In this section, we will attempt to categorise the very many analogue models researchers have investigated. Perhaps the most basic subdivision is into classical models and quantum models, but even then many other levels of refinement are possible. Consider for instance the following list:
-
Classical models:
-
Classical sound.
-
Sound in relativistic hydrodynamics.
-
Water waves (gravity waves).
-
Classical refractive index.
-
Normal modes.
-
Quantum models:
-
Bose-Einstein condensates (BECs).
-
The heliocentric universe.
(Helium as an exemplar for just about anything.)
-
Slow light.
We will now provide a few words on each of these topics.
Classical models
Classical sound
Sound in a non-relativistic moving fluid has already been extensively discussed in Section 2, and we will not repeat such discussion here. In contrast, sound in a solid exhibits its own distinct and interesting features, notably in the existence of a generalization of the normal notion of birefringence — longitudinal modes travel at a different speed (typically faster) than do transverse modes. This may be viewed as an example of an analogue model which breaks the “light cone” into two at the classical level; as such this model is not particularly useful if one is trying to simulate special relativistic kinematics with its universal speed of light, though it may be used to gain insight into yet another way of “breaking” Lorentz invariance (and the equivalence principle).
Sound in relativistic hydrodynamics
When dealing with relativistic sound, key historical papers are those of Moncrief [448] and Bilic [72], with astrophysical applications being more fully explored in [162, 161, 160], and with a more recent pedagogical follow-up in [639]. It is convenient to first quickly motivate the result by working in the limit of relativistic ray acoustics where we can safely ignore the wave properties of sound. In this limit we are interested only in the “sound cones”. Let us pick a curved manifold with physical spacetime metric g
μν
, and a point in spacetime where the background fluid 4-velocity is Vμ while the speed of sound is c
s
. Now (in complete direct conformity with our discussion of the generalised optical Gordon metric) adopt Gaussian normal coordinates so that g
μν
→ η
μν
, and go to the local rest frame of the fluid, so that \({V^\mu} \rightarrow (1;\overrightarrow 0)\) and
$${g_{\mu \nu}} \rightarrow \left[ {\begin{array}{*{20}c} {- 1} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}} \right];\quad {h_{\mu \nu}} = {g_{\mu \nu}} + {V_\mu}{V_\nu} \rightarrow \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}} \right].$$
(145)
In the rest frame of the fluid the sound cones are (locally) given by
$$- c_s^2{\rm{d}}{t^2} + {\left\Vert {{\rm{d}}\vec x} \right\Vert^2} = 0,$$
(146)
implying in these special coordinates the existence of an acoustic metric
$${{\mathcal G}_{\mu \nu}}\propto\left[ {\begin{array}{*{20}c} {- c_s^2} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}} \right].$$
(147)
That is, transforming back to arbitrary coordinates:
$${{\mathcal G}_{\mu \nu}} \propto - c_s^2{V_\mu}{V_\nu} + \left\{{{g_{\mu \nu}} + {V_\mu}{V_\nu}} \right\} \propto {g_{\mu \nu}} + \left\{{1 - c_s^2} \right\}{V_\mu}{V_\nu}.$$
(148)
Note again that in the ray acoustics limit, because one only has the sound cones to work with, one can neither derive (nor is it even meaningful to specify) the overall conformal factor. When going beyond the ray acoustics limit, seeking to obtain a relativistic wave equation suitable for describing physical acoustics, all the “fuss” is simply over how to determine the overall conformai factor (and to verify that one truly does obtain a d’Alembertian equation using the conformally-fixed acoustic metric).
One proceeds by combining the relativistic Euler equation, the relativistic energy equation, an assumed barotropic equation of state, and assuming a relativistic irrotational flow of the form [639]
$${V^\mu} = {{{g^{\mu \nu}}{\nabla _\nu}\Theta} \over {\left\Vert {\nabla \Theta} \right\Vert}}.$$
(149)
In this situation the relativistic Bernoulli equation can be shown to be
$$\ln \left\Vert {\nabla \Theta} \right\Vert = + \int\nolimits_0^p {{{{\rm{d}}p} \over {\varrho(p) + p}},}$$
(150)
where we emphasize that ϱ is now the energy density (not the mass density), and the total particle number density can be shown to be
$$n(p) = {n_{(p = 0)}}\exp \left[ {\int\nolimits_{\varrho(p = 0)}^{\varrho(p)} {{{{\rm{d\varrho}}} \over {\varrho + p(\varrho)}}}} \right].$$
(151)
After linearization around some suitable background [448, 72, 639], the perturbations in the scalar velocity potential Θ can be shown to satisfy a dimension-independent d’Alembertian equation
$${\nabla _\mu}\left\{{{{n_0^2} \over {{\varrho_0} + {p_0}}}\left[ {- {1 \over {c_s^2}}V_0^\mu V_0^\nu + {h^{\mu \nu}}} \right]{\nabla _\nu}{\Theta _1}} \right\} = 0,$$
(152)
which leads to the identification of the relativistic acoustic metric as
$$\sqrt {- {\mathcal G}} {{\mathcal G}^{\mu \nu}} = {{n_0^2} \over {{\varrho_0} + {p_0}}}\left[ {- {1 \over {c_s^2}}V_0^\mu V_0^\nu + {h^{\mu \nu}}} \right].$$
(153)
The dimension-dependence now comes from solving this equation for \({{\mathcal G}^{\mu \nu}}\). Therefore, we finally have the (contravariant) acoustic metric
$${{\mathcal G}^{\mu \nu}} = {\left({{{n_0^2c_s^{- 1}} \over {{\varrho_0} + {p_0}}}} \right)^{- 2/(d - 1)}}\left\{{- {1 \over {c_s^2}}V_0^\mu V_0^\nu + {h^{\mu \nu}}} \right\},$$
(154)
and (covariant) acoustic metric
$${{\mathcal G}_{\mu \nu}} = {\left({{{n_0^2c_s^{- 1}} \over {{\varrho _0} + {p_0}}}} \right)^{- 2/(d - 1)}}\left\{{- c_s^2{{\left[ {{V_0}} \right]}_\mu}{{\left[ {{V_0}} \right]}_\nu} + {h_{\mu \nu}}} \right\}.$$
(155)
In the non-relativistic limit p0 ≪ ϱ0 and \({\varrho _0} \approx \bar {m}\,{n_0}\), where \(\bar{m}\) is the average mass of the particles making up the fluid (which by the barotropic assumption is a time-independent and position-independent constant). So in the non-relativistic limit we recover the standard result for the conformal factor [639]
$${{n_0^2c_s^{- 1}} \over {{\varrho_0} + {p_0}}} \rightarrow {{{n_0}} \over {\bar m{c_s}}} = {1 \over {{{\bar m}^2}}}{{{\rho _0}} \over {{c_s}}}\propto{{{\rho _0}} \over {{c_s}}}.$$
(156)
Under what conditions is the fully general relativistic discussion of this section necessary? (The non-relativistic analysis is, after all, the basis of the bulk of the work in “analogue spacetimes”, and is perfectly adequate for many purposes.) The current analysis will be needed in three separate situations:
-
when working in a nontrivial curved general relativistic background;
-
whenever the fluid is flowing at relativistic speeds;
-
less obviously, when the internal degrees of freedom of the fluid are relativistic, even if the overall fluid flow is non-relativistic. (That is, in situations where it is necessary to distinguish the energy density from the mass density ρ; this typically happens in situations where the fluid is strongly self-coupled — for example in neutron star cores or in relativistic BECs [191]. See Section 4.2.)
Shallow water waves (gravity waves)
A wonderful example of the occurrence of an effective metric in nature is that provided by gravity waves in a shallow basin filled with liquid [560]. (See Figure 10.)Footnote 18 If one neglects the viscosity and considers an irrotational flow, v = ∇Φ, one can write Bernoulli’s equation in the presence of Earth’s gravity as
$${\partial _t}\phi + {1 \over 2}{(\nabla \phi)^2} = - {p \over \rho} - gz - {V_{\vert\vert}}.$$
(157)
Here ρ is the density of the fluid, p its pressure, g the gravitational acceleration and V‖ a potential associated with some external force necessary to establish an horizontal flow in the fluid. We denote that flow by \({\rm{v}}_{\rm{B}}^\parallel\). We must also impose the boundary conditions that the pressure at the surface, and the vertical velocity at the bottom, both vanish. That is, p(z = hB) = 0, and v⊥(z = 0) = 0.
Once a horizontal background flow is established, one can see that the perturbations of the velocity potential satisfy
$${\partial _t}\delta \phi + {\rm{v}}_{\rm{B}}^{\vert\vert} \cdot {{\bf{\nabla}} _{\vert\vert}}\delta \phi = - {{\delta p} \over \rho}.$$
(158)
If we now expand this perturbation potential in a Taylor series
$$\delta \phi = \sum\limits_{n = 0}^\infty {{{{z^n}} \over {n!}}\delta {\phi _n}} (x,y),$$
(159)
it is not difficult to prove [560] that surface waves with long wavelengths (long compared with the depth of the basin, λ ≫ hB), can be described to a good approximation by δΦ0(x, y) and that this field “sees” an effective metric of the form
$${\rm{d}}{s^2} = {1 \over {{c^2}}}\left[ {- ({c^2} - v_{\rm{B}}^{\vert\vert2}){\rm{d}}{t^2} - 2{\rm{v}}_{\rm{B}}^{\vert\vert} \cdot {\bf{dx}}\,{\rm{d}}t + {\bf{dx}} \cdot {\bf{dx}}} \right],$$
(160)
where \(c \equiv \sqrt {g{h_{\rm{B}}}}\). The link between small variations of the potential field and small variations of the position of the surface is provided by the following equation
$$\delta {v_ \bot} = - {h_B}\,\nabla _\parallel ^2\delta {\phi _0} = {\delta _t}\,\delta h + {\bf{v}}_{\rm{B}}^\parallel \, \cdot \,{{\bf{\nabla}} _\parallel}\delta h = {{\rm{d}} \over {{\rm{d}}t}}\delta h.$$
(161)
The entire previous analysis can be generalised to the case in which the bottom of the basin is not flat, and the background flow not purely horizontal [560]. Therefore, one can create different effective metrics for gravity waves in a shallow fluid basin by changing (from point to point) the background flow velocity and the depth, hB(x, y).
The main advantage of this model is that the velocity of the surface waves can very easily be modified by changing the depth of the basin. This velocity can be made very slow, and consequently, the creation of ergoregions should be relatively easier than in other models. As described here, this model is completely classical given that the analogy requires long wavelengths and slow propagation speeds for the gravity waves. Although the latter feature is convenient for the practical realization of analogue horizons, it is a disadvantage in trying to detect analogue Hawking radiation as the relative temperature will necessarily be very low. (This is why, in order to have a possibility of experimentally observing (spontaneous) Hawking evaporation and other quantum phenomena, one would need to use ultra-cold quantum fluids.) However, the gravity wave analogue can certainly serve to investigate the classical phenomena of mode mixing that underlies the quantum processes.
It is this particular analogue model (and its extensions to finite depth and surface tension) that underlies the experimental [532] and theoretical [531] work of Rousseaux et al., the historically-important experimental work of Badulin et al. [17], and the very recent experimental verification by Weinfurtner et al. of the existence of classical stimulated Hawking radiation [682].
More general water waves
If one moves beyond shallow-water surface waves the physics becomes more complicated. In the shallow-water regime (wavelength λ much greater than water depth d) the co-moving dispersion relation is a simple linear one ω = c
s
k, where the speed of sound can depend on both position and time. Once one moves to finite-depth (λ ∼ d) or deep (λ ≪ d) water, it is a standard result that the co-moving dispersion relation becomes
$$\omega = \sqrt {gk\,\tanh (kd)} = {c_s}k\,\sqrt {{{\tanh (kd)} \over {kd}}}.$$
(162)
See, for instance, Lamb [370] §228, p. 354, Equation (5). A more modern discussion in an analogue spacetime context is available in [643]. Adding surface tension requires a brief computation based on Lamb [370] §267 p. 459, details can be found in [643]. The net result is
$$\omega = {c_s}k\,\sqrt {1 + {k^2}/{K^2}} \,\sqrt {{{\tanh (kd)} \over {kd}}}.$$
(163)
Here K2 = gρ/σ is a constant depending on the acceleration due to gravity, the density, and the surface tension [643]. Once one adds the effects of fluid motion, one obtains
$$\omega = {\bf{v}}\,\cdot\,{\bf{k}} + {c_s}k\,\sqrt {1 + {k^2}/{K^2}} \,\sqrt {{{\tanh (kd)} \over {kd}}}.$$
(164)
All of these features, fluid motion, finite depth, and surface tension (capillarity), seem to be present in the 1983 experimental investigations by Badulin et al. [17]. All of these features should be kept in mind when interpreting the experimental [532] and theoretical [531] work of Rousseaux et al., and the very recent experimental work by Weinfurtner et al. [682].
A feature that is sometimes not remarked on is that the careful derivation we have previously presented of the acoustic metric, or in this particular situation the derivation of the shallow-water-wave effective metric [560], makes technical assumptions tantamount to asserting that one is in the regime where the co-moving dispersion relation takes the linear form ω ≈ c
s
k. Once the co-moving dispersion relation becomes nonlinear, the situation is more subtle, and based on a geometric acoustics approximation to the propagation of signal waves one can introduce several notions of conformal “rainbow” metric (momentum-dependent metric). Consider
and the inverse
At a minimum we could think of using the following notions of propagation speed
$$c({k^2}) \rightarrow \left\{{\begin{array}{*{20}c} {{c_{{\rm{phase}}}}({k^2});\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{c_{{\rm{group}}}}({k^2});\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{c_{{\rm{sound}}}} = \underset{k \rightarrow 0}{\lim} \,{c_{{\rm{phase}}}}({k^2})\,{\rm{provided}}\,{\rm{this}}\,{\rm{equals}}\,\underset{k \rightarrow 0}{\lim} \,{c_{{\rm{group}}}}({k^2});} \\ {{c_{{\rm{signal}}}} = \underset{k \rightarrow \infty}{\lim} \,{c_{{\rm{phase}}}}({k^2}).\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}} \right.$$
(167)
Brillouin, in his classic paper [92], identified at least six useful notions of propagation speed, and many would argue that the list can be further refined. Each one of these choices for the rainbow metric encodes different physics, and is useful for different purposes. It is still somewhat unclear as to which of these rainbow metrics is “best” for interpreting the experimental results reported in [17, 532, 682].
Classical refractive index
The macroscopic Maxwell equations inside a dielectric take the well-known form
$${\bf{\nabla}} \cdot {\bf{B}} = 0,\quad {\bf{\nabla}} \times {\bf{E}} + {\partial _t}{\bf{B}} = 0,$$
(168)
$${\bf{\nabla}} \cdot{\bf{D}} = 0,\quad {\bf{\nabla}} \times {\bf{H}} - {\partial _t}{\bf{D}} = 0,$$
(169)
with the constitutive relations H = μ−1 · B and D = ϵ · E. Here, ϵ is the 3 × 3 permittivity tensor and μ the 3 × 3 permeability tensor of the medium. These equations can be written in a condensed way as
$${\partial _\alpha}({Z^{\mu \alpha \nu \beta}}\,{F_{\nu \beta}}) = 0$$
(170)
where F
νβ
= A[μ, β] is the electromagnetic tensor,
$${F_{0i}} = - {F_{i0}} = - {E_i},\quad {F_{ij}} = {\varepsilon _{ijk}}{B^k},$$
(171)
and (assuming the medium is at rest) the non-vanishing components of the 4th-rank tensor are given by
$${Z^{0i0j}} = - {Z^{0ij0}} = {Z^{i0j0}} = - {Z^{i00j}} = - {1 \over 2}{\epsilon^{ij}};$$
(172)
$${Z^{ijkl}} = {1 \over 2}{\varepsilon ^{ijm}}\,{\varepsilon ^{kln}}\,\mu _{mn}^{- 1};$$
(173)
supplemented by the conditions that Z is antisymmetric on its first pair of indices and antisymmetric on its second pair of indices. Without significant loss of generality we can ask that Z also be symmetric under pairwise interchange of the first pair of indices with the second pair — thus Z exhibits most of the algebraic symmetries of the Riemann tensor, though this appears to merely be accidental, and not fundamental in any way.
If we compare this to the Lagrangian for electromagnetism in curved spacetime
$${\mathcal L} = \sqrt {- g} \,{g^{\mu \alpha}}\,{g^{\nu \beta}}\,{F_{\mu \nu}}\,{F_{\alpha \beta}}$$
(174)
we see that in curved spacetime we can also write the electromagnetic equations of motion in the form (170) where now (for some constant K):
$${Z^{\mu \nu \alpha \beta}} = K\,\sqrt {- g} \,\left\{{{g^{\mu \alpha}}\,{g^{\nu \beta}} - {g^{\mu \beta}}\,{g^{\nu \alpha}}} \right\}.$$
(175)
If we consider a static gravitational field we can always re-write it as a conformal factor multiplying an ultra-static metric
$${g_{\mu \nu}} = {\Omega ^2}\,\{- 1 \oplus \,{g_{ij}}\}$$
(176)
then
$${Z^{0i0j}} = - {Z^{0ij0}} = {Z^{i0j0}} = - {Z^{i00j}} = - K\,\sqrt {- g} \,{g^{ij}};$$
(177)
$${Z^{ijkl}} = K\,\sqrt {- g} \left\{{{g^{ik}}\,{g^{jl}} - {g^{il}}\,{g^{jk}}} \right\}.$$
(178)
The fact that Z is independent of the conformal factor Ω is simply the reflection of the well-known fact that the Maxwell equations are conformally invariant in (3+1) dimensions. Thus, if we wish to have the analogy (between a static gravitational field and a dielectric medium at rest) hold at the level of the wave equation (physical optics) we must satisfy the two stringent constraints
$$K\,\sqrt {- g} \,{g^{ij}} = {1 \over 2}\,{\epsilon^{ij}};$$
(179)
$$K\,\sqrt {- g} \,\,\left\{{{g^{ik}}\,{g^{jl}} - {g^{il}}\,{g^{jk}}} \right\} = {1 \over 2}{\varepsilon ^{ijm}}\,{\varepsilon ^{kln}}\,\mu _{mn}^{- 1}.$$
(180)
The second of these constraints can be written as
$$K\,\sqrt {- g} \,{\varepsilon _{ijm}}\,{\varepsilon _{kln}}\,\left\{{{g^{ik}}\,{g^{jl}}} \right\} = \mu _{mn}^{- 1}.$$
(181)
In view of the standard formula for 3 × 3 determinants
$${\varepsilon _{ijm}}\,{\varepsilon _{kln}}\left\{{{X^{ik}}\,{X^{jl}}} \right\} = 2\det \,X\,X_{mn}^{- 1},$$
(182)
this now implies
$$2K\,{{{g_{ij}}} \over {\sqrt {- g}}} = \mu _{ij}^{- 1},$$
(183)
whence
$${1 \over {2K}}\,\sqrt {- g} {g^{ij}} = {\mu ^{ij}}.$$
(184)
Comparing this with
$$2K\,\sqrt {- g} {g^{ij}} = {\epsilon^{ij}},$$
(185)
we now have:
$${\epsilon^{ij}} = 4\,{K^2}\,{\mu ^{ij}};$$
(186)
$${g^{ij}} = {{4\,{K^2}} \over {\det \,\epsilon}}\,{\epsilon ^{ij}} = {1 \over {4\,{K^2}\,\det \,\mu}}\,{\mu ^{ij}}.$$
(187)
To rearrange this, introduce the matrix square root [μ1/2]ij, which always exists because μ is real positive definite and symmetric. Then
$${g^{ij}} = {\left[ {{{\left\{{{{{\mu ^{1/2}}\, \epsilon\,{\mu ^{1/2}}} \over {\det (\mu \, \epsilon)}}} \right\}}^{1/2}}} \right]^{ij}}.$$
(188)
Note that if you are given the static gravitational field (in the form Ω, g
ij
) you can always solve it to find an equivalent analogue in terms of permittivity/permeability (albeit an analogue that satisfies the mildly unphysical constraint ϵ ∝ μ).Footnote 19 On the other hand, if you are given permeability and permittivity tensors ϵ and μ, then it is only for that subclass of media that satisfy ϵ ∝ μ that one can perfectly mimic all of the electromagnetic effects by an equivalent gravitational field. Of course, this can be done provided one only considers wavelengths that are sufficiently long for the macroscopic description of the medium to be valid. In this respect it is interesting to note that the behaviour of the refractive medium at high frequencies has been used to introduce an effective cutoff for the modes involved in Hawking radiation [523]. We shall encounter this model (which is also known in the literature as a solid state analogue model) later on when we consider the trans-Planckian problem for Hawking radiation. Let us stress that if one were able to directly probe the quantum effective photons over a dielectric medium, then one would be dealing with a quantum analogue model instead of a classical one.
Eikonal approximation
With a bit more work this discussion can be extended to a medium in motion, leading to an extension of the Gordon metric. Alternatively, one can agree to ask more limited questions by working at the level of geometrical optics (adopting the eikonal approximation), in which case there is no longer any restriction on the permeability and permittivity tensors. To see this, construct the matrix
$${C^{\mu \nu}} = {Z^{\mu \alpha \nu \beta}}\,{k_\alpha}\,{k_\beta}.$$
(189)
The dispersion relations for the propagation of photons (and therefore the sought for geometrical properties) can be obtained from the reduced determinant of C (notice that the [full] determinant of C is identically zero as Cμνk
ν
= 0; the reduced determinant is that associated with the three directions orthogonal to k
ν
=0). By choosing the gauge A0 = 0 one can see that this reduced determinant can be obtained from the determinant of the 3 × 3 sub-matrix Cij. This determinant is
$$\det ({C^{ij}}) = {1 \over 8}\det (- {\omega ^2}{\epsilon^{ij}} + {\varepsilon ^{ikm}}\,{\varepsilon ^{jln}}\,\mu _{mn}^{- 1}{k_k}{k_l})$$
(190)
or, after making some manipulations,
$$\det ({C^{ij}}) = {1 \over 8}\det \,[ - {\omega ^2}{\epsilon^{ij}} + {(\det \, \mu)^{- 1}}({\mu ^{ij}}{\mu ^{kl}}{k_k}{k_l} - {\mu ^{im}}{k_m}{\mu ^{jl}}{k_l})].$$
(191)
To simplify this, again introduce the matrix square roots [μ1/2]ij and [μ−1/2]ij, which always exist because the relevant matrices are real positive definite and symmetric. Then define
$${\tilde k^i} = {[{\mu ^{1/2}}]^{ij}}\,{k_j}$$
(192)
and
$${[\tilde \epsilon ]^{ij}} = \det (\mu)\,{[{\mu ^{- 1/2}}\,\epsilon \,{\mu ^{- 1/2}}]_{ij}}$$
(193)
so that
$$\det ({C^{ij}})\,\propto\,\det \left\{{- {\omega ^2}{{[ \tilde \epsilon ]}^{ij}} + {\delta ^{ij}}\,[{\delta _{mn}}\,{{\tilde k}^m}\,{{\tilde k}^n}] - {{\tilde k}^i}\,{{\tilde k}^j}} \right\}.$$
(194)
The behaviour of this dispersion relation now depends critically on the way that the eigenvalues of \(\tilde{\epsilon}\) are distributed.
3 degenerate eigenvalues
If all eigenvalues are degenerate then \(\tilde{\epsilon} = \tilde{\epsilon} \,{\bf{I}}\), implying ϵ ∝ μ but now with the possibility of a position-dependent proportionality factor (in the case of physical optics the proportionality factor was constrained to be a position-independent constant). In this case we now easily evaluate
$$\epsilon = {{{\rm{tr}}(\epsilon)} \over {{\rm{tr}}(\mu)}}\, \mu \quad {\rm{and}}\quad \tilde \epsilon = \det \, \mu \,{{{\rm{tr}}(\epsilon)} \over {{\rm{tr}}(\mu)}},$$
(195)
while
$$\det ({C^{ij}})\,\propto\,{\omega ^2}{\left\{{{\omega ^2} - [{{\tilde \epsilon}^{- 1}}\,{\delta _{mn}}\,{{\tilde k}^m}\,{{\tilde k}^n}]} \right\}^2}.$$
(196)
That is
$$\det ({C^{ij}})\,\propto\,{\omega ^2}{\{{\omega ^2} - [{g^{ij}}\,{k_i}\,{k_j}]\} ^2},$$
(197)
with
$${g^{ij}} = {1 \over {\tilde \epsilon}}{[ \mu ]^{ij}} = {{{\rm{tr}}(\mu)\,\,{{[ \mu ]}^{ij}}} \over {{\rm{tr}}(\epsilon)\,\,\det \, \mu}} = {{{\rm{tr}}(\epsilon)\,\,{{[ \epsilon]}^{ij}}} \over {{\rm{tr}}(\mu)\,\,\det \, \epsilon}}.$$
(198)
This last result is compatible with but more general than the result obtained under the more restrictive conditions of physical optics. In the situation where both permittivity and permeability are isotropic, (ϵij = ϵ δij and μij = μ δij) this reduces to the perhaps more expected result
$${g^{ij}} = {{{\delta ^{ij}}} \over {\epsilon\,\mu}}.$$
(199)
2 distinct eigenvalues
If \(\tilde{\epsilon}\) has two distinct eigenvalues then the determinant det(Cij) factorises into a trivial factor of ω2 and two quadratics. Each quadratic corresponds to a distinct effective metric. This is the physical situation encountered in uni-axial crystals, where the ordinary and extraordinary rays each obey distinct quadratic dispersion relations [82]. From the point of view of analogue models this corresponds to a two-metric theory.
3 distinct eigenvalues
If \(\tilde{\epsilon}\) has three distinct eigenvalues then the determinant det(Cij) is the product of a trivial factor of ω2 and a non-factorizable quartic. This is the physical situation encountered in bi-axial crystals [82, 638], and it seems that no meaningful notion of the effective Riemannian metric can be assigned to this case. (The use of Finsler geometries in this situation is an avenue that may be worth pursuing [306]. But note some of the negative results obtained in [573, 574, 575].)
Abstract linear electrodynamics
Hehl and co-workers have championed the idea of using the linear constitutive relations of electrodynamics as the primary quantities, and then treating the spacetime metric (even for flat space) as a derived concept. See [474, 276, 371, 277].
Nonlinear electrodynamics
In general, the permittivity and permeability tensors can be modified by applying strong electromagnetic fields (this produces an effectively nonlinear electrodynamics). The entire previous discussion still applies if one considers the photon as the linear perturbation of the electromagnetic field over a background configuration
$${F_{\mu \nu}} = F_{\mu \nu}^{{\rm{bg}}} + f_{\mu \nu}^{{\rm{ph}}}.$$
(200)
The background field \(F_{\mu \nu}^{{\rm{bg}}}\) sets the value of ϵij(Fbg), and μij(Fbg). Equation (170) then becomes an equation for \(f_{\mu \nu}^{{\rm{ph}}}\). This approach has been extensively investigated by Novello and co-workers [465, 469, 170, 468, 466, 467, 464, 214].
Summary
The propagation of photons in a dielectric medium characterised by 3 × 3 permeability and permittivity tensors constrained by ϵ ∝ μ is equivalent (at the level of geometric optics) to the propagation of photons in a curved spacetime manifold characterised by the ultra-static metric (198), provided one only considers wavelengths that are sufficiently long for the macroscopic description of the medium to be valid. If, in addition, one takes a fluid dielectric, by controlling its flow one can generalise the Gordon metric and again reproduce metrics of the Painlevé-Gullstrand type, and therefore geometries with ergo-regions. If the proportionality constant relating ϵ ∝ μ is position independent, one can make the stronger statement (187) which holds true at the level of physical optics. Recently this topic has been revitalised by the increasing interest in (classical) meta-materials.
Normal mode meta-models
We have already seen how linearizing the Euler-Lagrange equations for a single scalar field naturally leads to the notion of an effective spacetime metric. If more than one field is involved the situation becomes more complicated, in a manner similar to that of geometrical optics in uni-axial and bi-axial crystals. (This should, with hindsight, not be too surprising since electromagnetism, even in the presence of a medium, is definitely a Lagrangian system and definitely involves more than one single scalar field.) A normal mode analysis based on a general Lagrangian (many fields but still first order in derivatives of those fields) leads to a concept of refringence, or more specifically multi-refringence, a generalization of the birefringence of geometrical optics. To see how this comes about, consider a straightforward generalization of the one-field case.
We want to consider linearised fluctuations around some background solution of the equations of motion. As in the single-field case we write (here we will follow the notation and conventions of [45])
$${\phi ^A}(t,\vec x) = \phi _0^A(t,\vec x) + \epsilon\,\phi _1^A(t,\vec x) + {{{\epsilon^2}} \over 2}\,\phi _2^A(t,\vec x) + O({\epsilon^3}){.}$$
(201)
Now use this to expand the Lagrangian
$$\begin{array}{*{20}c} {{\mathcal L}({\partial _\mu}{\phi ^A},{\phi ^A}) = {\mathcal L}({\partial _\mu}\phi _0^A,\phi _0^A) + \epsilon \left[ {{{\partial {\mathcal L}} \over {\partial ({\partial _\mu}{\phi ^A})}}\,{\partial _\mu}\phi _1^A + {{\partial {\mathcal L}} \over {\partial {\phi ^A}}}\,\phi _1^A} \right]} \\ {+ {{{\epsilon ^2}} \over 2}\left[ {{{\partial {\mathcal L}} \over {\partial ({\partial _\mu}{\phi ^A})}}\,{\partial _\mu}\phi _2^A + {{\partial {\mathcal L}} \over {\partial {\phi ^A}}}\,\phi _2^A} \right]} \\ {\quad \,\,\, + {{{\epsilon ^2}} \over 2}\left[ {{{\partial ^2 {\mathcal L}} \over {\partial ({\partial _\mu}{\phi ^A})\,\partial ({\partial _\nu}{\phi ^B})}}\,{\partial _\mu}\phi _1^A\,{\partial _\nu}\phi _1^B} \right.} \\ {\left. {\quad \quad \quad \quad \quad \quad + 2{{{\partial ^2}{\mathcal L}} \over {\partial ({\partial _\mu}{\phi ^A})\,\partial {\phi ^B}}}\,{\partial _\mu}\phi _1^A\,\phi _1^B + {{{\partial ^2}{\mathcal L}} \over {\partial {\phi ^A}\,\partial {\phi ^B}}}\,\phi _1^A\,\phi _1^B} \right]} \\ + O({\epsilon} ^{3}){.}\quad \quad \quad \quad \quad \quad \quad \,\, \\ \end{array}$$
(202)
Consider the action
$$S[{\phi ^A}] = \int {{{\rm{d}}^{d + 1}}x\,{\mathcal L}({\partial _\mu}{\phi ^A},{\phi ^A}){.}}$$
(203)
Doing so allows us to integrate by parts. As in the single-field case we can use the Euler-Lagrange equations to discard the linear terms (since we are linearizing around a solution of the equations of motion) and so get
$$\begin{array}{*{20}c} {S[{\phi ^A}] = S[\phi _0^A]\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,\,} \\ {+ {{{{\epsilon}^2}} \over 2}\int {{{\rm{d}}^{d + 1}}x\left[ {\left\{{{{{\partial ^2}{\mathcal L}} \over {\partial ({\partial _\mu}{\phi ^A})\,\partial ({\partial _\nu}{\phi ^B})}}} \right\}\,{\partial _\mu}\phi _1^A\,{\partial _\nu}\phi _1^B} \right.}} \\ {\left. {\quad \quad \quad \quad \quad + 2\left\{{{{{\partial ^2}{\mathcal L}} \over {\partial ({\partial _\mu}{\phi ^A})\,\partial {\phi ^B}}}} \right\}\,{\partial _\mu}\phi _1^A\,\phi _1^B + \left\{{{{{\partial ^2}{\mathcal L}} \over {\partial {\phi ^A}\,\partial {\phi ^B}}}} \right\}\,\phi _1^A\,\phi _1^B} \right]} \\ {+ O({{\epsilon}^3}){.}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(204)
Because the fields now carry indices (AB) we cannot cast the action into quite as simple a form as was possible in the single-field case. The equation of motion for the linearised fluctuations are now read off as
$$\begin{array}{*{20}c} {{\partial _\mu}\left({\left\{{{{{\partial ^2}{\mathcal{L}}} \over {\partial ({\partial _\mu}{\phi ^A})\,\partial ({\partial _\nu}{\phi ^B})}}} \right\}{\partial _\nu}\phi _1^B} \right) + {\partial _\mu}\left({{{{\partial ^2}{\mathcal{L}}} \over {\partial ({\partial _\mu}{\phi ^A})\,\partial {\phi ^B}}}\,\phi _1^B} \right)} \\ {- {\partial _\mu}\phi _1^B{{{\partial ^2}{\mathcal{L}}} \over {\partial ({\partial _\mu}{\phi ^B})\,\partial {\phi ^A}}} - \left({{{{\partial ^2}{\mathcal{L}}} \over {\partial {\phi ^A}\,\partial {\phi ^B}}}} \right)\,\phi _1^B = 0.} \\ \end{array}$$
(205)
This is a linear second-order system of partial differential equations with position-dependent coefficients. This system of PDEs is automatically self-adjoint (with respect to the trivial “flat” measure dd+1x).
To simplify the notation we introduce a number of definitions. First
$${f^{\mu \nu}}_{AB}\, \equiv \,{1 \over 2}\left({{{{\partial ^2}{\mathcal{L}}} \over {\partial ({\partial _\mu}{\phi ^A})\,\partial ({\partial _\nu}{\phi ^B})}} + {{{\partial ^2}{\mathcal{L}}} \over {\partial ({\partial _\nu}{\phi ^A})\,\partial ({\partial _\mu}{\phi ^B})}}} \right).$$
(206)
This quantity is independently symmetric under interchange of μ, ν and A, B. We will want to interpret this as a generalization of the “densitised metric”, fμν, but the interpretation is not as straightforward as for the single-field case. Next, define
$$\begin{array}{*{20}c} {{\Gamma ^\mu}_{AB} \equiv + {{{\partial ^2}{\mathcal{L}}} \over {\partial ({\partial _\mu}{\phi ^A})\,\partial {\phi ^B}}} - {{{\partial ^2}{\mathcal{L}}} \over {\partial ({\partial _\mu}{\phi ^B})\,\partial {\phi ^A}}}\quad \quad \quad \quad \quad \quad \quad \quad \,\quad} \\ {+ {1 \over 2}{\partial _\nu}\left({{{{\partial ^2}{\mathcal{L}}} \over {\partial ({\partial _\nu}{\phi ^A})\,\partial ({\partial _\mu}{\phi ^B})}} - {{{\partial ^2}{\mathcal{L}}} \over {\partial ({\partial _\mu}{\phi ^A})\,\partial ({\partial _\nu}{\phi ^B})}}} \right).} \\ \end{array}$$
(207)
This quantity is anti-symmetric in A, B. One might want to interpret this as some sort of “spin connection”, or possibly as some generalization of the notion of “Dirac matrices”. Finally, define
$${K_{AB}} = - {{{\partial ^2}{\mathcal{L}}} \over {\partial {\phi ^A}\,\partial {\phi ^B}}} + {1 \over 2}{\partial _\mu}\left({{{{\partial ^2}{\mathcal{L}}} \over {\partial ({\partial _\mu}{\phi ^A})\,\partial {\phi ^B}}}} \right) + {1 \over 2}{\partial _\mu}\left({{{{\partial ^2}{\mathcal{L}}} \over {\partial ({\partial _\mu}{\phi ^B})\,\partial {\phi ^A}}}} \right).$$
(208)
This quantity is by construction symmetric in (AB). We will want to interpret this as some sort of “potential” or “mass matrix”. Then the crucial point for the following discussion is to realise that Equation (205) can be written in the compact form
$${\partial _\mu}({f^{\mu \nu}}{\,_{AB}}\,{\partial _\nu}\phi _1^B) + {1 \over 2}[\Gamma _{AB}^\mu \,{\partial _\mu}\phi _1^B + {\partial _\mu}(\Gamma _{AB}^\mu \,\phi _1^B)] + {K_{AB}}\,\phi _1^B = 0.$$
(209)
Now it is more transparent that this is a formally self-adjoint second-order linear system of PDEs. Similar considerations can be applied to the linearization of any hyperbolic system of second-order PDEs.
Consider an eikonal approximation for an arbitrary direction in field space; that is, take
$${\phi ^A}(x) = {{\epsilon}^A}(x)\,\,\exp [ - i\varphi (x)],$$
(210)
with ϵA(x) a slowly varying amplitude, and φ(x) a rapidly varying phase. In this eikonal approximation (where we neglect gradients in the amplitude, and gradients in the coefficients of the PDEs, retaining only the gradients of the phase) the linearised system of PDEs (209) becomes
$$\{{f^{\mu \nu}}_{AB}\,{\partial _\mu}\varphi (x)\,{\partial _\nu}\varphi (x) + {\Gamma ^\mu}_{AB}\,{\partial _\mu}\varphi (x) + {K_{AB}}\} \,{{\epsilon}^B}(x) = 0.$$
(211)
This has a nontrivial solution if and only if ϵA(x) is a null eigenvector of the matrix
$${f^{\mu \nu}}{\,_{AB}}\,{k_\mu}\,{k_\nu} + {\Gamma ^\mu}{\,_{AB}}\,{k_\mu} + {K_{AB}},$$
(212)
where k
μ
= ∂
μ
φ(x). Now, the condition for such a null eigenvector to exist is that
$$F(x,k) \equiv \det \{{f^{\mu \nu}}_{AB}\,{k_\mu}\,{k_\nu} + {\Gamma ^\mu}_{AB}\,{k_\mu} + {K_{AB}}\} = 0,$$
(213)
with the determinant to be taken on the field space indices AB. This is the natural generalization to the current situation of the Fresnel equation of birefringent optics [82, 375]. Following the analogy with the situation in electrodynamics (either nonlinear electrodynamics, or more prosaically propagation in a birefringent crystal), the null eigenvector ϵA(x) would correspond to a specific “polarization”. The Fresnel equation then describes how different polarizations can propagate at different velocities (or in more geometrical language, can see different metric structures). In the language of particle physics, this determinant condition F(x,k) = 0 is the natural generalization of the “mass shell” constraint. Indeed, it is useful to define the mass shell as a subset of the cotangent space by
$${\mathcal {F}}(x) \equiv \left\{{\left. {{k_\mu}} \right\vert F(x,k) = 0} \right\}.$$
(214)
In more mathematical language we are looking at the null space of the determinant of the “symbol” of the system of PDEs. By investigating F(x,k) one can recover part (not all) of the information encoded in the matrices \({f^{\mu \nu}}_{AB},\,{\Gamma ^\mu}_{AB}\), \({f^{\mu \nu}}_{AB}\), and K
AB
, or equivalently in the “generalised Fresnel equation” (213). (Note that for the determinant equation to be useful it should be non-vacuous; in particular one should carefully eliminate all gauge and spurious degrees of freedom before constructing this “generalised Fresnel equation”, since otherwise the determinant will be identically zero.) We now want to make this analogy with optics more precise, by carefully considering the notion of characteristics and characteristic surfaces. We will see how to extract from the the high-frequency high-momentum regime described by the eikonal approximation all the information concerning the causal structure of the theory.
One of the key structures that a Lorentzian spacetime metric provides is the notion of causal relationships. This suggests that it may be profitable to try to work backwards from the causal structure to determine a Lorentzian metric. Now the causal structure implicit in the system of second-order PDEs given in Equation (209) is described in terms of the characteristic surfaces, and it is for this reason that we now focus on characteristics as a way of encoding causal structure, and as a surrogate for some notion of a Lorentzian metric. Note that, via the Hadamard theory of surfaces of discontinuity, the characteristics can be identified with the infinite-momentum limit of the eikonal approximation [265]. That is, when extracting the characteristic surfaces we neglect subdominant terms in the generalised Fresnel equation and focus only on the leading term in the symbol (\({f^{\mu \nu}}_{AB}\)). In the language of particle physics, going to the infinite-momentum limit puts us on the light cone instead of the mass shell; and it is the light cone that is more useful in determining causal structure. The “normal cone” at some specified point x, consisting of the locus of normals to the characteristic surfaces, is defined by
$${\mathcal{N}}(x) \equiv \left\{{\left. {{k_\mu}} \right\vert \det \,({f^{\mu \nu}}_{AB}\,{k_\mu}\,{k_\mu}) = 0} \right\}.$$
(215)
As was the case for the Fresnel Equation (213), the determinant is to be taken on the field indices AB. (Remember to eliminate spurious and gauge degrees of freedom so that this determinant is not identically zero.) We emphasise that the algebraic equation defining the normal cone is the leading term in the Fresnel equation encountered in discussing the eikonal approximation. If there are N fields in total then this “normal cone” will generally consist of N nested sheets each with the topology (not necessarily the geometry) of a cone. Often several of these cones will coincide, which is not particularly troublesome, but unfortunately it is also common for some of these cones to be degenerate, which is more problematic.
It is convenient to define a function Q(x, k) on the co-tangent bundle
$$Q(x,k) \equiv \det \,({f^{\mu \nu}}_{AB}(x)\,{k_\mu}\,{k_\mu}){.}$$
(216)
The function Q(x, k) defines a completely-symmetric spacetime tensor (actually, a tensor density) with 2N indices
$$Q(x,k) = {Q^{{\mu _1}{\nu _1}{\mu _2}{\nu _2} \cdots {\mu _N}{\nu _N}}}(x)\,{k_{{\mu _1}}}\,{k_{{\nu _1}}}\,{k_{{\mu _2}}}\,{k_{{\nu _2}}} \cdots \,{k_{{\mu _N}}}\,{k_{{\nu _N}}}.$$
(217)
(Remember that \({f^{\mu \nu}}_{AB}\) is symmetric in both μν and AB independently.) Explicitly, using the expansion of the determinant in terms of completely antisymmetric field-space Levi-Civita tensors
$${Q^{{\mu _1}{\nu _1}{\mu _2}{\nu _2} \cdots {\mu _N}{\nu _N}}} = {1 \over {N!}}{{\epsilon}^{{A_1}{A_2} \cdots {A_N}}}\,{{\epsilon}^{{B_1}{B_2} \cdots {B_N}}}\,{f^{{\mu _1}{\nu _1}}}_{{A_1}{B_1}}{f^{{\mu _2}{\nu _2}}}_{{A_2}{B_2}} \cdots {f^{{\mu _N}{\nu _N}}}_{{A_N}{B_N}}.$$
(218)
In terms of this Q(x,k) function, the normal cone is
$${\mathcal{N}}(x) \equiv \left\{{\left. {{k_\mu}} \right\vert Q(x,k) = 0} \right\}.$$
(219)
In contrast, the “Monge cone” (aka “ray cone”, aka “characteristic cone”, aka “null cone”) is the envelope of the set of characteristic surfaces through the point x. Thus the “Monge cone” is dual to the “normal cone”, its explicit construction is given by (Courant and Hilbert [154, vol. 2, p. 583]):
$${\mathcal{M}}(x) = \left\{{{t^\mu} = \left. {{{\partial Q(x,k)} \over {\partial {k_\mu}}}} \right\vert {k_\mu} \in {\mathcal{N}}(x)} \right\}.$$
(220)
The structure of the normal and Monge cones encode all the information related with the causal propagation of signals associated with the system of PDEs. We will now see how to relate this causal structure with the existence of effective spacetime metrics, from the experimentally favoured single-metric theory compatible with the Einstein equivalence principle to the most complicated case of pseudo-Finsler geometries [306].
-
Suppose that \({f^{\mu \nu}}_{AB}\) factorises
$${f^{\mu \nu}}_{AB} = {h_{AB}}\,{f^{\mu \nu}}.$$
(221)
Then
$$Q(x,k) = \det ({h_{AB}})\,{[{f^{\mu \nu}}\,{k_\mu}\,{k_\nu}]^N}.$$
(222)
The Monge cones and normal cones are then true geometrical cones (with the N sheets lying directly on top of one another). The normal modes all see the same spacetime metric, defined up to an unspecified conformal factor by gμν ∝ fμν. This situation is the most interesting from the point of view of general relativity. Physically, it corresponds to a single-metric theory, and mathematically it corresponds to a strict algebraic condition on the \({f^{\mu \nu}}_{AB}\).
-
The next most useful situation corresponds to the commutativity condition:
$${f^{\mu \nu}}_{AB}\,{f^{\alpha \beta}}_{BC} = {f^{\alpha \beta}}_{AB}\,{f^{\mu \nu}}_{BC};\quad {\rm{that}}\,{\rm{is}}\quad [\,{f^{\mu \nu}},{f^{\alpha \beta}}\,] = 0.$$
(223)
If this algebraic condition is satisfied, then for all spacetime indices μν and αβ the \(g_A^{\mu \nu} \propto \bar f_A^{\mu \nu}\) can be simultaneously diagonalised in field space leading to
$${\bar f^{\mu \nu}}{\,_{AB}} = {\rm{diag}}\{\bar f_1^{\mu \nu},\bar f_2^{\mu \nu},\bar f_3^{\mu \nu}, \ldots, \bar f_N^{\mu \nu}\}$$
(224)
and then
$$Q(x,k) = \prod\limits_{A = 1}^N {[\bar f_A^{\mu \nu}\,{k_\mu}\,{k_\nu}]}.$$
(225)
This case corresponds to an N-metric theory, where up to an unspecified conformal factor \({f^{\mu \nu}}_{AB}\). This is the natural generalization of the two-metric situation in bi-axial crystals.
-
If \({f^{\mu \nu}}_{AB}\) is completely general, satisfying no special algebraic condition, then Q(x,k) does not factorise and is, in general, a polynomial of degree 2N in the wave vector k
μ
. This is the natural generalization of the situation in bi-axial crystals. (And for any deeper analysis of this situation one will almost certainly need to adopt pseudo-Finsler techniques [306]. But note some of the negative results obtained in [573, 574, 575].)
The message to be extracted from this rather formal discussion is that effective metrics are rather general and mathematically robust objects that can arise in quite abstract settings — in the abstract setting discussed here it is the algebraic properties of the object \(\hat{\Psi}\) that eventually leads to mono-metricity, multi-metricity, or worse. The current abstract discussion also serves to illustrate, yet again,
-
1.
that there is a significant difference between the levels of physical normal modes (wave equations), and geometrical normal modes (dispersion relations), and
-
2.
that the densitised inverse metric is in many ways more fundamental than the metric itself.
Quantum models
Bose-Einstein condensates
We have seen that one of the main aims of research in analogue models of gravity is the possibility of simulating semiclassical gravity phenomena, such as the Hawking radiation effect or cosmological particle production. In this sense systems characterised by a high degree of quantum coherence, very cold temperatures, and low speeds of sound offer the best test field. One could reasonably hope to manipulate these systems to have Hawking temperatures on the order of the environment temperature (∼ 100 nK) [48]. Hence it is not surprising that in recent years Bose-Einstein condensates (BECs) have become the subject of extensive study as possible analogue models of general relativity [231, 232, 45, 48, 47, 195, 194].
Let us start by very briefly reviewing the derivation of the acoustic metric for a BEC system, and show that the equations for the phonons of the condensate closely mimic the dynamics of a scalar field in a curved spacetime. In the dilute gas approximation, one can describe a Bose gas by a quantum field \(\hat{\Psi} = \psi + \hat{\varphi}\) satisfying
$${\rm{i}}\hbar {\partial \over {\partial t}}\hat \Psi = \left({- {{{\hbar ^2}} \over {2m}}{\nabla ^2} + {V_{{\rm{ext}}}}({\bf{x}}) + \kappa (a)\,{{\hat \Psi}^\dagger}\hat \Psi} \right)\,\hat \Psi.$$
(226)
Here κ parameterises the strength of the interactions between the different bosons in the gas. It can be re-expressed in terms of the scattering length as
$$\kappa (a) = {{4\pi a{\hbar ^2}} \over m}.$$
(227)
As usual, the quantum field can be separated into a macroscopic (classical) condensate and a fluctuation: \(\langle \hat {\Psi} \rangle = \psi\). Then, by adopting the self-consistent mean-field approximation (see, for example, [261])
$${\widehat\varphi ^\dagger}\widehat\varphi \widehat\varphi \simeq 2\left\langle {{{\widehat\varphi}^\dagger}\widehat\varphi} \right\rangle \,\widehat\varphi + \left\langle {\widehat\varphi \widehat\varphi} \right\rangle \,{\widehat\varphi ^\dagger},$$
(228)
one can arrive at the set of coupled equations:
$$\begin{array}{*{20}c} {{\rm{i}}\hbar {\partial \over {\partial t}}\psi (t,{\bf{x}}) = \left({- {{{\hbar ^2}} \over {2m}}{\nabla ^2} + {V_{{\rm{ext}}}}({\bf{x}}) + \kappa \,{n_c}} \right)\,\psi (t,{\bf{x}})} \\ {+ \kappa \,\{2\tilde n\psi (t,{\bf{x}}) + \tilde m\,\psi ^{\ast}(t,{\bf{x}})\};} \\ \end{array}$$
(229)
$$\begin{array}{*{20}c} {{\rm{i}}\hbar {\partial \over {\partial t}}\widehat\varphi (t,{\bf{x}}) = \left({- {{{\hbar ^2}} \over {2m}}{\nabla ^2} + {V_{{\rm{ext}}}}({\bf{x}}) + \kappa \,2{n_T}} \right)\widehat\varphi (t,{\bf{x}})\quad} \\ {+ \kappa \,{m_T}\,{{\widehat\varphi}^\dagger}(t,{\bf{x}}).} \\ \end{array}$$
(230)
Here
$${n_c} \equiv \,\vert \psi (t,{\bf{x}}){\vert ^2};\quad {m_c} \equiv {\psi ^2}(t,{\bf{x}});$$
(231)
$$\tilde n \equiv \left\langle {{{\hat \varphi}^\dagger}\,\hat \varphi} \right\rangle; \quad \tilde m \equiv \left\langle {\hat \varphi \hat \varphi} \right\rangle;$$
(232)
$${n_T} = {n_c} + \tilde n;\quad {m_T} = {m_c} + \tilde m.$$
(233)
The equation for the classical wave function of the condensate is closed only when the backreaction effect due to the fluctuations is neglected. (This backreaction is hiding in the parameters \(\tilde{m}\) and ñ.) This is the approximation contemplated by the Gross-Pitaevskii equation. In general, one will have to solve both equations simultaneously. Adopting the Madelung representation for the wave function of the condensate
$$\psi (t,{\bf{x}}) = \sqrt {{n_c}(t,{\bf{x}})} \,\exp [ - {\rm{i}}\theta (t,{\bf{x}})/\hbar ],$$
(234)
and defining an irrotational “velocity field” by v ≡ ∇θ/m, the Gross-Pitaevskii equation can be rewritten as a continuity equation plus an Euler equation:
$${\partial \over {\partial t}}{n_c} + \nabla \cdot ({n_c}{\bf{v}}) = 0,$$
(235)
$$m{\partial \over {\partial t}}{\bf{v}} + {\bf{\nabla}} \,\left({{{m{v^2}} \over 2} + {V_{{\rm{ext}}}}(t,{\bf{x}}) + \kappa {n_c} - {{{\hbar ^2}} \over {2m}}{{{\nabla ^2}\sqrt {{n_c}}} \over {\sqrt {{n_c}}}}} \right) = 0.$$
(236)
These equations are completely equivalent to those of an irrotational and inviscid fluid apart from the existence of the quantum potential
$${V_{{\rm{quantum}}}} = - {{{\hbar ^2}} \over {2m}}{{{\nabla ^2}\sqrt {{n_c}}} \over {\sqrt {{n_c}}}},$$
(237)
which has the dimensions of an energy. Note that
$${n_c}\,{\nabla _i}{V_{{\rm{quantum}}}} \equiv {n_c}\,{\nabla _i}\left[ {- {{{\hbar ^2}} \over {2m}}{{{\nabla ^2}\sqrt {{n_c}}} \over {\sqrt {{n_c}}}}} \right] = {\nabla _j}\left[ {- {{{\hbar ^2}} \over {4m}}{n_c}\,{\nabla _i}{\nabla _j}\,\ln \,{n_c}} \right],$$
(238)
which justifies the introduction of the quantum stress tensor
$$\sigma _{ij}^{{\rm{quantum}}} = - {{{\hbar ^2}} \over {4m}}\,{n_c}\,{\nabla _i}{\nabla _j}\ln \,{n_c}.$$
(239)
This tensor has the dimensions of pressure, and may be viewed as an intrinsically quantum anisotropic pressure contributing to the Euler equation. If we write the mass density of the Madelung fluid as ρ = m n
c
, and use the fact that the flow is irrotational, then the Euler equation takes the form
$$\rho \,\left[ {{\partial \over {\partial t}}{\bf{v}} + ({\bf{v}} \cdot {\bf{\nabla}}){\bf{v}}} \right] + \rho \,{\bf{\nabla}} \,\left[ {{{{V_{{\rm{ext}}}}(t,\,{\bf{x}})} \over m}} \right] + {\bf{\nabla}} \,\left[ {{{\kappa {\rho ^2}} \over {2{m^2}}}} \right] + {\bf{\nabla}} \cdot {\sigma ^{{\rm{quantum}}}} = 0.$$
(240)
Note that the term Vext/m has the dimensions of specific enthalpy, while κρ2/(2m) represents a bulk pressure. When the gradients in the density of the condensate are small one can neglect the quantum stress term leading to the standard hydrodynamic approximation. Because the flow is irrotational, the Euler equation is often more conveniently written in Hamilton-Jacobi form:
$$m{\partial \over {\partial t}}\theta + \left({{{{{[{\bf{\nabla}} \theta ]}^2}} \over {2m}} + {V_{{\rm{ext}}}}(t,{\bf{x}}) + \kappa {n_c} - {{{\hbar ^2}} \over {2m}}{{{\nabla ^2}\sqrt {{n_c}}} \over {\sqrt {{n_c}}}}} \right) = 0.$$
(241)
Apart from the wave function of the condensate itself, we also have to account for the (typically small) quantum perturbations of the system (230). These quantum perturbations can be described in several different ways, here we are interested in the “quantum acoustic representation”
$$\hat \varphi (t,{\bf{x}}) = {e^{- {\rm{i}}\theta/\hbar}}\left({{1 \over {2\sqrt {{n_c}}}}{{\hat n}_1} - {\rm{i}}{{\sqrt {{n_c}}} \over \hbar}{{\hat \theta}_1}} \right),$$
(242)
where \({\hat {n}_1},\,\hat {\theta}\) are real quantum fields. By using this representation, Equation (230) can be rewritten as
$${\partial _t}{\hat n_1} + {1 \over m}{\bf{\nabla}} \cdot \left({{n_1}\,{\bf{\nabla}} \theta + {n_c}{\bf{\nabla}} {{\hat \theta}_1}} \right) = 0,$$
(243)
$${\partial _t}{\hat \theta _1} + {1 \over m}{\bf{\nabla}} \theta \cdot {\bf{\nabla}} {\hat \theta _1} + \kappa (a)\,{n_1} - {{{\hbar ^2}} \over {2m}}\,{D_2}{\hat n_1} = 0.$$
(244)
Here D2 represents a second-order differential operator obtained by linearizing the quantum potential. Explicitly:
$${D_2}\,{\hat n_1} \equiv - {1 \over 2}n_c^{- 3/2}\,[{\nabla ^2}(n_c^{+ 1/2})]\,{\hat n_1} + {1 \over 2}n_c^{- 1/2}\,{\nabla ^2}(n_c^{- 1/2}\,{\hat n_1}){.}$$
(245)
The equations we have just written can be obtained easily by linearizing the Gross-Pitaevskii equation around a classical solution: \({n_c} \rightarrow {n_c} + {\hat n_1},\,\phi \rightarrow \phi + {\hat \phi _1}\). It is important to realise that in those equations the backreaction of the quantum fluctuations on the background solution has been assumed negligible. We also see in Equations (243) and (244) that time variations of Vext and time variations of the scattering length a appear to act in very different ways. Whereas the external potential only influences the background Equation (241) (and hence the acoustic metric in the analogue description), the scattering length directly influences both the perturbation and background equations. From the previous equations for the linearised perturbations it is possible to derive a wave equation for \(\hat {\theta}_1\) (or alternatively, for \(\hat {n}_1\)). All we need is to substitute in Equation (243) the \(\hat {n}_1\) obtained from Equation (244). This leads to a PDE that is second-order in time derivatives but infinite-order in space derivatives — to simplify things we can construct the symmetric 4 × 4 matrix
$${f^{\mu \nu}}(t,{\bf{x}}) \equiv \left[ {\begin{array}{*{20}c} {{f^{00}}} & \vdots & {{f^{0j}}} \\ {\ldots \ldots} & . & {\ldots \ldots \ldots \ldots} \\ {{f^{i0}}} & \vdots & {{f^{ij}}} \\ \end{array}} \right].$$
(246)
(Greek indices run from 0–3, while Roman indices run from 1–3.) Then, introducing (3+1)-dimensional space-time coordinates,
$${x^\mu} \equiv (t;\,{x^i})$$
(247)
the wave equation for θ1 is easily rewritten as
$${\partial _\mu}({f^{\mu \nu}}\,{\partial _\nu}{\hat \theta _1}) = 0{.}$$
(248)
Where the fμν are differential operators acting on space only:
$${f^{00}} = - {\left[ {\kappa (a) - {{{\hbar ^2}} \over {2m}}{D_2}} \right]^{- 1}}$$
(249)
$${f^{0j}} = - {\left[ {\kappa (a) - {{{\hbar ^2}} \over {2m}}{D_2}} \right]^{- 1}}{{{\nabla ^j}{\theta _0}} \over m}$$
(250)
$${f^{i0}} = - {{{\nabla ^i}{\theta _0}} \over m}{\left[ {\kappa (a) - {{{\hbar ^2}} \over {2m}}{D_2}} \right]^{- 1}}$$
(251)
$${f^{ij}} = {{{n_c}\,{\delta ^{ij}}} \over m} - {{{\nabla ^i}{\theta _0}} \over m}{\left[ {\kappa (a) - {{{\hbar ^2}} \over {2m}}{D_2}} \right]^{- 1}}{{{\nabla ^j}{\theta _0}} \over m}.$$
(252)
Now, if we make a spectral decomposition of the field \(\hat {\theta}_1\) we can see that for wavelengths larger than ℏ/mcs (this corresponds to the “healing length”, as we will explain below), the terms coming from the linearization of the quantum potential (the D2) can be neglected in the previous expressions, in which case the fμν can be approximated by (momentum independent) numbers, instead of differential operators. (This is the heart of the acoustic approximation.) Then, by identifying
$$\sqrt {- g} \,{g^{\mu \nu}} = {f^{\mu \nu}},$$
(253)
the equation for the field \(\hat {\theta}_1\) becomes that of a (massless minimally coupled) quantum scalar field over a curved background
$$\Delta {\theta _1} \equiv {1 \over {\sqrt {- g}}}{\partial _\mu}\left({\sqrt {- g} \,{g^{\mu \nu}}\,{\partial _\nu}} \right){\hat \theta _1} = 0,$$
(254)
with an effective metric of the form
$${g_{\mu \nu}}(t,{\bf{x}}) \equiv {{{n_c}} \over {m\,{c_{\rm{s}}}(a,\,{n_c})}}\left[ {\begin{array}{*{20}c} {- \{{c_{\rm{s}}}{{(a,\,{n_c})}^2} - {v^2}\}} & \vdots & {- {v_j}} \\ {\ldots \ldots \ldots \ldots} & . & {\ldots \ldots} \\ {- {v_i}} & \vdots & {{\delta _{ij}}} \\ \end{array}} \right].$$
(255)
Here, the magnitude cs(n
c
, a) represents the speed of the phonons in the medium:
$${c_{\rm{s}}}{(a,\,{n_c})^2} = {{\kappa (a)\,{n_c}} \over m}.$$
(256)
With this effective metric now in hand, the analogy is fully established, and one is now in a position to start asking more specific physics questions.
Lorentz breaking in BEC models — the eikonal approximation
It is interesting to consider the case in which the above “hydrodynamical” approximation for BECs does not hold. In order to explore a regime where the contribution of the quantum potential cannot be neglected we can use the eikonal approximation, a high-momentum approximation where the phase fluctuation \(\hat {\theta}_1\) is itself treated as a slowly-varying amplitude times a rapidly varying phase. This phase will be taken to be the same for both \(\hat {n}_1\) and \(\hat {\theta}_1\) fluctuations. In fact, if one discards the unphysical possibility that the respective phases differ by a time-varying quantity, any time-constant difference can be safely reabsorbed in the definition of the (complex) amplitudes. Specifically, we shall write
$${\hat \theta _1}(t,{\bf{x}}) = {\rm{Re}}\{{\mathcal{A}_\theta}\exp (- i\phi)\},$$
(257)
$${\hat n_1}(t,{\bf{x}}) = {\rm Re} \{{\mathcal{A}_\rho}\exp (- i\phi)\}.$$
(258)
As a consequence of our starting assumptions, gradients of the amplitude, and gradients of the background fields, are systematically ignored relative to gradients of Φ. (Warning: What we are doing here is not quite a “standard” eikonal approximation, in the sense that it is not applied directly on the fluctuations of the field ψ(t, x) but separately on their amplitudes and phases ρ1 and Φ1.) We adopt the notation
$$\omega = {{\partial \phi} \over {\partial t}};\quad {k_i} = {\nabla _i}\phi.$$
(259)
Then the operator D2 can be approximated as
$${D_2}\,{\hat n_1} \equiv - {1 \over 2}n_c^{- 3/2}[\Delta (n_c^{+ 1/2})]\,{\hat n_1} + {1 \over 2}n_c^{- 1/2}\,\Delta (n_c^{- 1/2}{\hat n_1})$$
(260)
$$\approx + {1 \over 2}n_c^{- 1}[\Delta {\hat n_1}]$$
(261)
$$= - {1 \over 2}n_c^{- 1}{k^2}{\hat n_1}.$$
(262)
A similar result holds for D2 acting on \(\hat {\theta}_1\). That is, under the eikonal approximation we effectively replace the operator D2 by the function
$${D_2} \rightarrow - {1 \over 2}n_c^{- 1}{k^2}.$$
(263)
For the matrix fμν this effectively results in the (explicitly momentum dependent) replacement
$${f^{00}} \rightarrow - {\left[ {\kappa (a) + {{{\hbar ^2}\,{k^2}} \over {4m\,{n_c}}}} \right]^{- 1}}$$
(264)
$${f^{0j}} \rightarrow - {\left[ {\kappa (a) + {{{\hbar ^2}\,{k^2}} \over {4m\,{n_c}}}} \right]^{- 1}}{{{\nabla ^j}{\theta _0}} \over m}$$
(265)
$${f^{i0}} \rightarrow - {{{\nabla ^i}{\theta _0}} \over m}{\left[ {\kappa (a) + {{{\hbar ^2}\,{k^2}} \over {4m\,{n_c}}}} \right]^{- 1}}$$
(266)
$${f^{ij}} \rightarrow {{{n_c}{\delta ^{ij}}} \over m} - {{{\nabla ^i}{\theta _0}} \over m}{\left[ {\kappa (a) + {{{\hbar ^2}\,{k^2}} \over {4m\,{n_c}}}} \right]^{- 1}}{{{\nabla ^j}{\theta _0}} \over m}.$$
(267)
As desired, this has the net effect of making fμν a matrix of (explicitly momentum dependent) numbers, not operators. The physical wave equation (248) now becomes a nonlinear dispersion relation
$${f^{00}}{\omega ^2} + ({f^{0i}} + {f^{i0}})\omega {k_i} + {f^{ij}}{k_i}{k_j} = 0.$$
(268)
After substituting the approximate D2 into this dispersion relation and rearranging, we see (remember: k2=‖k‖=δijk
i
k
j
)
$$- {\omega ^2} + 2v_0^i\omega {k_i} + {{{n_c}{k^2}} \over m}\left[ {\kappa (a) + {{{\hbar ^2}} \over {4m{n_c}}}{k^2}} \right] - {(v_0^i{k_i})^2} = 0.$$
(269)
That is:
$${(\omega - v_0^i{k_i})^2} = {{{n_c}{k^2}} \over m}\left[ {\kappa (a) + {{{\hbar ^2}} \over {4m{n_c}}}{k^2}} \right].$$
(270)
Introducing the speed of sound cs, this takes the form:
$$\omega = v_0^i{k_i} \pm \sqrt {c_{\rm{s}}^2{k^2} + {{\left({{\hbar \over {2m}}{k^2}} \right)}^2}.}$$
(271)
At this stage some observations are in order:
-
1.
It is interesting to recognize that the dispersion relation (271) is exactly in agreement with that found in 1947 by Bogoliubov [79] (reprinted in [508]; see also [374]) for the collective excitations of a homogeneous Bose gas in the limit T→0 (almost complete condensation). In his derivation Bogoliubov applied a diagonalization procedure for the Hamiltonian describing the system of bosons.
-
2.
Coincidentally this is the same dispersion relation that one encounters for shallow-water surface waves in the presence of surface tension. See Section 4.1.4.
-
3.
Because of the explicit momentum dependence of the co-moving phase velocity and co-moving group velocity, once one goes to high momentum the associated effective metric should be thought of as one of many possible “rainbow metrics” as in Section 4.1.4. See also [643]. (At low momentum one, of course, recovers the hydrodynamic limit with its uniquely specified standard metric.)
-
4.
It is easy to see that Equation (271) actually interpolates between two different regimes depending on the value of the wavelength λ = 2π/‖k‖ with respect to the “acoustic Compton wavelength” λ
c
= h/(mcs). (Remember that cs is the speed of sound; this is not a standard particle physics Compton wavelength.) In particular, if we assume v0 = 0 (no background velocity), then, for large wavelengths λ≫λ
c
, one gets a standard phonon dispersion relation ω ≈ c‖k‖. For wavelengths λ ≪ λ
c
the quasi-particle energy tends to the kinetic energy of an individual gas particle and, in fact, ω ≈ ℏ2k2/(2m).
We would also like to highlight that in relative terms, the approximation by which one neglects the quartic terms in the dispersion relation gets worse as one moves closer to a horizon where v0 = −cs. The non-dimensional parameter that provides this information is defined by
$$\delta \equiv {{\sqrt {1 + {{\lambda _c^2} \over {4{\lambda ^2}}}} - 1} \over {(1 - {v_0}/{c_{\rm{s}}})}} \simeq {1 \over {(1 - {v_0}/{c_{\rm{s}}})}}{{\lambda _c^2} \over {8{\lambda ^2}}}.$$
(272)
As we will discuss in Section 5.2, this is the reason why sonic horizons in a BEC can exhibit different features from those in standard general relativity.
-
5.
The dispersion relation (271) exhibits a contribution due to the background flow \(\upsilon _0^i\,{k_i}\), plus a quartic dispersion at high momenta. The group velocity is
$$v_g^i = {{\partial \omega} \over {\partial {k_i}}} = v_0^i \pm {{\left({{c^2} + {{{\hbar ^2}} \over {2{m^2}}}{k^2}} \right)} \over {\sqrt {{c^2}{k^2} + {{\left({{\hbar \over {2m}}{k^2}} \right)}^2}}}}{k^i}.$$
(273)
Indeed, with hindsight, the fact that the group velocity goes to infinity for large k was preordained: After all, we started from the generalised nonlinear Schrödinger equation, and we know what its characteristic curves are. Like the diffusion equation the characteristic curves of the Schrödinger equation (linear or nonlinear) move at infinite speed. If we then approximate this generalised nonlinear Schrödinger equation in any manner, for instance by linearization, we cannot change the characteristic curves: For any well-behaved approximation technique, at high frequency and momentum we should recover the characteristic curves of the system we started with. However, what we certainly do see in this analysis is a suitably large region of momentum space for which the concept of the effective metric both makes sense, and leads to finite propagation speed for medium-frequency oscillations.
Relativistic BEC extension
Bose-Einstein condensation can occur not only for non-relativistic bosons but for relativistic ones as well. The main differences between the thermodynamical properties of these condensates at finite temperature are due both to the different energy spectra and also to the presence, for relativistic bosons, of anti-bosons. These differences result in different conditions for the occurrence of Bose-Einstein condensation, which is possible, e.g., in two spatial dimensions for a homogeneous relativistic Bose gas, but not for its non-relativistic counterpart — and also, more importantly for our purposes, in the different structure of their excitation spectra.
In [191] an analogue model based on a relativistic BEC was studied. We summarise here the main results. The Lagrangian density for an interacting relativistic scalar Bose field \(\hat {\phi} ({\rm{x}},\,t)\) may be written as
$$\hat {\mathcal{L}} = {1 \over {{c^2}}}{{\partial {{\hat \phi}^\dagger}} \over {\partial t}}{{\partial \hat \phi} \over {\partial t}} - \nabla {\hat \phi ^\dagger} \cdot \nabla \hat \phi - \left({{{{m^2}{c^2}} \over {{\hbar ^2}}} + V(t,{\bf{x}})} \right){\hat \phi ^\dagger}\hat \phi - U({\hat \phi ^\dagger}\hat \phi; {\lambda _i}),$$
(274)
where V(t, x) is an external potential depending both on time and position x, m is the mass of the bosons and c is the light velocity. U is an interaction term and the coupling constant λ
i
(t, x) can depend on time and position too (this is possible, for example, by changing the scattering length via a Feshbach resonance [151, 175]). U can be expanded as
$$U({\hat \phi ^\dagger}\hat \phi; {\lambda _i}) = {{{\lambda _2}} \over 2}{\hat \rho ^2} + {{{\lambda _3}} \over 6}{\hat \rho ^3} + \cdots$$
(275)
where \(\hat \rho = {\hat \phi ^\dagger}\hat \phi\). The usual two-particle \({\lambda _2}{\hat {\phi}^4}\)-interaction corresponds to the first term \(({\lambda _2}/2){\hat \rho ^2}\), while the second term represents the three-particle interaction and so on.
The field \({\hat {\phi}}\) can be written as a classical field (the condensate) plus perturbation:
$$\hat \phi = \phi (1 + \hat \psi){.}$$
(276)
It is worth noticing now that the expansion in Equation (276) can be linked straightforwardly to the previously discussed expansion in phase and density perturbations \({\hat {\theta}_1},\,{\hat {\rho}_1}\) by noting that
$${{{{\hat \rho}_1}} \over \rho} = {{\hat \psi + {{\hat \psi}^\dagger}} \over 2},\quad {\hat \theta _1} = {{\hat \psi - {{\hat \psi}^\dagger}} \over {2i}}.$$
Setting ψ ∝ exp[i(k · x − ωt)] one then gets from the equation of motion [191]
$$\begin{array}{*{20}c} {\left({- {\hbar \over m}{\bf{q}} \cdot {\bf{k}} + {{{u^0}} \over c}\omega - {\hbar \over {2m{c^2}}}{\omega ^2} + {\hbar \over {2m}}{k^2}} \right)\left({{\hbar \over m}{\bf{q}} \cdot {\bf{k}} - {{{u^0}} \over c}\omega - {\hbar \over {2m{c^2}}}{\omega ^2} + {\hbar \over {2{m^2}}}{k^2}} \right)} \\ {- {{\left({{{{c_0}} \over c}} \right)}^2}{\omega ^2} + c_0^2{k^2} = 0,} \\ \end{array}$$
(277)
where, for convenience, we have defined the following quantities as
$${u^\mu} \equiv {\hbar \over m}{\eta ^{\mu \nu}}{\partial _\nu}\theta,$$
(278)
$$c_0^2 \equiv {{{\hbar ^2}} \over {2{m^2}}}U^{\prime\prime}(\rho; {\lambda _i})\rho,$$
(279)
$${\bf{q}} \equiv m{\bf{u}}/\hbar.$$
(280)
Here q is the speed of the condensate flow and c is the speed of light. For a condensate at rest (q = 0) one then obtains the following dispersion relation
$$\omega _ \pm ^2 = {c^2}\left\{{{k^2} + 2{{\left({{{m{u^0}} \over \hbar}} \right)}^2}\left[ {1 + {{\left({{{{c_0}} \over {{u^0}}}} \right)}^2}} \right] \pm 2\left({{{m{u^0}} \over \hbar}} \right)\sqrt {{k^2} + {{\left({{{m{u^0}} \over \hbar}} \right)}^2}{{\left[ {1 + {{\left({{{{c_0}} \over {{u^0}}}} \right)}^2}} \right]}^2}}} \right\}.$$
(281)
The dispersion relation (281) is sufficiently complicated to prevent any obvious understanding of the regimes allowed for the excitation of the system. It is much richer than the non-relativistic case. For example, it allows for both a massless/gapless (phononic) and massive/gapped mode, respectively for the ω− and ω+ branches of (281). Nonetheless, it should be evident that different regimes are determined by the relative strength of the the first two terms on the right-hand side of Equation (281) (note that the same terms enter in the square root). This can be summarised, in low and high momentum limits respectively, for k much less or much greater than
$${{m{u^0}} \over \hbar}\left[ {1 + {{\left({{{{c_0}} \over {{u^0}}}} \right)}^2}} \right] \equiv {{m{u^0}} \over \hbar}(1 + b),$$
(282)
where b encodes the relativistic nature of the condensate (the larger b the more the condensate is relativistic).
A detailed discussion of the different regimes would be inappropriately long for this review; it can be found in [191]. The results are summarised in Table 1. Note that μ ≡ mcu0 plays the role of the chemical potential for the relativistic BEC. One of the most remarkable features of this model is that it is a condensed matter system that interpolates between two different Lorentz symmetries, one at low energy and a different Lorentz symmetry at high energy.
Table 1 Dispersion relation of gapless and gapped modes in different regimes. Note that we have \(c_{\mathcal S}^2 = {c^2}b/(1 + b)\), and \(c_{{\mathcal S},{\rm{gap}}}^2 = {c^2}(2 + b)/(1 + b)\), while meff = 2(μ/c2)(1 + b)3/2/(2 + b). Finally, it is also possible to recover an acoustic metric for the massless (phononic) perturbations of the condensate in the low momentum limit (k ≪ mu0(1 + b)/ℏ):
$${g_{\mu \nu}} = {\rho \over {\sqrt {1 - {u_\sigma}{u^\sigma}/c_0^2}}}\left[ {{\eta _{\mu \nu}}\left({1 - {{{u_\sigma}{u^\sigma}} \over {c_0^2}}} \right) + {{{u_\mu}{u_\nu}} \over {c_0^2}}} \right].$$
(283)
As should be expected, it is just a version of the acoustic geometry for a relativistic fluid previously discussed, and in fact can be cast in the form of Equation (155) by suitable variable redefinitions [191].
The heliocentric universe
Helium is one of the most fascinating elements provided by nature. Its structural richness confers on helium a paradigmatic character regarding the emergence of many and varied macroscopic properties from the microscopic world (see [660] and references therein). Here, we are interested in the emergence of effective geometries in helium, and their potential use in testing aspects of semiclassical gravity.
Helium four, a bosonic system, becomes superfluid at low temperatures (2.17 K at vapour pressure). This superfluid behaviour is associated with condensation in the vacuum state of a macroscopically large number of atoms. A superfluid is automatically an irrotational and inviscid fluid, so, in particular, one can apply to it the ideas worked out in Section 2. The propagation of classical acoustic waves (scalar waves) over a background fluid flow can be described in terms of an effective Lorentzian geometry: the acoustic geometry. However, in this system one can naturally go considerably further, into the quantum domain. For long wavelengths, the quasiparticles in this system are quantum phonons. One can separate the classical behaviour of a background flow (the effective geometry) from the behaviour of the quantum phonons over this background. In this way one can reproduce, in laboratory settings, different aspects of quantum field theory over curved backgrounds. The speed of sound in the superfluid phase is typically on the order of cm/sec. Therefore, at least in principle, it should not be too difficult to establish configurations with supersonic flows and their associated ergoregions.
Helium three, the fermionic isotope of helium, in contrast, becomes superfluid at much lower temperatures (below 2.5 milli-K). The reason behind this rather different behaviour is the pairing of fermions to form effective bosons (Cooper pairing), which are then able to condense. In the 3He-A phase, the structure of the fermionic vacuum is such that it possesses two Fermi points, instead of the more typical Fermi surface. In an equilibrium configuration one can choose the two Fermi points to be located at {p
x
= 0, p
y
= 0, p
z
= ±p
F
} (in this way, the z-axis signals the direction of the angular momentum of the pairs). Close to either Fermi point the spectrum of quasiparticles becomes equivalent to that of Weyl fermions. From the point of view of the laboratory, the system is not isotropic, it is axisymmetric. There is a speed for the propagation of quasiparticles along the z-axis, c‖ ≃ cm/sec, and a different speed, c⊥ ≃ 10−5 c‖, for propagation perpendicular to the symmetry axis. However, from an internal observer’s point of view this anisotropy is not “real”, but can be made to disappear by an appropriate rescaling of the coordinates. Therefore, in the equilibrium case, we are reproducing the behaviour of Weyl fermions over Minkowski spacetime. Additionally, the vacuum can suffer collective excitations. These collective excitations will be experienced by the Weyl quasiparticles as the introduction of an effective electromagnetic field and a curved Lorentzian geometry. The control of the form of this geometry provides the sought for gravitational analogy.
Apart from the standard way to provide a curved geometry based on producing nontrivial flows, there is also the possibility of creating topologically nontrivial configurations with a built-in nontrivial geometry. For example, it is possible to create a domain-wall configuration [327, 326] (the wall contains the z-axis) such that the transverse velocity c⊥ acquires a profile in the perpendicular direction (say along the x-axis) with c⊥ passing through zero at the wall (see Figure 11). This particular arrangement could be used to reproduce a black-hole-white-hole configuration only if the soliton is set up to move with a certain velocity along the x-axis. This configuration has the advantage that it is dynamically stable, for topological reasons, even when some supersonic regions are created.
A third way in which superfluid helium can be used to create analogues of gravitational configurations is the study of surface waves (or ripplons) on the interface between two different phases of 3He [657, 659]. In particular, if we have a thin layer of 3He-A in contact with another thin layer of 3He-B, the oscillations of the contact surface “see” an effective metric of the form [657, 659]
$${\rm{d}}{s^2} = {1 \over {(1 - {\alpha _A}{\alpha _B}{U^2})}}\left[ {- (1 - {W^2} - {\alpha _A}{\alpha _B}{U^2}){\rm{d}}{t^2} - 2{\bf{W}} \cdot {\bf{dx}}\,{\rm{d}}t + {\bf{dx}} \cdot {\bf{dx}}} \right],$$
(284)
where
$${\bf{W}} \equiv {\alpha _A}{{\bf{v}}_{\bf{A}}} + {\alpha _B}{{\bf{v}}_{\bf{B}}},\quad {\bf{U}} \equiv {{\bf{v}}_{\bf{A}}} - {{\bf{v}}_{\bf{B}}},$$
(285)
and
$${\alpha _A} \equiv {{{h_B}{\rho _A}} \over {{h_A}{\rho _B} + {h_B}{\rho _A}}};\quad {\alpha _B} \equiv {{{h_A}{\rho _B}} \over {{h_A}{\rho _B} + {h_B}{\rho _A}}}.$$
(286)
(All of this provided that we are looking at wavelengths larger than the layer thickness, kh
A
≪ 1 and kh
B
≪ 1.)
The advantage of using surface waves instead of bulk waves in superfluids is that one could create horizons without reaching supersonic speeds in the bulk fluid. This could alleviate the appearance of dynamical instabilities in the system, that in this case are controlled by the strength of the interaction of the ripplons with bulk degrees of freedom [657, 659].
Slow light in fluids
The geometrical interpretation of the motion of light in dielectric media leads naturally to conjecture that the use of flowing dielectrics might be useful for simulating general relativity metrics with ergoregions and black holes. Unfortunately, these types of geometry require flow speeds comparable to the group velocity of the light. Since typical refractive indexes in non-dispersive media are quite close to unity, it is then clear that it is practically impossible to use them to simulate such general relativistic phenomena. However recent technological advances have radically changed this state of affairs. In particular the achievement of controlled slowdown of light, down to velocities of a few meters per second (or even down to complete rest) [617, 338, 96, 353, 506, 603, 565], has opened a whole new set of possibilities regarding the simulation of curved-space metrics via flowing dielectrics.
But how can light be slowed down to these “snail-like” velocities? The key effect used to achieve this takes the name of Electromagnetically Induced Transparency (EIT). A laser beam is coupled to the excited levels of some atom and used to strongly modify its optical properties. In particular one generally chooses an atom with two long-lived metastable (or stable) states, plus a higher energy state that has some decay channels into these two lower states. The coupling of the excited states induced by the laser light can affect the transition from a lower energy state to the higher one, and hence the capability of the atom to absorb light with the required transition energy. The system can then be driven into a state where the transitions between each of the lower energy states and the higher energy state exactly cancel out, due to quantum interference, at some specific resonant frequency. In this way the higher-energy level has null averaged occupation number. This state is hence called a “dark state”. EIT is characterised by a transparency window, centered around the resonance frequency, where the medium is both almost transparent and extremely dispersive (strong dependence on frequency of the refractive index). This in turn implies that the group velocity of any light probe would be characterised by very low real group velocities (with almost vanishing imaginary part) in proximity to the resonant frequency.
Let us review the most common setup envisaged for this kind of analogue model. A more detailed analysis can be found in [383]. One can start by considering a medium in which an EIT window is opened via some control laser beam which is oriented perpendicular to the direction of the flow. One then illuminates this medium, now along the flow direction, with some probe light (which is hence perpendicular to the control beam). This probe beam is usually chosen to be weak with respect to the control beam, so that it does not modify the optical properties of the medium. In the case in which the optical properties of the medium do not vary significantly over several wavelengths of the probe light, one can neglect the polarization and can hence describe the propagation of the latter with a simple scalar dispersion relation [390, 211]
$${k^2} = {{{\omega ^2}} \over {{c^2}}}[1 + \chi (\omega)],$$
(287)
where χ is the susceptibility of the medium, related to the refractive index via the simple relation \(n = \sqrt {1 + \chi}\).
It is easy to see that in this case the group and phase velocities differ
$${v_{\rm{g}}} = {{\partial \omega} \over {\partial k}} = {c \over {\sqrt {1 + \chi} + {\omega \over {2n}}{{\partial \chi} \over {\partial \omega}}}};\quad {v_{{\rm{ph}}}} = {\omega \over k} = {c \over {\sqrt {1 + \chi}}}.$$
(288)
So even for small refractive indexes one can get very low group velocities, due to the large dispersion in the transparency window, and in spite of the fact that the phase velocity remains very near to c. (The phase velocity is exactly at the resonance frequency ω0). In an ideal EIT regime the probe light experiences a vanishing susceptibility χ near the the critical frequency ω0, this allows us to express the susceptibility near the critical frequency via the expansion
$$\chi (\omega) = {{2\alpha} \over {{\omega _0}}}(\omega - {\omega _0}) + O\left[ {{{(\omega - {\omega _0})}^3}} \right],$$
(289)
where α is sometimes called the “group refractive index”. The parameter α depends on the dipole moments for the transition from the metastable states to the high energy one, and most importantly depends on the ratio between the probe-light energy per photon, ℏω0, and the control-light energy per atom [383]. This might appear paradoxical because it seems to suggest that for a dimmer control light the probe light would be further slowed down. However this is just an artificial feature due to the extension of the EIT regime beyond its range of applicability. In particular in order to be effective the EIT requires the control beam energy to dominate all processes and hence it cannot be dimmed at will.
At resonance we have
$${v_{\rm{g}}} = {{\partial \omega} \over {\partial k}} \rightarrow {c \over {1 + \alpha}} \approx {c \over \alpha};\quad {v_{{\rm{ph}}}} = {\omega \over k} \rightarrow c.$$
(290)
We can now generalise the above discussion to the case in which our highly dispersive medium flows with a characteristic velocity profile u(x, t). In order to find the dispersion relation of the probe light in this case we just need to transform the dispersion relation (287) from the comoving frame of the medium to the laboratory frame. Let us consider for simplicity a monochromatic probe light (more realistically a pulse with a very narrow range of frequencies ω near ω0). The motion of the dielectric medium creates a local Doppler shift of the frequency
$$\omega \rightarrow \gamma ({\omega _0} - {\bf{u}} \cdot {\bf{k}}),$$
(291)
where γ is the usual relativistic factor. Given that k2 − ω2/c2 is a Lorentz invariant, it is then easy to see that this Doppler detuning affects the dispersion relation (287) only via the susceptibility dependent term. Given further that in any realistic case one would deal with non-relativistic fluid velocities u ≪ c we can then perform an expansion of the dispersion relation up to second order in u/c. Expressing the susceptibility via Equation (289) we can then rewrite the dispersion relation in the form [390]
$${g^{\mu \nu}}{k_\mu}{k_\nu} = 0,$$
(292)
where
$${k_\nu} = \left({{{{\omega _0}} \over c}, - {\bf{k}}} \right),$$
(293)
and
(Note that most of the original articles on this topic adopt the opposite signature (+ − −−).) The inverse of this tensor will be the covariant effective metric experienced by the probe light, whose rays would then be null geodesics of the line element ds2 = g
μν
dxμdxν. In this sense the probe light will propagate as in a curved background. Explicitly one finds the covariant metric to be
where
$$A = {{1 - 4\alpha {u^2}/{c^2}} \over {1 + ({\alpha ^2} - 3\alpha){u^2}/{c^2} - 4{\alpha ^2}{u^4}/{c^4}}};$$
(296)
$$B = {1 \over {1 + ({\alpha ^2} - 3\alpha){u^2}/{c^2} - 4{\alpha ^2}{u^4}/{c^4}}};$$
(297)
$$C = {{1 - (4/\alpha + 4{u^2}/{c^2})} \over {1 + ({\alpha ^2} - 3\alpha){u^2}/{c^2} - 4{\alpha ^2}{u^4}/{c^4}}}.$$
(298)
Several comments are in order concerning the metric (295). First of all, it is clear that, although more complicated than an acoustic metric, it will still be possible to cast it into the Arnowitt-Deser-Misner-like form [627]
where the effective speed ueff is proportional to the fluid flow speed u and the three-space effective metric geff is (rather differently from the acoustic case) nontrivial.
In any case, the existence of this ADM form already tells us that an ergoregion will always appear once the norm of the effective velocity exceeds the effective speed of light (which for slow light is approximately c/α, where α can be extremely large due to the huge dispersion in the transparency window around the resonance frequency ω0). However, a trapped surface (and hence an optical black hole) will form only if the inward normal component of the effective flow velocity exceeds the group velocity of light. In the slow light setup so far considered such a velocity turns out to be \(u = c/(2\sqrt {\alpha)}\).
The realization that ergoregions and event horizons can be simulated via slow light may lead one to the (erroneous) conclusion that this is an optimal system for simulating particle creation by gravitational fields. However, as pointed out by Unruh in [470, 612], such a conclusion would turn out to be over-enthusiastic. In order to obtain particle creation through “mode mixing”, (mixing between the positive and negative norm modes of the incoming and outgoing states), an inescapable requirement is that there must be regions where the frequency of the quanta as seen by a local comoving observer becomes negative.
In a flowing medium this can, in principle, occur thanks to the tilting of the dispersion relation due to the Doppler effect caused by the velocity of the flow Equation (291); but this also tells us that the negative norm mode must satisfy the condition ω0 − u · k < 0, but this can be satisfied only if the velocity of the medium exceeds |ω0/k|, which is the phase velocity of the probe light, not its group velocity. This observation suggests that the existence of a “phase velocity horizon” is an essential ingredient (but not the only essential ingredient) in obtaining Hawking radiation. A similar argument indicates the necessity for a specific form of “group velocity horizon”, one that lies on the negative norm branch. Since the phase velocity in the slow light setup we are considering is very close to c, the physical speed of light in vacuum, not very much hope is left for realizing analogue particle creation in this particular laboratory setting.
However, it was also noticed by Unruh and Schützhold [612] that a different setup for slow light might deal with this and other issues (see [612] for a detailed summary). In the setup suggested by these authors there are two strong-background counter-propagating control beams illuminating the atoms. The field describing the beat fluctuations of this electromagnetic background can be shown to satisfy, once the dielectric medium is in motion, the same wave equation as that on a curved background. In this particular situation the phase velocity and the group velocity are approximately the same, and both can be made small, so that the previously discussed obstruction to mode mixing is removed. So in this new setup it is concretely possible to simulate classical particle creation such as, e.g., super-radiance in the presence of ergoregions.
Nonetheless, the same authors showed that this does not open the possibility for a simulation of quantum particle production (e.g., Hawking radiation). This is because that effect also requires the commutation relations of the field to generate the appropriate zero-point energy fluctuations (the vacuum structure) according to the Heisenberg uncertainty principle. This is not the case for the effective field describing the beat fluctuations of the system we have just described, which is equivalent to saying that it does not have a proper vacuum state (i.e., analogue to any physical field). Hence, one has to conclude that any simulation of quantum particle production is precluded.
Slow light in fibre optics
In addition to the studies of slow light in fluids, there has now been a lot of work done on slow light in a fibre-optics setting [505, 504, 64, 63], culminating in recent experimental detection of photons apparently associated with a phase-velocity horizon [66]. The key issue here is that the Kerr effect of nonlinear optics can be used to change the refractive index of an optical fibre, so that a “carrier” pulse of light traveling down the fibre carries with it a region of high refractive index, which acts as a barrier to “probe” photons (typically at a different frequency). If the relative velocities of the “carrier” pulse and “probe” are suitably arranged then the arrangement can be made to mimic a black-hole-white-hole pair. This system is described more fully in Section 6.4.
Lattice models
The quantum analogue models described above all have an underlying discrete structure: namely the atoms they are made of. In abstract terms one can also build an analogue model by considering a quantum field on specific lattice structures representing different spacetimes. In [310, 149, 322] a falling-lattice black-hole analogue was put forward, with a view to analyzing the origin of Hawking particles in black-hole evaporation. The positions of the lattice points in this model change with time as they follow freely falling trajectories. This causes the lattice spacing at the horizon to grow approximately linearly with time. By definition, if there were no horizons, then for long wavelengths compared with the lattice spacing one would recover a relativistic quantum field theory over a classical background. However, the presence of horizons makes it impossible to analyze the field theory only in the continuum limit, it becomes necessary to recall the fundamental lattice nature of the model.
Graphene
A very interesting addition to the catalogue of analogue systems is the graphene (see, for example, these reviews [127, 339]). Although graphene and some of its peculiar electronic properties have been known since the 1940s [672], only recently has it been specifically proposed as a system with which to probe gravitational physics [153, 152]. Graphene (or mono-layer graphite) is a two-dimensional lattice of carbon atoms forming a hexagonal structure (see Figure 13). From the perspective of this review, its most important property is that its Fermi surface has two independent Fermi points (see Section 4.2.2 on helium). The low-energy excitations around these points can be described as massless Dirac fields in which the light speed is substituted by a “sound” speed c
s
about 300 times smaller:
$${\partial _t}{\psi _j} = {c_s}\sum\limits_{k = 1,2} {{\sigma ^k}{\partial _{{x_k}}}} {\psi _j}.$$
(300)
Here ψ
j
with j = 1, 2 represent two types of massless spinors (one for each Fermi point), the σk are the Pauli sigma matrices, and c
s
= 3ta/2, with a = 1.4 Å being the interatomic distance, and t = 2.8 eV the hopping energy for an electron between two nearest-neighboring atoms.
From this perspective graphene can be used to investigate ultra-relativistic phenomena such as the Klein paradox [339]. On the other hand, graphene sheets can also acquire curvature. A nonzero curvature can be produced by adding strain fields to the sheet, imposing a curved substrate, or by introducing topological defects (e.g., some pentagons within the hexagonal structure) [669]. It has been suggested that, regarding the electronic properties of graphene, the sheet curvature promotes the Dirac equation to its curved space counterpart, at least on the average [153, 152]. If this proves to be experimentally correct, it will make graphene a good analogue model for a diverse set of spacetimes. This set, however, does not include black-hole spacetimes, as the curvatures mentioned above are purely spatial and do not affect the temporal components of the metric.
Going further
We feel that the catalogue we have just presented is reasonably complete and covers the key items. For additional background on many of these topics, we would suggest sources such as the books “Artificial Black Holes” [470] and “The Universe in a Helium Droplet” [660]. For more specific detail, check this review’s bibliography, and use SPIRES (or the beta version of INSPIRE) to check for recent developments.