# Closed strings in pp-wave spacetimes and the memory effect

Ayan Dey and Sayan Kar<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology, Kharagpur 721 302,
India

###### Abstract

We study closed string evolution in the pp-wave spacetime assuming different
pulse shapes (square and sech-squared) in the exact gravitational wave metric.
It is shown that the shape of a circular closed string changes permanently to
an elliptical form after the pulse has departed. The worldsheet geometry also
displays characteristic permanent changes caused by the gravitational wave
pulse. The above effects collectively demonstrate features akin to what is
known as ‘memory’ in gravitational wave physics.

##  I Introduction

In studies on the physics of gravitational waves, the gravitational wave
memory effect is often illustrated by considering the change of shape of a
ring of particles when it encounters a gravitational wave pulse for a short
duration of time [1, 2]. For example, if the circular shape of the ring of
particles gets permanently deformed to an elliptical shape, we say that the
pulse has ‘memory’. On the other hand, if the deformation is not permanent and
the ring reverts back to its circular shape we say that the pulse has ‘no
memory’. Such a visually appealing picture is very helpful in providing an
elegant way of appreciating what ‘memory’ really means (see Figure 1). This
illustrative example is by no means a toy example–in fact it has in it the
very basic idea of ‘permanent change’ which lies at the heart of all ways of
defining memory.

Figure 1: Qualitative idea of memory for a ring of particles experiencing a GW
pulse.

If we smoothen out the so-called ‘ring of particles’ what we get is a closed
string. Right now, we are yet to state whether we are dealing with fundamental
strings or cosmic strings. In our way of looking at the problem, it could be
anything – in fact, even a macroscopic, elastic, circular string. The simplest
description of such a string is known to be done, in a relativistic setting,
through the Nambu–Goto area action – the ensuing equations and physics [3].
The string configuration $x^{i}(\tau,\sigma)$, where $\tau$ and $\sigma$ are
the worldsheet surface coordinates, is an extremal (minimal) surface in a
background spacetime with a metric $g_{ij}(x^{k})$. It is given mathematically
through solutions of the Nambu–Goto equations of motion and constraints. For
closed strings, we have the condition–
$x^{i}(\tau,\sigma)=x^{i}(\tau,\sigma+2\pi)$. If such a closed string
encounters a gravitational wave pulse what happens to its circular shape? Do
we see a permanent change at later times, i.e. much after the pulse has
disappeared? We intend to ask and answer such a question in our work here.

How do we model the gravitational wave pulse ? It is possible to use the
perturbative approach and use a form of $h_{ij}$ which is pulse-like. This may
be observationally more relevant. However, here we would like to restrict
ourselves to the purely theoretical domain and consider a pp-wave geometry (an
exact gravitational wave solution of the vacuum Einstein equations) [4],[5].
In such a geometry we have a free function which we may choose appropriately,
in order to represent a specific pulse shape. Hence, we would be studying the
evolution of closed strings in a pp-wave spacetime through which we will try
to arrive at what is conventionally known as the ‘memory’ effect.

Studies on strings in pp-wave spacetimes are not new. The earliest works seems
to be due to Amati and Klimcik [6] and then by Horowitz and Steif [7]. Later
there have been articles by de Vega and Sanchez [8] and de Vega et al [9].
Very recently, Liška and von Unge [10] have revisited the topic in an
interesting paper which originally motivated us to pursue this work. On the
other hand, there has been some work on string memory in the past– for
example, note the papers by Afshar, Esmaeili and Sheikh-Jabbari [11] and by
Aldi, Bianchi and Firrotta [12].

In general, gravitational wave memory has been a topic of active research
interest over the last few years. The prospect of detection of memory in
future detectors has greatly influenced such studies [13], [14], [15]. The
origins of the idea of GW memory in linearised gravity are in the papers by
Zel’dovich, Polnarev [16] and Braginsky, Grishchuk [17] (the term ‘memory’ in
the context of gravitational waves was introduced in this paper).
Subsequently, Christodoulou[18], using full GR, showed how gravitational waves
travelling to null infinity can give rise to a contribution to memory. Later,
it was shown in [19], using gauge invariant observables that both the
abovementioned types of memory can arise within linearised gravity. This gave
rise to the notion of ordinary and null memory which are discussed to some
extent in [20],[21],[22],[23]. More recently, a theoretical link has been
conjectured involving the memory effects, the Bondi-van der Burg-Metzner-Sachs
(BMS) symmetries and soft theorems [24] [25]. Memory in pp-wave and related
spacetimes have been extensively studied in a large number of papers [26, 27,
28, 29, 30, 31, 32, 33, 34, 35, 36] where diverse aspects of displacement and
velocity memory have been thoroughly analysed. A novel ‘displacement within
velocity memory’ which happens for a specific choice of the wave parameters,
has recently been reported in [37]. On the other hand, an approach which aims
at defining various ‘persistent gravitational wave observables’ has been
pursued by Flanagan, Nichols, Grant and Harte in a series of papers [38],
[39], [40]. The idea of ${\cal B}$-memory (related to the ${\cal B}$ tensor
which encodes the behaviour of the gradient of the geodesic velocity field)
appeared in the work in [41] and was followed up in [31]. Other related work
on diverse aspects of memory appear in [42], [43], [44]. The review article
and the lecture by Favata [1] [2] chronicles the history till 2010. A very
recent review on the memory effect and its connections with asymptotic
symmetries is available in [45].

Our article is organised as follows. In the next section (II), we write down
the string equations of motion and constraints in a pp-wave spacetime, using
the conformal gauge. Thereafter, in III, assuming a square pulse gravitational
wave, we obtain solutions at late times, assuming a circular closed string
configuration before the arrival of the pulse. In this way, via the permanent
change of shape of the closed string we are able to arrive at a memory effect.
We also do a similar analysis for a continuous pulse shape (the sech-squared
pulse). In IV, we discuss some features such as the behaviour of the induced
metric, the worldsheet Ricci scalar and the worldsheet geometry as an
embedding. Finally in Section V we present our conclusions.

##  II Strings in pp-wave spacetimes

Let us begin by first recalling the string equations of motion and constraints
which follow from the variation (w.r.t. $x^{i}$) of the Nambu–Goto area action
($S_{NG}=-T_{0}\int d\tau d\sigma\sqrt{-\gamma}$, $\gamma$ being the
determinant of the induced metric
$\gamma_{ab}=g_{ij}\partial_{a}x^{i}\partial_{b}x^{j}$ with $a,b...$ being the
worldsheet indices $\tau$, $\sigma$ and $T_{0}$, the string tension). The
equations of motion are:

$\displaystyle\ddot{x}^{i}-{x^{i}}^{\prime\prime}+\Gamma_{jk}^{i}\left(\dot{x}^{j}\dot{x}^{k}-{x^{j}}^{\prime}{x^{k}}^{\prime}\right)=0$
(1)

and the constraints (conformal gauge) are:

$\displaystyle g_{ij}\dot{x}^{i}{x^{j}}^{\prime}=0$ (2) $\displaystyle
g_{ij}\dot{x}^{i}\dot{x}^{j}+g_{ij}{x^{i}}^{\prime}{x^{j}}^{\prime}=0$ (3)

where the dot and prime denote differentiation w.r.t. $\tau$ and $\sigma$,
respectively. The two constraints ensure that the induced metric $\gamma_{ab}$
(as defined above) is diagonal. We will need to solve the Eqn. (1) written in
a background geometry with metric $g_{ij}$, subject to the constraints in (2)
and (3).

The plane parallel ($pp$) wave geometry which is a vacuum solution of the
Einstein equations is given by the line element in Brinkmann coordinates as:

$\displaystyle ds^{2}=-2dudv+F(u,x,y)du^{2}+dx^{2}+dy^{2}$ (4)

where, $u$ and $v$ are null coordinates
($u=\frac{1}{\sqrt{2}}\left(t+z\right)$,
$v=\frac{1}{\sqrt{2}}\left(t-z\right)$), $x$ and $y$ are the transverse
spatial coordinates and the function $F(u,x,y)$ encodes specific properties of
the spacetime. Assuming the above line element as a solution of the vacuum
Einstein equations one ends up requiring
$\partial_{\alpha}\partial^{\alpha}F=0$ ($\alpha$ being the spatial index
representing $x$, $y$). It is easily seen that

$\displaystyle F(u,x,y)=W(u)(x^{2}-y^{2})$ (5)

is a solution for $\partial_{\alpha}\partial^{\alpha}F=0$, irrespective of the
form of $W(u)$. It is this free function $W(u)$ which we will choose to
specify the gravitational wave pulse.

The string equations of motion in the pp-wave spacetime [10], for coordinates
$u$, $x$, $y$ are,

$\displaystyle\left(\partial_{\tau}^{2}-\partial_{\sigma}^{2}\right)u=0$ (6)
$\displaystyle\left(\partial_{\tau}^{2}-\partial_{\sigma}^{2}\right)x=\frac{\partial_{x}F}{2}\left[(\partial_{\tau}u)^{2}-(\partial_{\sigma}u)^{2}\right]$
(7)
$\displaystyle\left(\partial_{\tau}^{2}-\partial_{\sigma}^{2}\right)y=\frac{\partial_{y}F}{2}\left[(\partial_{\tau}u)^{2}-(\partial_{\sigma}u)^{2}\right]$
(8)

and the equation for $v$ is

$\displaystyle\left(\partial_{\tau}^{2}-\partial_{\sigma}^{2}\right)v=\frac{1}{2}\left(\partial_{u}F\right)\left[(\partial_{\tau}u)^{2}-(\partial_{\sigma}u)^{2}\right]+\left(\partial_{x}F\right)\left[\partial_{\tau}u\,\partial_{\tau}x-\partial_{\sigma}u\,\partial_{\sigma}x\right]$
$\displaystyle+\left(\partial_{y}F\right)\left[\partial_{\tau}u\,\partial_{\tau}y-\partial_{\sigma}u\,\partial_{\sigma}y\right]$
(9)

Note that one can solve for the equations for $u$, $x$, $y$ independently and
then use the solutions in the R. H. S. of (9) to write the equation for $v$.
The $v$ equation can then be solved separately.

A simple solution of the $u$ equation is:

$\displaystyle u=p\tau$ (10)

This identifies the worldsheet time coordinate with the spacetime null
coordinate $u$. With this choice, the two gauge conditions in Eqns (2) and (3)
reduce to

$\displaystyle
p\,\partial_{\sigma}v=\partial_{\tau}x\,\partial_{\sigma}x+\partial_{\tau}y\,\partial_{\sigma}y$
(11) $\displaystyle
2p\,\partial_{\tau}v=p^{2}F+\left(\partial_{\tau}x\right)^{2}+\left(\partial_{\sigma}x\right)^{2}+\left(\partial_{\tau}y\right)^{2}+\left(\partial_{\sigma}y\right)^{2}$
(12)

It can easily be checked by that the Eqns. (10), (11) and (12) imply Eqn. (9),
provided one uses the following equations for $x$ and $y$.

$\displaystyle\left(\partial_{\tau}^{2}-\partial_{\sigma}^{2}\right)x=\frac{p^{2}}{2}\partial_{x}F$
(13)
$\displaystyle\left(\partial_{\tau}^{2}-\partial_{\sigma}^{2}\right)y=\frac{p^{2}}{2}\partial_{y}F$
(14)

Therefore, given $F$ one can find $x$ and $y$, which can then be used in (11)
and (12) to find $v$. We will now choose the $W$ in $F$ and try to solve for
$x$, $y$ and $v$. Given $F$ in the form in (5) and the assumption

$\displaystyle x(\tau,\sigma)=e^{ik_{1}\sigma}x(\tau)$ (15) $\displaystyle
y(\tau,\sigma)=e^{ik_{1}\sigma}y(\tau)$ (16)

we end up with the following equations for $x(\tau)$ and $y(\tau)$.

$\displaystyle\ddot{x}+\left(k_{1}^{2}-p^{2}W(\tau)\right)x=0$ (17)
$\displaystyle\ddot{y}+\left(k_{1}^{2}+p^{2}W(\tau)\right)y=0$ (18)

where the dot denotes differentiation w.r.t. $\tau$ (or, equivalently, u). In
the remote past when $W(\tau)$ is zero we have

$\displaystyle x(\tau,\sigma)=R\cos k_{1}\tau\,cosk_{1}\sigma$ (19)
$\displaystyle y(\tau,\sigma)=R\cos k_{1}\tau\,sink_{1}\sigma$ (20)

which yields a circle with radius $R\cos k_{1}\tau$ in the past. We will now
assume different types of pulse profiles and solve the Eqns.(17) and (18).
This will lead us to a ‘permanent change’ after the pulse has gone away–a
feature known as the memory effect.

##  III Closed string evolution and memory

As mentioned above, we will solve Eqns. (17) and (18) with specific chosen
inputs for $W(\tau)$, representing a pulse. Once the solutions for $x(\tau)$
and $y(\tau)$ subject to the initial conditions in (19), (20) are known, we
will use them in (15), (16) and extract out the real part to write down the
profile functions $x(\tau,\sigma)$ and $y(\tau,\sigma)$. Using the
$x(\tau,\sigma)$ and $y(\tau,\sigma)$ in (11), (12) and integrating once we
can obtain the profile $v(\tau,\sigma)$. Knowing all of $u$, $v$, $x$, $y$ as
functions of $\tau$ and $\sigma$ gives us the embedding of the string world
sheet in the background $pp-$wave spacetime. Let us now proceed with the above
scheme.

###  III.1 Square pulse

Figure 2: The square pulse $W$ as function of $\tau$.

To solve the equations for $x(\tau)$ and $y(\tau)$ we have to assume a
functional form for $W(\tau)$. Here we choose it to be the following function
(see Figure 2):

$\displaystyle W(\tau)=0,\hskip 28.90755pt-\infty\leq\tau\leq 0$ (21)
$\displaystyle=W_{0},\hskip 36.135pt0\leq\tau\leq T$ $\displaystyle=0\hskip
65.04256pt\tau\geq T$

This is the well-known square pulse shown in Figure 2. In order to solve the
equations for $x(\tau)$ and $y(\tau)$ with the square pulse as $W(\tau)$ we
break up the domain of $\tau$ into three regions, solve the equations in the
three regions and then match the function and its derivative at the two
boundaries $\tau=0$ and $\tau=T$. In the region I ($-\infty\leq\tau\leq 0$) we
will assume $x(\tau)=R\cos\,k_{1}\tau$ and $y(\tau)=R\cos\,k_{1}\tau$.

Working through this rather straightforward exercise one eventually arrives at
the expressions for $x(\tau)$ and $y(\tau)$ in the $\tau\geq T$ region. We
have:

$\displaystyle
x(\tau,\sigma)=R\left(cos(k_{2}T)cos[k_{1}(\tau-T)]-\frac{k_{2}}{k_{1}}sin(k_{2}T)sin[k_{1}(\tau-T)]\right)cos(k_{1}\sigma)$
(22) $\displaystyle
y(\tau,\sigma)=R\left(cos(k_{3}T)cos[k_{1}(\tau-T)]-\frac{k_{3}}{k_{1}}sin(k_{3}T)sin[k_{1}(\tau-T)]\right)sin(k_{1}\sigma)$
(23)

where,

$\displaystyle k_{2}=\sqrt{k_{1}^{2}-W_{0}p^{2}}\hskip 14.45377pt;\hskip
14.45377ptk_{3}=\sqrt{k_{1}^{2}+W_{0}p^{2}}$ (24)

It is easily evident from the expressions in (22) and (23) that the profile at
any $\tau\geq T$ is not a circle even though it was a circle of radius
‘$R\cos\,k_{1}\tau$’ before $\tau=0$. For example, if we look at the profile
at any value $\tau=T+\frac{2n\pi}{k_{1}}$ ($n=0,1,2....$) we always get

$\displaystyle x=R\cos(k_{2}T)\,\cos k_{1}\sigma\hskip 14.45377pt;\hskip
14.45377pty=R\cos(k_{3}T)\,\sin k_{1}\sigma$ (25)

Since $k_{2}$ and $k_{3}$ are never equal for $W_{0}\neq 0$, the profile of
the closed string is an ellipse. One may track the shape of the closed string
for different $\tau\geq T$ and obtain the behaviour with the progress of
$\tau$. It is easy to see that the ellipse changes its shape with the
evolution of $\tau$. It does degenerate to a line along the $x$ axis when the
$y$ coordinate becomes zero and vice versa. However, as is seen from the
expression, it is never possible for both $x$ and $y$ to be zero or equal
simultaneously. Therefore, we never see a circular string again. This is the
‘permanent change’ which is akin to a memory effect caused by a gravitational
wave pulse. Figure 3 shows the string profile at different increasing
$\tau\geq T$ values in order to illustrate the permanent deformation of the
closed circular string. Notice that the shape of the pulse plays a crucial
role in creating this deformation, through the product $W_{0}T$ (height times
the width).

Figure 3: The string profile on the $xy$ plane for a square pulse, at two
different values of $u$ or $\tau$, after the pulse has left. The $v$
coordinate is suppressed here. We have chosen $T=1$ unit, $k_{1}T=1$,
$k_{2}T=0.5$, $k_{3}T=\frac{\sqrt{7}}{2}$, $R=1$ unit. The $\tau$ values are
$\tau=1.5$ unit (left) and a later value, $\tau=1.8$ unit (right)

It is possible to rewrite the above discussion in a general matrix notation.
Let us consider column vectors ${\bf X}=(x\,\,\,y)$, ${\bf
Q}=(cos\,k_{1}\sigma\,\,\,sin\,k_{1}\sigma)$ and write the above $x$, $y$
profiles as a matrix equation,

$\displaystyle{\bf X}=R\,{\bf A^{-1}Q}$ (26)

where ${\bf A^{-1}}$ is the coefficient matrix. Then we have

$\displaystyle{\bf X^{T}\left(A^{T}A\right)X}=R^{2}$ (27)

The string will be a circle if

$\displaystyle{\bf A^{T}A}=h(\tau){\bf I}$ (28)

For $\tau<0$, we get $h(\tau)=(cos\,k_{1}\tau)^{-2}$. But, for $\tau\geq T$,
${\bf A^{T}A}$ is not proportional to the identity matrix, which indicates the
presence of a memory effect.

In order to complete this exercise, we must know the $v$ coordinate too. This
is a somewhat involved calculation and we would need to set
$p=\frac{k_{1}R}{\sqrt{2}}$ to obtain an expression for $v$ which will satisfy
(11) and (12) with the inputs for $x$ and $y$ from (22) and (23). We end up
with

$\displaystyle v(\tau,\sigma)=-\frac{R}{2\sqrt{2}}\frac{\cos
2k_{1}\sigma}{k_{1}^{2}}G(\tau)+H(\tau)$ (29)

where

$\displaystyle
G(\tau)=\frac{1}{2}\left(k_{1}^{2}\cos^{2}k_{2}T-k_{2}^{2}\sin^{2}k_{2}T\right)\sin\,2k_{1}(\tau-T)+k_{1}k_{2}\cos
k_{2}T\sin k_{2}T\cos 2k_{1}(\tau-T)$
$\displaystyle-\frac{1}{2}\left(k_{1}^{2}\cos^{2}k_{3}T-k_{3}^{2}\sin^{2}k_{3}T\right)\sin\,2k_{1}(\tau-T)-k_{1}k_{3}\cos
k_{3}T\sin k_{3}T\cos 2k_{1}(\tau-T)$

and

$\displaystyle
H(\tau)=\frac{R}{2\sqrt{2}k_{1}}\left(k_{1}^{2}\cos^{2}k_{2}T+k_{2}^{2}\sin^{2}k_{2}T+k_{1}^{2}\cos^{2}k_{3}T+k_{3}^{2}\sin^{2}k_{3}T\right)\tau$
(31)

It is important to state the embedding functions for $\tau<0$ too. These are

$\displaystyle u=\frac{k_{1}R}{\sqrt{2}}\tau\hskip 14.45377pt;\hskip
14.45377ptv=\frac{k_{1}R}{\sqrt{2}}\tau$ (32) $\displaystyle x=R\cos
k_{1}\tau\cos k_{1}\sigma\hskip 14.45377pt;\hskip 14.45377pty=R\cos
k_{1}\tau\sin k_{1}\sigma$ (33)

The above analysis shows that before the pulse arrives the centre of mass of
the string (given by $z=\frac{1}{\sqrt{2}}(u-v)$) is at $z=0$. However, after
the pulse arrives and leaves, the centre of mass develops a translational
motion in $\tau$ as well as an oscillation. One can go over to a Lorentz frame
(Lorentz transformation in the $t,z$ coordinates) and make sure that the
centre of mass has only oscillatory motion after the pulse leaves. With such a
Lorentz transformation the new $u$, $v$ coordinates (named as $u^{\prime}$,
$v^{\prime}$) become:

$\displaystyle u^{\prime}=\sqrt{a}\frac{k_{1}R}{\sqrt{2}}\tau$ (34)
$\displaystyle
v^{\prime}=\sqrt{a}\frac{k_{1}R}{\sqrt{2}}\tau-\frac{R}{2\sqrt{2a}}\frac{\cos
2k_{1}\sigma}{k_{1}^{2}}G(\tau)$ (35)

where $a=\frac{\sqrt{2}H(\tau)}{k_{1}R\,\tau}$. Since $H(\tau)$ is
proportional to $\tau$, $a$ is a number given by

$\displaystyle
a=\frac{1}{2k_{1}^{2}}\left(k_{1}^{2}\cos^{2}k_{2}T+k_{2}^{2}\sin^{2}k_{2}T+k_{1}^{2}\cos^{2}k_{3}T+k_{3}^{2}\sin^{2}k_{3}T\right)$
(36)

and the magnitude of the boost velocity between the two frames is given as
$v_{b}=\frac{1-a}{1+a}$.

The emerging picture of the closed string before and after it experiences the
pulse is as follows:

$\bullet$ Before the pulse arrives the closed string shape is circular and its
centre of mass is at $z=0$. The radius of the circle changes with $\tau$ (a
pulsating string) and can even become zero at specific $\tau$ values. The $v$
coordinate has the same value as the $u$ coordinate for any $\tau$. In fact
$u=v$ as in Eqn. (32).

$\bullet$ After the pulse departs, the closed string is elliptical in shape
and never becomes circular. If we include the behaviour of the $v$ coordinate,
we can see from (29) or (31) that for fixed $\tau$, $v$ has different values
at different $\sigma$. In fact, at fixed $\tau$, one can replace the spacetime
coordinate $v$ with the worldsheet coordinate $\sigma$, for the given
embedding. The shape of the string can degenerate to a line (along $x$ or $y$)
at specific $\tau$ values. The centre of mass
($z^{\prime}=\frac{1}{\sqrt{2}}(u^{\prime}-v^{\prime})$ value) is now purely
oscillatory with different amplitudes at different $\tau$ (construct
$z^{\prime}$ from (34), (35)) – it does not have any linear in $\tau$ motion
as long as we are in a properly chosen frame. Note that prior to the arrival
of the pulse, the centre of mass was at $z=0$ (see Eqn. (32)).

There are changes in the geometry of the worldsheet caused by the pulse. We
will discuss these aspects briefly in the penultimate section of this article.

###  III.2 Sech-squared pulse

It is easy to redo the above analysis for a delta function pulse
($W(\tau)=W_{0}\delta(\tau)$). The results are similar to the square pulse and
the mathematical details have been worked out in [10], though the authors in
[10] unfortunately missed the ‘memory’ aspect! The square and the delta
function pulses are largely idealistic. We therefore move on now to a pulse
shape which is smooth–the so–called sech-squared pulse. The $W(\tau)$ here is
given as:

$\displaystyle W(\tau)=\frac{\alpha^{2}}{4p^{2}}sech^{2}\alpha\tau$ (37)

Figure 4: The Sech-squared pulse $W$ as function of $\tau$ (or $u$). Here
$\alpha=1$.

The two differential equations which we need to solve to obtain $x(\tau)$ and
$y(\tau)$ are,

$\displaystyle\ddot{x}+\left(k_{1}^{2}-\frac{\alpha^{2}}{4}sech^{2}\alpha\tau\right)x=0$
(38)
$\displaystyle\ddot{y}+\left(k_{1}^{2}+\frac{\alpha^{2}}{4}sech^{2}\alpha\tau\right)y=0$
(39)

Both these equations can be solved exactly and the asymptotic forms of the
solutions are known. As we did earlier, we will impose the circular closed
string shape at negative infinity (i.e. $\tau\rightarrow-\infty$). Using the
known asymptotic forms we will then find out the shape of the closed string as
$\tau\rightarrow\infty$. A change in the shape, like before, will be
associated with a permanent deformation and a memory effect caused by the
pulse.

The Eqns. (38) and (39) fall in the general class of Poschl–Teller potential
problems usually studied in quantum mechanics [46]. The two linearly
independent solutions for equations of the type

$\displaystyle\ddot{U}(\tau)+\left(k^{2}+\frac{\alpha^{2}\lambda(\lambda-1)}{\cosh^{2}\alpha\tau}\right)U(\tau)=0$
(40)

are given as

$\displaystyle
U_{1}(\tau)=\left(1+\tanh\alpha\tau\right)^{\frac{ik}{2\alpha}}\left(1-\tanh\alpha\tau\right)^{-\frac{ik}{2\alpha}}{}^{2}F_{1}\left(\lambda;1-\lambda;\frac{ik}{\alpha}+1;\frac{1+\tanh\alpha\tau}{2}\right)$
$\displaystyle
U_{2}(\tau)=2^{\frac{ik}{\alpha}}\left(1-\tanh^{2}\alpha\tau\right)^{-\frac{ik}{2\alpha}}{}^{2}F_{1}\left(\lambda-\frac{ik}{\alpha};1-\lambda-\frac{ik}{\alpha};1-\frac{ik}{\alpha};\frac{1+\tanh\alpha\tau}{2}\right)$
(41)

where ${}^{2}F_{1}$ denotes the hypergeometric function. The general solution
therefore is,

$\displaystyle U(\tau)=AU_{1}(\tau)+BU_{2}(\tau)$ (42)

where $A$ and $B$ are constants to be determined by the initial conditions.
Eqn. (40) will match with (38) when $\lambda=\frac{1}{2}$. Similarly, when
$\lambda=\frac{1+\sqrt{2}}{2}$ we will get Eqn. (39).

We now need to write down the asymptotic form of the general solution in (42)
for $\tau\rightarrow-\infty$ and $\tau\rightarrow+\infty$.

Notice that for $\tau\rightarrow-\infty$, the argument of the hypergeometric
function goes to zero and the value of ${}^{2}F_{1}(a;b;c;0)$ is equal to one.
Hence, expanding the hyperbolic tangents as exponentials and taking the limit
$\tau\rightarrow-\infty$ we have

$\displaystyle U(\tau\rightarrow-\infty)=Ae^{ik\tau}+Be^{-ik\tau}$ (43)

The limit of $\tau\rightarrow+\infty$ is a little tricky to obtain. One will
have to use a linear transformation formula [47], 111Use is made of the
formula 15.3.6, Pg. 559 in [47]. and write the hypergeometric function with
the argument $z=\frac{1+\tanh\alpha\tau}{2}$ in terms of a sum (with a $z$
dependent coefficient) of two hypergeometric functions with arguments
$1-z=\frac{1-\tanh\alpha\tau}{2}$. This helps in getting the asymptotic form
easily because, now, as one takes the limit $\tau\rightarrow\infty$ one ends
up with the argument as zero in the two hypergeometric functions and their
value become equal to one.

Thus, as $\tau\rightarrow\infty$ one gets

$\displaystyle
U(\tau\rightarrow\infty)=A^{\prime}e^{ik\tau}+B^{\prime}e^{-ik\tau}$ (44)

where

$\displaystyle
A^{\prime}=\frac{\Gamma(\frac{ik}{\alpha}+1)\Gamma(\frac{ik}{\alpha})}{\Gamma(\frac{ik}{\alpha}+1-\lambda)\Gamma(\frac{ik}{\alpha}+\lambda)}\,A+\frac{\Gamma(1-\frac{ik}{\alpha})\Gamma(\frac{ik}{\alpha})}{\Gamma(1-\lambda)\Gamma(\lambda)}\,B=a_{1}\,A+b_{1}\,B$
(45) $\displaystyle
B^{\prime}=\frac{\Gamma(\frac{ik}{\alpha}+1)\Gamma(-\frac{ik}{\alpha})}{\Gamma(1-\lambda)\Gamma(\lambda)}\,A+\frac{\Gamma(1-\frac{ik}{\alpha})\Gamma(-\frac{ik}{\alpha})}{\Gamma(\lambda-\frac{ik}{\alpha})\Gamma(1-\lambda-\frac{ik}{\alpha})}\,B=a_{2}\,A+b_{2}\,B$
(46)

Armed with these asymptotic forms we will now write down the solutions for
Eqns. (38) and (39).

Let us first consider Eqn. (38), i.e. the $x$ equation. We keep in mind that
we need to use $\lambda=\frac{1}{2}$ while using the solutions quoted above.
Here, as $\tau\rightarrow-\infty$ we need $x(\tau)=R\cos k_{1}\tau$ (and
$x(\tau,\sigma)=R\cos k_{1}\tau\cos k_{1}\sigma$). Therefore, one has to
choose $A=B=\frac{R}{2}$. With this asymptotic condition at
$\tau\rightarrow-\infty$, one can easily write down the final solution for
$\tau\rightarrow\infty$. This is given as (taking the real part of the full
complex solution, since $a_{1},b_{1},a_{2},b_{2}$ can be complex):

$\displaystyle
x(\tau,\sigma)=\frac{R}{2}Re\left[\left(a_{1}+b_{1}+a_{2}+b_{2}\right)\cos
k_{1}\tau\cos k_{1}\sigma+\left(a_{2}+b_{2}-a_{1}-b_{1}\right)\sin
k_{1}\tau\sin k_{1}\sigma\right.$
$\displaystyle+\left.i\left(a_{1}+b_{1}-a_{2}-b_{2}\right)\sin k_{1}\tau\cos
k_{1}\sigma+i\left(a_{1}+b_{1}+a_{2}+b_{2}\right)\cos k_{1}\tau\sin
k_{1}\sigma\right]$

For Eqn. (39), we need to use $\lambda=\frac{1+\sqrt{2}}{2}$. The
$\tau\rightarrow-\infty$ asymptotic form of $y(\tau,\sigma)$ is given as
$R\cos k_{1}\tau\sin k_{1}\sigma$. To implement this form in the asymptotic
past, we will need to use $A=B=\frac{R}{2i}$ in the solutions quoted above. We
will write the final results using $a_{1}^{\prime}$, $b_{1}^{\prime}$,
$a_{2}^{\prime}$ and $b_{2}^{\prime}$ since we have a different equation and a
different $\lambda$ value. Thus, we have $y(\tau,\sigma)$ as:

$\displaystyle
y(\tau,\sigma)=\frac{R}{2}Re\left[-i\left(a_{1}^{\prime}+b_{1}^{\prime}+a_{2}^{\prime}+b_{2}^{\prime}\right)\cos
k_{1}\tau\cos
k_{1}\sigma-i\left(a_{2}^{\prime}+b_{2}^{\prime}-a_{1}^{\prime}-b_{1}^{\prime}\right)\sin
k_{1}\tau\sin k_{1}\sigma\right.$
$\displaystyle+\left.\left(a_{1}^{\prime}+b_{1}^{\prime}-a_{2}^{\prime}-b_{2}^{\prime}\right)\sin
k_{1}\tau\cos
k_{1}\sigma+\left(a_{1}^{\prime}+b_{1}^{\prime}+a_{2}^{\prime}+b_{2}^{\prime}\right)\cos
k_{1}\tau\sin k_{1}\sigma\right]$

It is evident from the above expressions that the circular string in the
asymptotic past, under the influence of the gravitational wave pulse will
change its shape and become elliptical in shape, in the asymptotic future. To
see this explicitly, we will now work out the details with explicit values for
the various parameters and then plot the profiles. We have chosen to retain
values upto four decimal places.

Let us consider the case where $k_{1}=1$, $\alpha=1$. Using the expressions
$a_{1}$, $a_{2}$, $b_{1}$, $b_{2}$ with $\lambda=\frac{1}{2}$ we obtain

$\displaystyle
x(\tau,\sigma)=\frac{R}{2}\left(1.9379\cos\,\tau+0.6971\sin\,\tau\right)\cos\,\sigma$
(49)

For $\lambda=\frac{1+\sqrt{2}}{2}$ and the same $k_{1}$, $\alpha$ one gets,
using $a_{1}^{\prime}$, $b_{1}^{\prime}$, $a_{2}^{\prime}$ and
$b_{2}^{\prime}$,

$\displaystyle
y(\tau,\sigma)=\frac{R}{2}\left(1.9461\cos\,\tau-0.5777\sin\,\tau\right)\sin\,\sigma$
(50)

Figure 5: The late time string configurations (blue curves) at
$\tau=\frac{103\pi}{7}$ (left) and at $\tau=\frac{101\pi}{4}$ (right). The
yellow circles correspond to the earlier configurations at
$\tau=-\frac{103\pi}{7}$ and $\tau=-\frac{101\pi}{4}$. We have assumed $R=2$
units and Eqns. (49) and (50).

The expressions are, by definition, the profiles at large values of $\tau$.
However, we can get an idea of the elliptical shape by plotting at a large,
finite value of $\tau$. Two representative plots are shown in Figure 5. The
initial profile (at $\tau\rightarrow-\infty$) is just a circle (
$x=2\cos\tau\cos\sigma$, $y=2\cos\tau\sin\sigma$).

A permanent change in the shape of the string appears to have taken place. The
cause of this is, like in the square pulse case, the gravitational wave pulse
– given here as a sech-squared function of $\tau$ or $u$.

It is possible to obtain $v(\tau,\sigma)$ in this case too though we do not
attempt to analyse it here. The qualitative result is not expected to be very
different from the square pulse case discussed earlier.

##  IV Worldsheet geometry and memory

Let us now focus on the geometry of the worldsheet. We will consider here the
case of the square pulse only. The worldsheet line element can be found using
the embedding functions and the worldsheet metric functions
$\gamma_{ab}=g_{ij}\partial_{a}x^{i}\partial_{b}x^{j}$. As expected, the line
element is diagonal (conformal gauge choice). We have, for $\tau\leq 0$, the
line element as given by,

$\displaystyle ds_{\tau\leq
0}^{2}=R^{2}k_{1}^{2}\cos^{2}\,k_{1}\tau\left(-d\tau^{2}+d\sigma^{2}\right)$
(51)

The metric is degenerate (zero determinant) at all
$\tau=\frac{(2n+1)\pi}{2k_{1}}$ ($n=0,1,2....)$.

On the other hand the line element for $\tau\geq T$ is given as

$\displaystyle
ds^{2}=\Omega^{2}(\tau,\sigma)\left(-d\tau^{2}+d\sigma^{2}\right)$ (52)

where

$\displaystyle\Omega^{2}(\tau,\sigma)=R^{2}\sin^{2}k_{1}\sigma\left[k_{1}\cos\,k_{2}T\cos\,k_{1}(\tau-T)-k_{2}\sin\,k_{2}T\sin
k_{1}(\tau-T)\right]^{2}+$ $\displaystyle
R^{2}\cos^{2}k_{1}\sigma\left[k_{1}\cos\,k_{3}T\cos\,k_{1}(\tau-T)-k_{3}\sin\,k_{3}T\sin
k_{1}(\tau-T)\right]^{2}$ (53)

The metric here is never degenerate because $\Omega^{2}(\tau,\sigma)$ cannot
be zero for any value of $\tau$, $\sigma$. Since $\Omega^{2}$ is a sum of two
squares, it can only be zero if the individual terms are both zero, which is
impossible ($k_{2}\neq k_{3}$).

Figure 6: The worldsheet geometry before the arrival of the pulse. The zero
radius locations are the worldsheet singularities. We have used Eqns. (32),
(33) with $R=1$. $k_{1}\tau$ and $k_{1}\sigma$ are the surface coordinates.

Figure 7: The worldsheet geometry as a $xyv$ embedding after the departure of
the pulse. Two different orientations of the worldsheet are shown. There are
no locations where the worldsheet shrinks to a point. Hence, no worldsheet
singularities. Eqns. (22), (23) and (29)-(31) are used with values for the
parameters $T$, $k_{1}T$, $k_{2}T$, $k_{3}T$ and $R$ as mentioned for Figure
3. The surface coordinates are $k_{1}\tau$ and $k_{1}\sigma$.

Figure 8: The worldsheet geometry as a $xyu$ embedding after the arrival of
the pulse. Two different orientations of the worldsheet are shown. There are
no locations where the worldsheet shrinks to a point. Hence, no worldsheet
singularities. Eqns. (22), (23) and $u=\frac{R}{\sqrt{2}}k_{1}\tau$ are used
with values for the parameters $T$, $k_{1}T$, $k_{2}T$, $k_{3}T$ and $R$, as
mentioned in Figure 3. The surface coordinates are $k_{1}\tau$ and
$k_{1}\sigma$. Notice the straightlines along which the worldsheet collapses,
at different $u$–these are the ‘degenerate ellipses’.

The Ricci scalar ${\cal R}$ of the two dimensional worldsheet also undergoes a
change in character after the passage of the pulse. Recall that the expression
for the Ricci scalar in terms of the conformal factor $\Omega^{2}$ is

$\displaystyle{\cal
R}=-\frac{2}{\Omega^{2}}\left(-\frac{\partial^{2}}{\partial\tau^{2}}+\frac{\partial^{2}}{\partial\sigma^{2}}\right)\ln\Omega$
(54)

The denominator factor in the above expression can lead to a worldsheet
singularity where $\Omega^{2}=0$. Thus, a singularity exists prior to the
arrival of the pulse, but gets removed after the passage of the pulse.

Both the above features– (a) metric becoming non-degenerate and (b) the
worldsheet singularites being removed — are ‘permanent changes’ caused by the
gravitational wave pulse. Thus, apart from the change of shape of the circular
string to an elliptical one, we also see changes in the worldsheet geometry
which seem to add on to the character of ‘memory’.

One can visualise the two dimensional surface of the worldsheet by embedding
it in a three dimensional background. The string worldsheet however is a two
dimensional surface in the four dimensional pp-wave spacetime, which, as we
noted before, is just flat spacetime before and after the duration of the
pulse. We show the embedded (in a three dimensional background) worldsheet
geometry in Figures 6,7 and 8. Figure 6 shows the embedding
($x(\tau,\sigma),y(\tau,\sigma),u(\tau)=\frac{1}{\sqrt{2}}\tau$) before the
arrival of the pulse. One can see the presence of singularities on the
worldsheet at locations where it collapses to a point. As mentioned before, it
is a pulsating string. Here, i.e. prior to the arrival of the pulse, the
embedding ($x(\tau,\sigma),y(\tau,\sigma),v(\tau)=\frac{1}{\sqrt{2}}\tau$) is
the same as the $xyu$ embedding.

Figure 7 shows the worldsheet after the pulse departs. One can see that here
the worldsheet is indeed nonsingular – it never collapses to a point–at best
it can collapse to a line. The plots are obtained by using the parametric
relations for $x(\tau,\sigma)$, $y(\tau,\sigma)$ and $v(\tau,\sigma)$. Since
$u$ is proportional to $\tau$, the spacetime null coordinate $u$ is
proportional to the worldsheet ‘time’ $\tau$. Before the pulse arrives $u=v$,
but later it it is not so. The embedding $x(\tau,\sigma)$, $y(\tau,\sigma)$
and $u(\tau)$ is also shown in Figure 8, where the nature of the worldsheet is
perhaps more clearly visible. For the embedding $xyv$, at any $\tau$ value,
points at different $\sigma$ do not lie on a plane and the string is spread
away from any $u=$constant plane. This does not happen for the $xyu$
embedding. Both visualisations are required in order to understand the nature
of the worldsheet. As stated earlier, since the string worldsheet is a
codimension two surface in a four dimensional background one needs to look at
different embeddings in order to develop a complete picture.

##  V Conclusions

Our main objective in this article was to show how a gravitational wave pulse
can affect the shape of string as it evolves to generate the worldsheet. As
mentioned in the Introduction, our motivation was to provide a ‘stringy’
realisation of the often used picture where memory is linked to the ‘permanent
change’ in the shape of a ring of particles. We achieve this goal by studying
strings in pp-wave spacetimes where the gravitational wave pulse is modeled by
the free function available in such spacetimes. Studying strings in the
presence of a square and a sech-squared pulse we show that a circular string
in the past (before the arrival of the pulse) evolves into an elliptical shape
in the asymptotic future. The satisfying fact is that we are able to
demonstrate this explicitly by solving all relevant equations analytically.

It may be argued that the shape of the string is a coordinate dependent
feature and one should look for quantities which do not depend on the choice
of background coordinates. We have shown, towards the end of our article, that
there exist such quantities which exhibit a permanent change – the behaviour
of the metric determinant and the Ricci scalar of the worldsheet metric being
two such quantities. Both of them exhibit a ‘permanent change’ – a fact we
have demonstrated clearly in the previous section. In addition, the embeddings
of the worldsheets in a three dimensional background provides a useful way of
visualising the changes that occur before and after the duration of the pulse.
One may also consider finding the energy (target space) of the string using
the expression $P^{0}=-T_{0}\int_{0}^{2\pi}\sqrt{-\gamma}\gamma^{\tau
a}\left(\partial_{a}x^{0}\right)d\sigma$. Working through this calculation
(not quoted here) we did find a difference between the expressions for
‘before’ and ‘after’ the pulse, which is also a signature of memory.

It is possible to work on this topic further and find more examples. We have
indeed found very similar effects for a triangular pulse as well as a
completely general pulse shape replacing the non-zero constant in the square
case. In all scenarios though we do find a ‘permanent change’. One is tempted
to ask therefore–what kind of pulse will show no change in shape of a string?
Another open question is related to the use of a more general Kundt wave
spacetime background. One can investigate strings in such Kundt spacetimes and
look for a similar memory effect [33], [34] and its consequences.

## Acknowledgements

AD thanks the Department of Physics, IIT Kharagpur, India for the opportunity
to do a part of this work during his fourth year bachelors project in 2023-24.

## References

  * [1] M. Favata, Class. Quant. Grav. 27, 084036 (2010).
  * [2] M. Favata, Lecture slides.
  * [3] B. Zweibach, A first course in string theory, Cambridge University Press, Cambridge, UK (2004).
  * [4] J. B. Griffiths and J. Podolsky, Exact Space-Times in Einstein’s General Relativity, Cambridge University Press, Cambridge, UK (2009).
  * [5] M. Blau, Lectures on ‘Plane waves and Penrose limits’ Lecture notes.
  * [6] D. Amati and C. Klimcik, Phys. Letts. B210, 92 (1988).
  * [7] G. T. Horowitz and A. Steif, Phys. Rev. D 42, 1950 (1990).
  * [8] H. J. de Vega and N. Sanchez, Phys. Rev. D45, 2783 (1992).
  * [9] H. J. de Vega, M. Ramon-Medrano and N. Sanchez, Class. Quant. Grav. 10, 2007 (1993).
  * [10] V. Liška and R. von Unge, Phys. Rev. D106, 044066 (2022).
  * [11] H. Afshar, E. Esmaeili and M. M. Sheikh-Jabbari, JHEP02, 053 (2019).
  * [12] A. Aldi, M. Bianchi and M. Firrotta, Phys. Letts. B 813,136037 (2021).
  * [13] O. M. Boersma, D. A. Nichols, P. Schmidt, Phys. Rev. D 101, 083026 (2020).
  * [14] M. Hübner, P. Lasky, E. Thrane, Phys. Rev. D 104, 023004 (2021).
  * [15] H. Inchauspé, S. Gasparotto, D. Blas, L. Heisenberg, J. Zosso, S. Tiwari, arXiv:2406.09228.
  * [16] Y. B. Zel’dovich and A. G. Polnarev, Sov. Astron 18, 17 (1974).
  * [17] V. B. Braginsky and L. P. Grishchuk, Sov. Phys. JETP 62, 427 (1985).
  * [18] D. Christodoulou, Phys. Rev. Lett. 67, 1486 (1991).
  * [19] L. Bieri and D. Garfinkle, Phys. Rev. D 89, 084039 (2014).
  * [20] A. Tolish, L. Bieri, D. Garfinkle, and R. M. Wald, Phys. Rev. D 90, 044060 (2014).
  * [21] A. Tolish and R. M. Wald, Phys. Rev. D 89, 064008 (2014).
  * [22] T. Mädler and J. Winicour, Class. Quant. Grav. 33, 175006 (2016).
  * [23] T. Mädler and J. Winicour, Class. Quant. Grav. 34, 115009 (2017).
  * [24] A. Strominger and A. Zhiboedov, J. High Energy Phys. 01 (2016), 86.
  * [25] A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory (2017) arXiv:1703.05448 [hep-th].
  * [26] P.-M. Zhang, C. Duval, G. W. Gibbons, and P. A. Horvathy, Phys. Lett. B 772, 743 (2017).
  * [27] P.-M. Zhang, C. Duval, G. W. Gibbons, and P. A. Horvathy, Phys. Rev. D 96, 064013 (2017).
  * [28] P. M. Zhang, C. Duval, G. W. Gibbons, and P. A. Horvathy, JCAP 05, 030.
  * [29] P.-M. Zhang, C. Duval, and P. A. Horvathy, Class. Qtm. Grav. 35, 065011 (2018).
  * [30] P. M. Zhang, M. Elbistan, G. W. Gibbons, and P. A. Horvathy, Gen. Rel. Grav. 50,107 (2018).
  * [31] I. Chakraborty and S. Kar, Phys. Rev. D 101, 064022 (2020).
  * [32] B. Cvetković and D. Simić, Eur. Phys. J. C 82, 127 (2022).
  * [33] I. Chakraborty and S. Kar, Phys. Lett. B 808, 135611 (2020).
  * [34] S. Siddhant, I. Chakraborty, and S. Kar, Eur. Phys. J. C 81, 350 (2021).
  * [35] I. Chakraborty and S. Kar, Eur. Phys. J. Plus 137, 418 (2022).
  * [36] J. Ben Achour and J-P Uzan, arXiv:2406.07106.
  * [37] P. -M. Zhang and P. A. Horvathy, arXiv:2405.12928.
  * [38] E. E. Flanagan, A. M. Grant, A. I. Harte, and D. A. Nichols.Phys. Rev. D 99, 084004 (2019).
  * [39] E. E. Flanagan, A. M. Grant, A. I. Harte, and D. A. Nichols, Phys. Rev. D 101, 104033 (2020).
  * [40] A. M. Grant and D. A. Nichols, Phys. Rev. D 105, 024056 (2022).
  * [41] M. O’Loughlin and H. Demirchian, Phys. Rev. D 99, 024031 (2019).
  * [42] A. K. Divakarla and B. F. Whiting, Phys. Rev. D 104, 064001 (2021).
  * [43] G. M. Shore, JHEP 12, 133 (2018).
  * [44] H. Hadi, A. R. Akbarieh, David F. Mota, Class. Quantum Grav. 41, 105005 (2024).
  * [45] K. Mitman, M. Boyle, L. C. Stein, N. Deppe, L. E. Kidder, J. Moxon, H. P. Pfeiffer, M. A. Scheel, S. A. Teukolsky, W. Throwe, N. L. Vu, A Review of Gravitational Memory and BMS Frame Fixing in Numerical Relativity, arXiv:2405.08868. See the references in this article for a chronological history of the memory effect.
  * [46] D. Cevik, M. Gadella, S. Kuru and J. Negro, Phys. Letts. A380,1600 (2016).
  * [47] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs and mathematical tables, National Bureau of Standards, Applied Mathematics Series (1964).


for $\\{i=0;i<\mathcal{T};i\\!++\\}$ do

assert $\sum\limits_{\scriptscriptstyle j=0}^{\scriptscriptstyle
k-1}u[i,j]=24$

$b\leftarrow b\;\|\;u[i,:]$

$\mathsf{m}\leftarrow\mathsf{m}\;\|\;b$

$\mathrm{y}\leftarrow\vec{0}$

if $|\mathsf{m}|\geq q$ then

for $u\in\mathsf{m}$ do

for $\\{i=0;i<\mathcal{T};i\\!++\\}$ do

$\mathrm{y}\leftarrow\mathrm{y}+u[i,:]$

return $\mathrm{y}$

The above pseudocode uses the following notation conventions:

  * •

Given matrix $z$, the notation $z[i,j]$ denotes accessing the $i$-th row and
$j$-th column of $z$.

  * •

$z[i,:]$ denotes the row vector corresponding to the $i$-th row of $z$;
$z[:,j]$ is the column vector corresponding to the $j$-th column

  * •

We denote by $\mathsf{CircularBuffer}(n)$ the creation of a new $n$-sized
circular buffer. Appending an item to the buffer is accomplished through
concatenation operator $\;\|\;$. After $n$ items have been appended to a
buffer, it will overwrite the first record in the buffer, and so on

##  Appendix 0.D Steel simulator

We now describe the simulator presented in [12], while adapting the message
syntax to fit with the messages sent by $DD\textrm{-}\mathrm{FESR}$ and
$DD\textrm{-}\mathsf{Steel}$.

Simulator
$\mathcal{S}_{\mathrm{FESR}}$[$\mathsf{PKE},\Sigma,\mathsf{N},\lambda,\mathtt{F}$]
State variables Description $\mathcal{H}[\cdot]\leftarrow\emptyset$ Table of
ciphertext and handles in public repository $\mathcal{K}\leftarrow[]$ List of
$\mathsf{prog}_{\mathsf{\mathsf{FE}^{\mathsf{VK}}[\mathrm{F}]}}$ enclaves and
their $\mathsf{eid}_{\mathsf{F}}$ $\mathcal{G}\leftarrow\\{\\}$ Collects all
messages sent to $G_{\mathsf{att}}$ and its response
$\mathcal{B}\leftarrow\\{\\}$ Collects all messages signed by
$G_{\mathsf{att}}$ $(\mathsf{crs},\tau)\leftarrow\mathsf{N}.\mathcal{S}_{1}$
Simulated reference string and trapdoor _On message
$(\textsc{setup},\allowbreak P)$ from
$\mathcal{S}_{DD\textrm{-}\mathrm{FESR}}$:_ if $\mathsf{mpk}=\bot$ then
$\mathsf{eid}_{\mathsf{\mathsf{KME}}}\leftarrow
G_{\mathsf{att}}.\mathsf{install}(\mathsf{C}.\mathsf{sid},\mathsf{prog}_{\mathsf{\mathsf{KME}^{\mathsf{VK}}}})$
$(\mathsf{mpk},\sigma_{\mathsf{\mathsf{KME}}})\leftarrow
G_{\mathsf{att}}.\mathsf{resume}(\mathsf{eid}_{\mathsf{\mathsf{KME}}},\mathrm{init},\mathsf{crs},\mathsf{C}.\mathsf{sid})$
if $P=\mathsf{A}$ then send
$(\textsc{setup},\allowbreak\mathsf{mpk},$$\sigma_{\mathsf{\mathsf{KME}}}$$)$
to $\mathcal{SC}_{\mathsf{A}}$ else if $P=\mathsf{B}$ then send
$(\textsc{setup},\allowbreak\mathsf{mpk},$$\sigma_{\mathsf{\mathsf{KME}}}$$,\mathsf{eid}_{\mathsf{\mathsf{KME}}})$
to $\mathcal{SC}^{\mathsf{C}}_{\mathsf{B}}$ and receive
$\left(\textsc{provision},\sigma_{\mathsf{}},\mathsf{eid}_{\mathsf{\mathsf{DE}}},\mathsf{pk}_{KD}\right)$
assert
$(\mathsf{C}.\mathsf{sid},\mathsf{eid}_{\mathsf{\mathsf{DE}}},\mathsf{prog}_{\mathsf{\mathsf{DE}^{\mathsf{VK}}}},\mathsf{pk}_{KD})\in\mathcal{B}[\sigma_{\mathsf{}}]$
$(\mathsf{ct}_{\mathsf{key}},\sigma_{\mathsf{\mathsf{KME}}})\leftarrow
G_{\mathsf{att}}.\mathsf{resume}(\mathsf{eid}_{\mathsf{\mathsf{KME}}},(\mathrm{provision},(\sigma_{\mathsf{,}}\mathsf{eid}_{\mathsf{\mathsf{DE}}},\mathsf{pk}_{KD},\mathsf{eid}_{\mathsf{\mathsf{KME}}},\mathsf{crs}))))$
send
$(\textsc{provision},\allowbreak\mathsf{ct}_{\mathsf{key}},\sigma_{\mathsf{\mathsf{KME}}})$
to $\mathcal{SC}_{\mathsf{B}}$ _On message
$(\textsc{read},\allowbreak\mathsf{h})$ from $\text{party }\mathsf{B}\text{ to
}\mathcal{REP}$:_ send
$(\textsc{decrypt},\allowbreak\mathrm{F}_{0},\mathsf{h})$ to
$\mathcal{S}_{DD\textrm{-}\mathrm{FESR}}\text{ on behalf of }\mathsf{B}$ and
receive $(\textsc{decrypted},|(\mathrm{m},$$k$$)|)$ assert
$|(\mathrm{m},$$k$$)|\neq\bot$
$\mathsf{ct}\leftarrow\mathsf{PKE}.\mathsf{Enc}(\mathsf{mpk},0^{|(\mathrm{m},k)|})$
$\pi\leftarrow\mathsf{N}.\mathcal{S}_{2}(\mathsf{crs},\tau,(\mathsf{mpk},\mathsf{ct}))$
$\mathsf{ct}_{\mathsf{msg}}\leftarrow(\mathsf{ct},\pi)$;
$\mathcal{H}[\mathsf{ct}_{\mathsf{msg}}]\leftarrow h$ send
$(\textsc{read},\allowbreak\mathsf{ct}_{\mathsf{msg}})$ to $\mathsf{B}$ _On
message $(\textsc{install},\allowbreak\mathsf{idx},\mathsf{prog}_{\mathsf{}})$
from $\text{party }P\in\\{\mathbf{B}\cup\mathsf{C}\\}\text{ to
}G_{\mathsf{att}}$:_ $\mathsf{eid}_{\mathsf{}}\leftarrow
G_{\mathsf{att}}.\mathsf{install}(\mathsf{idx},\mathsf{prog}_{\mathsf{}})$
$\mathcal{G}[\mathsf{eid}_{\mathsf{}}].\mathsf{install}\leftarrow(\mathsf{idx},\mathsf{prog}_{\mathsf{}},$$P$$)$
// $\mathcal{G}[\mathsf{eid}_{\mathsf{}}].install[1]$ is the program’s code
forward $\mathsf{eid}_{\mathsf{}}$ to $\mathsf{B}$ _On message
$(\textsc{resume},\allowbreak\mathsf{eid}_{\mathsf{}},\mathsf{input})$ from
$\text{party }P\in\\{\mathbf{B}\cup\mathsf{C}\\}\text{ to }G_{\mathsf{att}}$:_
assert $\mathcal{G}[\mathsf{eid}_{\mathsf{}}]$$.\mathsf{install}[2]=P$ if
$\mathcal{G}[\mathsf{eid}_{\mathsf{}}].\mathsf{install}[1]\neq\mathsf{prog}_{\mathsf{\mathsf{FE}^{\mathsf{VK}}[\cdot]}}\lor(\mathsf{input}[0]\neq\mathrm{run}\lor\mathsf{input}[-1]\neq\bot)$
then $(\mathsf{output},\sigma_{\mathsf{}})\leftarrow
G_{\mathsf{att}}.\mathsf{resume}(\mathsf{eid}_{\mathsf{}},\mathsf{input})$
$\mathcal{G}[\mathsf{eid}_{\mathsf{}}].\mathsf{resume}\leftarrow\mathcal{G}[\mathsf{eid}_{\mathsf{}}].\mathsf{resume}\;\|\;(\sigma_{\mathsf{}},\mathsf{input},\mathsf{output})$
$\mathcal{B}[\sigma_{\mathsf{}}]\leftarrow(\mathcal{G}[\mathsf{eid}_{\mathsf{}}].\mathsf{install}[0],\mathsf{eid}_{\mathsf{}},\mathcal{G}[\mathsf{eid}_{\mathsf{}}].\mathsf{install}[1],\mathsf{output})$
forward $(\mathsf{output},\sigma_{\mathsf{}})$ to $P$ else
$(\mathsf{idx},\mathsf{prog}_{\mathsf{\mathsf{FE}^{\mathsf{VK}}[\mathrm{F}]}},$$P$$)\leftarrow\mathcal{G}[\mathsf{eid}_{\mathsf{}}].\mathsf{install}$
$(\mathrm{run},\sigma_{\mathsf{\mathsf{DE}}},\mathsf{eid}_{\mathsf{\mathsf{DE}}},\mathsf{ct}_{\mathsf{key}},\mathsf{ct}_{\mathsf{msg}},$$k_{\mathrm{F}}$$,\mathsf{crs},\bot)\leftarrow\mathsf{input}$
assert
$(\sigma_{\mathsf{\mathrm{F}}},(\mathrm{init},\mathsf{mpk},\mathsf{idx}),(\mathsf{pk}_{\mathsf{FD}}))\in\mathcal{G}[\mathsf{eid}_{\mathsf{}}].\mathsf{resume}$
assert
$(\mathsf{idx},\mathsf{eid}_{\mathsf{}},\mathsf{prog}_{\mathsf{\mathsf{FE}^{\mathsf{VK}}[\mathrm{F}]}},\mathsf{pk}_{\mathsf{FD}})\in\mathcal{B}[\sigma_{\mathsf{\mathrm{F}}}]$
assert
$(\mathsf{idx},\mathsf{eid}_{\mathsf{\mathsf{DE}}},\mathsf{prog}_{\mathsf{\mathsf{DE}^{\mathsf{VK}}}},\mathsf{ct}_{\mathsf{key}},$$k_{\mathrm{F}}$$,\mathsf{crs}))\in\mathcal{B}[\sigma_{\mathsf{\mathsf{DE}}}]$
// If the ciphertext was not computed honestly and saved to $\mathcal{H}$ if
$\mathcal{H}[\mathsf{ct}_{\mathsf{msg}}]=\bot$ then
$(\mathsf{ct},\pi)\leftarrow\mathsf{ct}_{\mathsf{msg}}$
$((\mathrm{m},$$k$$),\mathrm{r})\leftarrow\mathsf{N}.\mathcal{E}(\tau,(\mathsf{mpk},\mathsf{ct}),\pi)$
send $(\textsc{encrypt},\allowbreak\mathrm{m},$$k$$)$ to
$\mathcal{S}_{DD\textrm{-}\mathrm{FESR}}\text{ on behalf of }P$ and receive
$\left(\textsc{encrypted},\mathsf{h}\right)$ if $\mathsf{h}\neq\bot$ then
$\mathcal{H}[\mathsf{ct}_{\mathsf{msg}}]\leftarrow\mathsf{h}$ else return
$\mathsf{h}\leftarrow\mathcal{H}[\mathsf{ct}_{\mathsf{msg}}]$ send
$(\textsc{decrypt},\allowbreak\mathrm{F},\mathsf{h})$ to
$\mathcal{S}_{DD\textrm{-}\mathrm{FESR}}\text{ on behalf of }P$ and receive
$\left(\textsc{decrypted},\mathrm{y}\right)$
$((\mathrm{computed},\mathrm{y}),\sigma_{\mathsf{}})\leftarrow
G_{\mathsf{att}}.\mathsf{resume}(\mathsf{eid}_{\mathsf{\mathrm{F}}},(\mathrm{run},\bot,\bot,\bot,\bot,$$\bot$$,\bot,\mathrm{y}))$
$\mathcal{G}[\mathsf{eid}_{\mathsf{}}].\mathsf{resume}\leftarrow\mathcal{G}[\mathsf{eid}_{\mathsf{}}].\mathsf{resume}\;\|\;(\sigma_{\mathsf{}},\mathsf{input},(\mathrm{computed},\mathrm{y})))$
$\mathcal{B}[\sigma_{\mathsf{}}]\leftarrow(\mathcal{G}[\mathsf{eid}_{\mathsf{}}].\mathsf{install}[0],\mathsf{eid}_{\mathsf{}},\mathcal{G}[\mathsf{eid}_{\mathsf{}}].\mathsf{install}[1],(\mathrm{computed},\mathrm{y}))$
forward $((\mathrm{computed},\mathrm{y}),\sigma_{\mathsf{}})$ to $P$