Gravitational waves and Black Hole perturbations in acoustic analogues

Coviello, Chiara; Chiofalo, Maria Luisa; Grasso, Dario; Liberati, Stefano; Mannarelli, Massimo; Trabucco, Silvia

doi:10.1116/5.0241559

Phonons in Bose–Einstein condensates propagate as massless scalar particles on top of an emergent acoustic metric. This hydrodynamics/gravity analogy can be exploited to realize acoustic black holes, featuring an event horizon that traps phonons. The authors show that by an appropriate external potential, gravitational wave-like perturbations of the acoustic metric can be produced. Such perturbations can be used to excite an acoustic black hole, which should then relax by phonon emission.

I. INTRODUCTION

In addition to their interest as prominent astrophysical objects, black holes (BHs) serve as perfect playgrounds for theoretical physicists interested to explore the deepest and possibly intertwined aspects of quantum mechanics and general relativity. Indeed, a full understanding of BHs is likely to require insight from both fields. This line of research is of great physical interest, because solving open problems regarding the nature of BHs could lead us to significantly improve our comprehension of gravity.

The detection of gravitational waves (GWs) originating from coalescing binary BHs has opened up new avenues for probing these objects.¹ This discovery holds also the potential to reveal, through future generation detectors, aspects of their quantum mechanical nature.² However, like other astronomical observations, it is inherently constrained by the challenge of studying uncontrolled physical systems in the cosmos.

Recently, rapid advancements in quantum technologies have begun to address this limitation by enabling the creation of analog (sonic) BHs in laboratory settings, offering a controlled environment to explore similar phenomena. These systems, typically created through quantum fluids, share the same kinematic properties as their astrophysical counterparts by mimicking the same metric, while the dynamics of the background will generally differ. The connection between hydrodynamics and gravity lies in the fact that the low-energy phonons propagate as massless scalar fields on an effective acoustic metric. Once the metric is determined, the comparison between sonic and astrophysical systems can provide valuable insight into specific gravitational theories.³ One of the most remarkable achievements of this approach has been the quantum simulation of the Hawking radiation from a sonic horizon in a Bose–Einstein condensate (BEC)⁴ implementing the idea proposed by Unruh for a conventional fluid.⁵ Although sonic BHs are clearly very different from astrophysical BHs—most importantly the former do not obey Einstein equations—they can, however, test universal properties common to both entities. For example, in Ref. 6, it has been shown that, similarly to what computed for astrophysical BHs, the expression of the Hawking temperature of a sonic BH in a BEC springs from quantum mechanics and the assumption of the area law for the horizon entropy.

Following that line of reasoning, we employ analog systems to explore open questions about sonic BHs and gravity. In particular, we show how specifically designed analog systems can be used to mimic the propagation of GWs in a BEC. In general relativity, GWs can be treated as classical perturbations of the background metric. Here, we mimic such metric perturbations employing appropriate external potentials. We will refer to the variations of the acoustic metric produced by such external potentials as GW-like perturbations. We discuss the propagation of GW-like perturbations onto two different metrics: vacuum and analog BH. The study of GW-like perturbations in hydrodynamic systems was first discussed in Ref. 7, which focused on the interaction, and possible detection, of real GWs with BECs. Here, we treat complex and interesting physical situations as that of a GW-like perturbation impinging onto a sonic BH. The systems discussed in this paper pave the way for investigating significant theoretical questions that can also be experimentally tested.

This paper is organized as follows. In Sec. II, we review the acoustic metric in the context of non-relativistic BECs. Section III demonstrates how to simulate a GW perturbation within a flat background acoustic metric in a BEC. We then extend this analysis in Sec. IV, where we examine how to create GW-like perturbations on the acoustic BH metric within a BEC environment. At the beginning of this section, we introduce a vortex geometry, which serves as the foundation for our model of an acoustic BH disturbed by GW-like perturbations. Finally, in Sec. V, we summarize our findings and discuss potential future research directions enabled by our models.

II. THEORETICAL SETUP

In this section, we briefly discuss the emergence of the acoustic metric in non-relativistic BECs at vanishing temperature.³ This is achieved by an appropriate scale separation between the background fluid and the low-energy (phonon) excitations.

For a Bose gas in the dilute gas approximation, where the average spacing between particles is greater than the scattering length a, the true interaction potential between particles can be approximated using the Fermi pseudo-potential. Under this approximation and employing the Hartree–Fock method, the total wavefunction of the system, consisting of N bosons, can be represented as a product of single-particle functions. This allows us to describe the diluted Bose gas through a single quantum field,

\hat{Ψ}

⁠, which satisfies the Gross–Pitaevskii equation⁸

i ℏ \frac{\partial}{\partial t} \hat{Ψ} = (- \frac{ℏ^{2}}{2 m} \nabla^{2} + V_{ext} (x) + γ (a) {\hat{Ψ}}^{†} \hat{Ψ}) \hat{Ψ},

(1)

where m is the mass of the bosons,

V_{ext}

is the external potential, and

γ (a) = \frac{4 π a ℏ^{2}}{m},

(2)

parameterizes the strength of the interaction between bosons. The quantum field can be divided into a macroscopic (classical) condensate

ψ (t, x) = ⟨ \hat{Ψ} (t, x) ⟩

⁠, and a fluctuation

\hat{φ} (t, x)

⁠,

\hat{Ψ} (t, x) = ψ (t, x) + \hat{φ} (t, x),

(3)

where the expectation value is defined as the trace of the density matrix times the operator itself. This splitting indicates a separation of scales, with the fluctuation

\hat{φ} (t, x)

being linked to rapid and short-distance quantum fluctuations of the wavefunction, while the background field

ψ (t, x)

accounts for the large-distance and long-time scale variations. Neglecting back-reactions of the quantum fluctuations on the background, the equation for the classical wave function is given by

i ℏ \frac{\partial}{\partial t} ψ = (- \frac{ℏ^{2}}{2 m} \nabla^{2} + V_{ext} (x) + γ (a) n_{c}) ψ,

(4)

with

n_{c} \equiv | ψ (t, x) |^{2}

the condensate density. Equation is the most commonly adopted expression to approximate the dynamics of the condensate wavefunction. The linearized quantum fluctuations, on the other hand, follow the Bogoliubov–de Gennes equation:

\begin{matrix} i ℏ \frac{\partial}{\partial t} \hat{φ} (t, x) = (- \frac{ℏ^{2}}{2 m} \nabla^{2} + V_{ext} (x) + γ (a) 2 n_{c}) \hat{φ} (t, x) \\ + γ (a) ψ^{2} (t, x) {\hat{φ}}^{†} (t, x), & , \end{matrix}

(5)

which depends on the solution of mean-field equation. This set of equations arises from neglecting back-reactions within the self-consistent mean-field approximation. Would we account for the back-reaction of quantum fluctuations on the dynamics of the condensate wavefunction, the equations governing the mean-field and quantum fluctuations become inseparable.

In order to describe the quantum fluctuations as massless scalar field propagating on top of an emergent acoustic metric, we use the Madelung representation for the condensate wavefunction

ψ (t, x) = \sqrt{n_{c} (t, x)} e^{- i θ (t, x) / ℏ},

(6)

and the quantum acoustic representation for the fluctuation

\hat{φ} (t, x) = e^{- i θ / ℏ} (\frac{1}{2 \sqrt{n_{c}}} {\hat{n}}_{1} - i \frac{\sqrt{n_{c}}}{ℏ} {\hat{θ}}_{1}),

(7)

where

{\hat{n}}_{1}

and

{\hat{θ}}_{1}

are two real scalar fields corresponding to phonon density and phase fluctuations, respectively.³ Upon substituting this expression in Eq. , we obtain two coupled differential equations for

{\hat{n}}_{1}

and

{\hat{θ}}_{1}

⁠. These equations can be cast as a single nonlocal differential equation for

{\hat{θ}}_{1}

⁠, see Ref. 3 for more details. In the hydrodynamic limit, that is for length scales much larger than the healing length,⁸ such equation can be written as that of a quantum scalar field propagating over a curved background

□ {\hat{θ}}_{1} = \frac{1}{\sqrt{| g |}} \partial_{μ} (\sqrt{| g |} g^{μ ν} \partial_{ν} {\hat{θ}}_{1}) = 0,

(8)

with the so-called acoustic metric

g_{μ ν} (t, x) = \frac{n_{c}}{m c_{s}} (\begin{matrix} - [c_{s}^{2} - v^{2}] & - v_{j} \\ - v_{i} & δ_{i j} \end{matrix}),

(9)

where

c_{s} = \sqrt{\frac{γ (a) n_{c}}{m}},

(10)

is the adiabatic speed of sound and

v = \frac{\nabla θ}{m},

(11)

is the irrotational velocity field of the condensate. The 00 component of the metric vanishes when the fluid velocity equals the speed of sound; this corresponds to the acoustic horizon.

III. GRAVITATIONAL WAVE ACOUSTIC METRIC

So far we have seen that the propagation of the phonon field is formally equivalent to the propagation of a massless scalar field in curved spacetime, with the acoustic metric given in Eq. (9). We now consider to which extent the perturbations of the acoustic metric can be described as GWs. Notice that we consider two different types of perturbations: background perturbations, which are externally driven and correspond to the acoustic metric variations, and phonons, which are the quantum fluctuations of the system and can be described as massless modes propagating on the emergent acoustic metric. The separation between these two types of perturbations is necessary and can be ensured because the frequency of the external perturbation, being controllable, can be chosen arbitrarily.

We begin with considering GWs propagating in flat spacetime (GWs in a BH-like metric are instead discussed in Sec. IV). We first briefly recap how GWs are introduced in general relativity, then we show that GWs expressed in a particular gauge can be emulated by appropriate fluid perturbations.

A. Gravitational waves in flat spacetime

The metric we aim to reproduce is that of GWs in vacuum:⁹

γ_{μ ν} = η_{μ ν} + ε h_{μ ν},

(12)

where

ε ≪ 1

and we indicate with

γ_{μ ν}

being the gravitational metric tensor,

η_{μ ν}

being the Minkowski metric, while

h_{μ ν}

represents the metric perturbation. GWs are metric perturbations solution of the Einstein's field equations in the linearized theory. An important symmetry in this linearized framework is represented by coordinate (or gauge) transformations of the following form:

x^{μ} \to x^{' μ} = x^{μ} + ε^{'} ζ^{μ} (x),

(13)

where

ε^{'} ≪ 1

and

ζ^{μ} (x)

is a generic four-vector. This gauge transformation induces a metric transformation, which can be cast as a metric perturbation. Taking

ε = ε^{'}

⁠, at the first order in

ε

⁠, this transformation is

\begin{matrix} η_{μ ν} + ε h_{μ ν} (x) \to η_{μ ν} + ε h_{μ ν}^{'} (x^{'}) \\ = η_{μ ν} + ε (h_{μ ν} - \partial_{μ} ζ_{ν} - \partial_{ν} ζ_{μ}) . \end{matrix}

(14)

Thus, slowly varying infinitesimal diffeomorphisms are symmetries of linearized theory of gravity. In contrast, full general relativity has complete invariance under all coordinate transformations, not limited to infinitesimal changes.

By an appropriate gauge transformation, it is possible to set

h^{μ}_{μ} = 0

⁠, then in the Lorentz gauge

\partial^{ν} h_{μ ν} = 0,

(15)

the in vacuum propagation of the GW is described by the wave equation

(- \partial_{t}^{2} + c^{2} \nabla^{2}) h_{μ ν} = 0,

(16)

indicating that GWs propagate at the speed of light c.

One may further simplify the treatment of GWs using the transverse-traceless (TT) gauge, in this case the metric perturbation satisfies the following conditions:

h^{0 μ} = 0, h^{i}_{i} = 0, \partial^{j} h_{i j} = 0,

(17)

hence in the TT gauge a GW propagating along the z axis is described by

\begin{matrix} h_{μ ν}^{T T} (t, z) = (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & h_{+}^{T T} (t, z) & h_{\times}^{T T} (t, z) & 0 \\ 0 & h_{\times}^{T T} (t, z) & - h_{+}^{T T} (t, z) & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) \\ = (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & h_{+} & h_{\times} & 0 \\ 0 & h_{\times} & - h_{+} & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) cos (ω (t - z / c)), \end{matrix}

(18)

where

h_{+}

and

h_{\times}

denote the amplitudes of the “plus” and “cross” polarizations of the wave. As we will see, it is not possible to express the acoustic metric perturbation in this form. However, given the simplicity of the GWs in TT-gauge, we will start from this gauge to obtain an expression of

h_{μ ν}

that can be matched to the acoustic metric perturbation.

B. Method

In order to emulate a GW propagating in flat spacetime with a BEC, we have to cast the acoustic metric in Eq. (9) as in Eq. (12), that is a Minkowski background with a propagating GW perturbation. To this end, we have to determine how the BEC background perturbations change the acoustic metric and then look for a gauge transformation of the space coordinates such that $h^{μ ν}$ takes a form that can be matched to the acoustic metric perturbation.

More in details, the emulation of the GW with an acoustic analog is obtained with the following steps:

Acoustic metric perturbation: Perturb the fluid background quantities to induce an acoustic metric perturbation. For small perturbations, the metric in Eq. (9) can then be written as a background $g_{μ ν}^{0}$ plus a perturbation ${\tilde{g}}_{μ ν}$ ⁠:

$g_{μ ν} = g_{μ ν}^{0} + ε {\tilde{g}}_{μ ν} .$
(19)
GWs in different gauges: Exploit the gauge symmetry of general relativity, to find the metric of a GW that propagates in vacuum $h_{μ ν}$ in a form that can be matched to ${\tilde{g}}_{μ ν}$ ⁠.
Comparison of GW with the perturbed acoustic metric: Identify the fluid characteristics needed to match $g_{μ ν}^{0}$ with $η_{μ ν}$ and ${\tilde{g}}_{μ ν}$ with $h_{μ ν}$ ⁠.
Physical systems: Check whether the choice of the condensate quantities needed to match the acoustic metric with the gravitational metric are physical, meaning that the continuity and the Euler equations, as well as the irrotational condition are all satisfied. To this aim, we introduce an external potential which mimics a GW.

This approach is inspired by the work in Ref. 7, where a propagating homogeneous GW is simulated in a BEC. The main difference is that they do not consider the spatial dependence of the GW, while we do. This introduces technical complications, since the space dependence does not make it possible to work in Fermi normal coordinates⁹ for the step 2, i.e., the inertial limit of the proper detector frame, as done in Ref. 7. Because of that, we need to introduce a different suitable gauge, which is not the usual coordinate reference frame.

1. Acoustic metric perturbation

The acoustic metric in Eq. depends on the number density, the fluid velocity and speed of sound. Now we assume that there is an external potential that determines a space-time variation of these quantities

\begin{matrix} n_{c} \to n_{c} + ε δ n_{c}, \\ v_{i} \to v_{i} + ε δ v_{i}, \\ c_{s} \to c_{s} + ε δ c_{s}, \end{matrix}

(20)

on top of their corresponding averages

n_{c}

⁠,

v_{i}

⁠, and

c_{s}

⁠. Upon plugging the varied quantities in Eq. , one obtains a slight change of the acoustic metric. Using the coordinates

x^{μ} = (c_{s} t, x, y, z)

⁠, the unperturbed metric is given by

g_{μ ν}^{0} = \frac{n_{c}}{m c_{s}} (\begin{matrix} - (1 - \frac{v^{2}}{c_{s}^{2}}) & - \frac{v_{j}}{c_{s}} \\ - \frac{v_{i}}{c_{s}} & δ_{i j} \end{matrix}),

(21)

while the metric perturbation at first order in

ε

is

{\tilde{g}}_{μ ν} (t, x) = \frac{n_{c}}{m c_{s}} (\begin{matrix} {\tilde{g}}_{00} & {\tilde{g}}_{01} & {\tilde{g}}_{02} & {\tilde{g}}_{03} \\ {\tilde{g}}_{01} & {\tilde{g}}_{11} & 0 & 0 \\ {\tilde{g}}_{02} & 0 & {\tilde{g}}_{22} & 0 \\ {\tilde{g}}_{03} & 0 & 0 & {\tilde{g}}_{33} \end{matrix}),

(22)

where the metric components are

{\tilde{g}}_{00} = 2 \frac{v^{2}}{c_{s}^{2}} \frac{δ v}{v} - \frac{1}{2} \frac{(c_{s}^{2} + v^{2})}{c_{s}^{2}} \frac{δ a}{a} - \frac{1}{2} \frac{3 c_{s}^{2} - v^{2}}{c_{s}^{2}} \frac{δ n_{c}}{n_{c}},

(23)

{\tilde{g}}_{0 i} = - \frac{1}{2} \frac{v_{i}}{c_{s}} \frac{δ n_{c}}{n_{c}} + \frac{1}{2} \frac{v_{i}}{c_{s}} \frac{δ a}{a} - \frac{v_{i}}{c_{s}} \frac{δ v_{i}}{v_{i}},

(24)

{\tilde{g}}_{i j} = \frac{1}{2} (\frac{δ n_{c}}{n_{c}} - \frac{δ a}{a}) δ_{i j},

(25)

with

δ c_{s} / c_{s} = δ a / (2 a) + δ n_{c} / (2 n_{c})

⁠. Apart from a prefactor,

g_{μ ν}^{0}

written in Eq. is the flat spacetime metric only if

v_{i} = 0

for all i. With this choice, the perturbation of the acoustic metric becomes

{\tilde{g}}_{μ ν} = \frac{n_{c}}{m c_{s}} (\begin{matrix} - \frac{3}{2} \frac{δ n_{c}}{n_{c}} - \frac{1}{2} \frac{δ a}{a} & - \frac{δ v_{j}}{c_{s}} \\ - \frac{δ v_{i}}{c_{s}} & (\frac{1}{2} \frac{δ n_{c}}{n_{c}} - \frac{1}{2} \frac{δ a}{a}) δ_{i j} \end{matrix}) .

(26)

At this point, we have to find a gauge in which the GW metric can be written in the same form of the metric in Eq. (26).

2. GWs in different gauges

In this section, our goal is to derive an expression for

h_{μ ν}

⁠, representing a GW metric, which can be matched to

{\tilde{g}}_{μ ν}

⁠. We use the symmetry of linearized general relativity, as described in Eq. , which induces the transformation of

h_{μ ν}

outlined in Eq. . The traditional TT gauge used for GWs [refer to Eq. ] does not align with

{\tilde{g}}_{μ ν}

[see Eq. ] nor do Fermi normal coordinates.⁹ Consequently, we have to adopt a different reference frame, distinct from both the TT gauge and the local detector frame. We start from the TT gauge and then perform a linearized gauge transformation of the type given in Eq. with

\begin{matrix} ζ_{0} = (l x + b y + d z) (f h_{+} + g h_{\times}) cos (ω (t - z / c)), \\ ζ_{x} = \frac{1}{2} (x h_{+} + y h_{\times}) cos (ω (t - z / c)), \\ ζ_{y} = \frac{1}{2} (x h_{\times} - y h_{+}) cos (ω (t - z / c)), \\ ζ_{z} = 0, \end{matrix}

(27)

where

h_{+}

and

h_{\times}

are the amplitudes of the GW in the TT gauge and l, b, d, f, and g are real numbers. In this gauge the linearized Einstein's equations do not in general take the simple form given in Eq. . However, we do not need to determine such equations, because we know that under a gauge transformation

h_{μ ν}

transforms according to Eq. . Hence, plugging Eq. in Eq. , we derive the metric perturbation at first order in

ε

⁠,

h_{μ ν}^{'} (x^{' ρ})

⁠. Its nonvanishing components are

h_{13}^{'} (x^{' ρ}) = - \frac{ω}{2 c} (x^{'} h_{+} + y^{'} h_{\times}) sin (ω (t^{'} - z^{'} / c)),

(28)

h_{23}^{'} (x^{' ρ}) = - \frac{ω}{2 c} (x^{'} h_{\times} - y^{'} h_{+}) sin (ω (t^{'} - z^{'} / c)),

(29)

\begin{matrix} {h^{'}}_{00} (x^{' ρ}) = \frac{2 ω}{c} (l x^{'} + b y^{'} + d z^{'}) (f h_{+} + g h_{\times}) \\ \times sin (ω (t^{'} - z^{'} / c)), \end{matrix}

(30)

\begin{matrix} {h^{'}}_{01} (x^{' ρ}) = \frac{ω}{2 c} (x^{'} h_{+} + y^{'} h_{\times}) sin (ω (t^{'} - z^{'} / c)) \\ - l (f h_{+} + g h_{\times}) cos (ω (t^{'} - z^{'} / c)), \end{matrix}

(31)

\begin{matrix} {h^{'}}_{02} (x^{' ρ}) = \frac{ω}{2 c} (x^{'} h_{\times} - y^{'} h_{+}) sin (ω (t^{'} - z^{'} / c)) \\ - b (f h_{+} + g h_{\times}) cos (ω (t^{'} - z^{'} / c)), \end{matrix}

(32)

\begin{matrix} {h^{'}}_{03} (x^{' ρ}) = - d (f h_{+} + g h_{\times}) cos (ω (t^{'} - z^{'} / c)) - \frac{ω}{c} \\ \times (l x^{'} + b y^{'} + d z^{'}) (f h_{+} + g h_{\times}) sin (ω (t^{'} - z^{'} / c))), \end{matrix}

(33)

with

x^{' ρ} = (c t^{'}, x^{'}, y^{'}, z^{'})

⁠. Therefore, in this gauge, the GW is given by

h_{μ ν}^{'} = (\begin{matrix} h_{00}^{'} & h_{01}^{'} & h_{02}^{'} & h_{03}^{'} \\ h_{01}^{'} & 0 & 0 & h_{13}^{'} \\ h_{02}^{'} & 0 & 0 & h_{23}^{'} \\ h_{03}^{'} & h_{13}^{'} & h_{23}^{'} & 0 \end{matrix}),

(34)

which should be compared with

{\tilde{g}}_{μ ν}

in Eq. . The difference is that in this gauge

h_{13}^{'}, h_{23}^{'} \neq 0

⁠, which should instead vanish, while

h_{11}^{'} = h_{22}^{'} = h_{33}^{'} = 0

⁠, while they should not vanish. We note that if we assume

\frac{x^{'} ω}{c} ≪ 1 and \frac{y^{'} ω}{c} ≪ 1,

(35)

which we henceforth refer to as the long wavelength limit,

h_{13}^{'}

and

h_{23}^{'}

are suppressed and the dominant terms are

h_{00}^{'} (x^{' ρ}) = 2 d z^{'} \frac{ω}{c} (f h_{+} + g h_{\times}) sin (ω (t^{'} - z^{'} / c)),

(36)

h_{01}^{'} (x^{' ρ}) = - l (f h_{+} + g h_{\times}) cos (ω (t^{'} - z^{'} / c)),

(37)

h_{02}^{'} (x^{' ρ}) = - b (f h_{+} + g h_{\times}) cos (ω (t^{'} - z^{'} / c)),

(38)

\begin{matrix} {h^{'}}_{03} (x^{' ρ}) = - d (f h_{+} + g h_{\times}) cos (ω (t^{'} - z^{'} / c)) \\ - d z^{'} \frac{ω}{c} (f h_{+} + g h_{\times}) sin (ω (t^{'} - z^{'} / c)) . \end{matrix}

(39)

This approach seems promising for meeting our objectives. Still, the diagonal space components of the GW vanish; therefore, we have to find the appropriate fluid perturbation such that ${\tilde{g}}_{11}$ ⁠, ${\tilde{g}}_{22}$ ⁠, and ${\tilde{g}}_{33}$ vanish or are suppressed. For more suitable gauge transformations, see Ref. 10.

3. Comparison of GW with the perturbed acoustic metric

In this section, we compare the GW metric in the new gauge introduced previously to the acoustic perturbation metric ${\tilde{g}}_{μ ν}$ given in Eq. (26). Recall that to express the background acoustic metric as a Minkowski metric, we assume $v_{i} = 0$ for all i [refer to Eq. (21)]. Now, we aim to identify additional system characteristics that make the acoustic metric perturbation resemble that of a GW. As anticipated in the Introduction, we refer to this perturbation of the acoustic metric, when expressed similarly to a GW in the new gauge, as the GW-like perturbation. We disregard the prefactor of ${\tilde{g}}_{μ ν}$ ⁠, which we consider uniform and time-constant. To align $h_{μ ν}^{'} (x^{' ρ})$ (the GW metric in the transformed frame) with the acoustic metric ${\tilde{g}}_{μ ν}$ ⁠, we make the following identifications:

The coordinates $t^{'}, x^{'}, y^{'}, z^{'}$ in the new GW reference frame correspond to the time and space coordinates of our BEC system. For simplicity, in the following, we drop the primes and refer to these coordinates as t, x, y, and z and we also use $h_{μ ν}$ instead of $h_{μ ν}^{'}$ ⁠.
The frequency $ω$ of the GW is identified as the frequency of the GW-like perturbation.
The speed of light c in the GW metric is replaced by the speed of sound $c_{s}$ in the analogue system, implying that the GW-like perturbation propagates at $c_{s}$ ⁠.

Next, let us discuss the conditions in Eq. in our physical system. Let

L_{x}

⁠,

L_{y}

⁠, and

L_{z}

represent the system's dimensions in the x, y, and z directions, respectively. Defining dimensionless coordinates

\tilde{x} = x / L_{x}

⁠,

\tilde{y} = y / L_{y}

⁠, and

\tilde{z} = z / L_{z}

we get

\begin{matrix} x \frac{ω}{c_{s}} = \tilde{x} \frac{L_{x}}{c_{s}} ω = 2 π \tilde{x} \frac{ω}{ω_{x}}, \\ y \frac{ω}{c_{s}} = \tilde{y} \frac{L_{y}}{c_{s}} ω = 2 π \tilde{y} \frac{ω}{ω_{y}}, \\ z \frac{ω}{c_{s}} = \tilde{z} \frac{L_{z}}{c_{s}} ω = 2 π \tilde{z} \frac{ω}{ω_{z}}, \end{matrix}

(40)

where

ω_{x} = \frac{2 π c_{s}}{L_{x}}, ω_{y} = \frac{2 π c_{s}}{L_{y}}, ω_{z} = \frac{2 π c_{s}}{L_{z}},

(41)

are the three characteristic frequencies. For a setting, say an optical trap, which elongates along the z–direction, the transverse frequencies,

ω_{x}

and

ω_{y}

are much larger then the longitudinal frequency,

ω_{z}

⁠. It follows that we can consider perturbation frequencies such that:

\begin{matrix} \frac{x ω}{c_{s}} ≪ 1 \Leftrightarrow \frac{ω}{ω_{x}} ≪ 1, \frac{y ω}{c_{s}} ≪ 1 \Leftrightarrow \frac{ω}{ω_{y}} ≪ 1, \end{matrix}

(42)

while

\frac{ω}{ω_{z}} \sim 1 .

(43)

This implies that the considered perturbation in the transverse direction is negligible, while along the longitudinal direction it is comparable with the system dimension, $L_{z}$ ⁠. This is different from Ref. 7 where only GW waveleghts much larger than the BEC size where considered.

At this point, we compare the GW metric

h_{μ ν}

given in Eq. , with its components detailed in Eqs. , with

{\tilde{g}}_{μ ν}

given in Eq. . In order to match the two expressions, the space diagonal components

{\tilde{g}}_{i i}

⁠, of the acoustic metric perturbation, should vanish. From Eq. , we have to impose that the relative density variation is equal to the relative scattering length change, that is

δ a / a = δ n_{c} / n_{c}

⁠. With this choice, from Eq. , we have that

{\tilde{g}}_{00} = - 2 δ a / a = - 2 δ n_{c} / n_{c}

⁠. The space-time dependence of

δ a / a = δ n_{c} / n_{c}

is given through the comparison with

h_{00}^{'}

in Eq. . Then, by equating

h_{0 i}^{'}

in Eqs. with the

{\tilde{g}}_{0 i}

components in Eq. , we find the required space-time dependence of

δ v_{i} / c_{s}

⁠. Combining all these constraints, we conclude that to simulate a GW the system's parameters should satisfy the following conditions:

\begin{matrix} \frac{δ a}{a} = \frac{δ n_{c}}{n_{c}} = - d z \frac{ω}{c_{s}} (f H_{+} + g H_{\times}) sin (ω (t - z / c_{s})), \\ \frac{δ v_{x}}{c_{s}} = l (f H_{+} + g H_{\times}) cos (ω (t - z / c_{s})), \\ \frac{δ v_{y}}{c_{s}} = b (f H_{+} + g H_{\times}) cos (ω (t - z / c_{s})), \\ \frac{δ v_{z}}{c_{s}} = d (f H_{+} + g H_{\times}) cos (ω (t - z / c_{s})) \\ + d z \frac{ω}{c_{s}} (f H_{+} + g H_{\times}) sin (ω (t - z / c_{s})) . \end{matrix}

(44)

Here,

H_{+}

and

H_{\times}

are the analog of the GW polarization amplitudes. The first of these equations requires equal spacetime modulations of the scattering length and density perturbations, which is experimentally challenging. If we set

d = 0

⁠, we greatly simply the system, removing all together the need of a Feshbach resonance to vary

δ a

⁠, of a density fluctuation

δ n_{c}

and of a velocity fluctuation along the z–direction

δ a = δ n_{c} = δ v_{z} = 0 .

(45)

Under these conditions, no modulation of the sound velocity and consequent phonon production is needed. This is different from the case studied in Ref. 7 (see also Ref. 11).

Equating the GW to the acoustic metric perturbation

{h^{'}}_{μ ν} \propto {\tilde{g}}_{μ ν} = \frac{n_{c}}{m c_{s}} (\begin{matrix} 0 & - \frac{δ v_{x}}{c_{s}} & - \frac{δ v_{y}}{c_{s}} & 0 \\ - \frac{δ v_{x}}{c_{s}} & 0 & 0 & 0 \\ - \frac{δ v_{y}}{c_{s}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}),

(46)

we have that

\begin{matrix} \frac{δ v_{x}}{c_{s}} = l (f H_{+} + g H_{\times}) cos (ω (t - z / c_{s})), \\ \frac{δ v_{y}}{c_{s}} = b (f H_{+} + g H_{\times}) cos (ω (t - z / c_{s})) . \end{matrix}

(47)

By imposing

d = 0

⁠, the GW has only two independent degrees of freedom, see Eqs. ; therefore, we have completely fixed the gauge. This is determined by

ζ_{μ}

in Eq. , with

d = 0

⁠. This gauge is somehow analogous to the TT gauge, but the only nonvanishing components of the GW are now

h_{01}^{'}

and

h_{02}^{'}

⁠. Since

□ ζ_{μ} = 0

and

h_{μ ν}^{'}

is traceless, the linearized Einstein's equations take the simple form given in Eq. .

Regarding the metric perturbation in Eq. (46), it has been obtained assuming that the scattering length is constant. Thus, the velocity fluctuations along the x and y directions should be induced by an additional interaction potential. Such potential will be explored in Sec. III B 4 to determine if a BEC with these characteristics is physically feasible, meaning it satisfies the continuity and Euler equations as well as the irrotational condition.

4. Physical systems

We have determined a simple expression of the perturbed acoustic metric that matches the form of a GW in a given gauge. We need to verify that it can be obtained by appropriate external perturbations of the hydrodynamic quantities, such that the continuity, the Euler equations, as well as the irrotational velocity conditions, are satisfied.

In the hydrodynamic limit, the Gross–Pitaevskii equation, see Eq. , can be written as³

\partial_{t} n_{c} = - \nabla \cdot (n_{c} v),

(48)

m \partial_{t} v = - \nabla (\frac{m v^{2}}{2} + V_{ext} + \frac{4 π a ℏ^{2}}{m} n_{c}),

(49)

which are the continuity and Euler equation, respectively. Since we focus on the case where

d = 0

⁠, according to Eq. , we can restrict to consider the deviations from the equilibrium state determined by

\begin{matrix} v_{i} \to v_{i} + ε δ v_{i}, \\ V_{ext} \to V_{ext} + ε δ V_{ext} . \end{matrix}

(50)

Regarding the background, as discussed before, we require $v = 0$ ⁠, so that the unperturbed acoustic metric resembles the Minkowski spacetime. In this case, the continuity equation is satisfied if $n_{c}$ is stationary. The Euler equation is instead satisfied by an appropriate choice of the external potential and scattering length.

At leading order in

ε

⁠, the continuity equation holds if

n_{c}

is spatially homogeneous. To satisfy the Euler equation with the perturbed velocities in Eq. , the external potential perturbation must be

δ V_{ext} = m ω c_{s} (l x + b y) (f H_{+} + g H_{\times}) sin (ω (t - z / c_{s})) .

(51)

Upon inserting such potential modulation in the Euler equation, one can see that it induces the velocity perturbation along the z-axis

\frac{δ v_{z}}{c_{s}} = \frac{ω}{c_{s}} (l x + b y) (f H_{+} + g H_{\times}) sin (ω (t - z / c_{s}),

(52)

which is, however,

O (ε ω / ω_{x, y})

and, thus, it is negligible within our approximation. As discussed before, such

δ v_{z}

is consistent with the velocity irrotational condition: introducing

δ v_{x}

and

δ v_{y}

to simulate a GW naturally induces a negligible velocity fluctuation along the z-axis.

In summary, we have demonstrated that by applying a coordinate frame transformation from the TT gauge with $ζ^{μ}$ given by Eq. (27), the GW metric aligns to the perturbation in the acoustic metric ${\tilde{g}}_{μ ν}$ ⁠. By setting, the external potential perturbation necessary to have the appropriate velocity perturbations given in Eq. (47) is the one in Eq. (51).

IV. ACOUSTIC BH WITH A GW–LIKE PERTURBATION

So far, we have discussed the propagation of a GW-like perturbation in flat spacetime. Now, we consider scenarios characterized by a different metric. Specifically, we focus on an acoustic metric featuring an event horizon and we examine how to appropriately extend a GW-like perturbation in this context. For our analysis, we adopt a cylindrical geometry.

Clearly, in this section, we are not interested to make contact with astrophysical objects: both the background metric and the perturbation cannot be mapped to any realistic BH and GW, respectively. Nevertheless, with a minor abuse of language, we will refer to them as the acoustic BH and the GW-like perturbation.

A. Vortex geometry

We consider the “draining bathtub” fluid model, representing a

(2 + 1)

-dimensional flow with a sink at the origin.¹² For simplicity, we neglect any background density variation: the fluid is homogeneous, with constant density, pressure, and speed of sound. We use this approximation because we are focusing on a region sufficiently far from the vortex core, where we can neglect the density variation. In this configuration, the continuity equation requires that the radial velocity is given by

v_{r} \propto \frac{1}{r},

(53)

where r is the distance from the sink. The condition of irrotational (vorticity-free) flow is compatible with the tangential velocity

v_{ϕ} \propto \frac{1}{r},

(54)

thus, the fluid velocity can be written as

v = \frac{(A \hat{r} + B \hat{ϕ})}{r},

(55)

with A and B two constants having dimensions of length²/time. The corresponding velocity potential, see Eq. , must be expressed as a linear combination of the logarithm of the radial coordinate,

ln r

⁠, and the angular coordinate

ϕ

⁠. Such potential is discontinuous as

ϕ

wraps around by

2 π

⁠, and hence, it is not monodromic: the velocity potential should be defined patch-wise on overlapping regions surrounding the vortex core at

r = 0

⁠. Neglecting any overall space-independent prefactor, the acoustic metric for this draining bathtub scenario is

d s^{2} = - c_{s}^{2} d t^{2} + {(d r - \frac{A}{r} d t)}^{2} + {(r d ϕ - \frac{B}{r} d t)}^{2},

(56)

and an acoustic event horizon forms at the radial distance

r_{H} = \frac{| A |}{c_{s}},

(57)

where the radial fluid velocity is equal to the speed of sound. The sign of A determines the nature of the horizon: we take

A < 0

⁠, thus, the horizon is a future acoustic horizon (acoustic BH), whereas for

A > 0

⁠, corresponds to a past acoustic horizon (acoustic white hole).

Although this construction is described in

(2 + 1)

–dimensions, it can be extended to

(3 + 1)

–dimensions by adding an extra spatial dimension, z, indicating the height of the cylinder. The condition of irrotational flow implies that the z–component of the velocity is constant; for simplicity, we assume that it vanishes. Hence, in

(3 + 1)

–dimensions, we have a vortex filament located at

r = 0

generating the metric

d s^{2} = - c_{s}^{2} d t^{2} + {(d r - \frac{A}{r} d t)}^{2} + {(r d ϕ - \frac{B}{r} d t)}^{2} + d z^{2},

(58)

for the phonon excitations. This extended metric will serve as the foundation for our exploration of a system where an acoustic BH is perturbed by a GW-like perturbation that propagates radially. A schematic representation of the system is shown in Fig. 1. This choice of geometry is entirely distinct from astrophysical BHs. The background metric given by Eq. , with

A < 0

⁠, has indeed a cylindrical event horizon located at

r_{H}

for all z. We have chosen this specific geometry for several reasons. While simpler configurations, such as a two-dimensional horizon, could be considered, the cylindrical case offers a more general approach and holds significant experimental potential. Laboratory experiments have successfully created box-like BECs,^13–15 suggesting the feasibility of confining a BEC within a cylindrically symmetric trapping potential. This geometry can, thus, serve as a practical model for experimental analogues of gravitational phenomena. Moreover, cigar-shaped or disk-shaped BECs, which have been used in analog gravity experiments,^16,17 can be obtained as limiting cases of the cylindrical BEC.

Fig. 1.

View large Download slide

Pictorial representation of a GW-like perturbation in the draining bathtub geometry. The dark cylinder represents the event horizon, while pink arrows indicate the propagation direction of the GW-like perturbation.

B. Method

We proceed to consider the various steps needed to realize a GW-like perturbation in the cylindrical geometry:

Cylindrical perturbations: Extend the externally driven GW-like perturbation obtained in Sec. III to the cylindrical geometry.
Physical requirements: Check the irrotational condition constraint, as well as the continuity and Euler equations.
Acoustic metric: Calculate the acoustic metric and its inverse to characterize the emergent spacetime.
Perturbed acoustic horizon: Investigate the position of the acoustic horizon upon interaction with the analogue GW.

As in Sec. III, it is important to distinguish between the externally driven metric perturbations and phonons. In the Appendix, the generators of the acoustic horizon are examined to elucidate their properties under perturbations.

1. Cylindrical perturbations

We extend the GW-like plane wave perturbation obtained in Sec. III to a cylindrical geometry. In the linear case discussed earlier, we identified the analog of a gravitational wave by perturbing the condensate velocity along the directions perpendicular to the wave's propagation. For the cylindrical geometry, we consider that the perturbation propagates radially, as depicted by pink arrows in Fig. 1. Consequently, the GW-like perturbation is characterized by velocity fluctuations along the angular and z-coordinates, denoted as

δ v_{ϕ}

and

δ v_{z}

⁠, respectively. These perturbations can be obtained from those in Eq. with the replacements

δ v_{x} \to δ v_{z}

and

δ v_{y} \to r / r_{0} δ v_{ϕ}

⁠, where

r_{0}

is a normalizing length. Due to the radial propagation direction, the term

cos (ω (t - z / c_{s})

in Eq. is replaced by

cos (ω (t - r / c_{s}))

⁠. In conclusion, the GW-like perturbation in the cylindrical case is identified by the following velocity fluctuations:

\frac{δ v_{z}}{c_{s}} = l (f H_{+} + g H_{\times}) cos (ω (t - r / c_{s})),

(59)

\frac{δ v_{ϕ}}{c_{s}} = \frac{r_{0}}{r} b (f H_{+} + g H_{\times}) cos (ω (t - r / c_{s})) .

(60)

The characteristic frequencies of this system are

ω_{z} = \frac{2 π c_{s}}{L_{z}}, ω_{r} = \frac{2 π c_{s}}{L_{r}},

(61)

where

L_{z}

and

L_{r}

represent the physical dimensions of the system along the z and r directions, respectively. Given that the system elongates along the z-direction, we assume that

ω_{r} > ω_{z},

(62)

and we consider perturbation frequencies

ω \sim ω_{z},

(63)

thus, with a wavelength of order the height of the cylinder.

2. Physical requirements

We now check whether the irrotational condition, as well as the continuity, and Euler equations hold for the velocity perturbations in Eqs. (59) and (60). As in the Minkowskian case, we assume that the unperturbed system is homogeneous: the density, pressure, and the speed of sound are taken constant. The unperturbed velocity is given by Eq. (55) with A and B two constants.

We begin by examining the irrotationality of the flow, checking that

\nabla \times δ v = 0

⁠. We have seen that in flat spacetime the irrotational condition requires the presence of an additional velocity component; likewise here the fluctuations

δ v_{z}

and

δ v_{ϕ}

satisfy the irrotational condition only if

\begin{matrix} δ v_{r} = l z ω (f H_{+} + g H_{\times}) sin (ω (t - r / c_{s})) \\ + b ϕ r_{0} ω (f H_{+} + g H_{\times}) sin (ω (t - r / c_{s})), \end{matrix}

(64)

which is, thus, the sum of two terms: one proportional to

l z

and the other one to

b ϕ

⁠. In order to have a single-valued function, we impose this second term to be zero taking

b = 0

⁠. This constraint, in turn, implies that

δ v_{ϕ} = 0

[see Eq. ]. Hence, we start with a GW-like perturbation that is represented only by the velocity perturbation

δ v_{z}

in Eq. and this induces a radial velocity perturbation

\frac{δ v_{r}}{c_{s}} = l z \frac{ω}{c_{s}} (f H_{+} + g H_{\times}) sin (ω (t - r / c_{s})) .

(65)

In the Minkowskian case, the induced velocity perturbation was negligible [see Eq. ], because it was of order

O (ε ω / ω_{x, y})

⁠. In cylindrical geometry, it is instead

O (ε ω / ω_{z})

and we cannot neglect it [see Eq. ]. Note that the radial velocity perturbation depends on the z coordinate; hence, it produces a time dependent bending of the acoustic horizon that changes with height.

In order to simplify the analysis, we turn to dimensionless coordinates: we express z as

\tilde{z} L_{z}

with

\bar{z} \in [0, 1]

and r as

\bar{r} L_{r}

with

\bar{r} \in [0, 1]

⁠. For convenience, we choose

l = 1 / 2 π

and so

\begin{matrix} \frac{δ v_{z}}{c_{s}} = \frac{1}{2 π} (f H_{+} + g H_{\times}) cos (ω (t - 2 π \bar{r} / ω_{r})), \\ \frac{δ v_{r}}{c_{s}} = \bar{z} \frac{ω}{ω_{z}} (f H_{+} + g H_{\times}) sin (ω (t - 2 π \bar{r} / ω_{r})) . \end{matrix}

(66)

In the Minkowskian case, the velocity perturbations in Eq. were solutions of the continuity and Euler equations with

δ a = δ n_{c} = 0

⁠. However, in the cylindrical case with an acoustic horizon, the velocity perturbations in Eq. remain solutions of these equations, provided we introduce appropriate additional terms: the continuity equation forces us to take

δ n_{c} \neq 0

⁠, and as a consequence

δ a \neq 0

⁠. Specifically, the continuity equation holds if we include the source term

\begin{matrix} δ S = \frac{2 π}{3} n_{c} \frac{\bar{z} {\bar{r}}^{3}}{{\bar{r}}_{H}^{2}} \frac{ω^{3}}{ω_{r} ω_{z}} (f H_{+} + g H_{\times}) \\ \times (- 1 - \frac{{\bar{r}}_{H}}{\bar{r}}) sin (ω (t - 2 π \bar{r} / ω_{r})), \end{matrix}

(67)

where

{\bar{r}}_{H} = \frac{| A |}{c_{s} L_{r}},

(68)

is the unperturbed horizon position, see Eq. , in dimensionless units. Such source term induces the density variation

\begin{matrix} \frac{δ n_{c}}{n_{c}} = \frac{\bar{z} \bar{r}}{{\bar{r}}_{H}} \frac{ω}{ω_{z}} (f H_{+} + g H_{\times}) sin (ω (t - 2 π \bar{r} / ω_{r})) \\ + \frac{2 π}{3} \frac{\bar{z} {\bar{r}}^{3}}{{\bar{r}}_{H}^{2}} \frac{ω^{2}}{ω_{r} ω_{z}} (f H_{+} + g H_{\times}) cos (ω (t - 2 π \bar{r} / ω_{r})) . \end{matrix}

(69)

Furthermore, to maintain

c_{s}

as the local speed of sound, we require

δ c_{s} = 0

⁠, which leads to the introduction of a Feshbach resonance [see Eq. ] such that

δ a / a = - δ n_{c} / n_{c}

⁠. With this choice of

δ a

⁠, the Euler equation holds if we include the external potential

\begin{matrix} δ V_{ext} = (\frac{3}{8 π} \frac{\bar{z} (\bar{r} - {\bar{r}}_{H})}{{\bar{r}}_{H}^{2}} \frac{ω^{2} ω_{r}}{ω_{z} c_{s}^{2}} - \frac{π}{6} \frac{\bar{z} {\bar{r}}^{3}}{{\bar{r}}_{H}^{2}} \frac{ω^{4}}{ω_{z} ω_{r} c_{s}^{2}}) \frac{ℏ^{2}}{m} \\ \times (f H_{+} + g H_{\times}) cos (ω (t - 2 π \bar{r} / ω_{r})) - (\bar{z} \frac{c_{s}^{2} ω}{ω_{z}} m \\ + \frac{1}{16 π^{2}} \frac{\bar{z}}{\bar{r} {\bar{r}}_{H}} \frac{ω ω_{r}^{2}}{ω_{z} c_{s}^{2}} \frac{ℏ^{2}}{m} + \frac{\bar{z} \bar{r}}{{\bar{r}}_{H}^{2}} (\frac{7}{12} \bar{r} - \frac{1}{4} {\bar{r}}_{H}) \frac{ω^{3}}{ω_{z} c_{s}^{2}} \frac{ℏ^{2}}{m}) \\ \times (f H_{+} + g H_{\times}) sin (ω (t - 2 π \bar{r} / ω_{r})) . \end{matrix}

(70)

In this way, we obtain a physical system (irrotationality, continuity, and Euler equations are satisfied), in the geometry depicted in Fig. 1, which represents an acoustic BH with a GW-like perturbation.

3. Acoustic metric

At this point, we compute the emergent acoustic metric tensor for phonons, given by

g_{μ ν} = g_{μ ν}^{0} + ε {\tilde{g}}_{μ ν}

⁠. For simplicity, we assume vanishing tangential flow, which is we take

B = 0

in Eq. . Using coordinates

(c_{s} t, \bar{r}, θ, \bar{z})

⁠, the unperturbed metric is

g_{μ ν}^{0} = \frac{n_{c}}{m c_{s}} (\begin{matrix} - (1 - \frac{v_{r}^{2}}{c_{s}^{2}}) & - \frac{v_{r}}{c_{s}} L_{r} & 0 & 0 \\ - \frac{v_{r}}{c_{s}} L_{r} & L_{r}^{2} & 0 & 0 \\ 0 & 0 & {\bar{r}}^{2} L_{r}^{2} & 0 \\ 0 & 0 & 0 & L_{z}^{2} \end{matrix}),

(71)

and the perturbation,

{\tilde{g}}_{μ ν} = \frac{n_{c}}{m c_{s}} (\begin{matrix} {\tilde{g}}_{00} & {\tilde{g}}_{01} & 0 & {\tilde{g}}_{03} \\ {\tilde{g}}_{01} & {\tilde{g}}_{11} & 0 & 0 \\ 0 & 0 & {\tilde{g}}_{22} & 0 \\ {\tilde{g}}_{03} & 0 & 0 & {\tilde{g}}_{33} \end{matrix}),

(72)

has components

{\tilde{g}}_{00} = - (1 - \frac{v_{r}^{2}}{c_{s}^{2}}) \frac{δ n_{c}}{n_{c}} + 2 \frac{v_{r} δ v_{r}}{c_{s}^{2}},

(73)

{\tilde{g}}_{01} = L_{r} (- \frac{δ v_{r}}{c_{s}} - \frac{v_{r}}{c_{s}} \frac{δ n_{c}}{n_{c}}),

(74)

{\tilde{g}}_{03} = - \frac{δ v_{z}}{c_{s}} L_{z}, {\tilde{g}}_{11} = \frac{δ n_{c}}{n_{c}} L_{r}^{2},

(75)

{\tilde{g}}_{22} = {\tilde{r}}^{2} L_{r}^{2} \frac{δ n_{c}}{n_{c}}, {\tilde{g}}_{33} = \frac{δ n_{c}}{n_{c}} L_{z}^{2},

(76)

that are the variations of the acoustic metric induced by the GW-like perturbation. Far from the horizon, they can be seen as perturbations of the Minkowski metric. Near the horizon, they represent a perturbation of the cylindrical acoustic metric. Next, we compute the inverse of the metric defined by the relation

g_{μ ν} g^{ν α} = δ_{μ}^{α}

⁠, and we write it as

g^{μ ν} = g_{0}^{μ ν} - ε {\tilde{g}}^{μ ν} .

(77)

In the coordinates

(c_{s} t, \tilde{r}, θ, \tilde{z})

⁠, the unperturbed inverse metric is

g_{0}^{μ ν} = \frac{c_{s} m}{n_{c}} (\begin{matrix} - 1 & - \frac{v_{r}}{c_{s} L_{r}} & 0 & 0 \\ - \frac{v_{r}}{c_{s} L_{r}} & \frac{1}{L_{r}^{2}} (1 - \frac{v_{r}^{2}}{c_{s}^{2}}) & 0 & 0 \\ 0 & 0 & \frac{1}{{\tilde{r}}^{2} L_{r}^{2}} & 0 \\ 0 & 0 & 0 & \frac{1}{L_{z}^{2}} \end{matrix}),

(78)

and the inverse perturbation is

{\tilde{g}}^{μ ν} = \frac{m c_{s}}{n_{c}} (\begin{matrix} {\tilde{g}}^{00} & {\tilde{g}}^{01} & 0 & {\tilde{g}}^{03} \\ {\tilde{g}}^{01} & {\tilde{g}}^{11} & 0 & {\tilde{g}}^{13} \\ 0 & 0 & {\tilde{g}}^{22} & 0 \\ {\tilde{g}}^{03} & {\tilde{g}}^{13} & 0 & {\tilde{g}}^{33} \end{matrix}),

(79)

with components

{\tilde{g}}^{00} = - \frac{δ n_{c}}{n_{c}},

(80)

{\tilde{g}}^{01} = (- \frac{δ n_{c}}{n_{c}} \frac{v_{r}}{c_{s}} + \frac{δ v_{r}}{c_{s}}) \frac{1}{L_{r}},

(81)

{\tilde{g}}^{03} = \frac{δ v_{z}}{c_{s}} \frac{1}{L_{z}},

(82)

{\tilde{g}}^{11} = (2 \frac{v_{r} δ v_{r}}{c_{s}^{2}} + \frac{δ n_{c}}{n_{c}} - \frac{v_{r}^{2}}{c_{s}^{2}} \frac{δ n_{c}}{n_{c}}) \frac{1}{L_{r}^{2}},

(83)

{\tilde{g}}^{13} = \frac{v_{r} δ v_{z}}{c_{s}^{2}} \frac{1}{L_{r} L_{z}},

(84)

{\tilde{g}}^{22} = \frac{1}{{\tilde{r}}^{2} L_{r}^{2}} \frac{δ n_{c}}{n_{c}},

(85)

{\tilde{g}}^{33} = \frac{δ n_{c}}{n_{c}} \frac{1}{L_{z}^{2}} .

(86)

From the inverse metric, we can determine the event horizon's characteristics and generators.

4. Perturbed acoustic horizon

We now determine the displacement of the horizon position determined by the impinging GW-like perturbation. In general, the horizon position is given by the condition

| v_{r} |_{r_{H}} = c_{s}

⁠. In the unperturbed system, the horizon is located in

{\bar{r}}_{H}

given in Eq. . The GW-like propagation is determined by the velocity fluctuation

δ v_{r}

in Eq. . Consequently, the acoustic horizon is displaced at

{\tilde{r}}_{H}^{new} = {\tilde{r}}_{H} + ε δ {\tilde{r}}_{H}

⁠, which satisfies

| v_{r} + ε δ v_{r} |_{{\tilde{r}}_{H}^{new}} = c_{s} .

(87)

Solving Eq. , we find the horizon's displacement is

\frac{δ {\tilde{r}}_{H}}{{\tilde{r}}_{H}} = - \tilde{z} \frac{ω}{ω_{z}} (f H_{+} + g H_{\times}) sin (ω (t - 2 π {\tilde{r}}_{H} / ω_{r})),

(88)

indicating that it is opposite to the sign of the radial velocity perturbation [see Eq. ], while

δ v_{z}

has no effect on the horizon's position. The horizon's displacement vanishes at

\tilde{z} = 0

and for

\tilde{z} \neq 0

it depends on the sign of

sin (ω (t - 2 π {\tilde{r}}_{H} / ω_{r}))

⁠. At a given time t, such that

sin (ω (t - 2 π {\tilde{r}}_{H} / ω_{r})) > 0

⁠, we have that

δ {\tilde{r}}_{H} < 0

and the displacement decreases as

\tilde{z}

increases (see Fig. 2).

Fig. 2.

View large Download slide

Schematic representation of the perturbation of the acoustic horizon. Left: unperturbed horizon. Right: perturbed acoustic horizon at a fixed time such that $sin (ω (t - 2 π {\tilde{r}}_{H} / ω_{r})) > 0$ ⁠. The size of the perturbation is enhanced for clarity.

At time t such that $sin (ω (t - 2 π {\tilde{r}}_{H} / ω_{r})) < 0$ ⁠, we have he opposite behavior: $δ {\tilde{r}}_{H}$ increases linearly with $\tilde{z}$ (see Fig. 3).

Fig. 3.

View large Download slide

Schematic representation of the perturbation of the acoustic horizon. Left: unperturbed horizon. Right: perturbed acoustic horizon at a fixed time such that $sin (ω (t - 2 π {\tilde{r}}_{H} / ω_{r})) < 0$ ⁠. The size of the perturbation is enhanced for clarity.

If we now examine the scenario at fixed $\tilde{z} \neq 0$ ⁠, and we observe the changes over time, we find that the horizon displacement fluctuates around zero as shown in Fig. 4: the event horizon expands or contracts due to the GW-like propagation. The magnitude of the dilatation depends on the considered height: it vanishes for $\bar{z} = 0$ and it is maximal for $\bar{z} = 1$ ⁠.

Fig. 4.

Schematic representation of the perturbation of the perturbed acoustic horizon in the x̃ỹ plane. At fixed z̃>0, the horizon is a circle that oscillates as time lapses.

View large Download slide

Schematic representation of the perturbation of the perturbed acoustic horizon in the $\tilde{x} \tilde{y}$ plane. At fixed $\tilde{z} > 0$ ⁠, the horizon is a circle that oscillates as time lapses.

Differently from the flat spacetime case where GWs do not cause phonon production, when GWs impinge onto an acoustic horizon they induce on it periodic deformations which turns into a stimulated amplification of the Hawking phonon radiation in a phenomenon resembling the dynamical Casimir effect.¹⁸ This process was described in details in Ref. 19 in the case of a planar acoustic horizon under the action of a small velocity shear in background flow. Something similar takes place for the system considered here with a cylindrical geometry and the shear perturbation being originated by the impinging GWs. The expansion or contraction observed at a fixed $\tilde{z}$ corresponds to an oscillation of the tilting of the horizon. Due to this tilt, the radiated phonons, which are predominantly emitted in the direction perpendicular to the horizon, experience a change in their propagation direction as compared to the unperturbed case.^20,21 Focusing for instance on the $\tilde{x} \tilde{z}$ plane at $\tilde{y} = 0$ and considering only the half-plane $\tilde{x} > 0$ ⁠, the situation closely resembles the analysis conducted in Sec. V of Ref. 19, where the shear viscosity to entropy density ratio $η / s$ at the acoustic horizon was calculated to be equal to $1 / 4 π$ ⁠, namely, the KSS lower bound.²² Although a planar horizon was considered in Ref. 19, the result holds here as well: in both cases, the relevant effect is that given by the bending of the horizon surface with respect to the $\tilde{z}$ axis.

V. CONCLUSIONS AND FUTURE PERSPECTIVES

Developing a system that enables an in-depth investigation of acoustic horizons through perturbations is crucial for addressing fundamental theoretical questions regarding the physics of BHs. In this paper, we have designed a perturbation that closely mimics a classical GW and applied it to excite an acoustic horizon. We first demonstrate that it is possible to configure a BEC such that the metric perturbation experienced by phonons resembles that of a GW in Minkowski space in a specific gauge. Subsequently, we considered the effect of this perturbation, adapted to the new symmetry of the system, on an acoustic cylindrical BH. With the proposed approach, we have established a laboratory-reproducible system where to study how an acoustic horizon responds to perturbations closely modeled after GWs. In particular, experiments with ultra-cold atoms can be designed to replicate these systems in order to test the theoretical predictions arising from this research. These models open avenues for diverse future studies.

First, the shear viscosity to entropy density ratio of the acoustic horizon can be assessed. One approach is to determine and compare its value with that of a real BH in general relativity.²³ In this case, it is important to check whether this ratio adheres the universal lower bound of $η / s$ ⁠, proposed in the AdS/CFT context.²² This bound is conjectured to be $1 / 4 π$ for any fluid in nature.^24,25 However, the conditions under which this bound can be saturated by a matter system, given the known symmetries, remain an open question.²⁶ Thus, investigating $η / s$ for acoustic BHs could be crucial to determine if it saturates this bound and to establish whether the ratio for acoustic BHs is universal or dependent on specific hydrodynamic conditions.

Additionally, the surface reflectivity of the acoustic horizon can be explored and related to the membrane fluid viscosity.²⁷

Moreover, the GW-like perturbation discussed in this paper can be analyzed in greater details. A particularly intriguing aspect is how, despite the absence of spin-2 modes in the system, we can still replicate a spin-2 perturbation of the metric. It might be worthwhile to study the quantization of this classical analog of GWs to investigate whether its quanta can be identified and to predict the equation of motion for this mode. Although quasi-normal modes of acoustic BHs have been studied previously,²⁸ examining this spin-2 perturbation could potentially provide a new avenue for approximating a spin-2 test field analysis.²⁹ Furthermore, we can explore gravitational memory in these systems to understand how the emergent spacetime responds when a GW-like perturbation passes through it and whether there are any symmetries of the system related to it.³⁰

ACKNOWLEDGMENTS

M.L.C. acknowledges the support from the National Centre on HPC, Big Data, and Quantum Computing-SPOKE 10 Code CN00000013 (Quantum Computing) and received funding from the European Union Next-GenerationEU-National Recovery and Resilience Plan (NRRP)-MISSION 4 COMPONENT 2, INVESTMENT N. 1.4-CUP N. I53C22000690001. This research has received funding from the European Union's Digital Europe Programme DIGIQ under Grant Agreement No. 101084035. M.L.C. also acknowledges the support from the project PRA_2022_2023_98 “IMAGINATION” and in part by Grant Nos. NSF PHY-1748958 and PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP). D.G. presently on leave of absence at Embassy of Italy, The Hague.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Chiara Coviello: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal). Maria Luisa Chiofalo: Conceptualization (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Dario Grasso: Conceptualization (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Stefano Liberati: Conceptualization (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Massimo Mannarelli: Conceptualization (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Silvia Trabucco: Conceptualization (equal); Writing – review & editing (equal).

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

APPENDIX: HORIZON'S GENERATORS

To characterize the kinematics of the horizon, we compute the null vector field that is simultaneously tangent and orthogonal to the null congruence of the geodesics generating the horizon. The vector field normal to the hypersurface

S = constant

has the general expression

k^{μ} = ℓ (x) g^{μ ν} \partial_{ν} S,

(A1)

with

ℓ (x)

an arbitrary non-vanishing function of the coordinates. Hereafter, we take, for simplicity,

ℓ (x) = 1

⁠.

We first compute

k^{μ}

in the unperturbed system. We consider the family of hypersurfaces

S = \tilde{r} - {\tilde{r}}_{H} = constant,

(A2)

where

{\bar{r}}_{H}

is given in Eq. ; the acoustic horizon corresponds to

S = 0

⁠. For the considered hypersurface

\partial_{ν} S = (0, 1, 0, 0),

(A3)

hence, using the inverse of the metric in Eq. , we obtain that the vector normal to the hypersurface S is given by

k^{μ} = (\frac{m}{2 π n_{c}} ω_{r} \frac{{\tilde{r}}_{H}}{\tilde{r}}, \frac{m}{4 π^{2} n_{c} c_{s}} ω_{r}^{2} (1 - \frac{{\tilde{r}}_{H}^{2}}{{\tilde{r}}^{2}}), 0, 0),

(A4)

and then

g_{μ ν}^{0} k^{μ} k^{ν} = \frac{m}{4 π^{2} n_{c} c_{s}} ω_{r}^{2} (1 - \frac{{\tilde{r}}_{H}^{2}}{{\tilde{r}}^{2}}) .

(A5)

Since at the horizon

v_{r}^{2} = c_{s}^{2}

⁠, we find that

{g_{μ ν}^{0} k^{μ} k^{ν} |}_{S = 0} = 0,

(A6)

meaning that the acoustic horizon is a null hypersurface.

For the computation of the null vector field in the system perturbed by the GW-like perturbation, we have to consider that the inverse of the metric is given by Eq. and that the position of the acoustic horizon is shifted to

{\tilde{r}}_{H} + ε δ {\tilde{r}}_{H}

⁠, where

δ {\tilde{r}}_{H}

is in Eq. . For these reasons, we define a new family of hypersurfaces

S_{new} = \tilde{r} - ({\tilde{r}}_{H} + ε δ {\tilde{r}}_{H}) = constant,

(A7)

and the acoustic horizon hypersurface is

S_{new} = 0

⁠. In this case,

\partial_{ν} S_{new} = (0, 1, 0, 0) - ε \partial_{ν} δ {\tilde{r}}_{H},

(A8)

thus, the components of the normal vector field, which we now call

k_{new}^{μ}

⁠, at the leading order in the perturbations are given by

k_{new}^{μ} = g_{0}^{μ ν} \partial_{ν} S - ε {\tilde{g}}^{μ ν} \partial_{ν} S - ε g_{0}^{μ ν} \partial_{ν} δ {\tilde{r}}_{H} .

(A9)

Therefore,

k_{new}^{μ}

is identical to that of the unperturbed system at zeroth order in

ε

[see Eq. ], with corrections at order

ε

arising from both metric perturbations and changes in the horizon's position. Its components are given by

\begin{matrix} k_{new}^{0} = k^{0} + ε (f H_{+} + g H_{\times}) [\frac{m}{n_{c}} \frac{\tilde{z}}{π} \frac{ω ω_{r}}{ω_{z}} sin (ω T_{eff}) \\ - (\frac{1}{3} \frac{m}{n_{c}} \frac{ω^{2}}{ω_{z}} \frac{{\tilde{r}}^{2} \tilde{z}}{{\tilde{r}}_{H}} + \frac{m}{n_{c}} \tilde{z} {\tilde{r}}_{H} \frac{ω^{2}}{ω_{z}}) cos (ω T_{eff})], \end{matrix}

(A10)

\begin{matrix} k_{new}^{1} = k^{1} + ε (f H_{+} + g H_{\times}) [(\frac{m}{2 π n_{c} c_{s}} \frac{ω^{2} ω_{r}}{ω_{z}} \frac{\tilde{z} {\tilde{r}}_{H}^{2}}{\tilde{r}} \\ + \frac{m}{6 π n_{c} c_{s}} \frac{ω^{2} ω_{r}}{ω_{z}} \frac{\tilde{z} \tilde{r}}{{\tilde{r}}_{H}^{2}} (- {\tilde{r}}^{2} + {\tilde{r}}_{H}^{2})) cos (ω T_{eff}) \\ - \frac{m}{4 π^{2} n_{c} c_{s}} \frac{ω ω_{r}^{2}}{ω_{z}} \frac{\tilde{z}}{\tilde{r} {\tilde{r}}_{H}} ({\tilde{r}}^{2} - 3 {\tilde{r}}_{H}^{2}) sin (ω T_{eff})], \end{matrix}

(A11)

k_{new}^{2} = 0,

(A12)

\begin{matrix} k_{new}^{3} = ε (f H_{+} + g H_{\times}) (\frac{m}{4 π^{2} n_{c} c_{s}} ω ω_{z} {\tilde{r}}_{H} sin (ω T_{eff}) \\ + \frac{m}{8 π^{3} n_{c} c_{s}} ω_{z} ω_{r} \frac{{\tilde{r}}_{H}}{\tilde{r}} cos (ω T_{eff})), \end{matrix}

(A13)

where, for simplicity, we have defined

T_{eff} = t - 2 π \tilde{r} / ω_{r}

⁠. Now, we compute

g_{μ ν} k_{new}^{ν} k_{new}^{μ}

at the acoustic horizon, so at

S_{new} = 0

⁠, and we call it K. We find

K = ε 4 π \frac{n_{c} c_{s}}{m} \frac{ω^{2}}{ω_{r} ω_{z}} {\tilde{r}}_{H} \tilde{z} (f H_{+} + g H_{\times}) cos (ω T_{eff}) .

(A14)

Unlike the unperturbed case, where

g_{μ ν} k^{μ} k^{ν}

at the acoustic horizon is equal to 0 [see Eq. ], here, we have a dynamical horizon and because of that it is not required to be a null hypersurface.³¹ Depending on the sign of the cosine, we can have a null, timelike, or spacelike hypersurface. However, for a Killing horizon, this quantity should be zero and indeed for sufficiently low fluctuations this term vanishes: it is of order

O (ε ω^{2} / ω_{r} ω_{z})

⁠. The null vector field computed in this appendix allows to evaluate the expansion rate, shear, and rotation of the null geodesic congruence.³² We will discuss this topic in a future publication.

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Published open access through an agreement withJISC Collections

Gravitational waves and Black Hole perturbations in acoustic analogues

I. INTRODUCTION

II. THEORETICAL SETUP

III. GRAVITATIONAL WAVE ACOUSTIC METRIC

A. Gravitational waves in flat spacetime

B. Method

1. Acoustic metric perturbation

2. GWs in different gauges

3. Comparison of GW with the perturbed acoustic metric

4. Physical systems

IV. ACOUSTIC BH WITH A GW–LIKE PERTURBATION

A. Vortex geometry

B. Method

1. Cylindrical perturbations

2. Physical requirements

3. Acoustic metric

4. Perturbed acoustic horizon

V. CONCLUSIONS AND FUTURE PERSPECTIVES

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX: HORIZON'S GENERATORS

REFERENCES

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Gravitational waves and Black Hole perturbations in acoustic analogues

I. INTRODUCTION

II. THEORETICAL SETUP

III. GRAVITATIONAL WAVE ACOUSTIC METRIC

A. Gravitational waves in flat spacetime

B. Method

1. Acoustic metric perturbation

2. GWs in different gauges

3. Comparison of GW with the perturbed acoustic metric

4. Physical systems

IV. ACOUSTIC BH WITH A GW–LIKE PERTURBATION

A. Vortex geometry

B. Method

1. Cylindrical perturbations

2. Physical requirements

3. Acoustic metric

4. Perturbed acoustic horizon

V. CONCLUSIONS AND FUTURE PERSPECTIVES

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX: HORIZON'S GENERATORS

REFERENCES

Citing articles via

Submit your article

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Resources

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