Analogue Spacetime Research Articles

Anisotropic inflation in dipolar Bose–Einstein condensates

Early during the era of cosmic inflation, rotational invariance may have been broken, only later emerging as a feature of low-energy physics. This motivates ongoing searches for residual signatures of anisotropic space-time, for example in the power spectrum of the cosmic microwave background. We propose that dipolar Bose–Einstein condensates (BECs) furnish a laboratory quantum simulation platform for the anisotropy evolution of fluctuation spectra during inflation, exploiting the fact that the speed of dipolar condensate sound waves depends on direction. We construct the anisotropic analogue space-time metric governing sound, by linking the time-varying strength of dipolar and contact interactions in the BEC to the scale factors in different coordinate directions. Based on these, we calculate the dynamics of phonon power spectra during an inflation that renders the initially anisotropic Universe isotropic. We find that the expansion speed provides an experimental handle to control and study the degree of final residual anisotropy. Gravity analogues using dipolar condensates can thus provide tuneable experiments for a field of cosmology that was until now confined to a single experiment, our Universe.

Petrov classification of analogue spacetimes

In an effort to invariantly characterize the conformal curvature structure of analogue spacetimes built from a nonrelativistic fluid background, we determine the Petrov type of a variety of laboratory geometries. Starting from the simplest examples, we increase the complexity of the background, and thereby determine how the laboratory fluid symmetry affects the corresponding Petrov type in the analogue spacetime realm of the sound waves. We find that for more complex flows isolated hypersurfaces develop, which are of a Petrov type differing from that of the surrounding fluid. Finally, we demonstrate that within the incompressible background approximation, as well as for all compressible quasi-one-dimensional flows, the only possible Petrov types are the algebraically general type I and the algebraically special types O and D.

Explicit formulae for surface gravities in stationary circular axi-symmetric spacetimes

Using minimalist assumptions we develop a natural functional decomposition for the spacetime metric, and explicit tractable formulae for the surface gravities, in arbitrary stationary circular (PT symmetric) axisymmetric spacetimes. We relate rigidity results, (the existence of a Killing horizon), and the zeroth law to the absence of curvature singularities at the would-be horizons. These observations are of interest to both observational astrophysicists (modelling the cold, dark, heavy objects at the center of most spiral galaxies), and to the analogue spacetime community, (wherein the presence of naked singularities is not necessarily deprecated, and the occurrence of non-Killing horizons is relatively common).

Circular motion analogue Unruh effect in a thermal bath: robbing from the rich and giving to the poor

The Unruh effect states that a uniformly linearly accelerated observer with proper acceleration experiences the Minkowski vacuum as a thermal state at temperature . An observer in uniform circular motion experiences a similar effective temperature, operationally defined in terms of excitation and de-excitation rates, and physically interpretable in terms of synchrotron radiation, but this effective temperature depends not just on the acceleration but also on the orbital speed and the excitation energy. In this paper we consider an observer in uniform circular motion when the Minkowski vacuum is replaced by an ambient thermal bath, and we address the interplay of ambient temperature, Doppler effect, acceleration, and excitation energy. Specifically, we consider a massless scalar field in spacetime dimensions, probed by an Unruh–DeWitt detector, in a Minkowski (rather than proper) time formulation: this setting describes proposed analogue spacetime systems in which the effect may become experimentally testable, and in which an ambient temperature will necessarily be present. We establish analytic results for the observer’s effective temperature in several asymptotic regions of the parameter space and provide numerical results in the interpolating regions, finding that an acceleration effect can be identified even when the Doppler effect dominates the overall magnitude of the response. We also identify parameter regimes where the observer sees a temperature lower than the ambient temperature, experiencing a cooling Unruh effect.

Non-linear effective field theory simulators in two-fluid interfaces

Analogue gravity offers an approach for testing the universality and robustness of quantum field theories in curved spacetimes and validating them using down-to-earth, laboratory-based experiments. Fluid interfaces are a promising framework for creating these gravity simulators and have successfully replicated phenomena such as Hawking radiation and black hole superradiance. Recent work has shown that hydrodynamical instabilities on the interface between two fluids can capture features of the post-inflationary thermalisation of the Early Universe. In this study, we extend fluid dynamics methods to develop an effective field theory for the interface between two fluids, demonstrating the equivalence between the governing equations and a relativistic scalar field in an analogue spacetime. We also show that the interfacial height field serves as the analogue relativistic field even in a nonlinear, interacting field theory. We propose that these mathematical equivalences can be extrapolated to probe regimes where calculations are challenging or impractical. Our work provides a new framework for simulating far-from-equilibrium cosmological and gravitational scenarios in the laboratory.

Painlevé–Gullstrand coordinates versus Kerr spacetime geometry

We discuss the tension between the possible existence of Painlevé–Gullstrand coordinate systems versus the explicit geometrical features of the Kerr spacetime; a subject of interest to Professor Thanu Padmanabhan in the weeks immediately preceding his unexpected death. We shall carefully distinguish strong and weak Painlevé–Gullstrand coordinate systems, and conformal variants thereof, cataloguing what we know can and cannot be done—sometimes we can make explicit global statements, sometimes we must resort to implicit local statements. For the Kerr spacetime the best that seems to be achievable is to set the lapse function to unity and represent the spatial slices with a 3-metric in factorized unimodular form; this arises from considering the Doran version of Kerr spacetime in Cartesian coordinates. We finish by exploring the (limited) extent to which this construction might possibly lead to implementing an “analogue spacetime” model suitable for laboratory simulations of the Kerr spacetime.

Carter-Penrose diagrams for emergent spacetime in axisymmetrically accreting black hole systems

For general relativistic, inviscid, axisymmetric flow around Kerr black hole one may choose different flow thickness. The stationary flow equations can be solved using methods of dynamical system to get transonic accretion flows , i.e, flow infalling in the blackhole that turns supersonic from subsonic with decreasing radial distance, or vice versa. This transonic flows are obtained by choosing the particular flow passing through critical points of phase portrait. For certain flow thickness like the one maintaining conical shape, the sonic point coincide with the critical point. But there are certain flows maintaining hydrostatic equilibrium, such as the one described by Novikov-Thorne, where the sonic point is not same as the critical point. We perturb the flow for both kind of flow and study the behaviour of linear perturbation which behaves like massless scalar field in some curved spacetime, known as, analogue space time. We draw the compactified causal structure, i.e, Penrose Carter diagram for both kind of analogue metric and prove that for both cases critical points are the acoustic horizons, whereas in the case where sonic points do not coincide with critical points, the sonic points are not the acoustic horizon, as one may expect from the definition of sound speed.

Unruh and analogue Unruh temperatures for circular motion in 3+1 and 2+1 dimensions

The Unruh effect states that a uniformly linearly accelerated observer with proper acceleration $a$ experiences Minkowski vacuum as a thermal state in the temperature $T_{\text{lin}} = a/(2\pi)$, operationally measurable via the detailed balance condition between excitation and de-excitation probabilities. An observer in uniform circular motion experiences a similar Unruh-type temperature $T_{\text{circ}}$, operationally measurable via the detailed balance condition, but $T_{\text{circ}}$ depends not just on the proper acceleration but also on the orbital radius and on the excitation energy. We establish analytic results for $T_{\text{circ}}$ for a massless scalar field in $3+1$ and $2+1$ spacetime dimensions in several asymptotic regions of the parameter space, and we give numerical results in the interpolating regions. In the ultrarelativistic limit, we verify that in $3+1$ dimensions $T_{\text{circ}}$ is of the order of $T_{\text{lin}}$ uniformly in the energy, as previously found by Unruh, but in $2+1$ dimensions $T_{\text{circ}}$ is significantly lower at low energies. We translate these results to an analogue spacetime nonrelativistic field theory in which the circular acceleration effects may become experimentally testable in the near future. We establish in particular that the circular motion analogue Unruh temperature grows arbitrarily large in the near-sonic limit, encouragingly for the experimental prospects, but the growth is weaker in effective spacetime dimension $2+1$ than in $3+1$.

Acoustic ray method derived with the concept of analogue gravity for the calculation of the sound field due to rotating sound sources

Acoustics in a moving fluid is a simple model of an analogue space–time. The analysis shows that the null geodesics of the metric gμν in that space–time correspond to the rays in the eikonal approximation of geometrical acoustics. The physical connection to general relativity is the propagation of sound along the null geodesics that corresponds to the propagation of a scalar field in a background gravitational field. In this framework, we derive a three-dimensional acoustic ray method in a rotating frame of reference and calculate an acoustic metric, acoustic rays, travel times, and amplitudes for the fluctuating part of the pressure. Further the application to rotating beamforming, i.e., localization and estimation of the strength of rotating sound sources with a microphone array, is presented. Analytical calculations are compared with the spherical harmonic series expansion method for the sound propagation in the rotating frame of reference. Two benchmark cases on rotating sound sources with generic time-data and measured time-data of an axial fan demonstrate its applicability to rotating beamforming.

Spinning gyroscope in an acoustic black hole: precession effects and observational aspects

The exact precession frequency of a freely-precessing test gyroscope is derived for a 2+1 dimensional rotating acoustic black hole analogue spacetime, without making the somewhat unrealistic assumption that the gyroscope is static. We show that, as a consequence, the gyroscope crosses the acoustic ergosphere of the black hole with a finite precession frequency, provided its angular velocity lies within a particular range determined by the stipulation that the Killing vector is timelike over the ergoregion. Specializing to the ‘Draining Sink’ acoustic black hole, the precession frequency is shown to diverge near the acoustic horizon, instead of the vicinity of the ergosphere. In the limit of an infinitesimally small rotation of the acoustic black hole, the gyroscope still precesses with a finite frequency, thus confirming a behaviour analogous to geodetic precession in a physical non-rotating spacetime like a Schwarzschild black hole. Possible experimental approaches to detect acoustic spin precession and measure the consequent precession frequency, are discussed.

Electromagnetic analogue space-times, analytically and algebraically

While quantum field theory could more aptly be called the ‘quantum field framework’—as it encompasses a vast variety of varying concepts and theories—in comparison, relativity, both special and general, is more commonly portrayed as less of a ‘general framework’. Viewed from this perspective, the paradigm of analogue space-times (also often called analogue gravity) is to promote the specific theory of general relativity (Einstein gravity) to a framework which covers relativistic phenomena at large. Ultimately, this then also gives rise to new proposals for experiments in the laboratory, as it allows one to move general features of the ‘relativistic framework’ from general relativity to entirely new new areas of physics. This allows one to experimentally look into analogies of currently unobservable phenomena of general relativity proper. The only requirement for this to work is the presence of a notion of an upper limit for propagation speeds in this new setting. Systems of such a kind abound in physics, as all hyperbolic wave equations fulfil this requirement.Consequently, models for analogue space-times can be found aplenty. We shall demonstrate this here in two separate analogue space-time models, both taken from electrodynamics in continuous media. First of all, one can distinguish between analytic analogue models (where the analogue is based on some specific hyperbolic differential equation), on the one hand, and algebraic models (where the analogue is fashioned from the more or less explicit appearance of a metric tensor), on the other hand. Yet this distinction is more than just a matter of taste: the nature of the analogue space-time model will also determine which physical concepts from general relativity can be taken easily into an experimental context. Examples of this will constitute the main aim of this paper, and the Hawking effect in one of the two models considered the example of most immediate experimental interest.

Thermality from a Rindler quench

Ultracold fermionic atoms in an optical lattice, with a sudden position-dependent change (a quench) in the effective dispersion relation, have been proposed by Rodríguez-Laguna et al as an analogue spacetime test of the Unruh effect. We provide new support for this analogue by analysing a massless scalar field on a (1+1)-dimensional continuum spacetime with a similar quench: an early time Minkowski region is joined at a constant time surface, representing the quench, to a late time static region in which left and right asymptotically Rindler domains are connected by a smooth negative curvature bridge. We show that the quench is energetically mild, and late time static observers, modelled as a derivative-coupling Unruh–DeWitt detector, see thermality, in a temperature that equals the Unruh temperature for observers in the asymptotic Rindler domains. The Unruh effect hence prevails, despite the energy injected into the field by the quench and despite the absence of a late time Killing horizon. These results strengthen the motivation to realise the experimental proposal.

Bespoke analogue space-times: meta-material mimics

Modern meta-materials allow one to construct electromagnetic media with almost arbitrary bespoke permittivity, permeability, and magneto-electric tensors. If (and only if) the permittivity, permeability, and magneto-electric tensors satisfy certain stringent compatibility conditions, can the meta-material be fully described (at the wave optics level) in terms of an effective Lorentzian metric—an analogue spacetime. We shall consider some of the standard black-hole spacetimes of primary interest in general relativity, in various coordinate systems, and determine the equivalent meta-material susceptibility tensors in a laboratory setting. In static black hole spacetimes (Schwarzschild and the like) certain eigenvalues of the susceptibility tensors will be seen to diverge on the horizon. In stationary black hole spacetimes (Kerr and the like) certain eigenvalues of the susceptibility tensors will be seen to diverge on the ergo-surface.

Effective metrics and a fully covariant description of constitutive tensors in electrodynamics

Using electromagnetism to study analogue space-times is tantamount to considering consistency conditions for when a given (meta-)material would provide an analogue space-time model or --- vice versa --- characterizing which given metric could be modelled with a (meta-)material. While the consistency conditions themselves are by now well known and studied, the form the metric takes once they are satisfied is not. This question is mostly easily answered by keeping the formalisms of the two research fields here in contact as close to each other as possible. While fully covariant formulations of the electrodynamics of media have been around for a long while, they are usually abandoned for (3+1)- or 6-dimensional formalisms. Here we shall use the fully unified and fully covariant approach. This enables us even to generalize the consistency conditions for the existence of an effective metric to arbitrary background metrics beyond flat space-time electrodynamics. We also show how the familiar matrices for permittivity $\epsilon$, permeability $\mu^{-1}$, and magneto-electric effects $\zeta$ can be seen as the three independent pieces of the Bel decomposition for the constitutive tensor $Z^{abcd}$, i.e., the components of an orthogonal decomposition with respect to a given observer with four-velocity $V^a$. Finally, we shall use the Moore--Penrose pseudo-inverse and the closely related pseudo-determinant to then gain the desired reconstruction of the effective metric in terms of the permittivity tensor $\epsilon^{ab}$, the permeability tensor $[\mu^{-1}]^{ab}$, and the magneto-electric tensor $\zeta^{ab}$, as an explicit function $g_{\text{eff}}(\epsilon,\mu^{-1},\zeta)$.

Inertial Frame Dragging in an Acoustic Analogue Spacetime

AbstractWe report an incipient exploration of the Lense‐Thirring precession effect in a rotating acoustic analogue black hole spacetime. An exact formula is deduced for the precession frequency of a gyroscope due to inertial frame dragging, close to the ergosphere of a ‘Draining Bathtub’ acoustic spacetime which has been studied extensively for acoustic Hawking radiation of phonons and also for ‘superresonance’. The formula is verified by embedding the two dimensional spatial (acoustic) geometry into a three dimensional one where the similarity with standard Lense‐Thirring precession results within a strong gravity framework is well known. Prospects of experimental detection of this new ‘fixed‐metric’ effect in acoustic geometries, are briefly discussed.

On the Use of the Analogue Transformation Acoustics in Aeroacoustics

The objective of the paper is the assessment of the Analogue Transformation Acoustics (ATA) in the design of acoustic metamaterial for aeronautical applications. The work focuses on the consistency of the background flow resulting from the application of the ATA with the equations governing the potential aerodynamics. Indeed, in case of acoustic perturbations propagating within moving media, the convective terms in the governing equations are responsible for the failure of formal invariance under the action of conformal mappings. The ATA approach overcomes this limitation, introducing the possibility of handling the convective form of the wave equation in a straightforward and elegant way. It is based on the concept of analogue space-time and fully relies on the analytical tools of Lorentzian differential geometry. The present paper analyses the relationship between the analogue velocity field with a realistic potential flow. The method is validated through numerical simulations using two widely assessed acoustic cloaking problems. The preliminary results obtained show that the use of numerical, quasi-conformal mappings can lead to transformed streamlines negligibly deviating from those of the potential velocity field satisfying the fluid-dynamic conservation laws, but with incompatible intensity of the local velocity.

On spin dependence of relativistic acoustic geometry

This work makes the first ever attempt to understand the influence of the black hole background spacetime in determining the fundamental properties of the embedded relativistic acoustic geometry. To accomplish such task, we investigate the role of the spin angular momentum of the astrophysical black hole (the Kerr parameter a—a representative feature of the background black hole metric) in estimating the value of the acoustic surface gravity (the representative feature of the corresponding analogue spacetime). Since almost all astrophysical black holes are supposed to posses some degree of intrinsic rotation, the influence of the Kerr parameter on classical analogue models is very important to understand. We study the general relativistic, axially symmetric, non-self-gravitating inflow of the hydrodynamic fluid onto a rotating astrophysical black hole from the dynamical systems point of view. In this work the location of the acoustic horizon inside such fluid flow is identified and the associated acoustic surface gravity is estimated. We study the dependence of such surface gravity as a function of the Kerr parameter as well as with other dynamical and thermodynamic variables governing the fluid flow under strong gravity, and demonstrate that for retrograde flow, the surface gravity (and hence the associated analogue Hawking temperature) correlates with the black hole spin in general, whereas for the prograde flow, the surface gravity as well as the analogue temperature correlates with the black hole spin for slow to moderately rotating holes, but anti-correlates with the spin for fast to extremely rotating holes. We found that for certain values of the initial boundary conditions, more than one acoustic horizons, namely two black hole types and one white hole type, may form, and the surface gravity may become formally infinite at the acoustic white hole. We discuss the possible connection between the corresponding analogue Hawking temperature and astrophysically relevant observables associated with the spectral signature of the black hole candidates. Our result indicates that the modified dispersion relation evaluated at the close proximity of the acoustic horizon (and hence the nonuniversal feature of Hawking-like effects) is a sensitive function of the spin angular momentum of the astrophysical black hole. We propose that the black hole spin dependence of such dispersion relation may be used to distinguish a corotating flow from a counter rotating flow for axisymmetric accretion onto a Kerr black hole.

Tailoring diffusion in analog spacetimes

Diffusive transport is characterized by the scaling law (length)^{2}∝(time). In this paper we show that this relationship is significantly altered in curved analog spacetimes. This circumstance provides an opportunity to tailor diffusion: by a suitable design of the analog metric, it is possible to create materials where diffusion is either faster or slower than in normal media, as desired. This prediction can, in principle, be tested experimentally with optical analogs, curved graphene sheets, and so on (indeed with any analog spacetime).

Birefringence in pseudo–Finsler spacetimes

Based on the analogue spacetime programme, and many other ideas currently mooted in "quantum gravity", there is considerable ongoing speculation that the usual pseudo-Riemannian (Lorentzian) manifolds of general relativity might eventually be modified at short distances. Two specific modifications that are often advocated are the adoption of Finsler geometries (or more specifically, pseudo-Finsler spacetimes) and the possibility of birefringence (or more generally, multi-refringence). We have investigated the possibility of whether it is possible to usefully and cleanly deal with these two possibilities simultaneously. That is, given two (or more) "signal cones": Is it possible to naturally and intuitively construct a "unified" pseudo-Finsler spacetime such that the pseudo-Finsler metric is null on these "signal cones", but has no other zeros or singularities? Our results are much less encouraging than we had originally hoped, and suggest that while pseudo-Finsler spacetimes are certainly useful constructs, it is physically more appropriate to think of physics as taking place in a single topological manifold that carries several distinct pseudo-Finsler metrics, one for each polarization mode.

Analogue Hawking Radiation in a dc-SQUID Array Transmission Line

We propose the use of a superconducting transmission line formed from an array of direct-current superconducting quantum interference devices for investigating analogue Hawking radiation. Biasing the array with a space-time varying flux modifies the propagation velocity of the transmission line, leading to an effective metric with a horizon. Being a fundamentally quantum mechanical device, this setup allows for investigations of quantum effects such as backreaction and analogue space-time fluctuations on the Hawking process.