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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 f whose dynamics is governed by some generic Lagrangian L(@mf, f), which is some arbitrary function of the field and its first derivatives (here we will follow the notation and ideas of [15Jump To The Next Citation Point]). 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 f is now much more general. We want to consider linearised fluctuations around some background solution f0(t,x) of the equations of motion, and to this end we write

e2 3 f(t,x) = f0(t,x) + ef1(t,x) + 2 f2(t,x) + O(e ). (103)
Now use this to expand the Lagrangian around the classical solution f0(t,x):
[ ] @L @L L(@mf, f) = L(@mf0, f0) + e @(@--f) @mf1 + @f- f1 [ m ] e2 --@L--- @L- + 2 @(@mf) @mf2 + @f f2 [ e2 -----@2L------ ---@2L---- + 2 @(@mf) @(@nf) @mf1 @nf1 + 2 @(@mf) @f @mf1 f1 ] @2L + ------ f1 f1 + O(e3). (104) @f @f
It is particularly useful to consider the action
integral S[f] = dd+1x L(@mf, f), (105)
since doing so allows us to integrate by parts. (Note that the Lagrangian L is taken to be a scalar density, not a true scalar.) We can now use the Euler-Lagrange equations for the background field
( ) @m --@L--- - @L- = 0, (106) @(@mf) @f
to discard the linear terms (remember we are linearizing around a solution of the equations of motion) and so we get
[ e2 integral { @2L } S[f] = S[f0] + -- dd+1x -------------- @mf1 @nf1 2 @(@mf) @(@nf) ( 2 { 2 } ) ] + -@-L--- @ ---@--L--- f f + O(e3). (107) @f @f m @(@mf) @f 1 1
Having set things up this way, the equation of motion for the linearised fluctuation is now easily read off as
({ } ) ( { } ) @2L @2L @2L @m -------------- @nf1 - ------ - @m ---------- f1 = 0. (108) @(@mf) @(@nf) @f @f @(@mf) @f
This is a second-order differential equation with position-dependent coefficients (these coefficients all being implicit functions of the background field f0).

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

{ } | V~ --- mn mn @2L | - g g =_ f =_ -------------- || . (109) @(@mf) @(@nf) f0
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 f0 except when necessary for clarity, this implies [in (d+1) dimensions, d space dimensions plus 1 time dimension]

{ 2 } (- g)(d-1)/2 = - det -----@-L------ . (110) @(@mf) @(@nf)
( { 2 } )- 1/(d-1)|| { 2 } || gmn(f0) = - det -----@-L------ || -----@--L----- | . (111) @(@mf) @(@nf) | @(@mf) @(@nf) |f0 f0
And, taking the inverse
( { 2 } )1/(d-1)|| { 2 } -1|| gmn(f0) = - det -----@-L------ || -----@-L------ || . (112) @(@mf) @(@nf) | @(@mf) @(@nf) | f0 f0
We can now write the equation of motion for the linearised fluctuations in the geometrical form
[D(g(f0)) - V (f0)]f1 = 0, (113)
where D is the d’Alembertian operator associated with the effective metric g(f0), and V (f0) is the background-field-dependent (and so in general position-dependent) “mass term”:
( { } ) --1-- -@2L-- ----@2L--- V (f0) = V~ --g @f @f - @m @(@mf) @f (114) ( { } )- 1/(d-1) ( { } ) -----@2L------ -@2L-- ---@2L---- = - det @(@mf) @(@nf) × @f @f - @m @(@mf) @f . (115)
Thus V (f0) is a true scalar (not a density). Note that the differential Equation (113View Equation) is automatically formally self-adjoint (with respect to the measure V~ -- - g dd+1x).

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 g mn has Lorentzian signature ± [- ,+d]. 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 v = - \~/ f 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 f and its derivatives. [15Jump To The Next Citation Point]

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