4.1 Classical models
4.1.1 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).
4.1.2 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
, and a point in spacetime where the background fluid
4-velocity is
while the speed of sound is
. Now (in complete direct conformity with
our discussion of the generalised optical Gordon metric) adopt Gaussian normal coordinates
so that
, and go to the local rest frame of the fluid, so that
and
In the rest frame of the fluid the sound cones are (locally) given by
implying in these special coordinates the existence of an acoustic metric
That is, transforming back to arbitrary coordinates:
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 conformal 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
]
In this situation the relativistic Bernoulli equation can be shown to be
where we emphasize that
is now the energy density (not the mass density), and the total particle
number density can be shown to be
After linearization around some suitable background [448, 72, 639
], the perturbations in the scalar
velocity potential
can be shown to satisfy a dimension-independent d’Alembertian equation
which leads to the identification of the relativistic acoustic metric as
The dimension-dependence now comes from solving this equation for
. Therefore, we finally have the
(contravariant) acoustic metric
and (covariant) acoustic metric
In the non-relativistic limit
and
, where
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]
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
; 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.)
4.1.3 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.)
If one neglects the viscosity and considers an irrotational flow,
, one can write Bernoulli’s
equation in the presence of Earth’s gravity as
Here
is the density of the fluid,
its pressure,
the gravitational acceleration and
a potential
associated with some external force necessary to establish an horizontal flow in the fluid. We
denote that flow by
. We must also impose the boundary conditions that the pressure at the
surface, and the vertical velocity at the bottom, both vanish. That is,
, and
.
Once a horizontal background flow is established, one can see that the perturbations of the velocity
potential satisfy
If we now expand this perturbation potential in a Taylor series
it is not difficult to prove [560
] that surface waves with long wavelengths (long compared with the depth of
the basin,
), can be described to a good approximation by
and that this field “sees” an
effective metric of the form
where
. The link between small variations of the potential field and small variations of the
position of the surface is provided by the following equation
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,
.
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
].
4.1.4 More general water waves
If one moves beyond shallow-water surface waves the physics becomes more complicated. In the
shallow-water regime (wavelength
much greater than water depth
) the co-moving dispersion
relation is a simple linear one
, where the speed of sound can depend on both position and time.
Once one moves to finite-depth (
) or deep (
) water, it is a standard result that the
co-moving dispersion relation becomes
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
Here
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
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
. 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
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
].
4.1.5 Classical refractive index
The macroscopic Maxwell equations inside a dielectric take the well-known form
with the constitutive relations
and
. Here,
is the
permittivity tensor
and
the
permeability tensor of the medium. These equations can be written in a condensed way
as
where
is the electromagnetic tensor,
and (assuming the medium is at rest) the non-vanishing components of the 4th-rank tensor
are given
by
supplemented by the conditions that
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
also be symmetric
under pairwise interchange of the first pair of indices with the second pair – thus
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
we see that in curved spacetime we can also write the electromagnetic equations of motion in the
form (172) where now (for some constant
):
If we consider a static gravitational field we can always re-write it as a conformal factor multiplying an
ultra-static metric
then
The fact that
is independent of the conformal factor
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
The second of these constraints can be written as
In view of the standard formula for
determinants
this now implies
whence
Comparing this with
we now have:
To rearrange this, introduce the matrix square root
, which always exists because
is real
positive definite and symmetric. Then
Note that if you are given the static gravitational field (in the form
,
) you can always solve it to find an equivalent
analogue in terms of permittivity/permeability (albeit an analogue that satisfies the mildly unphysical constraint
).
On the other hand, if you are given permeability and permittivity tensors
and
, then it is only for
that subclass of media that satisfy
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
The dispersion relations for the propagation of photons (and therefore the sought for geometrical
properties) can be obtained from the reduced determinant of
(notice that the [full] determinant of
is identically zero as
; the reduced determinant is that associated with the three directions
orthogonal to
). By choosing the gauge
one can see that this reduced determinant
can be obtained from the determinant of the
sub-matrix
. This determinant is
or, after making some manipulations,
To simplify this, again introduce the matrix square roots
and
, which always exist
because the relevant matrices are real positive definite and symmetric. Then define
and
so that
The behaviour of this dispersion relation now depends critically on the way that the eigenvalues of
are
distributed.
3 degenerate eigenvalues:
If all eigenvalues are degenerate then
, implying
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
while
That is
with
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,
(
and
) this reduces to the perhaps more expected result
2 distinct eigenvalues:
If
has two distinct eigenvalues then the determinant
factorises into a trivial factor of
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
has three distinct eigenvalues then the determinant
is the product of a trivial factor of
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
The background field
sets the value of
, and
. Equation (172) then
becomes an equation for
. 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
permeability and
permittivity tensors constrained by
is equivalent (at the level of geometric optics) to the
propagation of photons in a curved spacetime manifold characterised by the ultra-static metric (200),
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
is
position independent, one can make the stronger statement (189) which holds true at the level of
physical optics. Recently this topic has been revitalised by the increasing interest in (classical)
meta-materials.
4.1.6 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
])
Now use this to expand the Lagrangian
Consider the action
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
Because the fields now carry indices (
) 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
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
).
To simplify the notation we introduce a number of definitions. First
This quantity is independently symmetric under interchange of
,
and
,
. We will want to
interpret this as a generalization of the “densitised metric”,
, but the interpretation is not as
straightforward as for the single-field case. Next, define
This quantity is anti-symmetric in
,
. 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
This quantity is by construction symmetric in
. 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 (207) can be written in the compact form
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
with
a slowly varying amplitude, and
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 (211) becomes
This has a nontrivial solution if and only if
is a null eigenvector of the matrix
where
. Now, the condition for such a null eigenvector to exist is that
with the determinant to be taken on the field space indices
. 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
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
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
In more mathematical language we are looking at the null space of the determinant of the “symbol” of the
system of PDEs. By investigating
one can recover part (not all) of the information encoded in the
matrices
,
, and
, or equivalently in the “generalised Fresnel equation” (215). (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 (211) 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 (
). 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
, consisting of the locus of normals to the characteristic surfaces, is defined by
As was the case for the Fresnel Equation (215), the determinant is to be taken on the field indices
. (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
fields in total
then this “normal cone” will generally consist of
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
on the co-tangent bundle
The function
defines a completely-symmetric spacetime tensor (actually, a tensor density) with
indices
(Remember that
is symmetric in both
and
independently.) Explicitly, using the
expansion of the determinant in terms of completely antisymmetric field-space Levi–Civita tensors
In terms of this
function, the normal cone is
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
. Thus the “Monge cone” is dual to the
“normal cone”, its explicit construction is given by (Courant and Hilbert [154, vol. 2, p. 583]):
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
factorises
Then
The Monge cones and normal cones are then true geometrical cones (with the
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
. 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
.
- The next most useful situation corresponds to the commutativity condition:
If this algebraic condition is satisfied, then for all spacetime indices
and
the
can
be simultaneously diagonalised in field space leading to
and then
This case corresponds to an
-metric theory, where up to an unspecified conformal factor
. This is the natural generalization of the two-metric situation in bi-axial
crystals.
- If
is completely general, satisfying no special algebraic condition, then
does not
factorise and is, in general, a polynomial of degree
in the wave vector
. 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
that eventually leads to
mono-metricity, multi-metricity, or worse. The current abstract discussion also serves to illustrate, yet
again,
- that there is a significant difference between the levels of physical normal modes (wave
equations), and geometrical normal modes (dispersion relations), and
- that the densitised inverse metric is in many ways more fundamental than the metric itself.