aainstitutetext: Department of Physics, Arizona State University, Tempe, AZ
85281, USAbbinstitutetext: Walter Burke Institute for Theoretical Physics,
Caltech, Pasadena, CA 91125, USA

# Rindler Fluids from Gravitational Shockwaves

Sang-Eon Bak a Cynthia Keeler b Yiwen Zhang b and Kathryn M. Zurek
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study a correspondence between gravitational shockwave geometry and its
fluid description near a Rindler horizon in Minkowski spacetime. Utilizing the
Petrov classification that describes algebraic symmetries for Lorentzian
spaces, we establish an explicit mapping between a potential fluid and the
shockwave metric perturbation, where the Einstein equation for the shockwave
geometry is equivalent to the incompressibility condition of the fluid,
augmented by a shockwave source. Then we consider an Ansatz of a stochastic
quantum source for the potential fluid, which has the physical interpretation
of shockwaves created by vacuum energy fluctuations. Under such circumstance,
the Einstein equation, or equivalently, the incompressibility condition for
the fluid, becomes a stochastic differential equation. By smearing the quantum
source on a stretched horizon in a Lorentz invariant manner with a Planckian
width (similarly to the membrane paradigm), we integrate fluctuations near the
Rindler horizon to find an accumulated effect of the variance in the round-
trip time of a photon traversing the horizon of a causal diamond.

##  1 Introduction

Almost fifty years ago, Damour Damour3598  first connected general relativity
to fluid dynamics by demonstrating that when perturbations fall into a black
hole horizon, the spacetime near the horizon behaves like a fluid. This idea
led to the conception of the membrane paradigm thorne1986black ; Parikh:1997ma
; Eling:2009sj ; Eling:2009pb ; Gourgoulhon:2005ch , in which the fluid
resides on a stretched horizon, a timelike hypersurface very close to the
actual black hole horizon. Later on, the discovery of the AdS/CFT
correspondence formalized a version of fluid/gravity duality in which the
fluid arises as an effective description in the long distance, low frequency
regime of the dual gauge theory on the AdS boundary Policastro:2001yc ;
Policastro:2002se ; Kovtun:2003wp ; Kovtun:2004de ; Son:2007vk ;
Bhattacharyya:2008kq ; Faulkner:2010jy .

More recently, substantial progress was made in studying fluid/gravity duality
on a cutoff surface in flat Rindler spacetime Bredberg:2010ky ;
Bredberg:2011jq ; Compere:2011dx ; Compere:2012mt ; Lysov:2011xx ; Pinzani-
Fokeeva:2014cka . The flat space cutoff approach to fluid/gravity duality
formulates the map between the Einstein and Navier-Stokes equations in a
precise manner. In particular, Refs. Bredberg:2011jq ; Compere:2011dx ;
Pinzani-Fokeeva:2014cka  showed that the fluid is defined in terms of the
extrinsic curvature of a cutoff hypersurface outside of the horizon. This
cutoff surface is equipped with a flat induced metric. As in Bredberg:2011jq ,
in the leading order hydrodynamic expansion, the constraint equation from the
Einstein tensor becomes the incompressibility condition

$\partial_{i}v^{i}=0.$ (1)

Here $v^{i}$ is the $i$-component of the fluid velocity. Furthermore, the next
leading order constraint Einstein equation becomes the incompressible Navier-
Stokes equation Bredberg:2011jq

$\partial_{\tau}v_{i}-r_{c}\partial^{2}v_{i}+\partial_{i}P+v^{j}\partial_{j}v_{i}=0.$
(2)

Thus the cutoff surface fluid-gravity duality relates Einstein gravity, which
governs nonlinear gravitational interactions, to the incompressible Navier-
Stokes equations, which are an effective description of classical fluids.

In this paper, we aim to study the connection between gravitational shockwave
spacetimes and their fluid descriptions. Gravitational shockwave geometries
were studied first by Dray and ’t Hooft DRAY1985173  when they considered
ultra-relativistic matter falling into a black hole. More recently,
Cristofoli:2020hnk ; Gray:2021dfk ; Verlinde_2022 ; He:2023qha  have studied
shockwaves in flat spacetimes on the quantum level. Rewriting these
gravitational shockwaves on the fluid side of fluid/gravity duality will allow
us to consider quantum shocks as a fluid source.

Technically, we match gravitational shockwaves to their dual fluids by using
their Petrov type. Since the shockwave geometries we consider are Petrov type
N, their fluid dual must correspond to a spacetime of the same type. As shown
in Keeler:2020rcv , potential fluids with $v_{i}=\nabla_{i}\phi$ have dual
geometries of Petrov type N. Since Petrov Type N spacetimes only allow one
degree of freedom, we map between the cutoff fluid formalism and shockwave
geometries by identifying the fluid potential with the shockwave metric via

$\phi=\beta rH_{uu}.$ (3)

Here $\phi$ is the fluid potential and $H_{uu}$ characterizes the metric
fluctuations due to a shockwave. By choosing $\beta$ carefully, we find an
exact mapping such that the Einstein equation for the shockwave geometries is
precisely the incompressibility condition in Eq. (1) with the right hand side
replaced by the shockwave stress-energy tensor. Solving this differential
equation allows us to constrain the pressure term in the next leading order
Navier-Stokes equation.

Since we map the shockwaves to potential fluids, the next major goal of our
paper is to study the effects of quantum fluctuations on potential fluids. In
the leading order hydrodynamic expansion, we keep the left hand side of the
incompressibility equation unchanged, while replacing the right hand side with
a quantum noise term. This Ansatz is motivated by a set of commutation
relations first studied by ’t Hooft tHooft:1996rdg ; tHooft:2018fxg  in the
context of black hole perturbations, and later developed by Verlinde_2022  and
He:2023qha  for flat spacetime. Such fluctuations alter the classical fluid
(Einstein) equation to a stochastic differential equation, resembling the form
of a first order Langevin equation, which describes Brownian motion of a
massless particle Zhang:2023mkf . Quantum uncertainty arises from a Brownian
particle undergoing random walk in a stochastic background Banks:2021jwj ;
Zurek:2022xzl .

We argue in a similar fashion as in Zhang:2023mkf  that spacetime fluctuations
give rise to a “smeared out” horizon, analogous to a stretched horizon
Damour3598 . However, our present paper utilizes the cutoff surface fluid
approach to devise a Lorentz invariant method to smear out the horizon. We
will impose the condition that the cutoff surface intersects with the boundary
of a causal diamond at a distance of the reduced Planck length
$\tilde{\ell}_{p}$ Banks:2021jwj ; Zurek:2022xzl ; Verlinde_2022 . In four
dimensional spacetime, the reduced Planck length is simply the Planck length
$\tilde{\ell}_{p}=\ell_{p}$. Positing such a cutoff surface allows us to
compute the variance of photon traversal time measured by a free-falling
observer on the boundary of the causal diamond

$\Delta T_{\rm r.t.}^{2}\sim 2L\ell_{p}f(\bm{x},\bm{x}^{\prime}).$ (4)

Here $f(\bm{x},\bm{x}^{\prime})$ is the transverse response function, which is
a unique feature to the shockwave geometries, and is absent in classical
Brownian motion. Eq. (4) is consistent with the result from previous
calculations Verlinde:2019ade ; Verlinde:2019xfb ; Banks:2021jwj ;
Zurek:2022xzl ; Zhang:2023mkf  done via different means.

The outline of this paper is as follows. In Sec. 2 we review the cutoff fluid
formalism in flat spacetime. In Sec. 3 we compute the Petrov classification
for the shockwave spacetime and the fluid metric, respectively, and we show
that the shockwave metric is mapped to the potential of a potential flow in
hydrodynamics. In Sec. 4, we solve the incompressibility equation with a
source term posited from the ’t Hooft commutation relations, and use the
solution to compute the uncertainty in photon traversal time. In Sec. 5, we
summarize our results and point to some future directions. Throughout this
paper, we use $8\pi G_{N}=\ell_{p}^{2}=\kappa^{2}$.

##  2 Rindler Fluids in Einstein Gravity

In this section we review the fluid/gravity correspondence on a cutoff surface
of Minkowski space. Following Bredberg:2010ky ; Bredberg:2011jq ;
Compere:2011dx ; Compere:2012mt , we develop the duality by first performing
coordinate transformations to introduce a constant fluid velocity and constant
pressure. These transformations produce the “seed” metric, which is just
Minkowski space in unusual coordinates. Then, we allow the fluid velocity and
pressure to vary slowly with both the $x^{i}$ and $\tau$ coordinates; allowing
for higher-order corrections in the hydrodynamic parameter then produces the
true fluid dual metric. Resolving the Einstein equations for this more general
metric, we find the constraint Einstein equations can be written as
conservation equations for the Brown-York stress tensor. Writing these
conservations equations explicitly, the first and second-order equations
become incompressibility and the Navier-Stokes equations, respectively.
Following this review, we additionally review the specific case of potential
fluids for later use.

###  2.1 Geometric Setup

We will work throughout in ingoing Eddington-Finkelstein coordinates

$ds^{2}=-(\alpha r)d\tau^{2}+2d\tau dr+\delta_{ij}dx^{i}dx^{j},$ (5)

where $i,j$ indicate the transverse directions and thus run from $1\ldots d-2$
(these correspond to to the 2 transverse directions in $d=4$). $\alpha$
corresponds to the constant proper acceleration of a Rindler observer at fixed
$r$. Our formulation most closely follows Compere:2011dx ; Compere:2012mt ;
Eling:2012ni .

Our fluid will live on a timelike hypersurface called $\Sigma_{c}$. We choose
this surface to be $r=r_{c}$; its induced metric $\gamma_{ab}$ is

$ds^{2}|_{\Sigma_{c}}=\gamma_{ab}dx^{a}dx^{b}=-(\alpha
r_{c})d\tau^{2}+\delta_{ij}dx^{i}dx^{j}.$ (6)

Here $a,b$ are spacetime indices on $\Sigma_{c}$, so they run over $\tau$ as
well as $i,\,j$. Note that this metric is flat.

Starting from this background geometry, we introduce a constant pressure by
shifting $r$; since we want to keep the induced metric fixed, we must also
rescale $\tau$. The transformation is

$\displaystyle r\rightarrow r+\frac{1}{\alpha
p^{2}}-r_{c},\quad\tau\rightarrow\sqrt{\alpha r_{c}}p\tau\,.$ (7)

This coordinate transformation shifts the position of the Rindler horizon from
$r_{h}=0$ to $r_{h}=r_{c}-\frac{1}{\alpha p^{2}}$.

We then introduce a constant fluid velocity by performing the boost

$\displaystyle\sqrt{\alpha r_{c}}\tau\rightarrow-u_{a}x^{a},\quad
x^{i}\rightarrow x^{i}-u^{i}\sqrt{\alpha
r_{c}}\tau+(1+\gamma)^{-1}u^{i}u_{j}x^{j}\,.$ (8)

where $\gamma=\quantity(1-v^{2}/(\alpha r_{c}))^{-1/2}$, and we set the fluid
four-velocity vector $u_{a}$ to a constant.

Thus we arrive at the ‘seed’ metric for a relativistic fluid, given by

$\displaystyle ds^{2}=-2pu_{a}dx^{a}dr+\left[\gamma_{ab}-\alpha
p^{2}\left(r-r_{c}\right)u_{a}u_{b}\right]dx^{a}dx^{b}\,.$ (9)

As noted above, so far this metric is just a rewriting of Minkowski space in
fancy coordinates. However, we can already begin to study the constant ‘fluid’
on the cutoff surface. To characterize the physics on the cutoff surface, we
introduce the Brown-York stress-energy tensor Brown:1992br

$\kappa^{2}T_{ab}=(K\gamma_{ab}-K_{ab}),$ (10)

where $\kappa^{2}=8\pi G_{N}$. Here,
$K_{ab}=\frac{1}{2}\mathcal{L}_{n}\gamma_{ab}$ is the extrinsic curvature on
$\Sigma_{c}$, and $K=K_{ab}\gamma^{ab}$ is its trace. Here, $\mathcal{L}_{n}$
is the Lie derivative along the unit normal vector $n^{\mu}$ on the cutoff
surface $\Sigma_{c}$.

For the seed metric in Rindler spacetime (9), the Brown-York stress tensor on
the cutoff surface has a form of the perfect fluid in equilibrium with a
vanishing energy density:

$\displaystyle\kappa^{2}\,T_{ab}=\alpha p\,h_{ab}\,,$ (11)

where $h_{ab}=\gamma_{ab}+u_{a}u_{b}$ is the metric of a spacelike
hypersurface orthogonal to the velocity vector $u_{a}$ but still embedded in
$\Sigma_{c}$. Since the pressure is constant, there is no time evolution of
the momentum density of the fluid. Due to the vanishing energy density, the
divergence of the momentum density is vanishing as well by the continuity
equation. Thus, the Brown-York stress tensor (11) for the seed metric
describes the equilibrium fluid that does not flow as seen by a comoving
observer with $u^{a}$. Thus this background seed metric is equivalent to a
somewhat boring fluid.

Before we introduce slow variations in order to consider more interesting
fluids, we note that the Brown-York stress-energy tensor $T_{ab}$ is not
equivalent to the external matter stress-energy tensor $\mathcal{T}_{\mu\nu}$.
The matter stress-energy tensor serves the role of sourcing the geometry,
whereas the Brown-York stress-energy tensor describes the properties of a
hypersurface within the geometry. More specifically, the Brown-York stress
tensor provides a useful prescription for computing the energy and momentum
density of a cutoff surface within a gravitational system. In the present
case, the Brown-York stress tensor relates the energy and momentum on a cutoff
surface $\Sigma_{c}$ to the energy and moment of a fluid. We also note that
for the simple fluid (11), the spacetime stress-energy tensor
$\mathcal{T}_{\mu\nu}$ is clearly zero, since the spacetime is just empty
Minkowski space. By contrast, the fluid does have a nonzero Brown-York tensor
as we have shown.

###  2.2 The Hydrodynamic Limit and Near-Horizon Expansion

In order to obtain non-constant fluids, the next step is to perturb the seed
metric (9) by allowing the fluid degrees of freedom $u_{a},p$ to depend on
$(\tau,x^{i})$, following the procedure in Compere:2011dx ; Compere:2012mt ;
Pinzani-Fokeeva:2014cka . Then, near-equilibrium solutions $g^{(n)}_{\mu\nu}$
($n=0,1,2\dots$) can be constructed order-by-order in the relativistic
gradient expansion. However, in this work we are interested in the non-
relativistic hydrodynamic limit since it agrees with the near-horizon
expansion Bredberg:2011jq ; Keeler:2020rcv , and we will be working in the
near-horizon regime for the Rindler shockwaves below.

Accordingly, in terms of the nonrelativistic fluid velocity $v_{i}$, and the
nonrelativistic pressure $P$, we have

$\displaystyle u_{a}=\frac{1}{\sqrt{\alpha r_{c}-v^{2}}}\left(-\alpha
r_{c},v_{i}\right),\qquad p=(\alpha r_{c}-2P)^{-1/2},$ (12)

where the relativistic velocity is normalized so $u_{a}u^{a}=-1$. The
equivalent of the relativistic gradient expansion becomes explicitly the
hydrodynamic scaling

$v_{i}\sim\order{\varepsilon},\quad
P\sim\order{\varepsilon^{2}},\quad\partial_{i}\sim\order{\varepsilon},\quad\partial_{\tau}\sim\order{\varepsilon^{2}}.$
(13)

Using this non-relativistic scaling, and further solving the non-constraint
Einstein equations order-by-order, we construct the non-relativistic fluid
metric given in Appendix (80). Although the full fluid-dual geometry is quite
complicated, with its metric given by (80) up to $\order{\varepsilon^{3}}$, we
are mainly interested in the lowest order expansion of (80):111We ignore
$\order{\varepsilon^{2}}$ and higher order terms presently because we
anticipate the relation between the fluid-dual metric in just the leading
order of hydrodynamic expansion (14) and the shockwave geometry that solves
the linearized Einstein equation. This point will become clear in section 3.

$\displaystyle\begin{split}ds^{2}=&-(\alpha r)d\tau^{2}+2d\tau
dr+\delta_{ij}dx^{i}dx^{j}\\\
&-2\quantity(1-\frac{r}{r_{c}})v_{i}dx^{i}d\tau-2\frac{v_{i}}{\alpha
r_{c}}dx^{i}dr+\order{\varepsilon^{2}}.\end{split}$ (14)

As noted above, the fluid behavior is described via the Brown-York stress
tensor (10) and its conservation equations. Explicitly in terms of the non-
relativistic fluid metric, we obtain Bredberg:2011jq 222We used the
symmetrization $X_{(i}Y_{j)}=\frac{1}{2}\left(X_{i}Y_{j}+X_{j}Y_{i}\right).$

$\kappa^{2}\,T_{ab}dx^{a}dx^{b}=\frac{\alpha v^{2}}{\sqrt{\alpha
r_{c}}}d\tau^{2}-\frac{2\alpha v_{i}}{\sqrt{\alpha
r_{c}}}dx^{i}d\tau+\frac{\alpha}{\sqrt{\alpha
r_{c}}}dx^{i}dx_{i}+\frac{P\delta_{ij}+v_{i}v_{j}}{r_{c}\sqrt{\alpha
r_{c}}}dx^{i}dx^{j}-\frac{2\partial_{(i}v_{j)}}{\sqrt{\alpha
r_{c}}}dx^{i}dx^{j}+\order{\varepsilon^{3}}.$ (15)

where details of the derivation are included in Appendix A.

Since the constraint Einstein equations on $\Sigma_{c}$ match the conservation
equations for the Brown-York tensor, we find333This is the Gauss-Codazzi
equation including the integration constant in $r$ Pinzani-Fokeeva:2014cka .
We set the integration constant to zero by imposing the Brown-York stress
tensor to be conserved when we consider the source-free situation.

$\nabla^{b}T_{ab}|_{\Sigma_{c}}=-2n^{\mu}\mathcal{T}_{a\mu}\,.$ (16)

As before, $n^{\mu}$ is the unit normal vector (81) to the cutoff surface
$\Sigma_{c}$. The spacetime stress-energy tensor $\mathcal{T}_{a\mu}$ provides
the nonzero source for Einstein’s equations, and we again wish to
differentiate it from the Brown-York tensor we introduced in Sec. 2.1.

In section 3, we are interested in the spacetime stress-energy tensor
$\mathcal{T}_{\mu\nu}$ due to an ultra-relativistic source, also known as the
gravitational shockwave. Before turning on the external sources, we now review
the fluid behavior when no source is present, that is, when
$\mathcal{T}_{\mu\nu}=0$. At $\order{\varepsilon^{0}}$, $T_{ab}$ is constant,
and the Einstein equations are satisfied trivially. At
$\order{\varepsilon^{2}}$, we obtain the incompressibility condition
Bredberg:2011jq

$\nabla^{a}T_{\tau a}=0\quad\iff\quad
G_{\tau\tau}=\frac{\alpha}{2}\partial^{i}v_{i}=0\,.$ (17)

In writing out (17), we have identified the conservation of the Brown-York
tensor at $\order{\varepsilon^{2}}$ with the Einstein tensor $G_{\tau\tau}$ at
the same order Pinzani-Fokeeva:2014cka . We note that other components of the
Einstein tensor vanish identically at the current order.

In a similar fashion, we compute the conservation equation of the Brown-York
tensor at $\order{\varepsilon^{3}}$, and we obtain the incompressible Navier-
Stokes equation Bredberg:2011jq

$\nabla^{a}T_{ia}=0\quad\iff\quad G_{\tau
i}=-\frac{1}{2r_{c}}\quantity(\partial_{\tau}v_{i}-r_{c}\partial^{2}v_{i}+\partial_{i}P+v_{j}\partial^{j}v_{i})=0,$
(18)

which is equivalent to the vacuum constraint Einstein equation $G_{\tau i}=0$
evaluated at the same order Pinzani-Fokeeva:2014cka .

Later, when we add sources to the Einstein equations, these constraint
equations will instead become (16), adding sources to the fluid equations. In
section 3, we will see how a nonzero $\mathcal{T}_{\mu\nu}$ coming from
gravitational shockwaves influences the fluid on $\Sigma_{c}$.

#### Potential Fluids

Before considering shockwave spacetimes, we review potential fluids. The class
of fluids known as potential fluids satisfies

$\displaystyle v_{i}=\partial_{i}\phi\,.$ (19)

These fluids are also known as irrotational, since they have zero vorticity.
In our 4-dimensional gravity (dual to a 2+1 dimensional fluid), the vorticity
is $\partial_{x}v_{y}-\partial_{y}v_{x}=0$, since gradients are curl-free.

For a source-free potential fluid, the incompressibility condition (17) yields
Keeler:2020rcv

$\displaystyle\nabla^{a}T_{0a}=0\quad\implies\quad\partial^{2}\phi=0\,,$ (20)

In this case, the incompressible Navier-Stokes equation (18) implies the
expression of the pressure in terms of $\phi$

$\displaystyle\nabla^{a}T_{ia}=0\quad\implies\quad\partial_{i}P=-\partial_{i}\partial_{\tau}\phi-\partial^{j}\phi\partial_{i}\partial_{j}\phi\,.$
(21)

#### Rindler to Light Cone Coordinates

Lastly, we review the coordinate transformation between the background to the
fluid-gravity dual metrics, and light cone coordinates. We review this
relationship because both the fluid-dual metrics, and the shockwave metrics we
study in the next section, are perturbations away from Minkowski space.
However in the fluid case we work in ingoing Eddington-Finkelstein coordinates
(5), while in the shockwave case the background is written in light cone
coordinates.

For the shockwave, the background is Minkowski spacetime in light-cone
coordinates $u=T+Z,\,v=T-Z$ where $(T,Z)$ are Minkowski global coordinates,
with metric

$\displaystyle ds^{2}=-dudv+\delta_{ij}dx^{i}dx^{j}.$ (22)

Here again $i,j$ indicate the transverse directions. The coordinate
transformation

$u=\sqrt{\frac{r_{c}}{\alpha}}e^{\alpha\tau/2}\mbox{\quad
and\quad}v=-\frac{4r}{\sqrt{\alpha r_{c}}}e^{-\alpha\tau/2}\,$ (23)

yields the line element of Rindler spacetime (5). Thus, Eq. (23) is the zeroth
order coordinate transformation between the fluid metric (14) and shockwave
metric (24). In the next section, we construct the relation at the next order,
between the perturbed geometries.

##  3 Near Horizon Fluids from Gravitational Shockwaves

In this section, we establish the relation between the near-horizon fluids
constructed in section 2.2 and the gravitational shockwaves in ASmetric . We
first briefly review the shockwave geometry and the external source from the
shockwave. Then, we consider how the fluid equations deforms in the presence
of an external source. With those two setups constructed, we find the
connection between the fluids and shockwaves by using their Petrov
classifications.

###  3.1 Gravitational shockwave geometry

The gravitational shockwave geometry was originally proposed by Aichelburg and
Sexl ASmetric  to describe gravitational radiation from a massless point
particle. The Aichelburg-Sexl metric is given by DRAY1985173 ; Verlinde_2022
with Minkowski spacetime in the light-cone coordinates (22).

If we consider a fast particle falling towards the future horizon, the
$uu-$component of the metric is perturbed, which produces the shockwave
geometry

$ds^{2}=-dudv+H_{uu}du^{2}+\delta_{ij}dx^{i}dx^{j}.$ (24)

Here $H_{uu}$ is given by

$H_{uu}(u,x^{i})=p_{u}\kappa^{2}\delta(u-u_{0})f(\bm{x};\bm{x}^{\prime}),$
(25)

where $p_{u}$ is the constant momentum of the shock, $u_{0}$ denotes the
location of the shockwave, and $f(\bm{x};\bm{x}^{\prime})$ is the Green’s
function of the transverse Laplacian operator:

$-\partial^{2}f(\bm{x};\bm{x}^{\prime})=\delta^{2}(\bm{x}-\bm{x}^{\prime}).$
(26)

In fact, Eq. (26) is equivalent to the Einstein equation, which is exactly
solvable in this case. Only the $uu$ component of the Einstein equation
$G_{\mu\nu}=\kappa^{2}\mathcal{T}_{\mu\nu}$ is nontrivial. Explicitly, this
component becomes

$-\frac{1}{2}\partial^{2}H_{uu}=\kappa^{2}\mathcal{T}_{uu},$ (27)

where $\mathcal{T}_{uu}$ is the only non-vanishing component of the shockwave
stress-energy tensor. For $H_{uu}$ as in (25), we have

$\mathcal{T}_{uu}=\frac{1}{2}p_{u}\delta(u-u_{0})\delta^{2}(\bm{x}-\bm{x}^{\prime}).$
(28)

The shockwave geometry can be described in a different gauge, which involves
an off-diagonal component with transverse and longitudinal directions. The
metric in this gauge is

$\displaystyle ds^{2}=-dudv+\partial_{i}X^{v}dudx^{i}+\delta_{ij}dx^{i}dx^{j}$
(29)

where we introduced the shockwave degree of freedom $X^{v}$

$\displaystyle
X^{v}(u,v,x^{i})=v+\int_{0}^{u}du^{\prime}H_{uu}\left(u^{\prime},x^{i}\right).$
(30)

Now, we have the shockwave geometry sourced by the external matter stress-
energy tensor $\mathcal{T}_{uu}$. In the next section, we will construct a
fluid-dual geometry with the same external source in order to make a
connection between shockwaves and fluids.

###  3.2 Near Horizon Fluids with External Sources

Just as the stress-energy tensor $\mathcal{T}_{uu}$ in (28) sources the
shockwave geometry with nonzero $H_{uu}$, in fluid-dual geometries an external
stress-energy tensor can be thought of as a sourcing the fluid dynamics. We
anticipate that in the leading order of hydrodynamic expansion the fluid-dual
metric given by Eq. (14) can be mapped to the shockwave geometry given by Eq.
(24). In anticipation of this match, we first add a non-zero external source
$\mathcal{T}_{uu}$ to the fluid equations (16), generalizing the fluid/gravity
setup in Bredberg:2011jq  beyond vacuum solutions.

Since the shockwave metric and fluid metric use different coordinate systems,
we need to find the components of $\mathcal{T}_{\mu\nu}$ from (28) in
$(\tau,r,x^{i})$ coordinates. Since the only non-vanishing component of
$\mathcal{T}_{\mu\nu}$ in $(u,v,x^{i})$ coordinates is $\mathcal{T}_{uu}$,
under the background coordinate transformation (23) we find that the only non-
vanishing component in $(\tau,r,x^{i})$ coordinates is
$\mathcal{T}_{\tau\tau}$, given by

$\mathcal{T}_{\tau\tau}=\frac{\alpha r_{c}}{4}e^{\alpha\tau}\mathcal{T}_{uu}.$
(31)

In Sec. 2.1, we showed that the conservation Eq. (16) of the Brown-York stress
tensor at order $\order{\varepsilon^{2}}$ is equivalent to the
incompressibility condition of the dual fluid (17). However, in the presence
of a gravitational shockwave, the incompressibility condition instead becomes

$\nabla^{a}T_{\tau
a}|_{\Sigma_{c}}=-2n^{\mu}\mathcal{T}_{\tau\mu}\quad\implies\quad\partial^{i}v_{i}=\frac{1}{2}\kappa^{2}r_{c}e^{\alpha\tau}\mathcal{T}_{uu},$
(32)

where we have additionally made use of the unit normal as explicitly given in
(81).

At $\mathcal{O}(\varepsilon^{3})$, the conservation Eq. (16) of the Brown-York
stress tensor is given by

$\displaystyle\nabla^{a}T_{ia}|_{\Sigma_{c}}=-2n^{\mu}\mathcal{T}_{i\mu}\implies\partial_{\tau}v_{i}-r_{c}\partial^{2}v_{i}+v_{j}\partial^{j}v_{i}+\partial_{i}P+v_{i}\partial^{j}v_{j}-r_{c}\partial_{i}\partial^{j}v_{j}=0.$
(33)

There is no contribution from the external source since the only non-vanishing
component of $\mathcal{T}_{\mu\nu}$ in $(\tau,r,x^{i})$ coordinates is
$\mathcal{T}_{\tau\tau}$. However, the equation is not equivalent to the
incompressible Navier-Stokes equation (18); instead, it has acquired one
further term: $r_{c}\partial_{i}\partial^{j}v_{j}$, since incompressibility is
broken.

#### Potential Fluids

If we assume a potential fluid of the form $v_{i}=\partial_{i}\phi$, the
conservation equation of the Brown-York stress tensor in the leading order
(32) becomes

$\displaystyle\nabla^{a}T_{\tau
a}|_{\Sigma_{c}}=-2n^{\mu}\mathcal{T}_{\tau\mu}\quad\implies\quad\partial^{2}\phi=\frac{1}{2}\kappa^{2}r_{c}e^{\alpha\tau}\mathcal{T}_{uu}.\,$
(34)

Upon examining Eq. (34) closely with the Einstein equation of the shockwave
geometry (27), we are tempted to believe that the two geometries are
equivalent, at least in the leading order of hydrodynamic expansion.
Specifically, we compare Eqs. (34) and (27) term by term

$\partial^{2}\phi=\frac{1}{2}\kappa^{2}r_{c}e^{\alpha\tau}\mathcal{T}_{uu}\quad\stackrel{{\scriptstyle?}}{{\iff}}\quad-\frac{1}{2}\partial^{2}H_{uu}=\kappa^{2}\mathcal{T}_{uu}$
(35)

and deduce a relation between $H_{uu}$ and $\phi$ to be

$\phi=-\frac{r_{c}}{4}e^{\alpha\tau}H_{uu}.$ (36)

While this may suggest a mapping between a potential fluid and the shockwave
metric, it does not guarantee the two metrics are diffeomorphic. The most
reassuring way to see whether two spacetimes are equivalent is to construct an
explicit diffeomorphism that maps one to the other. However, in our case,
finding such a diffeomorphism is quite difficult (considering the form of the
fluid metric (14)), if not impossible. Thankfully, there is a more
straightforward method to connect the shockwave geometry with the potential
fluid, because both geometries have Weyl tensors that exhibit explicit
algebraic speciality stephaniExactSolutionsEinstein2003 ; Keeler:2020rcv . In
Sec. 3.3, we utilize the Petrov classification to find a correspondence
between the shockwave and potential fluid geometries, thereby substantiating
our claim in Eq. (36).

Last but not least, at $\mathcal{O}(\varepsilon^{3})$, the conservation
equation (33) becomes

$\displaystyle\nabla^{a}T_{ia}|_{\Sigma_{c}}=-2n^{\mu}\mathcal{T}_{i\mu}\quad\implies\quad\partial_{\tau}\partial_{i}\phi-2r_{c}\partial_{i}\partial^{2}\phi+\partial_{i}P+\partial_{j}\phi\,\partial^{j}\partial_{i}\phi+\partial_{i}\phi\,\partial^{2}\phi=0.$
(37)

Eqs. (32) and (33) determine fluid dynamics on the cutoff surface when we have
added the shockwave stress-energy tensor $\mathcal{T}_{uu}$ as an external
source. Since the shockwave geometry (29) and the fluid-dual geometry are both
perturbations away from Minkowski spacetime, when we set them to have the same
external source, we expect that they are two equivalent geometries in
different gauges. In the next section, we establish the explicit connection
between the shockwave metric and the fluid-dual metric.

###  3.3 Petrov Classification Connecting Fluids with Shockwaves

In this section, we derive the relation between the near-horizon fluid metric
(14) and the shockwave metric (24). Instead of finding the complicated
diffeomorphism between them, we will adopt the virtue of the Newman-Penrose
formalism to link the shockwave degrees of freedom into the fluid story. We
will follow the conventions of stephaniExactSolutionsEinstein2003  throughout.

The Newman-Penrose formalism will allow us to calculate the Petrov
classification of our spacetimes. The Petrov classification categorizes
spacetimes according to the multiplicities of the principal null directions of
the Weyl tensor $C_{abcd}$. Principal null directions $k^{\mu}$ are null
vectors satisfying

$\displaystyle
k_{[\mu}C_{\nu]\rho\sigma[\gamma}k_{\delta]}k^{\rho}k^{\sigma}=0\,.$ (38)

All four-dimensional spacetimes have four principal null vectors. Spacetimes
with four distinct principal null vectors for the Weyl tensor are not
algebraically special; they are said to be of Petrov type I. If two principal
null directions coincide, then the spacetime is Petrov type II. If the four
principal nulls coincide in pairs, then the spacetime is type D. If all four
principal null directions coincide, then the spacetime is type N.

Moreover, if the spacetime is type N, the geometry is determined by only one
degree of freedom as we will see explicitly in this section. As we will review
below, both the shockwave metric and the metric dual to a potential fluid are
type N, so we can match them by matching the single degree of freedom.

We begin with the shockwave metric, defining the complex coordinates

$\displaystyle z=x+iy,\quad\bar{z}=x-iy,$ (39)

where we are now explicitly restricting to four-dimensional spacetime. Then,
the shockwave metric (24) becomes

$\displaystyle ds^{2}=-dudv+H_{uu}(u,z,\bar{z})du^{2}+dzd\bar{z}\,.$ (40)

In the Newman-Penrose formalism, the metric is rewritten in terms of a null
tetrad, $(k_{\mu},\,l_{\mu},\,m_{\mu},\,\bar{m}_{\mu})$. Explicity,
$g_{\mu\nu}=-l_{(\mu}n_{\nu)}+m_{(\mu}\bar{m}_{\nu)}$. For the shockwave
geometry, we will use the null tetrad

$\displaystyle
m^{\mu}\partial_{\mu}=\sqrt{2}\partial_{z},\quad\bar{m}^{\mu}\partial_{\mu}=\sqrt{2}\partial_{\bar{z}},\quad
l^{\mu}\partial_{\mu}=\sqrt{2}\partial_{u}-\sqrt{2}H_{uu}\partial_{v},\quad
k^{\mu}\partial_{\mu}=\sqrt{2}\partial_{v}\,.$ (41)

Here $k^{\mu}$ is a principal null direction since it satisfies (38). Indeed,
since $k^{\mu}$ obeys the condition

$\displaystyle C_{\mu\nu\rho\sigma}k^{\sigma}=0,$ (42)

it is a 4-fold repeated principal null vector, so the shockwave geometry is
type N. Importantly, when the null tetrad is chosen so that the first vector
in the null tetrad is itself a 4-fold repeated principal null, the Weyl
scalars $\Psi_{i},i=0,\ldots,4$ take on a special form, namely
$\Psi_{0}=\Psi_{1}=\Psi_{2}=\Psi_{3}=0,\Psi_{4}\neq 0$.

In the tetrad formalism, these Weyl scalars contain all the information in the
Weyl tensor. In general,

$\displaystyle\begin{array}[]{ll}\Psi_{0}\equiv
C_{\mu\nu\rho\sigma}k^{\mu}m^{\nu}k^{\rho}m^{\sigma},&\Psi_{1}\equiv
C_{\mu\nu\rho\sigma}k^{\mu}l^{\nu}k^{\rho}m^{\sigma},\quad\Psi_{2}\equiv
C_{\mu\nu\rho\sigma}k^{\mu}m^{\nu}\bar{m}^{\rho}l^{\sigma}\\\ \Psi_{3}\equiv
C_{\mu\nu\rho\sigma}k^{\mu}l^{\nu}\bar{m}^{\rho}l^{\sigma},&\Psi_{4}\equiv
C_{\mu\nu\rho\sigma}\bar{m}^{\mu}l^{\nu}\bar{m}^{\rho}l^{\sigma}\,.\end{array}$
(45)

These scalars do depend on the choice of null tetrad, but if there is any null
tetrad in which only $\Psi_{4}$ is nonzero, then the spacetime must be type N.

For fluid dual metrics, the non-relativistic fluid geometry (80) is
algebraically special, specifically of type II, as first shown in
Bredberg:2011jq . More recently, Keeler:2020rcv  showed that two special types
of fluid have higher algebraic speciality. For a constant vorticity fluid, the
fluid-dual metric is type D, while for a potential fluid it is type N.

Since both the fluid-dual metric for a potential fluid, and the shockwave
geometry (24), are type N, their Weyl tensors are described by the single
degree of freedom $\Psi_{4}$ (in a tetrad choice where all other components
are zero). Indeed, a potential fluid is described by only the scalar $\phi$,
while a shockwave is described by the single function $H_{uu}$, so we should
not be too surprised that they are both type N.

In order to relate the shockwave geometry (24) and the fluid metric (14), we
will choose a tetrad for each where only $\Psi_{4}$ is nonzero. If two Petrov
type N geometries actually represent the same spacetime, then if tetrads are
chosen so that all other $\Psi_{i}$ vanish, the $\Psi_{4}$’s for the two
geometries must match up to an overall scaling.

For the shockwave metric (24), the non-vanishing Weyl scalar is given by

$\displaystyle\Psi_{4}=2\partial_{\bar{z}}\partial_{\bar{z}}H_{uu}(u,z,\bar{z})\,.$
(46)

For the fluid metric (14) dual to a potential fluid, the tetrad can be chosen
so the non-vanishing Weyl scalar is given by Keeler:2020rcv

$\displaystyle\Psi_{4}=\frac{2}{r}\partial_{\bar{z}}\partial_{\bar{z}}\phi(\tau,z,\bar{z})\,.$
(47)

Indeed, we find that we can relate the potential fluid $\phi$ to the shockwave
metric component $H_{uu}$ by equating these two $\Psi_{4}$’s, up to an overall
scale $\beta$:

$\displaystyle\phi=\beta rH_{uu}\,.$ (48)

Note that we have one more constraint when we relate the fluid degrees of
freedom with the shockwave one: the leading-order conservation equation of the
Brown-York stress tensor (32) should be consistent with the Einstein equation
(27). This additional constraint enables us to determine the overall scaling
ambiguity $\beta$. Inserting (48) into the leading-order fluid equation (32),
we obtain

$\beta
r\partial^{2}H_{uu}=\frac{1}{2}\kappa^{2}r_{c}e^{\alpha\tau}\mathcal{T}_{uu}.$
(49)

By judiciously choosing the value of $\beta$ to be

$\beta=-\frac{r_{c}e^{\alpha\tau}}{4r}=\frac{u}{v},$ (50)

we discover that $\order{\varepsilon^{2}}$ conservation equation in the fluid-
dual metric matches exactly with the shockwave equation of motion (27).

In terms of the shockwave degree of freedom $X^{v}$ (30), the conservation
equations of the Brown-York stress tensor become

$\displaystyle\order{\varepsilon^{2}}:\qquad-\frac{1}{2}\partial^{2}\partial_{u}X^{v}=\kappa^{2}\mathcal{T}_{uu}$
(51) $\displaystyle\order{\varepsilon^{3}}:\qquad\alpha
u^{2}\partial_{i}\partial_{u}^{2}X^{v}+2\alpha
u\partial_{i}\partial_{u}X^{v}-4r_{c}u\partial_{i}\partial_{u}\partial^{2}X^{v}\approx\frac{8}{\alpha
u}\partial_{i}P$ (52)

In the second line, we have used
$\partial_{\tau}H_{uu}=\frac{\alpha}{2}u\partial_{u}H_{uu}$, since
$\partial_{v}H_{uu}=0$. Additionally, we have ignored terms with $X_{v}$
appearing twice. Regardless, the $\mathcal{O}(\varepsilon^{3})$ equation here
has no contribution from the stress-energy tensor sourced by the shockwave, as
we already anticipated in (37). Note that the $\mathcal{O}(\varepsilon^{2})$
equation contributes to the third term in the $\mathcal{O}(\varepsilon^{3})$
equation, leading to

$\displaystyle\order{\varepsilon^{3}}:\qquad
2e^{-\alpha\tau/2}P\approx\sqrt{\frac{r_{c}}{\alpha}}\partial_{\tau}^{2}X^{v}+\frac{\sqrt{\alpha
r_{c}}}{2}\partial_{\tau}X^{v}+2(\kappa
r_{c})^{2}e^{\alpha\tau/2}\mathcal{T}_{uu},$ (53)

where we used $(\tau,r,x^{i})$ coordinates, and the external source
$\mathcal{T}_{uu}$ for the shockwave (40) is given by (28). Note that we have
omitted nonlinear coupling terms in (52) and (53). Because time derivatives do
not act on the nonlinear coupling terms, coordinate transformation from
$(u,v,x^{i})$ to $(\tau,r,x^{i})$ coordinates leave them invariant.

So far, all of the calculations are performed at the classical level. However,
by instead promoting our source to a quantum operator, we will find a
structure quite similar to Verlinde_2022 ; Zhang:2023mkf . Specifically, we
introduce quantum light-ray operators associated with the shockwave degrees of
freedom and impose the t’Hooft commutation relation. Our goal is to study the
two-point correlation functions on the cutoff hypersurface $\Sigma_{c}$, which
is timelike and boost invariant. In the next section, we will thus generalize
the spacetime fluctuations on the null hypersurface Verlinde_2022  to a
timelike hypersurface, by utilizing the cutoff fluid approach that we
constructed in the previous sections.

##  4 Spacetime Fluctuations from Quantum Sources

In this section, we explore how spacetime fluctuations from quantum sources
influence the behavior of fluids living on $\Sigma_{c}$. In particular, we
will utilize the ’t Hooft commutation relations tHooft:1996rdg  to modify the
classical incompressibility equation to a stochastic differential equation.
Then we follow the procedure in Zhang:2023mkf  to solve the differential
equation and to compute the variance in photon traversal time.

Our critical assumption is that due to vacuum energy fluctuations,
$\mathcal{T}_{uu}$ is no longer the stress-energy tensor of a classical
shockwave. Rather, $\mathcal{T}_{uu}$ encapsulates features of quantum
fluctuations. We first utilize a commutation relation between $H_{vv}$ and
$\mathcal{T}_{uu}$, in close analogy to those proposed by ’t Hooft
Verlinde_2022 ; Zhang:2023mkf :

$\commutator{\mathcal{T}_{uu}(x)}{H_{vv}(x^{\prime})}=i\delta^{4}(x-x^{\prime}),$
(54)

where $x$ denotes coordinates in full spacetime dimensions. In writing Eq.
(54) we have assumed that $\mathcal{T}_{uu}$ and $H_{vv}$ are quantum
operators.

Here we have introduced an additional degree of freedom $H_{vv}$, which
classically lives only on the past horizon. Although it is true that along
either the future or past light front, only $H_{uu}$ or $H_{vv}$ is non-
vanishing classically, in the present case $\mathcal{T}_{uu}$ and $H_{vv}$ act
as quantum mechanical conjugate operators, as discussed in Refs. Verlinde_2022
; He:2023qha . Their commutation relation (54) is formally evaluated at the
bifurcate horizon. Nevertheless, we can heuristically argue Banks:2021jwj ;
Zurek:2022xzl ; Zhang:2023mkf  that a single causal diamond is foliated by a
series of nested causal diamonds (see Fig. 1), each separated by a microscopic
length scale $\tilde{\ell}_{p}$, known as the decoherence length. Beyond this
length scale, the subsequent diamonds become uncorrelated Banks:2021jwj ;
Zurek:2022xzl ; Zhang:2023mkf . This argument allows us to identify a series
of bifurcate horizons along the future and past light front, enabling us to
introduce $H_{vv}$ as an ultra local quantum degree-of-freedom. Crucially, the
notion of a nested causal diamond gives rise to an accumulated quantum
uncertainty in the photon traversal time Banks:2021jwj ; Zurek:2022xzl ;
Zhang:2023mkf . Next, we proceed to compute this uncertainty in detailed
steps.

Figure 1: Minkowski Rindler space that represents a slice of a Minkowski
diamond of size $L$. Blue and red dashed lines represent a series of nested
causal diamonds along the past (future) light front. The shaded blue/red
region is the “smearing” of the diamond due to quantum fluctuations modeled by
gravitational shockwaves. This procedure is discussed in detail in Ref.
Zhang:2023mkf . In this paper, we aim to describe the fuzzing of the light
front in a Lorentz invariant manner. This leads to a hyperbolic cutoff surface
(green curve), denoted as $\Sigma_{c}$. The proper distance between
$\Sigma_{c}$ and the bifurcate Rindler horizon is given by the relation
$-uv=4r_{c}/\alpha$.

Applying the commutator (54), we immediately discover that the classical
shockwave equation of motion (27), or equivalently, the incompressibility
equation for fluids (34) is now a differential equation involving quantum
commutators

$-\partial^{2}\commutator{H_{uu}(u,\bm{x})}{H_{vv}(v,\bm{x}^{\prime})}=i\ell_{p}^{2}\delta(u-u_{0})\delta(v-v_{0})\delta^{2}(\bm{x}-\bm{x}^{\prime}),$
(55)

where we have expressed $\delta^{4}(x-x^{\prime})$ in the $(u,v,x^{i})$
coordinates and expressed $\kappa^{2}=\ell_{p}^{2}$ explicitly. Note that the
operator $\partial^{2}$ only acts on the un-primed quantity $H_{uu}$. Eq. (55)
is the Minkowski limit of the AdS spacetime studied in Ref. Zhang:2023mkf . By
going through an analogous procedure, we obtain the following commutation
relations

$\commutator{H_{uu}(u,\bm{x})}{H_{vv}(v,\bm{x}^{\prime})}=i\ell_{p}^{2}\delta(u-u_{0})\delta(v-v_{0})f(\bm{x};\bm{x}^{\prime}),$
(56)

where $f(\bm{x},\bm{x}^{\prime})$ is given by Eq. (26).

Because $\mathcal{T}_{uu}$ is stochastic, it naturally implies that
$\expectationvalue{H_{uu}}$ and $\expectationvalue{H_{vv}}$ vanish. However,
the variance $\expectationvalue{H_{uu}^{2}}$ (similarly for $u\to v$) in
general does not vanish. To compute the variance, we invoke the Robertson
uncertainty principle Verlinde_2022 ; Zhang:2023mkf

$\expectationvalue{H_{uu}^{2}}\expectationvalue{H_{vv}^{2}}\geq\absolutevalue{\frac{1}{2i}\expectationvalue{\commutator{H_{uu}}{H_{vv}}}}^{2}=\quantity(\frac{\ell_{p}^{2}}{2})^{2}\quantity[\delta(u-u_{0})\delta(v-v_{0})f(\bm{x};\bm{x}^{\prime})]^{2}.$
(57)

Assuming that both are minimum uncertainty states, then the quantities
$\expectationvalue{H_{uu}^{2}}$ and $\expectationvalue{H_{vv}^{2}}$ are equal:

$\expectationvalue{H_{uu}^{2}}=\expectationvalue{H_{vv}^{2}}=\frac{\ell_{p}^{2}}{2}\delta(u-u_{0})\delta(v-v_{0})f(\bm{x};\bm{x}^{\prime}).$
(58)

In order to further evaluate $\expectationvalue{H_{uu}^{2}}$ and
$\expectationvalue{H_{vv}^{2}}$, we follow the heuristic argument of nested
causal diamonds and statistical independence once again. This line of
reasoning follows precisely the logic presented in details in Refs.
Banks:2021jwj ; Zhang:2023mkf , and we will not reproduce it in this paper.

However, one crucial distinction sets our current paper apart from
Zhang:2023mkf . With the help of the cutoff fluid approach, we implement a
“smeared-out” horizon (Fig. 1), using $\Sigma_{c}$ so we can regularize the
appropriate delta function in a manifestly Lorentz invariant manner. The first
step in our regularization procedure is to transform the lightcone coordinates
$(u,v,x^{i})$ into the ingoing coordinates $(\tau,r,x^{i})$ naturally adapted
to describe $\Sigma_{c}$. The two dimensional delta function in the $(u,v)$
coordinates are written explicitly in the $(\tau,r)$ coordinates as:

$\displaystyle\delta(\tau(u)-\tau_{0})$
$\displaystyle=\frac{\delta(u-u_{0})}{\absolutevalue{\tau^{\prime}(u)}}=\frac{\alpha
u}{2}\delta(u-u_{0}),$ (59) $\displaystyle\delta(r-r_{0})$
$\displaystyle=\delta\quantity(-\frac{\alpha uv}{4}-r_{0})=\delta(\alpha
uv/4+r_{0})=\frac{4}{\alpha u}\delta(v-v_{0}),$ (60)

where $\tau^{\prime}(u)=\partialderivative*{\tau}{u}$. The function
$\delta(\tau-\tau_{0})$ keeps track of the causal development of an observer
traversing on $\Sigma_{c}$, while $\delta(r-r_{0})$ sets the width of the
cutoff surface $\Sigma_{c}$. Therefore, we will replace $\delta(r-r_{0})$ with
a suitably regularized kernel. After first rescaling the argument of
$\delta(r-r_{0})$ we obtain

$\delta(r-r_{0})=\delta\quantity(\frac{\alpha
uv}{4}+r_{0})=\frac{4}{\alpha}\delta\quantity(uv+\frac{4r_{0}}{\alpha}).$ (61)

Next, we regularize the delta function with a Poisson kernel444The delta
function can be represented by various kernels, but choosing the Poisson
kernel allows us to keep consistency with the previous work Zhang:2023mkf .

$\delta\quantity(uv+\frac{4r_{0}}{\alpha})=\lim_{\epsilon\to
0}\frac{2}{\pi}\frac{\epsilon}{\epsilon^{2}+(uv+4r_{0}/\alpha)^{2}}\approx\frac{2}{\pi}\frac{1}{\epsilon}.$
(62)

In obtaining the last quantity, we have evaluated Eq. (62) on $\Sigma_{c}$,
where $r_{0}=r_{c}$ and $uv$ satisfies the relation

$uv|_{\Sigma_{c}}=-\frac{4r_{c}}{\alpha}.$ (63)

The parameter $\epsilon$ represents the width of $\delta(uv+4r_{c}/\alpha)$.
In order to satisfy the Lorentz invariance condition, a particularly
convenient candidate555There is an $\order{1}$ number uncertainty in choosing
$\epsilon$; however, this number can be absorbed into a redefinition of the
reduced Planck length $\tilde{\ell}_{p}$. Also note that this $\epsilon$ is
independent of the hydrodynamic expansion parameter $\varepsilon$. for
$\epsilon$ is that $\epsilon=4r_{c}/\alpha$. Setting $\epsilon=4r_{c}/\alpha$
gives us

$\frac{4}{\alpha}\delta\quantity(uv+\frac{4r_{c}}{\alpha})\approx\frac{2}{\pi
r_{c}}.$ (64)

Now we have all the ingredients to evaluate $\expectationvalue{H_{uu}^{2}}$.
First, we apply the definition of the “light ray” operator (30) as in
Verlinde_2022 ; Zhang:2023mkf , and perform a coordinate transformation to the
ingoing Rindler coordinates $(\tau,r,x^{i})$

$\displaystyle\begin{split}\expectationvalue{H_{uu}^{2}}&\equiv\expectationvalue{\partial_{u}X^{v}(u,\bm{x})\partial_{u^{\prime}}X^{v}(u^{\prime},\bm{x}^{\prime})}\\\
&=\expectationvalue{\quantity(\partialderivative{\tau}{u})\partial_{\tau}X^{v}(\tau,\bm{x})\quantity(\partialderivative{\tau^{\prime}}{u^{\prime}})\partial_{\tau^{\prime}}X^{v}(\tau,\bm{x}^{\prime})}\\\
&=\frac{4}{\alpha
r_{c}}e^{-\alpha(\tau+\tau^{\prime})/2}\expectationvalue{\partial_{\tau}X^{v}(\tau,\bm{x})\partial_{\tau^{\prime}}X^{v}(\tau^{\prime},\bm{x}^{\prime})},\end{split}$
(65)

where we have used $\partialderivative*{\tau}{u}=2/(\alpha u)$ and the
explicit relation between $u$ and $\tau$ (23).

In the next step, we substitute in the delta function of the ingoing Rindler
time $\tau$ (59), as well as the regularized delta function of the radial
coordinate $r$ in terms of a Poisson kernel of a width $\pi r_{c}/2$ (64)

$\expectationvalue{\partial_{\tau}X^{v}(\tau,\bm{x})\partial_{\tau^{\prime}}X^{v}(\tau^{\prime},\bm{x}^{\prime})}=\frac{\ell_{p}^{2}\alpha
r_{c}}{8\pi
r_{c}}e^{\alpha(\tau+\tau^{\prime})/2}\delta(\tau-\tau^{\prime})f(\bm{x};\bm{x}^{\prime}).$
(66)

Eq. (66) takes on the form of a first order Langevin equation. Although the
factor of $r_{c}$ seems to cancel out between the chosen delta function
regularization scheme and the coordinate transformation between $u$ and
$\tau$, we will soon discover that another hidden factor of $r_{c}$ will show
up once we relate the ingoing Rindler time $\tau$ back to the Minkowski global
time $T$ relevant for a laboratory observer.

In order to compute the two-point correlation function of $X^{v}$, we
integrate both sides of (66) over $\tau^{\prime}$ and $\tau$ readily to obtain

$\displaystyle\begin{split}\expectationvalue{X^{v}(\bm{x})X^{v}(\bm{x}^{\prime})}&=\frac{\ell_{p}^{2}\alpha
r_{c}}{8\pi
r_{c}}f(\bm{x};\bm{x}^{\prime})\int_{\tau_{i}}^{\tau_{f}}\differential{\tau}\int_{-\infty}^{\tau}\differential{\tau^{\prime}}e^{\alpha(\tau+\tau^{\prime})/2}\delta(\tau-\tau^{\prime})\\\
&=\frac{\ell_{p}^{2}}{8\pi}\quantity(\frac{\alpha\Delta
u^{2}}{r_{c}})f(\bm{x};\bm{x}^{\prime}),\end{split}$ (67)

where the Lorentz invariant combination $\alpha/r_{c}$ will be fixed in terms
of the size of the causal diamond and the scale of quantum fluctuations
shortly. The quantity

$\Delta
u^{2}\equiv\frac{r_{c}}{\alpha}\quantity(e^{\alpha\tau_{f}}-e^{\alpha\tau_{i}})$
(68)

denotes the square of the total elapsed coordinate time in $u$ according to
the coordinate transformation (23). The light cone coordinate666Note that our
convention of the light-cone coordinates differs from Zhang:2023mkf . As a
result, $(u,v)$ directions in Fig. 1 are flipped, compared to the
corresponding figure in Zhang:2023mkf . $u$ is related to Minkowski
coordinates $(T,Z)$ via $u=T+Z$. We immediately see that the elapsed time
$\Delta u=\Delta T$ for an observer at a fixed location, e.g. $Z=L$. Photon
traversal time measured by a free falling observer on the boundary of a causal
diamond of size $L$ is

$\Delta T_{\rm r.t.}=2L=\Delta u$ (69)

in the absence of any spacetime fluctuations.

Quantum fluctuations give rise to a “smeared-out” horizon, $\Sigma_{c}$,
represented by the green curve Fig. 1. This cutoff surface intersects with the
boundary of the causal diamond at a distance $\tilde{\ell}_{p}$ from either
bottom or top of the diamond. As in Refs. Banks:2021jwj ; Zurek:2022xzl ;
Verlinde_2022 ; Zhang:2023mkf , $\tilde{\ell}_{p}$ characterizes the scale of
quantum fluctuations. From Fig. 1, it is clear that the Minkowski global time
difference between the light front and $\Sigma_{c}$ is $\delta
T=\tilde{\ell}_{p}$. This implies that $\delta u=\tilde{\ell}_{p}$ for $Z=L$.
Moreover, $\delta v=\delta u=\tilde{\ell}_{p}$ by symmetry. We choose to
evaluate the uncertainty at the future tip of the diamond, at which future
light ray intersects with the worldline of a free-falling observer at $Z=L$.
These constraints along with Eq. (63) imply that

$\frac{4r_{c}}{\alpha}=-(u_{f}-\delta u)(v_{f}-\delta v)\approx u_{f}\delta
v\approx 2L\tilde{\ell}_{p},$ (70)

where $(u_{f},v_{f})=(2L,0)$ corresponds to the location at the top of the
diamond. Putting Eq. (70) back into Eq. (67) gives us the final answer

$\expectationvalue{X^{v}(\bm{x})X^{v}(\bm{x}^{\prime})}=\frac{\ell_{p}^{2}}{\pi}\quantity(\frac{L}{\tilde{\ell}_{p}})f(\bm{x};\bm{x}^{\prime}).$
(71)

Since $X^{v}$ and $X^{u}$ are related via time-reversal symmetry
$X^{v}\leftrightarrow X^{u}$, we immediately write down the two-point function
of $\expectationvalue{X^{u}(\bm{x})X^{u}(\bm{x}^{\prime})}$

$\expectationvalue{X^{u}(\bm{x})X^{u}(\bm{x}^{\prime})}=\frac{\ell_{p}^{2}}{\pi}\quantity(\frac{L}{\tilde{\ell}_{p}})f(\bm{x};\bm{x}^{\prime}).$
(72)

The total time delay is the sum of Eqs. (71) and (72)

$\displaystyle\begin{split}\Delta T_{\rm
r.t.}^{2}(\bm{x};\bm{x}^{\prime})&\equiv\expectationvalue{X^{v}(\bm{x})X^{v}(\bm{x}^{\prime})}+\expectationvalue{X^{u}(\bm{x})X^{u}(\bm{x}^{\prime})}\\\
&=2\frac{\ell_{p}^{2}}{\pi}\quantity(\frac{L}{\tilde{\ell}_{p}})f(\bm{x};\bm{x}^{\prime}).\end{split}$
(73)

In four dimensions, there is no distinction between $\tilde{\ell}_{p}$ and
$\ell_{p}$ Zurek:2022xzl ; Verlinde_2022 ; in other words,
$\tilde{\ell}_{p}\cong\ell_{p}$, and Eq. (73) reduces to

$\Delta T_{\rm r.t.}^{2}(\bm{x};\bm{x}^{\prime})\stackrel{{\scriptstyle
d=4}}{{=}}\frac{2\ell_{p}L}{\pi}f(\bm{x};\bm{x}^{\prime}).$ (74)

This equation is the main result of this section, and it is consistent with
the findings from past literature Verlinde:2019xfb ; Verlinde:2019ade ;
Banks:2021jwj ; Zurek:2022xzl ; Verlinde_2022 ; Zhang:2023mkf  that used
different and complementary means. Note that the variance in the photon
traversal time (74) has an overall scaling dependence on both UV ($\ell_{p}$)
and IR ($L$) scales. This kind of UV/IR mixing is most frequently encountered
in the context of Brownian motion feynman1964 . The hallmark of Brownian
motion is that the variance of a Brownian particle depends on two quantities:
1) a diffusion coefficient $D$, an inter-molecular length scale proportional
to the mean free path between collisions, and 2) total time elapsed $t$,
usually set by laboratory measurement, takes on a macroscopic scale. In our
case, we have found that quantum fluctuations near the light front of a
Minkowski causal diamond also exhibit Brownian-like behavior. This result
offers an exciting new avenue to test quantum fluctuations in gravity using
precise laser interferometer measurement Verlinde:2019ade ; Bub:2023bfi .

##  5 Summary and Outlook

We have studied the relation between fluid and shockwave geometries, showing
via Petrov classification that a potential fluid contains the same physical
information as a shockwave geometry. We then proceeded to add a source to the
fluid equation. The stress energy source is quantum in nature, with an
amplitude given by a fundamental uncertainty in spacetime. We solved the
equation of motion for spacetime and found an uncertainty in light travel time
that depends both on the UV scale (a Planck length), as well as the light-
crossing time of the causal diamond.

We hope the relation between the shockwave and potential fluid dual geometries
will allow for direct calculation in the sourced potential fluid itself,
_e.g._ of the two point function for the potential fluid, including its
dependence on time and transverse directions. Although technically more
difficult, extending beyond the transverse planar limit should allow for
understanding the angular dependence within finite causal diamonds. Ideally we
could also use the fluctuation-dissipation theorem to compute the fluid
diffusion constant from the quantum source. We leave these and other
extensions to future work.

###### Acknowledgements.

We thank Tom Banks for collaboration on related directions Banks:2023wua , as
well as Lars Aalsma, Samarth Chawla, Tom Hartman, Temple He, Maulik Parikh,
Jude Pereira, Allic Sivaramakrishnan, Marika Taylor, and Raphaela Wutte for
insightful discussions. YZ and KZ are supported by the Heising-Simons
Foundation “Observational Signatures of Quantum Gravity” collaboration grant
2021-2817, the U.S. Department of Energy grant DE-SC0011632, and the Walter
Burke Institute for Theoretical Physics. KZ is also supported by a Simons
Investigator award. C.K. and S.-E.B. are supported by the U.S. Department of
Energy under grant number DE-SC0019470 and by the Heising-Simons Foundation
Observational Signatures of Quantum Gravity collaboration grant 2021-2818.

##  Appendix A Derivation of the Brown-York Stress Tensor

In this appendix, we derive the Brown-York stress tensor in the non-
relativistic hydrodynamic expansion (15).

To end up with this result, we first describe the near-equilibrium fluid
metric by allowing the fluid variables $u_{a},p$ to depend on $(\tau,x^{i})$
Compere:2011dx ; Compere:2012mt ; Pinzani-Fokeeva:2014cka . In the derivative
expansion777The derivative expansion is defined as an order of different
scaling $\tilde{\varepsilon}$ such that
$\partial_{r}\sim\mathcal{O}(1),\,\partial_{a}\sim\mathcal{O}(\tilde{\varepsilon})$.,
the zeroth order seed metric is written as

$\displaystyle ds^{2}=-2p(x)u_{a}(x)dx^{a}dr+\left[\gamma_{ab}-\alpha
p^{2}(x)\left(r-r_{c}\right)u_{a}(x)u_{b}(x)\right]dx^{a}dx^{b}$ (75)

where $x$ denote $(\tau,x^{i})$.

In this work, we are interested in the non-relativistic hydrodynamic limit,
which agrees with the near-horizon expansion Bredberg:2011jq ; Keeler:2020rcv

$v_{i}\sim\order{\varepsilon},\quad
P\sim\order{\varepsilon^{2}},\quad\partial_{i}\sim\order{\varepsilon},\quad\partial_{\tau}\sim\order{\varepsilon^{2}}.$
(76)

In order to do that, consider the non-relativistic expansion of the pressure
and four-velocity

$\displaystyle p=\frac{1}{\sqrt{\alpha r_{c}-2P}}\approx\frac{1}{\sqrt{\alpha
r_{c}}}+\frac{P}{(\alpha
r_{c})^{3/2}}+\mathcal{O}\left(\varepsilon^{4}\right)\,,$ (77) $\displaystyle
u_{0}=\frac{-\alpha r_{c}}{\sqrt{\alpha r_{c}-v^{2}}}\approx\sqrt{\alpha
r_{c}}\left(1+\frac{v^{2}}{2\alpha
r_{c}}\right)+\mathcal{O}\left(\varepsilon^{4}\right)\,,$ (78) $\displaystyle
u_{i}=\frac{v_{i}}{\sqrt{\alpha r_{c}-v^{2}}}\approx\frac{v_{i}}{\sqrt{\alpha
r_{c}}}+\mathcal{O}\left(\varepsilon^{3}\right)\,,$ (79)

Here, $v_{i}$ is the non-relativistic fluid velocity in the transverse
direction, and $v^{2}\equiv v^{i}v_{i}$. $P$ is the non-relativistic pressure
of the fluid on $\Sigma_{c}$. We perform the non-relativistic hydrodynamic
expansion in terms of the parameter $\varepsilon$, which keeps track of the
scaling of various quantities. Combining these and expanding the metric in
power of $\varepsilon$, the resulting metric becomes Bredberg:2011jq ;
Keeler:2020rcv

$\displaystyle\begin{split}ds^{2}=&-(\alpha r)d\tau^{2}+2d\tau
dr+\delta_{ij}dx^{i}dx^{j}\\\
&-2\quantity(1-\frac{r}{r_{c}})v_{i}dx^{i}d\tau-2\frac{v_{i}}{\alpha
r_{c}}dx^{i}dr\\\
&+\quantity(1-\frac{r}{r_{c}})\quantity[(v^{2}+2P)d\tau^{2}+\frac{v_{i}v_{j}}{\alpha
r_{c}}dx^{i}dx^{j}]+\quantity(\frac{v^{2}}{\alpha r_{c}}+\frac{2P}{\alpha
r_{c}})d\tau dr\\\ &-\frac{\left(r^{2}-r_{c}^{2}\right)}{\alpha
r_{c}}\partial^{2}v_{i}dx^{i}d\tau+\order{\varepsilon^{3}}.\end{split}$ (80)

The unit normal vector on $\Sigma_{c}$ is given by Bredberg:2011jq

$n^{\mu}\partial_{\mu}=\frac{1}{\sqrt{\alpha
r_{c}}}\partial_{\tau}+\sqrt{\alpha r_{c}}\quantity(1-\frac{P}{\alpha
r_{c}})\partial_{r}+\frac{v^{i}}{\sqrt{\alpha
r_{c}}}\partial_{i}+\order{\varepsilon^{3}},$ (81)

and after a short computation, we obtain the extrinsic curvature
$K_{ab}=\frac{1}{2}\mathcal{L}_{n}\gamma_{ab}$ on $\Sigma_{c}$ Bredberg:2011jq

$K_{ab}dx^{a}dx^{b}=-\frac{\alpha\sqrt{\alpha r_{c}}}{2}d\tau^{2}+\frac{\alpha
v_{i}}{\sqrt{\alpha r_{c}}}dx^{i}d\tau-\frac{\alpha(v^{2}+P)}{2\sqrt{\alpha
r_{c}}}d\tau^{2}-\frac{\alpha(v_{i}v_{j}-2r_{c}\partial_{(i}v_{j)})}{2(\alpha
r_{c})^{3/2}}dx^{i}dx^{j}+\mathcal{O}(\varepsilon^{3})\,.$ (82)

Evaluating the Brown-York stress tensor defined in (10), we obtain the Eq.
(15)

$\kappa^{2}\,T_{ab}dx^{a}dx^{b}=\frac{\alpha v^{2}}{\sqrt{\alpha
r_{c}}}d\tau^{2}-\frac{2\alpha v_{i}}{\sqrt{\alpha
r_{c}}}dx^{i}d\tau+\frac{\alpha}{\sqrt{\alpha
r_{c}}}dx^{i}dx_{i}+\frac{P\delta_{ij}+v_{i}v_{j}}{r_{c}\sqrt{\alpha
r_{c}}}dx^{i}dx^{j}-\frac{2\partial_{(i}v_{j)}}{\sqrt{\alpha
r_{c}}}dx^{i}dx^{j}+\order{\varepsilon^{3}}\,.$ (83)

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