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  (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 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 . 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
If we compare this to the Lagrangian for electromagnetism in curved spacetime
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 constraints21 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 . We shall encounter this model later on when we shall consider the trans-Planckian problem for Hawking radiation.
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
The 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 )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. Firstsystem 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[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
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 . 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 bundleaka “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 .
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 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,
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