=\frac{\hat{m}^{I}}{r}\partial_{I},\ \
\bar{m}=\frac{\bar{\hat{m}}^{I}}{r}\partial_{I}.$ (B.3)

The scalar products of the tetrad vectors vanishes apart from

$\displaystyle l^{\mu}n_{\mu}=-m^{\mu}\bar{m}_{\mu}=-1.$ (B.4)

The coordinate basis can be read as

$\displaystyle\partial_{u}=e^{2\beta}n+\frac{1}{2}Fl-rh_{IJ}C^{I}\
(m\bar{\hat{m}}^{J}+\bar{m}\hat{m}^{J}),\ \ \partial_{r}=l,\ \
\partial_{I}=rh_{IJ}\ (m\bar{\hat{m}}^{J}+\bar{m}\hat{m}^{J}).$

We will use the above basis for writing the Riemann tensor in terms of the
Weyl scalars. In general the geodesic equation shows how the inertial
particles undergo a mutual acceleration due to the curvature of the space
time. The relative acceleration between two time like geodesics is descibed by
the equation of geodesic as follows

$\displaystyle\frac{D^{2}\xi^{\alpha}}{du^{2}}=-R_{\beta\gamma\delta}^{\alpha}v^{\beta}v^{\delta}\xi^{\gamma},$
(B.6)

where, $v^{\alpha}=(1,0,0,0)$ is the four vector velocity of a timelike
observer and $\xi^{\alpha}$ is the deviation vector between two neighbouring
geodesics.
We now find the large-$r$ expansion of the gravitational displacement due to
the passage of the gravitational burst. We consider the following large-r
expansion for the $R_{\beta\gamma\delta}^{\alpha}$ and the four-velocity of
the inertial observer151515One may consider the corrections to the four-
velocity of the observer (See exercise 13 of [11]), but actually the only
important component of the four-vector velocity of the inertial observer is
$v^{u(0)}$ to remain inertial during the gravitational energy burst. as

$\displaystyle
R_{\beta\gamma\delta}^{\alpha}=\sum_{n=1}^{n}\frac{R_{\beta\gamma\delta}^{\alpha(n)}}{r^{n}},\
\ \ \ v^{\alpha}=\sum_{m=0}^{m}\frac{v^{\alpha(m)}}{r^{m}}.$ (B.7)

The Riemann tensor component which is interesting for the leading memory reads
as

$\displaystyle
R_{uIuJ}=R_{\mu\nu\rho\sigma}(\partial_{u})^{\mu}(\partial_{u})^{\rho}(\partial_{I})^{\nu}(\partial_{J})^{\sigma}.$
(B.8)

In vacuum $R_{\mu\nu}=0$ and the Riemann tensor becomes as same as the Weyl
tensor. Of course, that is not the general case and one may investigate the
problem in the presence of matter. So Eq. (B.8) can be written as

$\displaystyle
R_{uIuJ}=C_{\mu\nu\rho\sigma}(\partial_{u})^{\mu}(\partial_{u})^{\rho}(\partial_{I})^{\nu}(\partial_{J})^{\sigma}.$
(B.9)

Five complex Weyl scalars161616See [27] for the physical interpretation of the
Weyl scalars in asymptotic region at large distance from the source. In
particular, $\Psi_{4}$ describes the gravitational radiation at null infinity.
This is also clear from the non-integrable part of the leading charge (B.25)
as $\Psi_{4}=-\bar{\ddot{\sigma}}^{0}=-\ddot{h}_{+}+i\ddot{h}_{\times}$.  are
as follows

$\displaystyle\Psi_{0}=l^{a}m^{b}l^{c}m^{d}C_{abcd},\ \
\Psi_{1}=l^{a}n^{b}l^{c}m^{d}C_{abcd},\ \
\Psi_{2}=l^{a}m^{b}\bar{m}^{c}n^{d}C_{abcd},$ $\displaystyle\ \
\Psi_{3}=l^{a}n^{b}\bar{m}^{c}n^{d}C_{abcd},\ \
\Psi_{4}=n^{a}\bar{m}^{b}n^{c}\bar{m}^{d}C_{abcd}.$ (B.10)

Assuming

$\displaystyle\partial_{u}^{\mu}=e^{2\beta}n^{\mu}+\frac{1}{2}Fl^{\mu}-rh_{KL}C^{K}(m^{\mu}\bar{\hat{m}}^{L}+\bar{m}^{\mu}\hat{m}^{L}),\
\
\partial_{I}^{\nu}=rh_{IN}(m^{\nu}\bar{\hat{m}}^{N}+\bar{m}^{\nu}\hat{m}^{N}).$

After a long but straightforward calculation, the $R_{uIuJ}$ component of the
Riemann tensor can be written in terms of Newman-Penrose charges as

$\displaystyle R_{uIuJ}$
$\displaystyle=\sum_{n=0}^{4}C_{n}\Psi_{n}+\sum_{n=0}^{4}C_{n}^{\prime}\bar{\Psi}_{n}$
$\displaystyle=C_{0}\Psi_{0}+C_{0}^{\prime}\bar{\Psi}_{0}+C_{1}\Psi_{1}+C_{1}^{\prime}\bar{\Psi}_{1}+C_{2}\Psi_{2}+C_{2}^{\prime}\bar{\Psi}_{2}+C_{3}\Psi_{3}+C_{3}^{\prime}\bar{\Psi}_{3}+C_{4}\Psi_{4}+C_{4}^{\prime}\bar{\Psi}_{4}.$

For $R_{uIuJ}$ to be real-valued, $C_{n}=\bar{C_{n}^{\prime}}$. The $C_{n}$s
coefficients read as

$\displaystyle C_{0}$
$\displaystyle=\bar{C}_{0}^{\prime}=\sum_{n=-2}^{n}\frac{C_{0}^{(n)}}{r^{n}}=\frac{1}{4}r^{2}F^{2}h_{IN}h_{JP}\bar{\hat{m}}^{N}\bar{\hat{m}}^{P},$
(B.13) $\displaystyle C_{1}$
$\displaystyle=\bar{C}_{1}^{\prime}=\sum_{n=-1}^{n}\frac{C_{1}^{(n)}}{r^{n}}=\frac{1}{2}r^{3}Fh_{IJ}h_{KL}C^{K}\bar{\hat{m}}^{L}-r^{3}Fh_{IN}h_{JP}h_{KL}C^{K}\bar{\hat{m}}^{N}\bar{\hat{m}}^{P}\hat{m}^{L},$
(B.14) $\displaystyle C_{2}$
$\displaystyle=\bar{C}_{2}^{\prime}=\sum_{n=-2}^{n}\frac{C_{2}^{(n)}}{r^{n}}=-\frac{1}{2}r^{2}e^{2\beta}Fh_{IJ}+r^{4}h_{IN}h_{JP}h_{KL}h_{RS}C^{R}C^{K}\hat{m}^{N}\hat{m}^{P}\bar{\hat{m}}^{L}\bar{\hat{m}}^{S}$
$\displaystyle+r^{4}h_{IN}h_{JP}h_{KL}h_{RS}C^{K}C^{R}\bar{\hat{m}}^{N}\bar{\hat{m}}^{P}\hat{m}^{L}\hat{m}^{S}-r^{4}h_{IN}h_{JP}h_{KL}C^{K}C^{L}\bar{\hat{m}}^{N},$
(B.15) $\displaystyle C_{3}$
$\displaystyle=\bar{C}_{3}^{\prime}=\sum_{n=-1}^{n}\frac{C_{3}^{(n)}}{r^{n}}=-2r^{3}e^{2\beta}h_{IN}h_{JP}h_{KL}C^{K}\bar{\hat{m}}^{N}\bar{\hat{m}}^{P}\hat{m}^{L}+r^{3}e^{2\beta}h_{IJ}C^{K}\bar{\hat{m}}^{L},$
(B.16) $\displaystyle C_{4}$
$\displaystyle=\bar{C}_{4}^{\prime}=\sum_{n=-2}^{n}\frac{C_{4}^{(n)}}{r^{n}}=r^{2}e^{4\beta}h_{IN}h_{JP}\hat{m}^{N}\hat{m}^{P}\hat{m}^{P}.$
(B.17)

In order to write the large-$r$ expansion of $R_{uIuJ}$, we use the peeling
property of the Weyl tensor and find the

$\displaystyle\Psi_{0}=\sum_{n=0}^{n}\frac{\Psi_{0}^{n}}{r^{5+n}},\ \
\Psi_{1}=\sum_{n=0}^{n}\frac{\Psi_{1}^{n}}{r^{4+n}},\ \
\Psi_{2}=\sum_{n=0}^{n}\frac{\Psi_{2}^{n}}{r^{3+n}},\ \
\Psi_{3}=\sum_{n=0}^{n}\frac{\Psi_{3}^{n}}{r^{2+n}},\ \
\Psi_{4}=\sum_{n=0}^{n}\frac{\Psi_{4}^{n}}{r^{1+n}}.$

At $\mathcal{O}(r)$ the component of the Riemann tensor reads as

$\displaystyle
R_{uIuJ}^{(-1)}=C_{4}^{(-2)}\Psi_{4}^{0}+C_{4}^{\prime(-2)}\bar{\Psi}_{4}^{0}=-\gamma_{IN}\gamma_{JP}(\hat{m}^{N}\hat{m}^{P}\bar{\ddot{\sigma}}^{0}+\bar{\hat{m}}^{N}\bar{\hat{m}}^{P}{\ddot{\sigma}}^{0}),$

where, the asymptotic shear $\sigma^{0}$ is

$\displaystyle\sigma^{0}=\frac{1+i}{2}(f_{0}+ig_{0}).$ (B.19)

$\hat{m}^{1}$ and $\hat{m}^{2}$ at the leading order should be defined such
that $h^{IJ}=\hat{m}^{I}\bar{\hat{m}}^{J}+\hat{m}^{J}\bar{\hat{m}}^{I}$.
Therefore, we define

$\displaystyle\hat{m}^{1}=\frac{(1+i)}{2},\ \ \ \ \ \
\hat{m}^{2}=\frac{(1-i)}{2\sin\theta}.$ (B.20)

Using Eqs. (B.19) and (B.20), (B) can be written as

$\displaystyle R_{u\theta u\theta}^{(-1)}$
$\displaystyle=-\gamma_{\theta\theta}\gamma_{\theta\theta}(\hat{m}^{1}\hat{m}^{1}\bar{\ddot{\sigma}}^{0}+\bar{\hat{m}}^{1}\bar{\hat{m}}^{1}{\ddot{\sigma}}^{0})=-\frac{1}{2}(\partial_{u}^{2}f_{0}+\partial_{u}^{2}g_{0})=-\frac{1}{2}\partial_{u}^{2}C_{\theta\theta},$
(B.21) $\displaystyle R_{u\phi u\phi}^{(-1)}$
$\displaystyle=-\gamma_{\phi\phi}\gamma_{\phi\phi}(\hat{m}^{2}\hat{m}^{2}\bar{\ddot{\sigma}}^{0}+\bar{\hat{m}}^{2}\bar{\hat{m}}^{2}{\ddot{\sigma}}^{0})=\frac{1}{2}\sin^{2}\theta\
(\partial_{u}^{2}f_{0}+\partial_{u}^{2}g_{0})=-\frac{1}{2}\partial_{u}^{2}C_{\phi\phi},$
(B.22) $\displaystyle R_{u\theta u\phi}^{(-1)}$
$\displaystyle=-\gamma_{\theta\theta}\gamma_{\phi\phi}(\hat{m}^{1}\hat{m}^{2}\bar{\ddot{\sigma}}^{0}+\bar{\hat{m}}^{1}\bar{\hat{m}}^{2}{\ddot{\sigma}}^{0})=-\frac{1}{2}\sin\theta(\partial_{u}^{2}f_{0}-\partial_{u}^{2}g_{0})=-\frac{1}{2}\partial_{u}^{2}C_{\theta\phi}.$
(B.23)

Therefore, at the leading order, $R_{uIuJ}$ is found to be

$\displaystyle R_{uIuJ}^{(-1)}=-\frac{1}{2}\partial_{u}^{2}C_{IJ}.$ (B.24)

And the gravitational displacement reads as the standard gravitational memory
displacement in [12]. At this order the standard BMS charge in terms of NP
quantities is defined as [17]

$\displaystyle\mathcal{Q}_{0}=-\frac{1}{2\pi G}\int d\Omega
Y_{lm}\mathfrak{R}(\Psi_{2}^{0}+\sigma^{0}\partial_{u}\bar{\sigma}^{0}).$
(B.25)

If $\Psi_{2}^{0}$ is conserved, then $\partial_{u}\Psi_{2}^{0}=0$ and as

$\displaystyle\psi_{2}^{0}-\bar{\psi}_{2}^{0}=\bar{\sigma}^{0}\partial_{u}\sigma^{0}-\sigma^{0}\partial_{u}\bar{\sigma}^{0}+\bar{\eth}^{2}\sigma^{0}-\eth^{2}\bar{\sigma}^{0},$
(B.26)

in order of $\Psi_{2}^{0}$ to be conserved, $\sigma^{0}$ should be a free data
at null infinity $\dot{\sigma^{0}}=0$. In this case, we can see from (B.21)
that gravitational displacement is zero due to the existence of a conserved
charge at this order. If $\Psi_{2}^{0}$ is not conserved then there is a shift
in the relative distance of two inertial observer which is related to the non-
integrable part of the charge (B.25).

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# Water-Wave Vortices and Skyrmions

Daria A. Smirnova  Theoretical Quantum Physics Laboratory, Cluster for
Pioneering Research, RIKEN, Wako-shi, Saitama 351-0198, Japan  Research School
of Physics, Australian National University, Canberra, ACT 2601, Australia
Franco Nori  Theoretical Quantum Physics Laboratory, Cluster for Pioneering
Research, RIKEN, Wako-shi, Saitama 351-0198, Japan  Center for Quantum
Computing (RQC), RIKEN, Wako-shi, Saitama 351-0198, Japan  Physics Department,
University of Michigan, Ann Arbor, MI 48109-1040, USA  Konstantin Y. Bliokh
Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research,
RIKEN, Wako-shi, Saitama 351-0198, Japan  Centre of Excellence ENSEMBLE3 Sp. z
o.o., 01-919 Warsaw, Poland  Donostia International Physics Center (DIPC),
Donostia-San Sebastián 20018, Spain

###### Abstract

Topological wave structures – phase vortices, skyrmions, merons, etc. – are
attracting enormous attention in a variety of quantum and classical wave
fields. Surprisingly, these structures have never been properly explored in
the most obvious example of classical waves: water-surface (gravity-capillary)
waves. Here we fill this gap and describe: (i) water-wave vortices of
different orders carrying quantized angular momentum with orbital and spin
contributions, (ii) skyrmion lattices formed by the instantaneous
displacements of the water-surface particles in wave interference, (iii) meron
(half-skyrmion) lattices formed by the spin density vectors, as well as (iv)
spatiotemporal water-wave vortices and skyrmions. We show that all these
topological entities can be readily generated in linear water-wave
interference experiments. Our findings can find applications in microfluidics
and show that water waves can be employed as an attainable playground for
emulating universal topological wave phenomena.

Introduction.—Wave vortices are universal physical entities with nontrivial
topological and dynamical properties: quantized phase increments around point
phase singularities and quantum-like angular momentum (AM). Examples of wave
vortices are known since the 19th century, these has been observed and
explored in tidal Nye  _et al._ (1988), quantum-fluid Yarmchuk  _et al._
(1979); Fetter (2009), optical Berry (1981); Allen  _et al._ (2003); Soskin
and Vasnetsov (2001), sound Hefner and Marston (1999); Volke-Sepulveda  _et
al._ (2008); Guo  _et al._ (2022), elastic Chaplain  _et al._ (2022), surface-
plasmon Gorodetski  _et al._ (2008); Prinz  _et al._ (2023), exciton-polariton
Dall  _et al._ (2014), quantum electron Bliokh  _et al._ (2017), neutron Clark
_et al._ (2015), and atom Luski  _et al._ (2021) waves.

Strikingly, wave vortices have not been properly studied in the most obvious
example of classical waves: water-surface (gravity-capillary) waves. Only a
recent series of experiments Filatov  _et al._ (2016a, b); Francois  _et al._
(2017); Bliokh  _et al._ (2022a) described the generation of a square lattice
of alternating vortices in the interference of orthogonal standing water
waves.

However, the theoretical description of these experiments lacks the
identification with wave vortices, very different from the usual
hydrodynamical vortices. It was indicated that the hydrodynamical vorticity
appears due to nonlinearity Filatov  _et al._ (2016a, b), and that these
vortices are closely related to the Stokes drift and AM Francois  _et al._
(2017); Bliokh  _et al._ (2022a), but no quantized topological and dynamical
properties have been indicated. Furthermore, only the simplest first-order
vortices were produced (cf., e.g., quantum-electron vortices of higher orders
$\sim 10^{2}\\!-\\!10^{3}$ McMorran  _et al._ (2011); Tavabi  _et al._
(2022)).

In this work, we describe water-wave vortices (WWVs) in gravity-capillary
waves. We reveal their topological properties and show that circularly-
symmetric vortices are eigenmodes of the total AM operator, including the spin
and orbital parts. In the linear approximation, WWVs have zero vorticity.
Nonetheless, the quadratic Stokes drift produces slow orbital motion of water
particles and nonzero nonlinear vorticity. Importantly, water particles
experience two kinds of circular motions with different spatial and temporal
scales: (i) local linear-amplitude-scale circular motion with the wave
frequency in the linear regime and (ii) slow wavelength-scale circular motion
due to the nonlinear Stokes drift. These two motions are responsible for the
spin and orbital contributions to the quantized total AM.

Moreover, water waves have inherent vector properties: the local Eulerian
displacement of water-surface particles is a counterpart of the 3D
polarization in optical or acoustic wavefields Sugic  _et al._ (2020); Bliokh
_et al._ (2021). Therefore, following great recent progress in the generation
of topological vector entities – skyrmions Nagaosa and Tokura (2013) – in
classical electromagnetic Tsesses  _et al._ (2018); Du  _et al._ (2019); Dai
_et al._ (2020); Gao  _et al._ (2020); Shen  _et al._ (2021); Lei  _et al._
(2021); Deng  _et al._ (2022), sound Ge  _et al._ (2021); Muelas-Hurtado  _et
al._ (2022), and elastic Cao  _et al._ (2023) waves, here we describe water-
wave skyrmions. We show that the interference of three plane water waves can
generate a hexagonal lattice of: (i) WWVs; (ii) skyrmions of the instantaneous
water-particle displacements and (iii) merons (half-skyrmions) of the local
spin density. This field configuration is just one step from the recent
experiments Filatov  _et al._ (2016a, b); Francois  _et al._ (2017); Bliokh
_et al._ (2022a), and is quite feasible for the experimental implementation.

Finally, following enormous current interest in spacetime structured waves
Yessenov  _et al._ (2022); Shen  _et al._ (2023), in particular spatiotemporal
vortices Sukhorukov and Yangirova (2005); Bliokh and Nori (2012); Hancock  _et
al._ (2019); Chong  _et al._ (2020); Bliokh (2021), we show that detuning the
frequency of one of the interfering waves, one can readily produce moving
lattices of spatiotemporal WWVs and spatiotemporal skyrmions.

Thus, we reveal new structures with remarkable topological and dynamical
properties in linear water waves. We argue that water waves offer a perfect
classical platform for emulating universal quantum and topological wave
phenomena, which can also find useful applications in microfluidics Ding  _et
al._ (2013); Wu  _et al._ (2019).

Figure 1: (a) Instantaneous water surfaces ${\mathcal{Z}}(x,y,0)$ and Eulerian
water-surface particle trajectories $\mathbfcal{R}(x,y,t)$ for circular WWVs
with different topological charges $\ell$, Eqs. (2) and (3). The spin density
${\bf S}$ is directed normally to the elliptical particle trajectories and
quantifies the AM of this elliptical motion. (b) The plane-wave spectrum of a
circular WWV with color-coded phases for $\ell=1$. (c) The complex vertical-
displacement field $Z(x,y)$ for WWVs from panel (a), with the phases and
amplitudes coded by the colors and brightness, respectively. The white arrows
indicate the second-order Stokes drift ${\bf U}$, Eq. (6), characterizing the
wave momentum density.

Water-wave vortices.—We first consider monochromatic gravity-capillary waves
on a deep-water surface. The 3D Eulerian displacement of the water particles
from the $z=0$ surface is $\mathbfcal{R}({\bf r}_{2},t)={\rm Re}[{\bf R}({\bf
r}_{2})e^{-i\omega t}]=(\mathcal{X},\mathcal{Y},\mathcal{Z})$, where ${\bf
R}=(X,Y,Z)$ is the complex displacement wavefield, ${\bf r}_{2}=(x,y)$ and
$\omega$ is the frequency. Separating the vertical and in-plane components of
3-vectors as ${\bf a}\equiv(a_{x},a_{y},a_{z})=({\bf a}_{2},a_{z})$, the wave
equations of motion can be written as Bliokh  _et al._ (2022a); See
Supplemental Material at … for the derivation of Eqs. , consideration of
finite-depth effects, as well as for the water-wave vortices and skyrmions
formed by standing waves in closed reservoirs.() (1)

${\color[rgb]{0,0,0}{\omega^{2}{\bf
R}_{2}\\!=\\!\left(g-\frac{\alpha}{\rho}\Delta_{2}\right)\\!{\bm{\nabla}}_{2}Z,\leavevmode\nobreak\
\omega^{2}Z\\!=\\!-\\!\left(g-\frac{\alpha}{\rho}\Delta_{2}\right)\\!{\bm{\nabla}}_{2}\cdot{\bf
R}_{2}.}}$ (1)

Here $g$ is the gravitational acceleration, $\alpha$ is the surface-tension
coefficient, $\rho$ is the water density,
$\Delta_{2}={\bm{\nabla}}_{2}\cdot{\bm{\nabla}}_{2}$, and Eqs. (1) with the
plane-wave ansatz ${\bm{\nabla}}_{2}\to i{\bf k}$ (${\bf k}$ is the wave
vector) yield the dispersion relation $\omega^{2}=gk+(\alpha/\rho)k^{3}$.

The vortex solutions of Eqs. (1) are obtained as a superposition of plane
waves with wavevectors uniformly distributed along the
$k_{x}^{2}+k_{y}^{2}=k^{2}$ circle with the azimuthal phase increment
$2\pi\ell$, $\ell\in{\mathbb{Z}}$, Fig. 1(b). Constructing the complex
vertical displacement in this way, we obtain:

$Z=\frac{A}{2\pi}\int_{0}^{2\pi}e^{i{\bf k}\cdot{\bf
r}_{2}+i\ell\phi}d\phi=AJ_{\ell}(kr)e^{i\ell\varphi}\,.$ (2)

Here $A$ is a constant wave amplitude, $J_{\ell}$ is the Bessel function of
the first kind, $\phi$ is the azimuthal angle in the $(k_{x},k_{y})$ plane,
whereas $(r,\varphi)$ are the polar coordinates in the $(x,y)$-plane.

Equation (2) describes 2D scalar cylindrical Bessel waves, Fig. 1(c). However,
water waves have a vectorial nature, and the other two components of the
wavefield can be found from the first Eq. (1). It is convenient to write these
in the basis of ‘circular polarizations’ Picardi  _et al._ (2018); Bliokh
(2022):

$R^{\pm}\equiv\frac{X\mp iY}{\sqrt{2}}=\pm\frac{A}{\sqrt{2}}J_{\ell\mp
1}(kr)\,e^{i(\ell\mp 1)\varphi}.$ (3)

In this basis, the $z$-component of the spin-1 operator, universal for
classical vector waves, reads $\hat{S}_{z}={\rm diag}(1,-1,0)$, while the
$z$-component of the orbital AM (OAM) operator is
$\hat{L}_{z}=-i\partial_{\varphi}$ Allen  _et al._ (2003). Introducing the
‘wavefunction’ $|\psi\rangle=(R^{+},R^{-},Z)$, one can see that the WWVs (2)
and (3), are not the OAM eigenmodes, but these are eigenomodes of the
$z$-component of the total AM with the quantized eigenvalue $\ell$:

$\hat{J}_{z}|\psi\rangle=(\hat{L}_{z}+\hat{S}_{z})|\psi\rangle=\ell\,|\psi\rangle\,.$
(4)

Such behavior (which can be interpreted as the inherent spin-orbit coupling)
is a common feature of all cylindrical vector waves: optical van Enk and
Nienhuis (1994); Bliokh  _et al._ (2010); Picardi  _et al._ (2018), quantum
Bliokh  _et al._ (2011), acoustic Bliokh and Nori (2019), and elastic Bliokh
(2022).

Figure 1(a) shows instantaneous water surfaces ${\mathcal{Z}}({\bf r}_{2},0)$
and water-particle trajectories $\mathbfcal{R}({\bf r}_{2},t)$ for WWVs with
different $\ell$. The water-particle trajectories are 3D ellipses, entirely
similar to the electric-field polarization in optical fields Bliokh  _et al._
(2021). The normal to the ellipse and its ellipticity determine the cycle-
averaged AM of the particle, i.e., spin density in water waves Jones (1973);
Bliokh  _et al._ (2022a): ${\bf S}=(\rho\omega/2){\rm Im}({\bf
R}^{*}\\!\times{\bf R})$. One can see that WWVs are characterized by
inhomogeneous polarization textures. In the vortex center $r=0$, the
polarization is purely vertical, $|\psi\rangle\propto(0,0,1)$, for $\ell=0$;
it is purely circular, $|\psi\rangle\propto(1,0,0)$ and
$|\psi\rangle\propto(0,1,0)$, for $\ell=\pm 1$; and the vector wavefield
vanishes, $|\psi\rangle\propto(0,0,0)$, for $|\ell|>1$ (the vanishing of all
vector wavefield components requires a higher-order degeneracy Picardi  _et
al._ (2018); Bliokh (2022); Bliokh  _et al._ (2010, 2011); Bliokh and Nori
(2019); Vernon  _et al._ (2023)).

Importantly, WWVs are not the usual hydrodynamical vortices, which are formed
by steady water motion with a nonzero circulation of the velocity
$\mathbfcal{V}=\partial_{t}\mathbfcal{R}$ and vorticity
${\bm{\nabla}}\times\mathbfcal{V}\neq{\bf 0}$ Landau and Lifshitz (1987). In
contrast, linear monochromatic gravity-capillary waves have zero vorticity:
${\bm{\nabla}}\times{\bf V}={\bf 0}$, where ${\bf V}=-i\omega{\bf R}$ is the
complex velocity field. This follows from Eqs. (1) and the incompressibility
equation ${\bm{\nabla}}\cdot{\bf V}=0$. Wave vortices are topological entities
with quantized phase singularities in the center. The ‘topological charge’ can
be defined in two equivalent ways Berry and Dennis (2001); Bliokh  _et al._
(2019):

$\frac{1}{2\pi}\\!\oint{\bm{\nabla}}_{2}{\rm Arg}(Z)\cdot d{\bf
r}_{2}=\frac{1}{4\pi}\\!\oint{\bm{\nabla}}_{2}{\rm Arg}({\bf R}\cdot{\bf
R})\cdot d{\bf r}_{2}=\ell\,,$ (5)

where the contour integral is taken along a circuit enclosing the vortex
center. These relations show that the center of the first-order $|\ell|=1$ WWV
can be considered as the first-order phase singularity in the scalar field
$Z(x,y)$ or the second-order polarization singularity (C-point of circular
polarization) in the vector field ${\bf R}(x,y)$ Nye and Hajnal (1987); Berry
and Dennis (2001); Bliokh  _et al._ (2019, 2021). Any perturbation breaking
the cylindrical symmetry splits the second-order C-point into a pair of the
first-order C-points, with topologically-robust Möbius-strip orientations of
the polarization ellipses around these points Freund (2010); Bauer  _et al._
(2015); Bliokh  _et al._ (2019, 2021); Muelas-Hurtado  _et al._ (2022).

Nonzero vorticity and circulation do appear in WWVs, but in the quadratic
corrections to linear wave solutions. Namely, water particles experience a
slow Stokes drift, i.e., the difference between the Lagrangian and Euler
velocities van den Bremer and Breivik (2017); Falkovich (2018); Bliokh  _et
al._ (2022a):

${\bf U}=\frac{\omega}{2}{\rm Im}[{\bf R}^{*}\cdot({\bm{\nabla}}_{2}){\bf
R}]\,.$ (6)

Multiplied by the mass density, it yields the canonical wave momentum
(‘pseudomomentum’) density Bliokh  _et al._ (2022a); McIntyre (1981); Peierls
(1979, 1991); Bliokh  _et al._ (2022b): ${\bf P}=\rho{\bf U}$.

Figure 1(c) shows the azimuthal Stokes-drift flow in WWVs. It is mostly
localized near the first radial maximum of the Bessel function $J_{\ell}(kr)$
and determines the $z$-directed OAM density: ${\bf L}={\bf r}_{2}\times{\bf
P}$, $L_{z}=(\rho\omega/2){\rm Im}({\bf R}^{*}\cdot\partial_{\varphi}{\bf
R})$. Notably, the local circular motion of water particles (spin) and the
global Stokes-drift circulation (OAM) have very different space and time
scales: the linear-wave amplitude $A$ and angular frequency $\omega$ vs. the
wavelength $k^{-1}\gg A$ and angular velocity $U/r\sim\omega
k^{2}A^{2}\ll\omega$. The spin and OAM densities in the WWVs (2) and (3)
satisfy the relation following from Eq. (4) Bliokh and Nori (2019); Bliokh
(2022):

$J_{z}=L_{z}+S_{z}=\frac{\rho\omega}{2}\ell|{\bf
R}|^{2}=2\frac{\ell}{\omega}T\,,$ (7)

where $T=\rho|{\bf V}|^{2}/4$ is the cycle-averaged kinetic energy density.

Thus, WWVs are naturally described by a quantum-like formalism and possess
nontrivial topological properties. Recent experiments Filatov  _et al._
(2016a, b); Francois  _et al._ (2017); Bliokh  _et al._ (2022a) generated
square lattices of alternating first-order vortices with $\ell=\pm 1$ by
interfering orthogonal standing waves with the $\pi/2$ phase difference. The
orbital Stokes drift and circular polarization (spin) in the vortex centers
were clearly observed in Refs. Francois  _et al._ (2017); Bliokh  _et al._
(2022a), but quantized topological properties of these vortices have not been
described. Higher-order WWVs with $|\ell|>1$, which have never been observed,
could provide areas of unperturbed water surface surrounded by intense
circular waves and orbital Stokes flows.

Figure 2: Hexagonal lattice produced by the interference of three waves with
equal frequencies, amplitudes, and color-coded phases shown in (c). (a)
Instantaneous water surface ${\mathcal{Z}}(x,y,0)$ and water-surface particle
displacements $\mathbfcal{R}(x,y,0)$ for the field (9). The displacement
directions in the unit hexagonal cell is mapped onto the unit sphere,
providing a skyrmion with the topological charge $Q=1$, Eq. (10). (b) The unit
displacement-direction field $\bar{\mathbfcal{R}}(x,y,0)$ represented by
colors (vertical component $\bar{\mathcal{Z}}$) and black arrows (in-plane
components $\bar{\mathbfcal{R}}_{2}$). (d) The complex vertical-displacement
field $Z(x,y)$ and the Stokes drift ${\bf U}$ indicating the lattice of
alternating WWVs with $\ell=\pm 1$. (e) The unit spin-density field
$\bar{\bf{S}}(x,y)$ represented similar to (b). The hexagonal unit cell is
split into triangular zones of spin merons (half-skyrmions) with topological
charges $Q_{S}=\pm 1/2$ and centers with $\bar{S}_{z}=\pm 1$ corresponding to
the $\ell=\pm 1$ vortices in (d).

Water-wave skyrmions and merons.—The 3D vector nature of water waves allows
the generation of topological vector textures, such as skyrmions or merons
Nagaosa and Tokura (2013); Tsesses  _et al._ (2018); Du  _et al._ (2019); Dai
_et al._ (2020); Shen  _et al._ (2021); Lei  _et al._ (2021); Deng  _et al._
(2022); Ge  _et al._ (2021); Muelas-Hurtado  _et al._ (2022); Cao  _et al._
(2023). Such textures can be produced by interfering several plane waves with
the same frequency and wavevectors ${\bf
k}_{j}=k(\cos{\phi}_{j},\sin{\phi_{j}},0)$, $j=1,...,N$:

${\bf R}=\sum_{j=1}^{N}{\bf R}_{0j}e^{{i{\bf k}_{j}\cdot{\bf
r}+i\Phi_{j}}},\leavevmode\nobreak\ {\bf
R}_{0j}=A_{j}(i\cos{\phi}_{j},i\sin{\phi_{j}},1),$ (8)

where $A_{j}$ and $\Phi_{j}$ are the real-valued amplitudes and phases of the
interfering waves.

Consider, for example, $N=3$ waves, uniformly distributed with
$\phi_{j}=2\pi(j-1)/N$, $A_{j}=A$, and vortex phases $\Phi_{j}=\phi_{j}$, Fig.
2(c). These waves form a hexagonal periodic lattice with the displacement
field

$\begin{pmatrix}X\\\ Y\\\ Z\\\ \end{pmatrix}\propto
A\\!\begin{pmatrix}ie^{ik{x}}+ie^{-i\frac{k{x}}{2}}\sin\\!\left(\frac{\sqrt{3}k{y}}{2}+\frac{\pi}{6}\right)\\\
-\sqrt{3}e^{-i\frac{k{x}}{2}}\cos\\!\left(\frac{\sqrt{3}k{y}}{2}+\frac{\pi}{6}\right)\\\
e^{ik{x}}-2e^{-i\frac{k{x}}{2}}\sin\\!\left(\frac{\sqrt{3}k{y}}{2}+\frac{\pi}{6}\right)\end{pmatrix}\\!.$
(9)

This field exhibits a number of nontrivial topological features. First, it
contains a lattice of WWVs with alternating topological charges $\ell=\pm 1$,
Fig. 2(d). Such vortex lattices are well known in optics Masajada and Dubik
(2001).

Second, Fig. 2(a) shows the instantaneous water surface ${\mathcal{Z}}({\bf
r}_{2},0)$ and the surface-particle displacements $\mathbfcal{R}({\bf
r}_{2},0)$ for the field (9). The displacements in a hexagonal unit cell of
the lattice contain all possible directions and can be mapped onto a unit
sphere. This is a signature of a skyrmion, which can be characterized by the
topological number

$Q=\frac{1}{4\pi}\iint_{\text{u.c.}}\bar{\mathbfcal{R}}\cdot[\partial_{x}\bar{\mathbfcal{R}}\times\partial_{y}\bar{\mathbfcal{R}}]\,dx\,dy,$
(10)

where $\bar{\mathbfcal{R}}={\mathbfcal{R}}/|{\mathbfcal{R}}|$. In the case
under consideration, $Q=1$ at $t=0$, but it can change its sign over time,
because the displacement evolves and becomes opposite after half a period,
$t=\pi/\omega$ Muelas-Hurtado  _et al._ (2022). Figure 2(b) displays another
representation of the skyrmion lattice, where colors and black vectors
indicate the $z$ and $(x,y)$ components of the displacement-direction field
$\bar{\mathbfcal{R}}$. Moving from the center of the cell towards its
boundary, the vector $\bar{\mathbfcal{R}}$ undergoes a rotation, where its
$z$-component changes sign, resulting in a nontrivial winding captured by the
nonzero skyrmion charge $Q$. Similar skyrmion lattices have been observed in
electromagnetic Tsesses  _et al._ (2018), sound Ge  _et al._ (2021); Muelas-
Hurtado  _et al._ (2022), and elastic Cao  _et al._ (2023) vector wavefields.

Third, instead of the instantaneous vector field ${\mathbfcal{R}}$, one can
trace the spin density vector ${\bf S}$ (normal to the local polarization
ellipse). Figure 2(e) displays the distribution of the unit spin vector
$\bar{\bf S}={\bf S}/|{\bf S}|$ in the field (9). The unit hexagonal cell is
split into triangular zones with $\bar{S}_{z}>0$ and $\bar{S}_{z}<0$ separated
by $\bar{S}_{z}=0$ lines and singular ${\bf S}={\bf 0}$ vertices. The centers
of these triangles with $\bar{S}_{z}=\pm 1$ (i.e., circular in-plane
polarizations) correspond to the centers of WWVs with $\ell=\pm 1$, Fig. 2(d)
Bliokh  _et al._ (2022a). Calculating the topological charges (10) for the
spin field $\bar{\bf S}$, we obtain $Q_{S}=\mp 1/2$ for the triangular zones
with $\bar{S}_{z}\lessgtr 0$. Such topologically nontrivial textures are
called merons or half-skyrmions, because the spin directions in each zone
covers the upper or lower semisphere. Similar spin merons have been observed
in electromagnetic waves Dai  _et al._ (2020); Lei  _et al._ (2021); Ghosh
_et al._ (2021); Król  _et al._ (2021).

Here we showed only one simple example of the water-wave interference field.
WWVs, field skyrmions, and spin merons are rather universal topological
entities and appear in many other fields. A square lattice formed by two
standing waves Filatov  _et al._ (2016a, b); Francois  _et al._ (2017); Bliokh
_et al._ (2022a) contains vortices and spin merons (cf. Lei  _et al._ (2021);
Ghosh  _et al._ (2021)), a hexagonal lattice formed by three standing waves
produces field skyrmions See Supplemental Material at … for the derivation of
Eqs. , consideration of finite-depth effects, as well as for the water-wave
vortices and skyrmions formed by standing waves in closed reservoirs.() (1),
and the zero-order $\ell=0$ Bessel mode, Eqs. (2) and (3) and Fig. 1, contains
a field skyrmion (cf. Tsesses  _et al._ (2018); Deng  _et al._ (2022)).

Figure 3: Same as in Figs. 2(a,c,d) but for the lattice of spatiotemporal WWVs
and skyrmions. The frequency of one of the interfering waves is detuned by
$\delta\omega$ and the corresponding $\delta k$. The complex vertical-
displacement field $Z$ and the real field envelope ${\mathbfcal
R}^{\prime}={\rm Re}[{\bf R}]$ (without fast oscillations $e^{-i\omega t}$)
are plotted over the spacetime domain $(t,y)$ at the fixed coordinate $x=0$.
The temporal component of the ‘spatiotemporal Stokes drift’ ${\bf
U}^{\prime}=(U_{t},U_{y})$ is defined as
$U_{t}=(k{\omega}/{2}\delta\omega){\rm Im}[{\bf R}^{*}\cdot(\partial_{t}){\bf
R}]$.

Spatiotemporal vortices and skyrmions.—Finally, we demonstrate another class
of topological entities which can be readily generated in water waves:
spatiotemporal vortices Sukhorukov and Yangirova (2005); Bliokh and Nori
(2012); Hancock  _et al._ (2019); Chong  _et al._ (2020); Bliokh (2021) and
skyrmions. It is sufficient to slightly detune the frequency of one of the
three interfering plane waves in Fig. 2: $\omega_{1}\to\omega+\delta\omega$,
$k_{1}\to k+\delta k=(\omega+\delta\omega)^{2}/g$ (for simplicity, here we
neglect capillarity, $\alpha\to 0$), Fig. 3(b). This transforms the wavefield
(8) as $\Phi_{1}\to\Phi_{1}-i\delta\omega t$, so that the spatial lattice in
Fig. 2 becomes moving along the $x$-axis, and the field becomes a function of
space and time: ${\bf R}({\bf r}_{2},t)$.

The real displacement field is $\mathbfcal{R}({\bf r}_{2},t)={\rm Re}[{\bf
R}({\bf r}_{2},t)e^{-i\omega t}]$, but we will analyze the field
$\mathbfcal{R}^{\prime}({\bf r}_{2},t)={\rm Re}[{\bf R}({\bf r}_{2},t)]$
subtracting the common fast oscillations $e^{-i\omega t}$. Plotting the
complex field $Z$ and real field $\mathbfcal{R}^{\prime}$ in the spacetime
domain $(t,y)$ at fixed $x=0$, we find that they exhibit a scaled hexagonal
lattice of vortices and skyrmions, Fig. 3. These spatiotemporal WWVs and
skyrmions have opposite topological charges $\ell$ and $Q$ compared to their
spatial counterparts in Fig. 2.

Conclusions.—We have analyzed the fundamental topologically nontrivial objects
in linear water-surface (gravity-capillary) waves, namely: WWVs, surface-
particle displacement skyrmions, spin-density merons, as well as
spatiotemporal WWVs and skyrmions. All these objects are universal across
different types of waves and only require standard wave-interference
ingredients: relative phases/amplitudes, polarizations, and spectral detuning,
to control the geometry and topology of the field. For simplicity, we
considered the deep-water approximation, the finite-depth effects in
monochromatic water waves simply produce global scaling of the vertical
component on the surface: $Z\to\tanh(kH)Z$, where $H$ is the water depth See
Supplemental Material at … for the derivation of Eqs. , consideration of
finite-depth effects, as well as for the water-wave vortices and skyrmions
formed by standing waves in closed reservoirs.() (1).

Notably, the vector features of water waves (displacement fields) are directly
observable, while in other fields these are usually measured via various
indirect methods. Therefore, water waves offer a highly attractive platform
for emulating topologically nontrivial field structures and wave phenomena in
a unified fashion. Furthermore, nontrivial dynamical properties of topological
water-wave objects — circulating Stokes-drift currents, fast circular motions
(spin) in the centers of the first-order WWVs, vanishing fields in the centers
of higher-order WWVs, etc. — can be attractive for fluid-mechanical
applications, such as manipulations of particles Ding  _et al._ (2013); Wu
_et al._ (2019). Finally, we note that while most of the attention in water-
wave physics has focused on nonlinear and high-amplitude effects Cazaubiel
_et al._ (2019); Birkholz  _et al._ (2016); McAllister  _et al._ (2022), our
study shows that wave structures around field zeros and linear-wave
interference exhibit a rich variety of largely unexplored phenomena.

###### Acknowledgements.

This work is supported in part by the Japan Society for the Promotion of
Science (JSPS); Nippon Telegraph and Telephone Corporation (NTT) Research; the
Asian Office of Aerospace Research and Development (AOARD) [Grant No.
FA2386-20-1-4069]; the Foundational Questions Institute Fund (FQXi) [Grant No.
FQXi-IAF19-06]; the International Research Agendas Programme (IRAP) of the
Foundation for Polish Science co-financed by the European Union under the
European Regional Development Fund and Teaming Horizon 2020 programme of the
European Commission [ENSEMBLE3 Project MAB/2020/14]; the TEAM program of the
Foundation for Polish Science co-financed by the European Union under the
European Regional Development Fund [Grant TEAM/2016-3/29]; and the Australian
Research Council [FT230100058].

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