Geometrical optics approximation in General Relativity - Expansion parameter In General Relativity when studying wave propagation on curved spacetimes one usually employs the geometrical optics approximation. This is described on the book  "Gravity: Newtonian, Post-Newtonian, Relativistic" by Eric Poisson as follows (Box 5.6, Chapter 5):

Maxwell’s equations imply that photons move on geodesics of a curved spacetime. In this context the term “photon” is employed in a classical sense, and designates a fictitious particle that follows the path of light rays. This is the domain of the geometric-optics approximation to Maxwell’s theory, which is applicable when the characteristic wavelength associated with a field configuration is much smaller than any other scale of relevance, including the curvature scale set by the Riemann tensor. In our analysis we take these external scales to be of order unity, while the wavelength is taken to be much smaller than this.
The geometric-optics approximation is built into the following ansatz for the vector potential: $$A^\alpha = [a^\alpha + i\epsilon b^\alpha + O(\epsilon^2)]e^{iS/\epsilon}$$
The prefactor within square brackets is a slowly-varying, complex amplitude, while the exponential factor contains a rapidly-varying, real phase $S/\epsilon$. The constant $\epsilon$ is a book-keeping parameter that we take to be small during our manipulations; at the end of our calculations we reset it to $\epsilon = 1$, so that $S$ becomes the actual phase function.

Now this is quite confusing for me for a few reasons:

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*What is the point of this arbitrary parameter being introduced? I could understand an expansion in some physically meaningful parameter, but this seems totally ad-hoc.


*Taking $\epsilon$ to be small during manipulations and later putting in $\epsilon = 1$ seem to be utterly wrong. I mean, during derivations if you use that $\epsilon$ is small you are basically making implications of the form "if $\epsilon$ is small this holds". If in the end we take $\epsilon = 1$ we are out of the hypothesis for that derivation to hold. What am I missing here?


*Where this ansatz comes from anyway? I mean, supposing $A^\alpha = C^\alpha e^{iS}$ is fine with me, because any complex number may be written in this form. But then one introduces that weird ad-hod parameter and supposes $C^\alpha$ is a power series on it. How one could conclude this is reasonable anyway?


*How all of this connects with the idea that this is meant to be an approximation when the wavelength of a field configuration is smaller of other scales of relevance? Moreover what is this "wavelenght of field configuration" for a general superposition of waves? I mean, for a plane wave we have well defined frequency and wavelength but for a general wavepacket we have a mixing of several such well defined frequencies and wavelengths.
 A: 
  
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*What is the point of this arbitrary parameter being introduced? I could understand an expansion in some physically meaningful parameter, but this seems totally ad-hoc.
  

The reason is stated in the quoted passage: book-keeping. Since we have multiple functions entering our equations at different orders of approximation it is useful to have a single parameter that controls the order of approximation we are considering. Terms without $\epsilon$ are the zeroth order of approximation, terms linear in $\epsilon$ are the first order, etc. In case of test EM field the gains may not be obvious, but a general perturbation equation can have a lot of symbols: background metric, background EM field, background matter, metric perturbation, EM field perturbation … so looking at a given term and deciding which order of perturbation it belongs to might be a time-consuming task. Another area where this is useful is analysis with computer algebra systems: by expanding equations in powers of $\epsilon$ we obtain equations for different orders of approximation as coefficients.


  
*Taking $ϵ$ to be small during manipulations and later putting in $ϵ=1$ seem to be utterly wrong.
  

Another viewpoint: instead of saying “$\epsilon$ is small” and then setting $ϵ=1$ we could absorb $\epsilon$ into the definitions of variables (like $\epsilon B \to B'$), so the statement “if $ϵ$ is small this holds” is really a shorthand for “if all first order terms are much smaller (with regard to appropriate norms) than corresponding zeroth order terms, and also if all second order terms are much smaller than first order terms … then this holds”.
Note, that such manipulations are mostly useful heuristics for obtaining appropriate equations, and careful analysis of convergence or applicability of approximation is still required for rigorous results.


  
*Where this ansatz comes from anyway?  I mean, supposing  $A^\alpha = C^\alpha e^{iS}$ is fine with me …
  

The important part here is that $S$ is assumed to satisfy an equation that is independent on $C$, but this could only be true in the limit of slowly changing metric or fast changing phase. So the phase multiplier around a given point would have a slowly varying correction (say $e^{i S_1}$) sensitive to changes in amplitude and it could be expanded at a given point producing the imaginary term at the first order of approximation.


  
*How all of this connects with the idea that this is meant to be an approximation when the wavelength of a field configuration is smaller of other scales of relevance? 
  

This is all about scales separation. For slowly changing amplitude we would have  for gradients orders of magnitude $\nabla C \sim C/l$ (where $l$ is a length scale of the metric), while for the phase $\nabla e^{iS} \sim k e^{iS} $. Our ability to write an separate equation for $S$ is connected with the possibility of discarding terms $1/l$ relative to $k \sim 1/\lambda $.

 … Moreover what is this “wavelength of field configuration” for a general superposition of waves?

Obviously, for general superposition of waves no single wavelength could be defined. But the equations for test EM field are linear, so one could check the validity of geometrical optics approximation for each comprising wave. For example, one could perform Fourier analysis in the asymptotic regions (say $\scr I^\pm$) and make statements about frequency ranges.
