Sound in a 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.

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 [345] (see
Figure 7).^{20}
If one neglects the viscosity and considers an irrotational flow, , one can write Bernoulli’s
equation in the presence of Earth’s gravity as

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 [345] 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 [345]. 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 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.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 (126) 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 thenThe 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 , ) one can always solve to find an equivalent analogue in terms of permittivity/permeability (albeit an analogue that satisfies the mildly unphysical constraint ).

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 thatThe behaviour of this dispersion relation now depends critically on the way that the eigenvalues of are distributed.

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

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 [16])

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 connexion”, 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 (162) 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 (166) 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 bi-refringent optics [39, 224]. Following the analogy with the situation in electrodynamics (either nonlinear electrodynamics, or more prosaically propagation in a bi-refringent 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 particle physics language 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” (170). (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 (166) 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 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 [155]. 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 particle physics language 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 (170), 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 generically 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 [92, volume 2, page 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 [184].

- 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 [184].)

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.

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