### 2.8 Simple Lagrangian meta-model

As a first (and rather broad) example of the very abstract ways in which the notion of an acoustic metric can be generalised, we start from the simple observation that irrotational barotropic fluid mechanics can be described by a Lagrangian, and ask if we can extend the notion of an acoustic metric to all (or at least some wide class of) Lagrangian systems?

Indeed, suppose we have a single scalar field whose dynamics is governed by some generic Lagrangian , which is some arbitrary function of the field and its first derivatives (here we will follow the notation and ideas of [15]). In the general analysis that follows the previous irrotational and inviscid fluid system is included as a particular case; the dynamics of the scalar field is now much more general. We want to consider linearised fluctuations around some background solution of the equations of motion, and to this end we write

Now use this to expand the Lagrangian around the classical solution :
It is particularly useful to consider the action
since doing so allows us to integrate by parts. (Note that the Lagrangian is taken to be a scalar density, not a true scalar.) We can now use the Euler-Lagrange equations for the background field
to discard the linear terms (remember we are linearizing around a solution of the equations of motion) and so we get
Having set things up this way, the equation of motion for the linearised fluctuation is now easily read off as
This is a second-order differential equation with position-dependent coefficients (these coefficients all being implicit functions of the background field ).

This can be given a nice clean geometrical interpretation in terms of a d’Alembertian wave equation - provided we define the effective spacetime metric by

Note that this is another example of a situation in which calculating the inverse metric density is easier than calculating the metric itself.

Suppressing the except when necessary for clarity, this implies [in (d+1) dimensions, d space dimensions plus 1 time dimension]

Therefore
And, taking the inverse
We can now write the equation of motion for the linearised fluctuations in the geometrical form
where is the d’Alembertian operator associated with the effective metric , and is the background-field-dependent (and so in general position-dependent) “mass term”:
Thus is a true scalar (not a density). Note that the differential Equation (113) is automatically formally self-adjoint (with respect to the measure ).

It is important to realise just how general the result is (and where the limitations are): It works for any Lagrangian depending only on a single scalar field and its first derivatives. The linearised PDE will be hyperbolic (and so the linearised equations will have wave-like solutions) if and only if the effective metric has Lorentzian signature . Observe that if the Lagrangian contains nontrivial second derivatives you should not be too surprised to see terms beyond the d’Alembertian showing up in the linearised equations of motion.

As a specific example of the appearance of effective metrics due to Lagrangian dynamics we reiterate the fact that inviscid irrotational barotropic hydrodynamics naturally falls into this scheme (which is why, with hindsight, the derivation of the acoustic metric presented earlier in this review was so relatively straightforward). In inviscid irrotational barotropic hydrodynamics the lack of viscosity (dissipation) guarantees the existence of a Lagrangian; which a priori could depend on several fields. Since the flow is irrotational is a function only of the velocity potential, and the Lagrangian is a function only of this potential and the density. Finally the equation of state can be used to eliminate the density leading to a Lagrangian that is a function only of the single field and its derivatives. [15]