aainstitutetext: Perimeter Institute for Theoretical Physics,
Waterloo, Ontario N2L 2Y5, Canadabbinstitutetext: Department of Physics and
Astronomy, University of Waterloo,
Waterloo, Ontario, N2L 3G1, Canadaccinstitutetext: Institute for Quantum
Computing, University of Waterloo,
Waterloo, Ontario, N2L 3G1, Canada

# Quantum imprints of gravitational shockwaves

Finnian Gray a,b David Kubizňák a,b Taillte May a,b
Sydney Timmerman b,c Erickson Tjoa<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(May 20, 2021)

###### Abstract

Gravitational shockwaves are simple exact solutions of Einstein equations
representing the fields of ultrarelativistic sources and idealized
gravitational waves (shocks). Historically, much work has focused on
shockwaves in the context of possible black hole formation in high energy
particle collisions, yet they remain at the forefront of research even today.
Representing hard modes in the bulk, shocks give rise to the gravitational
memory effect at the classical level and implant supertranslation (BMS) hair
onto a classical spacetime at the quantum level. The aim of this paper is to
further our understanding of the ‘information content’ of such
supertranslations. Namely, we show that, contrary to the several claims in the
literature, a gravitational shockwave does leave a quantum imprint on the
vacuum state of a test quantum field and that this imprint is accessible to
local observers carrying Unruh–DeWitt (UDW) detectors in this spacetime.

##  1 Introduction

Quantum field theory (QFT) in curved spacetime is an infrared (IR) low-energy
effective field theory designed to probe certain features of the currently-
unknown quantum theory of gravity. It is constructed by lifting standard QFT
to curved spacetime, via changing the background metric, and is expected to be
valid in the regime where quantum fluctuations are small enough, so that the
semi-classical Einstein field equations sourced by quantum matter are valid.
Even in the test field approximation where there is no backreaction to the
geometry, QFT in curved spacetime has helped to uncover a number of physical
phenomena, such as Unruh and Hawking radiation, and provided firm ground for
black hole thermodynamics, while giving rise to one of the greatest puzzles in
theoretical physics—the black hole information paradox.

More recent developments have uncovered universal and surprising relationships
between the classical gravitational displacement memory effect
Zeldovich:1974gvh ; Favata2010memory ; Tolish2014memory ;
Christodoulou1991memory ; Winicour2014memory , leading soft graviton theorems
Weinberg1965graviton ; Saha2020softgraviton ; Laddha2020softgraviton , and
supertranslation asymptotic symmetries sachs1962gravitational ;
bondi1962gravitational ; Strominger2014scattering ; Ashtekar1981asymptotic ;
he2014newQEDsymm ; campiglia2015asymptotic ; Strominger2014yangmills ;
Compere2019towermemory ; zhang2017memory : the so-called IR triangle. One
implication of the IR triangle is that the vacuum state of a quantum field in
an asymptotically flat background is invariant under the infinite-dimensional
Bondi–Metzner–Sachs (BMS) group, which contains the Poincaré group as a
finite-dimensional subgroup. The BMS group was investigated in the 1960s in an
attempt to better understand gravitational radiation sachs1962gravitational ;
bondi1962gravitational , but it was not until recently that the connection to
QFT has become more apparent.

To demonstrate this connection one needs to have a physical mechanism for
implanting a supertranslation hair onto a classical spacetime. This can be
achieved with a gravitational shockwave Hawking:2016sgy , which is a localized
(distributional) gravitational wave that coincides with the background
(without the wave) everywhere except at the wavefront (shock) located at a
single value of the null coordinate. The shockwave is sourced by a singular
stress energy tensor along the null direction and can be described by a series
of exact solutions of the Einstein equations going back to the Aichelburg–Sexl
metric Aichelburg:1970dh , and, more generally, the Dray and ’t Hooft solution
Dray1985shockwave ; tHooft:1987vrq . Such metrics have played an important
role in studying gravitational scattering and the conditions for black hole
formation in high energy particle collisions tHooft:1987vrq ; deVega:1988wp ;
Amati:1992zb ; Kabat:1992tb ; Eardley:2002re , and as discussed in
Kolekar2017memoryaccel ; Shore2018memory ; Donnay2018memory , they give rise
to the classical gravitational memory effect. Moreover, shockwaves are at the
forefront of research even today, e.g. Liu:2021kay , and also have
implications for the black hole information paradox Strominger:2017aeh ;
Gaddam:2020mwe . In this context the following two natural questions arise:
are there any quantum imprints of gravitational shockwaves (or
supertranslation hairs) on test quantum scalar fields living on the background
geometry? And, if so, are they readily accessible by localized observers?

Similar questions have been investigated by a number of recent studies
Kolekar2018quantummemory ; Compere:2019rof ; Majhi2020shock ; Ferreira:2020whz
. Our work is mainly motivated by Compere:2019rof ; Majhi2020shock  where the
authors studied the impact of a ‘matter induced’ supertranslation on the
Bogoliubov coefficients between the two asymptotic states. Therein, the
Bogoliubov coefficients were shown to have physical manifestation only for
non-vacuum states of the field; in particular, it was shown that the vacuum
expectation values of the (global) number operators associated to both
Minkowski and Rindler observers are not altered by the shockwave. However,
since there is no localized number operator that agrees with the global number
operator for the vacuum state, the Bogoliubov coefficients are arguably not a
true local observable in this context, a result deeply connected to the
Reeh–Schlieder theorem, see Dowker:2011ga ; Kubicki:2016ngz ;
Colosi2008localization ; Licht1963localization ; Witten2018entanglement ;
Strohmaier2002curvedRS  for further discussions. Therefore, the results
obtained in Kolekar2018quantummemory ; Compere:2019rof ; Ferreira:2020whz  are
global results that no localized observers can easily access. The method
employed in Majhi2020shock  is more suitable for local calculations, since it
relies on a computation of the Wightman function of the quantum field on the
shockwave background, which is known to be accessible to local observers, e.g.
Jose2018measure . However, we shall show in this paper that the study in
Majhi2020shock  is incomplete, as the Wightman function was only calculated
for spacetime events located on the same side of the shockwave.

In summary, the studies Compere:2019rof ; Majhi2020shock ; Ferreira:2020whz
have reached the same conclusion: for asymptotic vacua both Unruh and Hawking
spectra are unchanged by the presence of gravitational shockwaves, and, while
gravitational shockwaves can leave imprints on test quantum fields, this will
only happen for non-vacuum states. It is the purpose of the present paper to
refute this conclusion.

Intimately tied to the notion of local and _measureable_ observables in QFT is
the Unruh–Dewitt (UDW) detector. The detector models the local measurement of
a QFT via its interaction with a two-level non-relativistic quantum system, a
qubit Unruh1979evaporation ; DeWitt1979 . This simple setup is known to
reproduce essential aspects of light-matter interactions pozas2016entanglement
; Lopp2021deloc . The UDW detector has also been shown to be sensitive to
global properties of the spacetime as well as local curvature effects,
including the passage of gravitational waves smith2016topology ;
Smith2020harvestingGW ; VerSteeg2009entangling ; Tjoa2020vaidya ;
henderson2018harvestingBH ; henderson2019AdSharvesting ; cong2019entanglement
; Cong2021inertialdrag . New effects such as the anti-Unruh and anti-Hawking
effects, only accessible to local and finite time interaction observers, have
also been observed within this framework Garay2016anti-unruh ; Brenna2016anti-
unruh ; Henderson2020anti-hawking ; Dappiaggi2021anti-hawking .

In this paper we show that, contrary to the conclusions of Compere:2019rof ;
Majhi2020shock ; Ferreira:2020whz , a gravitational shockwave _does_ leave a
quantum imprint on the vacuum state of a test quantum scalar field, and, this
imprint is accessible to local observers carrying UDW detectors in the
shockwave spacetime. Namely, we show that the Wightman function contains an
additional term that depends on the shockwave profile and that this term is
nonzero between two spacetime events that are located on different sides of
the shockwave null plane. Using the entanglement harvesting protocol from
relativistic quantum information pozas2015harvesting ; pozas2016entanglement ;
Valentini1991nonlocalcorr ; reznik2003entanglement , we show that this
additional term has operational consequences: two UDW detectors can harvest
more entanglement from the quantum vacuum even though locally each detector
sees a strictly Minkowski vacuum (the excitation probability is equal to the
one in Minkowski space). In fact, we demonstrate that even a single UDW
detector can see the shockwave as it passes by. This is in contrast to the
recent study Smith2020harvestingGW  where it was shown that a single UDW
detector cannot detect (linearized) gravitational waves. As we shall discuss,
this negative result can be traced to the fact that the quantum detection was
only investigated at the linear order in the wave amplitude, while the
measurable effect appears at the second order.

Our paper is organized as follows. In Sec. 2 we introduce the classical
geometry of shockwave spacetime. In Sec. 3 we review quantum field theory on
the shockwave background, and construct the corresponding Wightman function
(to be contrasted with partial results of Majhi2020shock ). Sec. 4 reviews the
Unruh–DeWitt detector formalism and the entanglement harvesting protocol. In
Sec. 5 we present our results regarding the single detector response to the
gravitational shockwave and the associated entanglement harvesting. Sec. 6 is
devoted to the final discussion and conclusions. The technical Appendix A
contains the derivation of the Wightman function, while Appendix B describes
geodesics in the shockwave spacetime. In this paper we adopt the mostly-plus
signature for the metric and use natural units $\hbar=c=1$, while keeping the
gravitational constant $G$ explicit.

##  2 Gravitational shockwaves

In what follows we shall consider the Dray and ’t Hooft Dray1985shockwave
generalization of the Aichelburg–Sexl Aichelburg:1970dh  shockwave spacetime
in $D$ spacetime dimensions. For a wave propagating in the $z$-direction, the
corresponding Brinkmann form (see e.g. zhang2017memory ) of the metric reads

$ds^{2}=-dudv+f(\vec{x})\delta(u-u_{0})du^{2}+\delta_{ij}dx^{i}dx^{j}\,.$
(2.1)

Here, we have employed ‘Minkowski’-like coordinates
$\mathsf{x}\equiv(t,z,\vec{x})$: $u=t-z$ and $v=t+z$ are the ‘standard’ null
coordinates. $\vec{x}$ or $x^{i}$, $i\in\\{2,\dots,D-2\\}$, denote the
transverse directions (coordinates on the wavefront). The wavefront is
localised at $u=u_{0}$; on either side of $u_{0}$ the spacetime is exactly
Minkowski. The stress-energy tensor has only one non-zero component:

$T_{uu}=\delta(u-u_{0})\rho(\vec{x})\,,$ (2.2)

and is zero everywhere but at $u=u_{0}$.

The Einstein field equations reduce to

$\Delta f(\vec{x})=-16\pi G\rho(\vec{x})\,,$ (2.3)

where $\Delta=\delta^{ij}\partial_{i}\partial_{j}$ is the flat Laplacian in
the transverse direction. The shockwave profile $f(\vec{x})$ completely
determines the nature of the wave. The only restrictions on $f(\vec{x})$ are
that it is ‘smooth enough’ and that suitable energy conditions are satisfied
(see e.g. Kontou:2020bta ; Curiel:2014zba ). For concreteness, we shall impose
the null energy condition which implies $\rho(\vec{x})\geq 0$, i.e., $\Delta
f(\vec{x})\leq 0$.

The metric (2.1) describes a large variety of physical spacetimes. In the
original work of Aichelburg–Sexl Aichelburg:1970dh  the metric was obtained by
‘boosting’ the Schwarzschild spacetime to the speed of light while keeping the
energy constant, thus obtaining a gravitational field of a massless point
particle characterized by the shockwave profile

$f(\vec{x})=-4GP\log\left(\frac{|{\vec{x}}|}{x_{0}}\right)\,,$ (2.4)

where $P>0$ corresponds to the energy of the point particle and $x_{0}$ is
some arbitrary reference scale.

Another interesting example of a shockwave profile (form factor) is given by

$f(\vec{x})=-\vec{x}\cdot A\cdot\vec{x}=-\sum_{i}a_{i}(x^{i})^{2}\,,$ (2.5)

where $A$ is a symmetric constant matrix, where, without loss of generality,
we can work in transverse coordinates corresponding to the eigenvectors with
eigenvalues $a_{i}$ of $A$. Physically this form factor corresponds to a
domain wall boosted to the speed of light Vilenkin:1984hy ; Lousto:1990wn ;
Barrabes:2002hn , while the energy density of the shockwave is111Before we
proceed further let us pause and make a few comments on dimensional analysis.
In our units $G=1/m_{pl}^{2}$, so that $[G]=-2$. Given its relation, (2.2), to
the stress-energy tensor the dimensions of $\rho$ satisfy $[\rho]+[\delta]=4$,
and since $[\delta]=1$, we have $[\rho]=3$ and $[f]=-1$. Thus (2.6) means that
the matrix $A$ has dimensions $[A]=1=\text{Energy}=1/\text{Length}$.

$\rho=\frac{\Tr(A)}{8\pi G}\,.$ (2.6)

Obviously, when $\Tr(A)=0$, the corresponding energy momentum tensor vanishes,
and we have a sourceless (arbitrarily polarized) gravitational wave.

Another perspective on shockwaves, due to Penrose, views them as a ‘scissor
and paste’ of two copies of Minkowski spacetime Penrose:1972xrn . This is
easiest to understand by introducing a new coordinates $\hat{v}$ related to
the old by a ‘planar supertranslation’ Compere:2019rof

$\hat{v}=v-\Theta(u-u_{0})f(\vec{x})\,,$ (2.7)

where $\Theta$ is the Heaviside step function. In these coordinates the metric
(2.1) takes the form

$\displaystyle ds^{2}$
$\displaystyle=-dud\hat{v}-\Theta(u-u_{0})\partial_{i}f(\vec{x})\,dudx^{i}+d\vec{x}^{2}$
$\displaystyle=\eta+\Theta(u-u_{0}){\cal L}_{\xi}\eta\,,$ (2.8)

where $\eta=-dud\hat{v}+d\vec{x}^{2}$ is the flat metric, and

$\xi=f(\vec{x})\partial_{\hat{v}}$ (2.9)

is the supertranslation vector field, which reduces to a time-translation
vector field when $f(\vec{x})$ is independent of $\vec{x}$. The form (2.8)
makes it clear that the Minkowski spacetimes are diffeomorphic since $\xi$
implements a diffeomorphism. The key difference between the shockwave
spacetime and more general supertranslations typically considered in the IR
triangle program (see e.g. strominger2018lectures ) is that shockwaves
correspond to ‘hard’ (finite energy, or equivalently, matter-induced)
processes in the bulk, rather than the ‘soft’ (zero energy) supertranslations
Hawking:2016sgy ; Donnay2018memory ; Compere:2019rof .

For completeness, let us mention that, in $D=4$, the supertranslation
manifests itself as a change in the gravitational radiation data
strominger2018lectures . This supertranslation provides an infinitely
degenerate label for the Minkowski space—two Minkowski spacetimes labeled by
different supertranslations are physically inequivalent from the perspective
of gravitational scattering problem. In our case, two vacuum states defined on
the ‘in’ and ‘out’ Minkowski regions can be unitarily equivalent in the
Poincaré sense but nonetheless physically distinct in the BMS sense (see e.g.
campiglia2015asymptotic ; he2014newQEDsymm ; Strominger2014yangmills ).

We close this section by briefly commenting on the geodesics in this geometry.
It is well-known that geodesics in shockwave geometry are highly non-trivial
due to the distributional nature of the $\delta$-localized shockwave. Much
work has been done to address this from both physical and mathematical
perspectives for arbitrary shockwave profiles, see e.g. ferrari1988beam ;
balasin1997geodesics ; kunzinger1999rigorous . As we are interested in
localized observers who carry particle detectors (to be discussed in Section
4), we will need to calculate the timelike geodesics on this background. A key
feature of such geodesics is the distributional nature of the trajectories
that crucially depend on the impact parameter. As shown in Appendix B, for the
shockwave profile we consider in the subsequent sections, there is a natural
choice of impact parameter that ‘eliminates’ the need to deal with these
complications.

##  3 Quantum field theory in shockwave geometry

Having described the classical aspects of shockwave geometry we now turn to
review the quantization of a massless scalar field in this spacetime. While
quantum fields in gravitational wave spacetimes have been studied since the
70’s, e.g. Gibbons:1975jb ; Garriga:1990dp , QFT in the shockwave geometry was
first studied by Klimčík Klimcik1988shock  and more recently in
Compere:2019rof ; Majhi2020shock . In this section, we reproduce the essential
features for shockwaves and explicitly calculate the two-point (Wightman)
function for an arbitrary profile $f(\vec{x})$.

###  3.1 Klein–Gordon equation and its solutions

Following Klimcik1988shock , we consider the massless Klein–Gordon equation

$\Box\Phi=0\,,$ (3.1)

find its mode functions, and canonically quantize them in null coordinates.
Given that $\partial_{v}$ is a Killing vector of (2.1) we can make the
following separation ansatz for plane wave modes in the $\partial_{v}$
direction: $\phi_{k_{-}}=e^{-ik_{-}v}\psi(u,\vec{x})$, labeling the momentum
with the wavevector $k_{\mu}=(k_{t},k_{z},\vec{k})$ and defining
$k_{\pm}=\frac{1}{2}(k_{t}\pm k_{z})$. This reduces the wave equation (3.1) to
the following Schrödinger-like equation:

$i\partial_{u}\psi=-\left(\frac{\Delta}{4k_{-}}+f(\vec{x})k_{-}\delta(u-u_{0})\right)\psi\,.$
(3.2)

Clearly, on either side of the shockwave (in the ‘in’ ($u<u_{0}$) and ‘out’
($u>u_{0}$) regions) the solutions are simple plane waves that provide a
complete basis for quantization. Thus, we have two mode expansions at our
disposal $\phi^{\mathrm{in}}_{k_{-},\vec{k}}$ and
$\phi^{\mathrm{out}}_{k_{-},\vec{k}}$, for which we require that the mode
functions reduce to plane waves in the ‘in’ or ‘out’ regions, respectively.

Let us first consider the ‘in’ modes. Before the shockwave we have the
standard plane wave solution

$\phi^{\mathrm{in}}_{k_{-},\vec{k}}\Big{|}_{u<u_{0}}=e^{-ik_{-}v}\psi_{<}(u,\vec{x})\,,\quad\psi_{<}(u,\vec{x})=N_{k_{-}}e^{-i(k_{+}(u-u_{0})-\vec{k}\cdot\vec{x}))}\,,$
(3.3)

where $N_{k_{-}}=([2\pi]^{D-1}2k_{-})^{-1/2}$ and
$k_{+}=\vec{k}^{2}/(4k_{-})$. This remains a solution until $u=u_{0}^{-}$. The
only effect of the shockwave on the mode functions is to introduce a junction
condition at $u=u_{0}$ (much like the ‘scissor and paste’ idea on the
spacetime itself). Right after the shockwave, one finds Klimcik1988shock

$\phi^{\mathrm{in}}_{k_{-},\vec{k}}\,\Big{|}_{u=u_{0}^{+}}=N_{k_{-}}e^{-ik_{-}v}e^{i(\vec{k}\cdot\vec{x}+k_{-}f(\vec{x}))}\,,$
(3.4)

which can be derived by regularizing the Dirac $\delta$-function. In other
words, the shockwave has just ‘supertranslated’ the solution, changing
$v\rightarrow v-f(\vec{x})$. Eq. (3.4) provides an initial condition for the
time evolution via the Schrödinger-like equation (3.2), which for
$u>u_{0}^{+}$ yields (see Klimcik1988shock  for details)

$\psi_{>}(u,\vec{x})=N_{k_{-}}e^{-ik_{-}v}e^{i\vec{k}\cdot\vec{x}}\int\frac{d\vec{k^{\prime}}d\vec{x^{\prime}}}{(2\pi)^{D-2}}\,e^{i(\vec{k}-\vec{k^{\prime}})(\vec{x^{\prime}}-\vec{x})}e^{\frac{-i\vec{k^{\prime}}^{2}}{4k_{-}}(u-u_{0})+ik_{-}f(\vec{x^{\prime}})}\,.$
(3.5)

Putting this together, we can write the full ‘in’ mode as222In these
expressions we are implicitly ignoring the zero measure set of modes localized
to travel along the shockwave $u=u_{0}$.

$\displaystyle\phi^{\mathrm{in}}_{k_{-},\vec{k}}=N_{k_{-}}e^{-ik_{-}v}e^{i\vec{k}\cdot\vec{x}}\int\frac{d\vec{x^{\prime}}d\vec{k^{\prime}}}{(2\pi)^{D-2}}e^{i(\vec{k}-\vec{k^{\prime}})(\vec{x^{\prime}}-\vec{x})-\frac{i\vec{k^{\prime}}^{2}}{4k_{-}}(u-u_{0})+ik_{-}\Theta(u-u_{0})f(x^{\prime})}\,.$
(3.6)

In order to obtain the ‘out’ modes we use the fact the Schrödinger equation
(3.2) is symmetric under Compere:2019rof

$u\rightarrow-u,\qquad v\rightarrow-v,\qquad k_{-}\rightarrow-k_{-},\qquad
k_{+}\rightarrow-k_{+},\qquad u_{0}\rightarrow-u_{0}.$ (3.7)

Applying this yields for the ‘out’ mode

$\displaystyle\phi^{\mathrm{out}}_{k_{-},\vec{k}}=N_{k_{-}}e^{-ik_{-}v}e^{i\vec{k}\cdot\vec{x}}\int\frac{d\vec{x^{\prime}}d\vec{k^{\prime}}}{(2\pi)^{D-2}}e^{i(\vec{k}-\vec{k^{\prime}})(\vec{x^{\prime}}-\vec{x})-\frac{i\vec{k^{\prime}}^{2}}{4k_{-}}(u-u_{0})-ik_{-}\Theta(u_{0}-u)f(x^{\prime})}\,.$
(3.8)

###  3.2 Mode decomposition

It can be shown Compere:2019rof  that the two sets of mode functions (3.6) and
(3.8) are orthonormal with respect to the standard Klein–Gordon inner product
evaluated on a $t=const.$ Cauchy hypersurface:

$\displaystyle\left(\phi_{1},\phi_{2}\right)$
$\displaystyle=i\int_{\Sigma}d\Sigma^{\mu}\left(\phi_{1}^{*}\partial_{\mu}\phi_{2}-\phi_{2}\partial_{\mu}\phi_{1}^{*}\right)$
$\displaystyle=i\int_{\Sigma}dzd{\vec{x}}\Bigl{(}\phi_{1}^{*}(\partial_{u}+[1+2\delta(u-u_{0})f(\vec{x})]\partial_{v})\phi_{2}-(1\leftrightarrow
2)\Bigr{)}\,.$ (3.9)

That is, for $A\in\\{\mathrm{in},\mathrm{out}\\}$ we have333Note that we could
normalize with respect to $k_{t}=\omega$ by redefining $N_{k_{-}}\mapsto
N_{k_{t}}=\sqrt{(k_{-}/k_{t})}N_{k_{-}}$.

$\displaystyle\left(\phi^{A*}_{k_{-},\vec{k}},\,\phi^{A}_{l_{-},\vec{l}}\right)$
$\displaystyle=0\,,\hskip
14.22636pt\left(\phi^{A}_{k_{-},\vec{k}},\,\phi^{A}_{l_{-},\vec{l}}\right)=\left(\phi^{A*}_{k_{-},\vec{k}},\,\phi^{A*}_{l_{-},\vec{l}}\right)=\delta(k_{-}-l_{-})\delta(\vec{k}-\vec{l})\,.$
(3.10)

Thus we have two equivalent quantizations based of the global decomposition
using ‘in’ and ‘out’ modes

$\displaystyle\Phi=\int
dk_{-}d\vec{k}\,\left(a_{k_{-},\vec{k}}^{\mathrm{in}}\,\phi_{k_{-},\vec{k}}^{\mathrm{in}}+a_{k_{-},\vec{k}}^{\mathrm{in}\dagger}\,\phi_{k_{-},\vec{k}}^{\mathrm{in}*}\right)=\int
dk_{-}d\vec{k}\left(a_{k_{-},\vec{k}}^{\mathrm{out}}\,\phi_{k_{-},\vec{k}}^{\mathrm{out}}+a_{k_{-},\vec{k}}^{\mathrm{out}\dagger}\,\phi_{k_{-},\vec{k}}^{\mathrm{out}*}\right)\,,$
(3.11)

where the annihilation operators define the ‘in’ and ‘out’ vacua,
$\ket{0_{\mathrm{in}}}$ and $\ket{0_{\mathrm{out}}}$, respectively:

$\displaystyle a_{k_{-},\vec{k}}^{\mathrm{in}}\ket{0_{\mathrm{in}}}\equiv
0\,,\qquad a_{k_{-},\vec{k}}^{\mathrm{out}}\ket{0_{\mathrm{out}}}\equiv 0\,.$
(3.12)

These operators of course obey the standard commutation relations

$\left[a_{k_{-},\vec{k}}^{A},a_{l_{-},\vec{l}}^{A\,\dagger}\right]=\delta(k_{-}-l_{-})\delta(\vec{k}-\vec{l})\,$
(3.13)

with all others vanishing.

###  3.3 Bogoliubov coefficients

Now we want to find a relation between $\phi^{\mathrm{in}}_{k_{-},\vec{k}}$
and $\phi^{\mathrm{out}}_{k_{-},\vec{k}}$. In fact, since the positive and
negative frequencies are not mixed, c.f. (3.6) and (3.8), we have

$\phi^{\mathrm{in}}_{k_{-},\vec{k}}=\int
dl_{-}d\vec{l}\;\alpha_{k_{-},l_{-}}(\vec{k}-\vec{l})\;\phi^{\mathrm{out}}_{l_{-},\vec{l}}\;\,.$
(3.14)

Equivalently this implies that the creation/annihilation operators in the two
vacua are related by

$a^{\mathrm{out}}_{k_{-},\vec{k}}=\int
dl_{-}\,d\vec{l}\,\,\alpha_{k_{-},l_{-}}(\vec{k}-\vec{l})\,a^{\mathrm{in}}_{l_{-},\vec{l}}\;\,.$
(3.15)

Thus, we only have one set of Bogoliubov coefficients
$\alpha_{k_{-},l_{-}}(\vec{k}-\vec{l})$ which means there is no particle
production induced by the shockwave in the sense that,

$\bra{0_{\mathrm{in}}}n^{\mathrm{out}}_{k_{-},\vec{k}}\ket{0_{\mathrm{in}}}=\bra{0_{\mathrm{in}}}a^{\mathrm{out}\dagger}_{k_{-},\vec{k}}\,a^{\mathrm{out}}_{k_{-},\vec{k}}\ket{0_{\mathrm{in}}}=0\,.$
(3.16)

This calculation means that the two ‘Minkowski’ vacua defined on the ‘in’ and
‘out’ regions are unitarily equivalent.

In Compere:2019rof  these statements were generalized to consider the
shockwave in Rindler coordinates. Therein, it was found that the shockwave
only affected the phase of the Bogoliubov coefficients between the Rindler
vacuum and the stationary vaccuum as defined here. Hence the Unruh radiation
spectrum is left invariant under these shockwaves—only excited states directly
see the effect of the profile. Similar results were also obtained for Hawking
radiation, see Ferreira:2020whz .

To find the Bogoliubov coefficients explicitly, we note that, since
$\partial_{v}$ is a Killing vector, there can be no change of momentum in the
$v$ direction. That is

$\alpha_{k_{-},l_{-}}(\vec{k}-\vec{l})=\delta(k_{-}-l_{-})\,\widetilde{\alpha}_{k_{-}}(\vec{k}-\vec{l})\,,$
(3.17)

where $\widetilde{\alpha}$ is known as the mixing factor. Comparing the ‘in’
and ‘out’ modes yields

$\displaystyle\widetilde{\alpha}_{k_{-}}(\vec{k}-\vec{l})\equiv\int\frac{d\vec{x^{\prime}}}{(2\pi)^{D-2}}e^{i(\vec{k}-\vec{l})\vec{x}^{\prime}+ik_{-}[\Theta(u-u_{0})-\Theta(u_{0}-u)]f(\vec{x}^{\prime})}\,.$
(3.18)

While these statements certainly hold with regards to the creation and
annihilation operators and the number operator
$N^{A}_{k}=a^{A\,\dagger}_{k_{-},\vec{k}}a^{A}_{k_{-},\vec{k}}$, it is not
clear what this means for the local observers. To answer this we will
calculate the Wightman function, which plays a central role in the
Unruh–DeWitt paradigm, as outlined in Sec. 4.

###  3.4 Wightman function

The Wightman function was calculated in Majhi2020shock  on either side of the
shockwave, and was found to reduce to the standard Minkowski one. However, in
Majhi2020shock , the most interesting case—with the events on opposite sides
of the shockwave—was omitted. In this section we will calculate the Wightman
function in the full spacetime. Using the mode expansion (3.11) and assuming
that we are in the ‘in’ vacuum state, we have

$\displaystyle
W(\mathsf{x},\mathsf{y})=\bra{0_{\mathrm{in}}}\Phi(\mathsf{x})\Phi(\mathsf{y})\ket{0_{\mathrm{in}}}=\int
dk_{-}d\vec{k}\,\phi^{\mathrm{in}}_{k_{-},\vec{k}}(\mathsf{x})\,\phi^{\mathrm{in}\,*}_{k_{-},\vec{k}}(\mathsf{y})\,.$
(3.19)

Denoting $\mathsf{x}=(u,v,\vec{x})$ and $\mathsf{y}=(U,V,\vec{X})$ and
plugging in the expressions for the mode functions (3.6), we obtain

$\displaystyle W(\mathsf{x},\mathsf{y})$
$\displaystyle=\int\frac{dk_{-}\,d\vec{k}}{2(2\pi)^{D-1}k_{-}}e^{-ik_{-}(\Delta
v-i\epsilon)}\int\frac{d\vec{x}^{\prime}d\vec{X}^{\prime}}{(2\pi)^{2D-4}}e^{i\vec{k}\cdot(\vec{x}^{\prime}-\vec{X}^{\prime})+ik_{-}\left[\Theta_{u}f(\vec{x}^{\prime})-\Theta_{U}f(\vec{X}^{\prime})\right]}$
$\displaystyle\hskip 14.22636pt\times\frac{(4\pi
k_{-})^{D-2}}{[(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)]^{\frac{D-2}{2}}}\exp\left[\frac{ik_{-}(\vec{x}-\vec{x}^{\prime})^{2}}{u-u_{0}-i\epsilon}-\frac{ik_{-}(\vec{X}-\vec{X}^{\prime})^{2}}{U-u_{0}+i\epsilon}\right]\,,$
(3.20)

where we have used the shorthands $\Delta v\equiv v-V$ and
$\Theta_{u}\equiv\Theta(u-u_{0})$, and have performed one Gaussian integral
along $\vec{k}^{\prime}$ for $\phi^{\mathrm{in}}_{k_{-},\vec{k}}$. We have
also introduced an $i\epsilon$ to avoid a branch cut. Relegating the rest of
the calculation to Appendix A, the end result is

$\displaystyle W(\mathsf{x},\mathsf{y})$
$\displaystyle=\frac{\Gamma(D-2)(i)^{2-D}}{2(2\pi)^{D-1}}\frac{(4\pi)^{D-2}}{[(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)]^{\frac{D-2}{2}}}$
$\displaystyle\times\int\frac{d\vec{x}^{\prime}}{(2\pi)^{D-2}}\left(\left[\Delta
v-i\epsilon-f(\vec{x}^{\prime})\Delta\Theta_{u}\right]-\frac{(\vec{x}-\vec{x}^{\prime})^{2}}{u-u_{0}-i\epsilon}+\frac{(\vec{X}-\vec{x}^{\prime})^{2}}{U-u_{0}+i\epsilon}\right)^{2-D}\,,$
(3.21)

where we have defined $\Delta\Theta_{u}:=\Theta_{u}-\Theta_{U}$. In Appendix A
we also show that this reduces to the standard flat space form if either
$f(\vec{x})=0$ or $\Delta\Theta_{u}=0$. In the former there is no shockwave at
all, while the latter means that the events $u$ and $U$ are localized at the
same side of the shockwave.

This is as far we can go without picking a particular shockwave profile. For
simplicity we shall focus on the planar form in (2.5). In this case the
integrals can be performed analytically (see Appendix A) and we obtain

$\displaystyle W(\mathsf{x},\mathsf{y})$
$\displaystyle=\frac{(-i)^{D-2}\Gamma(D/2-1)}{4\pi^{D/2}}\prod_{i=1}^{D-2}\left(1+a_{i}\Delta\Theta_{u}\frac{(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)}{\Delta
u-i\epsilon}\right)^{-1/2}$ $\displaystyle\times\left((\Delta
v-i\epsilon)(\Delta
u-i\epsilon)-\Delta\vec{x}\,^{2}+\sum_{i=1}^{D-2}\frac{a_{i}\Delta\Theta_{u}\left([U-u_{0}+i\epsilon]x^{i}-[u-u_{0}-i\epsilon]X^{i}\right)^{2}}{\Delta
u-i\epsilon+a_{i}\Delta\Theta_{u}(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)}\right)^{(2-D)/2}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!.$
(3.22)

This expression should be contrasted to Majhi2020shock , where the Wightman
function was not evaluated across the shockwave and thus all the components
that depend on $\Delta\Theta_{u}$ drop out. Consequently, the main claim of
Majhi2020shock  does not directly corroborate the result in Compere:2019rof
which properly accounts for observer trajectories that cross the shock. One of
the reasons why the vacuum Wightman function encodes information about the
shock and the number operator calculations do not, is that the latter neglect
the spatial vacuum entanglement between the two regions of the spacetime
separated by the shock, which is an intrinsic property of the vacuum for any
relativistic QFT summers1985bell ; summers1987bell .

As we shall see, the Wightman function (3.22) will allow us to detect the
passing shockwave with a single UDW detector, and show that two localized
quantum systems (UDW detectors) interacting with shockwave vacuum can extract
more entanglement than from a global flat space, even though each detector is
completely localized in the respective Minkowski region.

##  4 Unruh–DeWitt detectors and entanglement harvesting

In this section we will briefly outline the basic framework of the
Unruh–DeWitt (UDW) particle detector and the entanglement harvesting protocol
from relativistic quantum information, as well as discuss the ‘experimental
setup’ for the quantum detection of the gravitational shockwave by a single
UDW detector and by entanglement harvesting.

###  4.1 Single UDW detector

A UDW detector is a pointlike two-level quantum system (a qubit) interacting
locally with an underlying quantum field along its trajectory in spacetime
Unruh1979evaporation ; DeWitt1979 . It provides a simplified model of light-
matter interaction where the usual atomic dipole coupled to an electromagnetic
field is replaced with a monopole interaction with a scalar field. This
approximation is good as long as there is no exchange of angular momentum
involved pozas2016entanglement . The pointlike model has the advantage of
being simple and covariant—the excitation probabilities are invariant under
arbitrary diffeomorphisms Tales2020GRQO ; Bruno2020time-ordering .

A single UDW detector couples locally with a scalar field $\Phi$ via the
interaction Hamiltonian

$\displaystyle H_{I}(\tau)$
$\displaystyle=\lambda\chi(\tau)\mu(\tau)\otimes\Phi(\mathsf{x}[\tau])\,,$
(4.1)

where $\lambda$ is the coupling constant, $\tau$ is the proper time of the
detector, $\mathsf{x}[\tau]$ is its spacetime trajectory, and $\chi(\tau)$ is
the switching function governing the duration of interaction. The operator
$\mu$ is the monopole moment of the detector given by

$\displaystyle\mu(\tau)=\ket{e}\\!\bra{g}e^{i\Omega\tau}+\ket{g}\\!\bra{e}e^{-i\Omega\tau}\,,$
(4.2)

where $\ket{g}$ and $\ket{e}$ are the ground and excited states of the
detector, separated by an energy gap $\Omega$. We will assume that the
detector switching function is a Gaussian function

$\displaystyle\chi(\tau)$
$\displaystyle=\exp\left[{-\frac{(\tau-{\tau_{0})^{2}}}{T^{2}}}\right]\,,$
(4.3)

where the Gaussian width $T$ controls the duration of interaction and
$\tau_{0}$ is the time when the interaction is strongest.444For our purposes
we can think of the detector as being effectively turned on within the time
interval $[-bT+\tau_{0},bT+\tau_{0}]$, where $b>0$ is sufficiently large. In
fact, already for $b=2$, one recovers $99.5\%$ of the Gaussian area. The
compact switching can thus be well-approximated by truncating the exponential
tails at $\pm bT+\tau_{0}$ for some finite $b\geq 2$.

We can perform a measurement with the UDW detector by preparing it in an
initial state, time-evolving it together with the field using the interaction
Hamiltonian, and computing the reduced density matrix. We begin by preparing
the detector-field system in the product state

$\displaystyle\rho_{0}$
$\displaystyle=\ket{g}\\!\bra{g}\otimes\ket{0}\\!\bra{0}\,,$ (4.4)

where $\ket{0}$ is the vacuum state of the field in the asymptotic past.555 In
our setup, the initial vacuum state $\ket{0}$ is unitarily equivalent to the
vacuum state $\ket{0^{\prime}}$ in the asymptotic future with respect to the
Poincaré subgroup, as the Bogoliubov coefficients of both vacua have vanishing
‘beta coefficients’, see previous section. However, since the gravitational
shockwave implants supertranslation ‘hair’, we can think of this as the soft
charges providing ‘soft’ quantum numbers $\alpha$ (which is uncountably
degenerate), and thence the two ‘Minkowski’ vacua (the in/out shockwave
geometry vacua) are not unitarily equivalent with respect to the full BMS
group: schematically, we have
$\ket{0^{\prime}}\coloneqq\ket{0,\alpha^{\prime}}\neq\ket{0,\alpha}\equiv\ket{0}$.

We time-evolve the initial state $\rho_{0}$ perturbatively by performing the
Dyson series expansion of the time-evolution operator to second order in
$\lambda$,

$\displaystyle U$ $\displaystyle=\mathds{1}+U^{(1)}+U^{(2)}+O(\lambda^{3})\,,$
(4.5a) $\displaystyle U^{(1)}$
$\displaystyle=-i\int_{-\infty}^{\infty}d\tau\,H_{I}(\tau)\,,$ (4.5b)
$\displaystyle U^{(2)}$
$\displaystyle=-\int_{-\infty}^{\infty}dt\int_{-\infty}^{\infty}dt^{\prime}\,\Theta(\tau-\tau^{\prime})H_{I}(\tau)H_{I}(\tau^{\prime})\,.$
(4.5c)

Tracing out the field degrees of freedom gives the reduced density matrix of
the qubit

$\displaystyle\rho_{d}=\Tr_{\phi}\left(U\rho_{0}U^{\dagger}\right)=\rho_{0}+\rho^{(1)}+\rho^{(2)}+O(\lambda^{3})\,,$
(4.6)

where $\rho^{(k)}$ is the correction of order $\lambda^{k}$:

$\displaystyle\rho^{(1)}$
$\displaystyle=\Tr_{\phi}\left(\rho_{0}U^{(1)\dagger}\right)+\Tr_{\phi}\left(U^{(1)}\rho_{0}\right)\,,$
(4.7a) $\displaystyle\rho^{(2)}$
$\displaystyle=\Tr_{\phi}\Bigl{(}\rho_{0}U^{(2)\dagger}\Bigr{)}+\Tr_{\phi}\left(U^{(2)}\rho_{0}\right)+\Tr_{\phi}\left(U^{(1)}\rho_{0}U^{(1)\dagger}\right)\,.$
(4.7b)

Since the field is initially in the vacuum state, $\rho^{(1)}=0$ due to the
vanishing of vacuum one-point functions (in fact $\rho^{(2k+1)}=0$ for non-
negative integer $k$). The leading order contribution is therefore of order
$\lambda^{2}$, which in the ordered basis $\\{\ket{g},\ket{e}\\}$ reads

$\rho_{d}=\begin{pmatrix}1-P&0\\\ 0&P\end{pmatrix}+O(\lambda^{4})\,,$ (4.8)

where

$P=\lambda^{2}\int
d\tau\,d\tau^{\prime}\,\chi(\tau)\chi(\tau^{\prime})e^{-i\Omega(\tau-\tau^{\prime})}W(\mathsf{x}[\tau],\mathsf{x}[\tau^{\prime}])\,,$
(4.9)

and $W(\mathsf{x}[\tau],\mathsf{x}[\tau^{\prime}])$ is the pullback of the
Wightman function along the detector’s trajectory. Here, $P$ can be understood
as the detector transition probability from the ground state to the excited
state.

###  4.2 Entanglement harvesting protocol

Implementing the entanglement havesting protocol requires generalizing to two
pointlike UDW detectors, denoted $A$ and $B$, which have proper times
$\tau_{j}$ and spacetime trajectories $\mathsf{x}_{j}[\tau_{j}],\,j=A,B$. In
general, the detectors may have different coupling constants $\lambda_{j}$,
switching functions $\chi_{j}(\tau_{j})$, and energy gaps $\Omega_{j}$.
However, in what follows, we consider a simplified scenario with identical
detectors such that $\lambda_{j}=\lambda$ and $\Omega_{j}=\Omega$. As in the
single detector case, we will also assume that the switching functions are
Gaussian functions peaked about $\tau_{j,0}$ with Gaussian width $T$.

Since the spacetime has a global coordinate time $t$, it is convenient to use
this to define the full interaction Hamiltonian of the detector-field system:

$\displaystyle H^{t}_{I}(t)$
$\displaystyle=H^{t}_{I,A}(t)\otimes\mathds{1}_{B}+\mathds{1}_{A}\otimes
H^{t}_{I,B}(t)$
$\displaystyle=\frac{d\tau_{A}(t)}{dt}H^{\tau_{A}}_{I,A}(\tau_{A}(t))\otimes\mathds{1}_{B}+\mathds{1}_{A}\otimes\frac{d\tau_{B}(t)}{dt}H^{\tau_{B}}_{I,B}(\tau_{B}(t))\,,$
(4.10)

where $H^{t}_{I}$ generates time translation with respect to $t$ (similarly
for $H^{\tau_{j}}_{I}$). The second equality follows from time-
reparametrization invariance of the Hamiltonian Tales2020GRQO  which allows us
to evolve the system with respect to this one time $t$.

The calculation involved in the entanglement harvesting protocol proceeds now
similarly to the previous section. First, we prepare the joint detector-field
system in the initially-uncorrelated state

$\displaystyle\rho_{0}$
$\displaystyle=\ket{g}_{A}\\!\bra{g}_{A}\otimes\ket{g}_{B}\\!\bra{g}_{B}\otimes\ket{0}\\!\bra{0}\,,$
(4.11)

where we take $\ket{0}$ to be the ‘in’ vacuum state. Evolving the state
perturbatively using the Dyson series (4.5) and tracing out the field degrees
of freedom gives the reduced density matrix of the bipartite qubits
$\rho_{AB}$. As before, the leading order contribution is of order
$\lambda^{2}$, which in the ordered basis
$\\{\ket{g}_{A}\ket{g}_{B},\ket{e}_{A}\ket{g}_{B},\ket{g}_{A}\ket{e}_{B},\ket{e}_{A}\ket{e}_{B}\\}$
reads

$\displaystyle\rho_{AB}=\begin{pmatrix}1-P_{A}-P_{B}&0&0&\mathcal{M}^{*}\\\
0&P_{B}&C^{*}&0\\\ 0&C&P_{A}&0\\\
\mathcal{M}&0&0&0\end{pmatrix}+O(\lambda^{4})\,,$ (4.12)

where

$\displaystyle C$ $\displaystyle=\lambda^{2}\int
d\tau_{A}\,d\tau_{B}\,\chi_{A}(\tau_{A})\chi_{B}(\tau_{B})e^{-i\Omega(\tau_{A}-\tau_{B})}W(\mathsf{x}_{A}[\tau_{A}],\mathsf{x}_{B}[\tau_{B}])\,,$
(4.13) $\displaystyle\mathcal{M}$ $\displaystyle=-\lambda^{2}\int
d\tau_{A}\,d\tau_{B}\,\chi_{A}(\tau_{A})\chi_{B}(\tau_{B})e^{i\Omega(\tau_{A}+\tau_{B})}\Theta[t(\tau_{A})-t(\tau_{B})]W(\mathsf{x}_{A}[\tau_{A}],\mathsf{x}_{B}[\tau_{B}])$
$\displaystyle\phantom{=}-\lambda^{2}\int
d\tau_{A}\,d\tau_{B}\,\chi_{B}(\tau_{B})\chi_{A}(\tau_{A})e^{i\Omega(\tau_{A}+\tau_{B})}\Theta[t(\tau_{B})-t(\tau_{A})]W(\mathsf{x}_{B}[\tau_{B}],\mathsf{x}_{A}[\tau_{A}])\,,$
(4.14)

and $P_{A}$ and $P_{B}$ are the transition probabilities of each detector
(4.9).

Entanglement can be harvested because $\rho_{AB}$ will be an entangled state
for suitable range of detector parameters. This can be verified using
computable measures of entanglement such as the negativity or concurrence
Wotters1998entanglementmeasure ; Horodecki996separable ; Vidal2002negativity .
For the purpose of this paper, we concentrate on the concurrence which (for
the joint density matrix of the detectors (4.12)) is given by
smith2016topology

$\mathcal{C}[\rho_{AB}]=2\max\\{0,|\mathcal{M}|-\sqrt{P_{A}P_{B}}\\}+O(\lambda^{4})\,.$
(4.15)

In this form, we see that entanglement between the two detectors admits a
simple interpretation, namely it is a competition between non-local quantum
correlations coming from $\mathcal{M}$ and the noise terms $\sqrt{P_{A}P_{B}}$
coming from each detector’s excitations as they interact locally with the
field.

Figure 1: Single detector and entanglement harvesting protocol. Left: Alice
and her detector cross the shockwave whilst interacting with the field for
time ${\cal T}_{A}$. (For example, ${\cal T}_{A}$ could correspond to the
interval $[-4T+\tau_{0},4T+\tau_{0}]$ around the peak of the Gaussian
switching (4.3).) Right. Alice and Bob are located on either side of the
shockwave and never cross the shockwave while interacting with the field for a
duration ${\cal T}_{A/B}$. This interaction leads to entanglement between
Alice and Bob.

###  4.3 Detectors’ worldlines

Here we are going to analyze the detector’s response and entanglement
harvesting in the shockwave spacetime, specializing to the physical case
$D=4$. For this we need to specify the trajectories of the detectors in this
spacetime and calculate the corresponding pullbacks of the Wightman functions.
However, there is a complication in that the presence of the shock alters the
geodesics—in general both spacelike and timelike geodesics will suffer from
discontinuities. Since our calculation involves comparing entanglement
harvesting in the shockwave geometry with that in Minkowski space for the
special case when the detectors are static at constant $x^{j}$, we need to
find comparable trajectories.

As shown in the Appendix B, it turns out that in the pointlike regime and for
the special case of the planar shockwave with the ‘Gaussian’ form factor
(2.5), there is a special choice of the impact parameter $\vec{b}=0$ (in the
given coordinates) so that the geodesics on the corresponding codimension-2
$(t,z)$-plane are identical to those in Minkowski space. This yields the two
natural scenarios for our calculations displayed in Fig. 1. Namely, to discuss
the response of a single detector to the shockwave, we consider a static
detector, placed at

$\mathsf{x}(\tau)=(t(\tau),z(\tau),\vec{x}(\tau))=(\tau,0,\vec{0})\,,$ (4.16)

which remains geodesic even when hit by the shockwave, as displayed on the
left in Fig. 1. For the entanglement harvesting, we shall consider two
(geodesic) static detectors666In our setup, we say that we have arranged two
detectors so that neither of them crosses the shock if the Gaussian tails of
the switching functions crossing the shock are negligible. In practice, we
truncate the exponential tails of the switching and declare that the detectors
do not touch the shock if the probabilities are equal to that in Minkowski
space up to the fifth significant figure.,

$\displaystyle\mathsf{x}_{A}(\tau_{A})$
$\displaystyle=(t(\tau_{A}),z(\tau_{A}),\vec{x}(\tau_{A}))=(\tau_{A},z_{A},\vec{0})\,,$
(4.17) $\displaystyle\mathsf{x}_{B}(\tau_{B})$
$\displaystyle=(t(\tau_{A}),z(\tau_{A}),\vec{x}(\tau_{B}))=(\tau_{B},z_{B},\vec{0})\,,$
(4.18)

as displayed on the right in Fig. 1. The important feature of this choice is
that the proper spatial distance between these detectors, $L=|z_{B}-z_{A}|$,
remains the same as in the Minkowski space, since the constant-$t$ slices
restricted to $\vec{x}=0$ are not affected by the shock. It is worth
mentioning that in our setup the two detectors are placed in the longitudinal
direction rather than the transverse direction, as was done for the linearized
gravity case in Smith2020harvestingGW .

##  5 Results

###  5.1 Single detector’s response to a shockwave

Before we turn to entanglement harvesting we will discuss the response of a
single detector to the shockwave. Previous studies Compere:2019rof ;
Majhi2020shock  have suggested that an observer who probes a scalar field in
the ground state will only detect the Minkowski results. Moreover, in
Smith2020harvestingGW  it was shown that a single stationary UDW detector will
not detect a linearized plane gravitational wave. These results seem intuitive
on the grounds that a single test particle (classical observer) cannot reveal
the presence/absence of gravitational waves.

On the other hand, it is clear from the Wightman function (3.21) that, despite
the local flatness, a single detector should be able to detect the
gravitational shockwave as it passes by. Indeed, if we allow the detector’s
switching function $\chi(\tau)$ to cross the shockwave, then the detector
response will _not_ reduce to the flat space one. Ultimately, this is because
we are integrating over the Wightman function where the shockwave form factor
does not cancel (essentially $\Delta\Theta_{u}\neq 0$).

Figure 2: Excitation probability of a single detector. We display the detector
transition probability for three shockwave profiles with the same energy
density $\alpha$: $a_{x}=2\alpha,a_{y}=-\alpha$ (red curve),
$a_{x}=3\alpha,a_{y}=-2\alpha$ (green curve), $a_{x}=a_{y}=\frac{\alpha}{2}$
(orange curve) and we compare them to the Minkowski response (dashed blue
curve). The scenario follows Left Fig. 1: the detector is located at the
origin, with the switching peak at $t=0$, and energy gap $\Omega T=2$, the
shockwave is localized at $u=0$. Note that, in this parametrization, when
$\alpha=0$, $f(\vec{x})=0$ identically and we recover Minkowski space.

To demonstrate this effect, we plot the excitation probability of the detector
as a function of the strength of the planar shockwave in Fig. 2. This is
characterized by various eigenvalues of the matrix $A$, which we parameterize
in terms of the wave energy density

$\alpha=\Tr(A)=8\pi G\rho\,.$ (5.1)

The response for Minkowski space is displayed by the blue dashed curve and
compared to three wave profiles depicted by solid orange, green, and red
curves. Interestingly, for small enough $\alpha$ the excitation probability of
the detector actually decreases compared to the flat space result. However,
increasing the strength/energy density of the shockwave ultimately leads to a
larger excitation probability. In particular, this means that for a given
profile of the wave, there exists an amplitude for which a single detector
cannot distinguish the wave from the flat space. Let us also note that the
symmetric shockwave ($a_{x}=a_{y}=\alpha/2$) has a much smaller effect than
the other cases. The reason for this can be understood as follows: if one
holds the energy density, $\alpha$ of the shockwave constant, then the
deviation from the Minkowski Wightman function in $\eqref{eq: Wightman-Gauss}$
is (when $\vec{x}=0=\vec{X}$) minimized for $a_{x}=a_{y}$.

To summarize, we have clearly illustrated that the gravitational shockwave
passing through the UDW detector leaves an observable, at least in principle,
imprint on the detector.

###  5.2 Comparison to linearized gravitational waves

Our results should be contrasted with the conclusions found in the context of
linearized gravity: how did we avoid the conclusions of Smith2020harvestingGW
? To begin answering this question, we recall that the metric of the
linearized plane gravitational wave considered in Smith2020harvestingGW  reads

$\displaystyle ds^{2}=-dudv+(1+B\cos\omega u)dx^{2}+(1-B\cos\omega
u)dy^{2}\,,$ (5.2)

where $B$ is the wave amplitude and $\omega$ its frequency.777Note that the
employed coordinates are not the Brinkmann coordinates we used in (2.1),
rather, they are a special case of the so called BJR coordinates
zhang2017memory . This metric is a solution to the vacuum linearized Einstein
equations, that is, it is only a solution to the full non-linear vacuum
equations to order $O(B)$. Furthermore, the authors showed that the first-
order correction $\Delta_{1}W_{\text{GW}}(\mathsf{x},\mathsf{x}^{\prime})$ to
the Wightman function restricted to the trajectory $\Delta x=\Delta y=0$
vanishes, hence the probability is not affected by the gravitational wave at
order $O(B)$.

We can in fact show that for this metric there is a closed-form expression for
the Wightman function888In fact, the integral of (A3) in Smith2020harvestingGW
is just Gaussian and so the techniques of Appendix A in this paper can
straightforwardly be applied. such that the $O(B^{2})$ correction to the
Minkowski Wightman function
$\Delta_{2}W_{\text{GW}}(\mathsf{x},\mathsf{x}^{\prime})$ for $\Delta x=\Delta
y=0$ does not vanish. It is given by

$\displaystyle\Delta_{2}W_{\text{GW}}(\mathsf{x},\mathsf{x}^{\prime})=-B^{2}\frac{\omega^{2}(\Delta
u-i\epsilon)^{2}[\cos^{2}(\omega u)+\cos^{2}(\omega u^{\prime})]-2[\sin(\omega
u)-\sin(\omega u^{\prime})]^{2}}{16\pi^{2}\omega^{2}(\Delta
u-i\epsilon)^{3}(\Delta v-i\epsilon)}\,.$ (5.3)

While the second-order correction cannot be used on the grounds that it goes
beyond the given (linear) approximation, it suggests that a full non-linear
gravitational wave solution to the Einstein equations could in general have
nontrivial signature to the detector response. This is exactly the case for
the ‘Gaussian’ shockwave considered in this paper with the Wightman function
(3.22).

To see how this arises, let us consider an expansion in small shockwave
amplitude, that is, $||AT||\ll 1$ (using switching duration $T$ as a reference
scale). Specifying to $\Delta x=\Delta y=0$, the general expression (3.22)
yields the following expansion in $A$:

$\displaystyle W(\mathsf{x},\mathsf{x}^{\prime})$
$\displaystyle=-\frac{1}{4\pi^{2}(\Delta u-i\epsilon)(\Delta
v-i\epsilon)}\left(1-\frac{\Tr(A)\Delta\Theta_{u}(u-u_{0}-i\epsilon)(u^{\prime}-u_{0}+i\epsilon)}{2(\Delta
u-i\epsilon)}\right.$
$\displaystyle\left.+\frac{(2\Tr(A^{2})+\Tr(A)^{2})[\Delta\Theta_{u}(u-u_{0}-i\epsilon)(u^{\prime}-u_{0}+i\epsilon)]^{2}}{8(\Delta
u-i\epsilon)^{2}}+O(A^{3})\right)\,,$ (5.4)

which has a first-order correction to the flat-space Wightman function
proportional to the trace of $A$. This result also explains why a slight
increase in the strength of the shock can actually decrease the transition
probability—the first order correction is negative.

Figure 3: ‘GW amplitude expansion’ of the excitation probability for source-
less waves. Considering the source-less shockwaves with $a_{x}=-a_{y}=a$, we
display the exact (orange curve), the $O(a^{2})$ (green curve), and the
$O(\alpha^{4})$ (red curve) amplitude expansions of the transition probability
and compare them to that of the Minkowski space (blue dashed curve). As
previously, the detector is located at the origin, with the switching peak at
$t=0$, the shockwave follows $u=0$, and $\Omega T=2$.

Suppose now that we want to mimic the setup of the vacuum plane gravitational
wave in $D=4$, obtained via linearized gravity. For this we choose a
sourceless, but _non-trivial_ , shockwave; $\alpha=\tr(A)=0$ with non-
vanishing

$a_{1}=-a_{2}=a\,,\quad aT\ll 1\,.$ (5.5)

Then, as in Smith2020harvestingGW , we see that the contribution of the
shockwave comes in only at second-order. Furthermore, it can be checked that
all corrections with odd powers of $a$ vanish so that the subleading
corrections are of $O(a^{2n})$. We plot the excitation probability as a
function of the shock profile diagonal element $a$ in Fig. 3 up to fourth
order in $a$.

This result shows that if we regard $a$ as taking the role analogous to
gravitational wave amplitude, then we obtain ‘similar results’ to
Smith2020harvestingGW , in that, at linear order $O(a)$ the gravitational
(shock)wave does not modify the excitation probability of a single detector
restricted to a static trajectory. On the other hand, since the shockwave is a
solution to the full non-linear Einstein equations, higher order corrections
always exist and these allow a single static detector to see the shockwave as
it passes by. We expect this to be a generic feature of non-linear
gravitational waves.

###  5.3 Entanglement harvesting

The single detector result shows that if we allow the shockwave to pass
through the (switched on) detector, then there is a clear signature that
changes the excitation probability of the detector. This is the case even if
the shockwave is sourceless. A natural question that arises is whether two
detectors, localized on two different sides of the shock (and thence in the
flat space for the entire duration of the measurement), are still able to
detect the presence of the shockwave. The entanglement harvesting protocol is
well-suited to address this type of question: the spirit is similar to
VerSteeg2009entangling , where it was shown that two inertial detectors can
distinguish the QFT vacuum on de Sitter background versus the thermal state in
Minkowski space, even though a single detector will register an identical
response in both cases.

Figure 4: Concurrence, non-local correlation and local noise contributions for
two detectors. The plots show the concurrence between Alice and Bob’s
detectors and the corresponding local/non-local contributions as a function of
Alice’s position. We show the concurrence for Minkowski space (orange dashed
curve) as a reference. For both plots, the shockwave is localized at $u=0$,
Bob is fixed at $z_{B}=7T$, and Alice’s position $z_{A}$ is varied. We set
$\Omega T=3.75$. Top: Shockwave with nonzero trace, i.e, $a_{x}T=a_{y}T=1$.
Bottom: Traceless shockwave $a_{x}T=-a_{y}T=1$. In each case the vertical
dashed line in red marks the position of Alice’s detector where the shock
begins to alter the excitation probability $P_{A}$, thus Alice has not crossed
the shock for $z_{A}\lesssim-2T$. The vertical dashed line in black marks the
position of Alice’s detector when the (effectively compact) support of
Gaussian switching begins to cross causal future/past of Bob’s detector, thus
allowing both detectors to communicate via the quantum field. This manifests
as increase in concurrence for $z_{A}\gtrsim 3T$.

In Fig. 4 we show that the entanglement harvesting protocol indeed has a
specific quantum signature in terms of concurrence. Namely, therein, we plot
the concurrence $\mathcal{C}[\rho_{AB}]$ for two detectors $A$ and $B$
associated with two static observers Alice and Bob along the worldlines in
(4.17) and (4.18). We also display the corresponding nonlocal $|{\cal M}|$
term and the local noise contributions (given by the probabilities of each
detector, $P_{A}$ and $P_{B}$). The top figure corresponds to a shockwave with
trace $a_{x}T=a_{y}T=1$ and the bottom one is traceless $a_{x}T=-a_{y}T=1$.

In Fig. 4, Bob is fixed at constant $z_{B}=7T$ located on the $u<u_{0}=0$ part
of the shockwave geometry. We study how concurrence varies when Alice is
placed on different static trajectories $z_{A}$, starting from $z_{A}=-7T$ in
the $u>u_{0}=0$ region. We also include the concurrence for the same setup in
full Minkowski spacetime as a reference. Note that, when compared to Figs. 2
and 3, we have considered here a much bigger detector energy gap, setting it
to $\Omega T=3.75$. This is because, in general, the excitation probability is
highly suppressed with higher $\Omega$ (physically it is harder to excite with
larger energy gap), and $\Omega$ needs to be sufficiently large for the non-
local contribution $|\mathcal{M}|$ to eventually dominate over local noise
$\sqrt{P_{A}P_{B}}$. Consequently, the noise contribution for $\Omega T=3.75$
is an order of magnitude smaller than in the previous figures where $\Omega
T=2$.

Fig. 4 shows that when the shock passes through Alice’s detector, the
concurrence is greatly amplified relative to the Minkowski equivalent, with a
larger amount of entanglement extracted for the shock with nonzero stress-
energy. Therefore, the shockwave increases the non-local quantum correlations
encoded in $|\mathcal{M}|$ more than the local excitation encoded in the noise
term $\sqrt{P_{A}P_{B}}$ (effectively $P_{A}$ since we hold $P_{B}$ fixed).
However, notice that the concurrence is amplified even before Alice reaches
$z_{A}\approx-2T$ (shown as vertical dashed line) where $P_{A}$ starts to
differ from the Minkowski value. In other words, there is a small window
$z_{A}\lesssim-2T$ where the detector responses are indistinguishable from the
Minkowski value but, nonetheless, the two detectors extract greater quantum
correlations than in true Minkowski space. Therefore, two spacelike separated
detectors999The vertical black dashed line in Fig. 4 indicates where Alice and
Bob become causally connected. also can detect quantum imprint of
gravitational shockwave even though locally each detector perceives the
background geometry to be true Minkowski space.

Figure 5: Concurrence as a function of $\alpha=\Tr(A)$. By fixing $a_{x}T$
(here we set $a_{x}T=1$ for concreteness), the trace characterizes the
relative magnitude of $a_{y}$. Here we set $\Omega T=3.75$, and have the
detectors fixed at $z_{A}=-2T$ and $z_{B}=2T$. Note that from Fig. 4 we know
that the detectors do not encounter the shockwave during their switched-on
period.

It is worth checking how the concurrence varies with the shock profile and
hence the stress-energy tensor. For concreteness, we fix one component of the
matrix $A=\text{diag}(a_{x},a_{y})$ to some value $a_{x}T=1$ and express
$a_{y}=\alpha-a_{x}$ where as before $\alpha=\Tr(A)$, and plot this setup in
Fig. 5. The two detectors are fixed at $x_{A}=-2T$ and $x_{B}=2T$, thus they
have not yet crossed the shock at $u=u_{0}=0$ (as can be checked from Fig. 4).
We include the Minkowski value (in blue) as a baseline. We see that first of
all the traceless case, which corresponds to sourceless shockwave, has lower
concurrence than the case with nonzero trace (nonzero stress-energy). This
result is consistent with what we found in Fig. 4. We also see that the amount
of entanglement extracted by the two detectors is able to probe a constant DC
shift (reminiscent of the clasical memory effect) due to the step function
$\Theta(u-u_{0})$ induced by the shockwave, which is not observable for a
single detector localized away from the shockwave (since
$\Delta\Theta_{u}=0$).

##  6 Conclusions

We have considered local observers carrying quantum-mechanical detectors
interacting with quantized scalar fields on a gravitational shockwave
background geometry. Our work was motivated by a series of studies
Kolekar2018quantummemory ; Compere:2019rof ; Majhi2020shock ; Ferreira:2020whz
which probed the effect of these shockwaves on the global states and
properties of the quantum field. In this context, it was observed that a
matter-induced supertranslation due to a propagating shockwave only has
physical consequences on non-vacuum states of the field, and that the spectrum
of Hawking/Unruh radiation remains unaffected Compere:2019rof ; Majhi2020shock
; Ferreira:2020whz . These results complimented the notion of ‘soft hairs’
encoded in BMS supertranslations and their connection with the infrared
triangle Hawking:2016sgy ; strominger2018lectures .

Contrary to Compere:2019rof ; Majhi2020shock ; Ferreira:2020whz , we have
demonstrated that local observers interacting with the scalar field vacuum via
Unruh–DeWitt detectors _can_ see non-trivial results. A single detector has a
non-trivial probability of getting excited from a gravitational shockwave
passing through its worldline, in contrast to the linearized gravity results
presented in Smith2020harvestingGW . We have also shown that if Alice and Bob,
each carrying a UDW detector, are placed on different sides of the shockwave,
they can use the entanglement harvesting protocol to discover the non-trivial
impact of the shockwave on the scalar field vacuum. This happens even when
their detectors do not cross the shockwave and remain in flat space for the
entire duration for which they are switched on.

So why does Alice see the shockwave? As mentioned above, in the first work
Compere:2019rof  only the globally defined number operators and Bogoliubov
coeffecients were considered. These global features do not necessarily capture
all the details of a local finite time interaction with the field
Garay2016anti-unruh ; Brenna2016anti-unruh ; Henderson2020anti-hawking ;
Dappiaggi2021anti-hawking . In particular, more physical interactions with the
field can lead to unintuitive results like the anti-Hawking/Unruh effect
Garay2016anti-unruh ; Brenna2016anti-unruh ; Henderson2020anti-hawking ;
Dappiaggi2021anti-hawking  and the ‘standard’ results are only recovered by
letting the interaction happen over a long time scale. Moreover, the second
work Majhi2020shock  is only concerned with an observer completely localized
on the Minkowski part in either side of the shockwave. Since the spacetime is
flat on either side, the Wightman function is necessarily equal to the
Wightman function in the standard Poincaré-invariant vacuum in Minkowski
space. This emphasizes that a concrete notion of local observers can be
important in the context of extracting information from quantum fields.

Finally our results provide another situation where certain gravitational wave
effects manifest at the level of Wightman functions and hence affect localized
interactions with an external probe. In contrast to linearized gravitational
waves Smith2020harvestingGW  we have seen that a single detector is sensitive
to a shockwave. This is a manifestation of the nonlinear characteristics of
gravity and we expect it to be a generic feature of quantum gravitational wave
detection.

Thus, local detection and entanglement harvesting can physically extract
quantum imprints of gravitational shockwaves.

## Acknowledgment

We thank Maïté Dupuis for organising the PSI Winterschool wherein this project
was initiated and our minds and bodies stretched. F.G. is funded from the
Natural Sciences and Engineering Research Council of Canada (NSERC) via a
Vanier Canada Graduate Scholarship. E.T. acknowledges support from
Mike–Ophelia Lazaridis Fellowship. This work was also partially supported by
NSERC and partially by the Perimeter Institute for Theoretical Physics.
Research at Perimeter Institute is supported in part by the Government of
Canada through the Department of Innovation, Science and Economic Development
Canada and by the Province of Ontario through the Ministry of Colleges and
Universities. Perimeter Institute and the University of Waterloo are situated
on the Haldimand Tract, land that was promised to the Haudenosaunee of the Six
Nations of the Grand River, and is within the territory of the Neutral,
Anishnawbe, and Haudenosaunee peoples.

##  Appendix A Wightman in gravitational shockwave spacetimes

In this appendix we present the explicit calculation of some of the integrals
necessary to arrive at the forms of the Wightman function in the main text
(3.21) and (3.22).

###  A.1 From Gaussian integrals to the Wightman function

We begin by recalling the familiar Gaussian integral: provided $\Im{a}>0$,

$\int_{-\infty}^{+\infty}\,dx\,e^{iax^{2}+ibx}=\sqrt{\frac{i\pi}{a}}e^{-i\frac{b^{2}}{4a}}\,.$
(A.1)

This result directly applies to the mode functions

$\displaystyle\phi^{\mathrm{in}}_{k_{-},\vec{k}}=N_{k_{-}}e^{-ik_{-}v}e^{i\vec{k}\cdot\vec{x}}\int\frac{d\vec{x^{\prime}}d\vec{k^{\prime}}}{(2\pi)^{D-2}}e^{i(\vec{k}-\vec{k^{\prime}})(\vec{x^{\prime}}-\vec{x})-\frac{i\vec{k^{\prime}}^{2}}{4k_{-}}(u-u_{0})+ik_{-}\Theta(u-u_{0})f(x^{\prime})}\,,$
(A.2)

where we can perform the integral over $\vec{k}^{\prime}$ using (A.1) by
adding an infinitesimal $i\epsilon$ to avoid branch cuts

$\displaystyle\int
d\vec{k}^{\prime}\,e^{i\vec{k}^{\prime}\cdot(\vec{x}-\vec{x^{\prime}})}e^{-i\frac{|\vec{k}^{\prime}|^{2}}{4k_{-}}(u-u_{0}-i\epsilon)}$
$\displaystyle=\left(\frac{-4\pi
ik_{-}}{u-u_{0}-i\epsilon}\right)^{\frac{D-2}{2}}\exp\left[\frac{ik_{-}(\vec{x}-\vec{x}^{\prime})^{2}}{u-u_{0}-i\epsilon}\right]\,.$
(A.3)

Thus

$\phi^{\mathrm{in}}_{k_{-},\vec{k}}=\left(\frac{-4\pi
ik_{-}}{u-u_{0}-i\epsilon}\right)^{\frac{D-2}{2}}N_{k_{-}}e^{-ik_{-}v}e^{i\vec{k}\cdot\vec{x}}\\\
\times\int\frac{d\vec{x^{\prime}}}{(2\pi)^{D-2}}\exp\left[\frac{ik_{-}(\vec{x}-\vec{x}^{\prime})^{2}}{u-u_{0}-i\epsilon}+ik_{-}\Theta(u-u_{0})f(x^{\prime})\right]\,.$
(A.4)

Plugging this expression into the mode sum version of the Wightman function

$W(\mathsf{x},\mathsf{y})=\int
dk_{-}d\vec{k}\,\phi^{\mathrm{in}}_{k_{-},\vec{k}}(\mathsf{x})\,\phi^{\mathrm{in}\,*}_{k_{-},\vec{k}}(\mathsf{y})\,,$
(A.5)

and integrating over the $\vec{k}$ directions we find (3.4) of the main text:

$\displaystyle W(u,v,\vec{x};U,V,\vec{X})$
$\displaystyle=\int\frac{dk_{-}}{2(2\pi)^{D-1}}\,e^{-ik_{-}(\Delta
v-i\epsilon)}\int\frac{d\vec{x}^{\prime}}{(2\pi)^{D-2}}\,e^{ik_{-}\left[\Theta_{u}f(\vec{x}^{\prime})-\Theta_{U}f(\vec{x}^{\prime})\right]}$
$\displaystyle\times\frac{(4\pi)^{D-2}k_{-}^{D-3}}{[(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)]^{\frac{D-2}{2}}}\exp\left[\frac{ik_{-}(\vec{x}-\vec{x}^{\prime})^{2}}{u-u_{0}-i\epsilon}-\frac{ik_{-}(\vec{X}-\vec{x}^{\prime})^{2}}{U-u_{0}+i\epsilon}\right]\,.$
(A.6)

Here we have introduced $\Delta v=v-V$ and
$\Delta\Theta_{u}=\Theta(u-u_{0})-\Theta(U-u_{0})$. Next we integrate over
$k_{-}$:

$\displaystyle\int_{0}^{\infty}dk_{-}\,k_{-}^{D-3}\exp\left(-ik_{-}\left[\Delta
v-i\epsilon-f(\vec{x}^{\prime})\Delta\Theta_{u}\right]\right)\exp\left[\frac{ik_{-}(\vec{x}-\vec{x}^{\prime})^{2}}{u-u_{0}-i\epsilon}-\frac{ik_{-}(\vec{X}-\vec{x}^{\prime})^{2}}{U-u_{0}+i\epsilon}\right]$
$\displaystyle=\Gamma(D-2)(i)^{2-D}\left(\left[\Delta
v-i\epsilon-f(\vec{x}^{\prime})\Delta\Theta_{u}\right]-\frac{(\vec{x}-\vec{x}^{\prime})^{2}}{u-u_{0}-i\epsilon}+\frac{(\vec{X}-\vec{x}^{\prime})^{2}}{U-u_{0}+i\epsilon}\right)^{2-D}\,,$
(A.7)

and thence recover (3.21)

$\displaystyle W(u,v,\vec{x};U,V,\vec{X})$
$\displaystyle=\frac{\Gamma(D-2)(i)^{2-D}}{2(2\pi)^{D-1}}\frac{(4\pi)^{D-2}}{[(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)]^{\frac{D-2}{2}}}\int\frac{d\vec{x}^{\prime}}{(2\pi)^{D-2}}$
$\displaystyle\hskip 14.22636pt\times\left(\left[\Delta
v-i\epsilon-f(\vec{x}^{\prime})\Delta\Theta_{u}\right]-\frac{(\vec{x}-\vec{x}^{\prime})^{2}}{u-u_{0}-i\epsilon}+\frac{(\vec{X}-\vec{x}^{\prime})^{2}}{U-u_{0}+i\epsilon}\right)^{2-D}\,,$
(A.8)

which appeared in the main text. This is as far as we can go without
specifying $f(\vec{x})$.

To check the flat space limit, we will set $f(\vec{x})=0$ and use standard QFT
integrals to evaluate this (see, e.g., Appendix B.3.2 of Schwartz:2013pla ).
Defining

$\displaystyle a$ $\displaystyle=\frac{\Delta
u-i\epsilon}{(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)}\,,\quad\vec{b}=\frac{(U-u_{0}+i\epsilon)\vec{x}-(u-u_{0}-i\epsilon)\vec{X}}{(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)}\,,$
$\displaystyle c$ $\displaystyle=\Delta
v-i\epsilon+\frac{(u-u_{0}-i\epsilon)\vec{X}^{2}-(U-u_{0}+i\epsilon)\vec{x}\,^{2}}{(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)}\,,$
(A.9)

we have

$\displaystyle\int\frac{d\vec{x}\,^{\prime}}{(2\pi)^{D-2}}\,$
$\displaystyle\left(\Delta
v-i\epsilon-\frac{(\vec{x}-\vec{x}\,^{\prime})^{2}}{u-u_{0}-i\epsilon}+\frac{(\vec{X}-\vec{x}\,^{\prime})^{2}}{U-u_{0}+i\epsilon}\right)^{2-D}$
$\displaystyle=\int\frac{d\vec{x}\,^{\prime}}{(2\pi)^{D-2}}\,\left(a\vec{x}\,^{\prime}\,{}^{2}+2\vec{b}\cdot\vec{x}\,^{\prime}+c\right)^{2-D}=\frac{\pi^{\frac{D-2}{2}}\Gamma(D/2-1)}{(2\pi)^{D-2}\Gamma(D-2)}(ac-\vec{b}^{2})^{(2-D)/2}$
$\displaystyle=\frac{\pi^{\frac{D-2}{2}}\Gamma(D/2-1)}{(2\pi)^{D-2}\Gamma(D-2)}\left(\frac{(\Delta
v-i\epsilon)(\Delta
u-i\epsilon)-(\vec{x}-\vec{X})^{2}}{(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)}\right)^{(2-D)/2}\,.$
(A.10)

The factor $(u-u_{0})(U-u_{0})$ cancels with the same prefactor in (A.8). Thus
we get the flat space limit

$\displaystyle
W(u,v,\vec{x};U,V,\vec{X})=\frac{\Gamma(D/2-1)}{4\pi^{D/2}}\bigr{[}i\sigma(\mathsf{x},\mathsf{y})\bigr{]}^{(2-D)}\,,$
(A.11)

where $\sigma({\sf x},{\sf y})=\sqrt{-({\sf x-y})^{2}}=\sqrt{([\Delta
u-i\epsilon][\Delta v-i\epsilon])-(\vec{x}-\vec{X})^{2}}$, recalling our
convention ${\sf x}=(u,v,\vec{x})$ and ${\sf y}=(U,V,\vec{X})$. Indeed in
$D=4$ this simply becomes

$\displaystyle
W(u,v,\vec{x};U,V,\vec{X})=-\frac{1}{4\pi^{2}}\frac{1}{\sigma(x,y)^{2}}=-\frac{1}{4\pi^{2}}\frac{1}{(\Delta
u-i\epsilon)(\Delta v-i\epsilon)-(\vec{x}-\vec{X})^{2}}\,.$

###  A.2 Planar shockwave profile

Let us now calculate explicitly the Wightman function for the planar shockwave
(2.5). Starting from (A.8) and substituting the profile
$f(\vec{x})=-\sum_{i}a_{i}(x^{i})^{2}$ we are left with the following
integral:

$\displaystyle\int\frac{d\vec{x}\,^{\prime}}{(2\pi)^{D-2}}\,\left(\Delta
v-i\epsilon+\left(\sum_{i=1}^{D-2}a_{i}\Delta\Theta_{u}([x^{\prime}]^{i})^{2}\right)-\frac{(\vec{x}-\vec{x}\,^{\prime})^{2}}{u-u_{0}-i\epsilon}+\frac{(\vec{X}-\vec{x}\,^{\prime})^{2}}{U-u_{0}+i\epsilon}\right)^{2-D}$
$\displaystyle=\prod_{i=1}^{D-2}\left(1+a_{i}\Delta\Theta_{u}\frac{(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)}{\Delta
u-i\epsilon}\right)^{-1/2}\
\int\frac{d\vec{r}}{(2\pi)^{D-2}}\,\left(a\vec{r}\,^{2}+2\vec{\beta}\cdot\vec{r}+c\right)^{2-D}\,.$
(A.12)

Here we have defined $a$, $\vec{b}$ and $c$ as in (A.1), and

$\displaystyle r^{i}$
$\displaystyle=\left(1+a_{i}\Delta\Theta_{u}\frac{(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)}{\Delta
u-i\epsilon}\right)^{1/2}(x^{\prime})^{i}\,,$ $\displaystyle\beta^{i}$
$\displaystyle=\left(1+a_{i}\Delta\Theta_{u}\frac{(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)}{\Delta
u-i\epsilon}\right)^{-1/2}b^{i}\,,$ (A.13)

where there is no summation over the indices. Thus we can directly apply (A.1)
up to the final simplification. Then we are left with the term
$(ac-\vec{\beta}^{2})$ which simplifies as follows:

$\displaystyle(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)(ac-\vec{\beta}^{2})$
$\displaystyle=(\Delta v-i\epsilon)(\Delta
u-i\epsilon)-\sum_{i=1}^{D-2}\left(1+a_{i}\Delta\Theta_{u}\frac{(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)}{\Delta
u-i\epsilon}\right)^{-1}$
$\displaystyle\times\left[(x^{i}-y^{i})^{2}-a_{i}\Delta\Theta_{u}\frac{(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)}{\Delta
u-i\epsilon}\left(\frac{(x^{i})^{2}}{u-u_{0}-i\epsilon}-\frac{(y^{i})^{2}}{U-u_{0}+i\epsilon}\right)\right]$
$\displaystyle=(\Delta v-i\epsilon)(\Delta
u-i\epsilon)-\Delta\vec{x}\,^{2}+\sum_{i=1}^{D-2}\frac{a_{i}\Delta\Theta_{u}\left([U-u_{0}+i\epsilon]x^{i}-[u-u_{0}-i\epsilon]X^{i}\right)^{2}}{\Delta
u+a_{i}\Delta\Theta_{u}(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)}\,.$ (A.14)

Finally, putting everything together, we obtain (3.22) of the main text

$W(u,v,\vec{x};U,V,\vec{X})=\frac{(-i)^{D-2}\Gamma(D/2-1)}{4\pi^{D/2}}\prod_{i=1}^{D-2}\left(1+a_{i}\Delta\Theta_{u}\frac{(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)}{\Delta
u-i\epsilon}\right)^{-1/2}\\\ \times\left((\Delta v-i\epsilon)(\Delta
u-i\epsilon)-\Delta\vec{x}\,^{2}\\!+\\!\sum_{i=1}^{D-2}\frac{a_{i}\Delta\Theta_{u}\left([U-u_{0}+i\epsilon]x^{i}-[u-u_{0}-i\epsilon]X^{i}\right)^{2}}{\Delta
u+a_{i}\Delta\Theta_{u}(u-u_{0}-i\epsilon)(U-u_{0}+i\epsilon)}\right)^{\frac{2-D}{2}}\,.$
(A.15)

##  Appendix B Geodesics

In this appendix we present a solution to the geodesic equations on the
shockwave geometry. While this problem is now well understood from both
physical and mathematical perspectives, see e.g. Dray1985shockwave ;
ferrari1988beam ; balasin1997geodesics ; kunzinger1999rigorous , here we
present a ‘simplified derivation’ based on standard calculus, augmented with
well-known properties of distributional derivatives and ‘natural’ choice of
products of distributions. For the choice of metric we are considering, it
turns out that these ‘heuristic’ calculations can be given rigorous
justification, e.g. kunzinger1999rigorous .

Starting from the shockwave geometry in Brinkmann coordinates, (2.1), the
nonzero Christoffel symbols are

$\displaystyle\Gamma^{v}_{uu}=-f\delta^{\prime}(u-u_{0})\,,\quad\Gamma^{v}_{ui}=-\delta(u-u_{0})\partial_{i}f\,,\quad\Gamma^{i}_{uu}=-\frac{1}{2}\delta(u-u_{0})\partial^{i}f\,,$
(B.1)

where $\delta^{\prime}(u-u_{0})$ is the distributional derivative of the delta
function with respect to $u$. Denoting an affine parameter by $\tau$, the
geodesic equation yields

$\displaystyle\frac{d^{2}u}{d\tau^{2}}$ $\displaystyle=0\,,$ (B.2)
$\displaystyle\frac{d^{2}v}{d\tau^{2}}$
$\displaystyle=f\delta^{\prime}(u-u_{0})\left(\frac{du}{d\tau}\right)^{2}+2\delta(u-u_{0})\partial_{i}f\frac{du}{d\tau}\frac{dx^{i}}{d\tau}\,,$
(B.3) $\displaystyle\frac{d^{2}x^{i}}{d\tau^{2}}$
$\displaystyle=\frac{1}{2}\delta(u-u_{0})\partial^{i}f\left(\frac{du}{d\tau}\right)^{2}\,,$
(B.4)

subject to the norm constraint

$\displaystyle
g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}=f\delta(u-u_{0})\left(\frac{du}{d\tau}\right)^{2}+\frac{du}{d\tau}\frac{dv}{d\tau}+\left(\frac{dx^{i}}{d\tau}\right)^{2}=e\,,$
(B.5)

where $e=0,-1,1$ for null, timelike, spacelike geodesics, respectively. It
follows that $u(\tau)=A\tau+B$ for some constants $A,B$, thus $u$ itself is a
valid affine parameter for the geodesics. This allows us to rewrite
$\frac{d}{d\tau}=A\frac{d}{du}$. To solve the remaining geodesic equations, we
shall use the following properties of distributional
derivatives:101010Strictly speaking this is only a heuristic argument because
we are not directly dealing with subtle issues with products of distributions
very carefully. Instead, we will be integrating using rules of calculus but
augmented with ‘distributional (anti-)derivatives’ and reasonable assumptions
on ‘mild’ product of distributions, such as
$\Theta(u-u_{0})^{2}=\Theta(u-u_{0})$.

$\displaystyle\frac{d}{du}\Theta(u-u_{0})$ $\displaystyle=\delta(u-u_{0})\,,$
(B.6) $\displaystyle\frac{d}{du}\left[(u-u_{0})\Theta(u-u_{0})\right]$
$\displaystyle=\Theta(u-u_{0})\,.$ (B.7)

By integrating the geodesic equation twice (taking into account
‘distributional’ antiderivatives), we get

$\displaystyle\frac{dx^{i}}{du}$
$\displaystyle=\frac{1}{2}\Theta(u-u_{0})\partial^{i}f+C^{i}\,,$ (B.8)
$\displaystyle x^{i}(u)$
$\displaystyle=C^{i}(u-u_{0})+D^{i}+K^{i}(u-u_{0})\Theta(u-u_{0})\,,$ (B.9)

where $C^{i},D^{i},K^{i}$ are integration constants that depend on initial
data $x_{0}^{i}$ and the shockwave profile $f$; in particular $D^{i}=x_{0}$,
$C^{i}=\dot{x}^{i}_{0}$, and $K^{i}=\frac{1}{2}\partial^{i}f(x_{0})$.

The solution for $v(u)$ is more complicated if we attempt to directly
integrate the geodesic equation due to products of distributions (which are
ill-defined outside of the Colombeau algebra of generalized distributions). To
simplify things we use the norm constraint, since the equation of motion for
$x^{i}(u)$ is distributionally well-defined. That is, we substitute (B.8) into
the norm constraint, to get

$\displaystyle\frac{dv}{du}$
$\displaystyle=\frac{e}{A^{2}}-f\delta(u-u_{0})-\left(\frac{1}{2}\Theta(u-u_{0})\partial_{i}f+C_{i}\right)^{2}$
$\displaystyle=\frac{e}{A^{2}}-f\delta(u-u_{0})-\frac{1}{4}(\partial_{i}f)(\partial^{i}f)\Theta(u-u_{0})^{2}-C^{i}\partial_{i}f\Theta(u-u_{0})-C_{i}C^{i}\,.$
(B.10)

At this point we are again faced with products of distributions, however this
is not difficult to deal with. A reasonable assumption would be to take
$\Theta^{2}=\Theta$ (which in calculus is a natural choice to make). By direct
integration and collecting the constants together, we can express the solution
as

$\displaystyle
v(u)=v_{0}+\dot{v}_{0}(u-u_{0})-f\Theta(u-u_{0})-\left[C^{i}\partial_{i}f+\frac{1}{4}(\partial_{i}f)^{2}\right](u-u_{0})\Theta(u-u_{0})\,.$
(B.11)

Here $v_{0}$ is arbitrary and $\dot{v}_{0}=e/A^{2}-C_{i}C^{i}$.

The main problem we need to address in this paper is the fact that, in the
presence of the shockwave, one cannot in general pick a nice codimension-1
spacelike surface that serves as a simultaneity hyperplane because the
spacetime does not admit a smooth spacelike Cauchy surface. The next best
thing we can ask for is to see if by restricting observers (timelike
geodesics) to be on a certain codimension-2 plane, we can at least have a
‘simultaneity curve’ on this plane. On any other plane, the geodesics will be
‘refracted’ by the shock (including the spacelike ones), complicating any
attempts to measure proper distances between two events across the shock.

Figure 6: Geodesics in shockwave spacetime. We display timelike geodesics
corresponding to a static trajectory at fixed $z_{0}=2T$ for different impact
parameters $\vec{b}$. We consider the case where the trajectory has no initial
velocity before encountering the shockwave; this corresponds to
$dx/du=dy/du=0$, so that $C^{i}=0$, and $dv/du=-1$ for $u<u_{0}$. For this
trajectory the proper time $\tau$ is related to the affine parameter $u$ by a
constant shift $u=\tau-z_{0}$. The plot for $y(u)$ is similar to the $x(u)$
plot on the right. Due to the choice of the shockwave profile and coordinate
system $(u,v,\vec{x})$, we see that from the metric (2.1) the static
trajectories that lie in the $\vec{b}=0$ plane are not affected by the
shockwave.

It turns out that for our choice of planar shockwave given by the profile
$f(\vec{x})=\sum_{i=1}^{D-2}a_{i}(x^{i})^{2}$, we are very lucky. For this to
work, we can choose the impact parameter of the trajectories of both detectors
to be $\vec{b}=0$, corresponding to $x^{i}=0$ for all $i=1,2,...,D-2$, then
the contributions to the geodesics due to the shockwave for these trajectories
vanish:

$\displaystyle f(\vec{b}=\vec{0})=\partial_{i}f\Bigr{|}_{\vec{b}=0}=0\,.$
(B.12)

What this means is that if we consider only timelike geodesics of the form

$\displaystyle\mathsf{x}(\tau)$ $\displaystyle=(u(\tau),v(\tau),\vec{0})\,,$
(B.13)

which corresponds to trajectories constrained to the $(t,z)$-plane, then the
solution of the geodesic equation is precisely equal to the Minkowski one,
independently of the shockwave profile. Therefore, the shockwave does not
alter the behaviour of the geodesics on the plane $\vec{x}=0$. We show how
timelike geodesics can be altered by the impact parameter in Fig. 6; the
results for other geodesics are similar by virtue of the solution (B.11) and
(B.9).

In particular, it means that on this plane alone, we can take any two points
labeled by $(t,z_{A})$ and $(t,z_{B})$, situated on different sides of the
shockwave, and still argue that the proper separation is simply given by
$|z_{B}-z_{A}|$ since the spacelike geodesic $t=\text{constant}$ is well-
defined, thus defining a simultaneity curve on the subspace $(t,z,\vec{0})$.

## References

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