Linearized gravity: Derivatives of the metric perturbation

Question

In linearized gravity, the metric is given by the Minkowski metric and a small perturbation, \begin{equation} g^{\mu\nu} = \eta^{\mu\nu}+h^{\mu\nu},\quad |h^{\mu\nu}|\ll 1. \end{equation} Plugging this expression into the expression for our Christoffel symbols, we obtain \begin{equation} \Gamma^{\alpha}_{\mu\nu} = \frac{1}{2} \color{red}{g^{\alpha\lambda}} (\partial_\nu h_{\mu\lambda} + \partial_\mu h_{\nu\lambda} - \partial_\lambda h_{\mu\nu}). \end{equation} Various textbooks and lecture notes like these notes from Carroll (eq. 6.4) simplify this to \begin{equation} \Gamma^{\alpha}_{\mu\nu} = \frac{1}{2} \color{red}{\eta^{\alpha\lambda}} (\partial_\nu h_{\mu\lambda} + \partial_\mu h_{\nu\lambda} - \partial_\lambda h_{\mu\nu}) \end{equation} by asserting that $\partial_\alpha h_{\beta\gamma}$ is also first order in $h_{\mu\nu}$.

How is that justified? Couldn't the perturbation be small, but rapidly oscillating, for example?

Answer 1

This is a common issue that comes up in any classical perturbation theory, not just gravity. At the end of the day, the systematic method for doing any perturbative expansion is to take the would-be true solution for a field, say $\phi$, and expand it in powers of some expansion parameter, say $\alpha$: $$ \phi=\phi_0+\alpha\phi_1+\alpha^2\phi_2+\cdots. $$ The expansion is then in this parameter $\alpha$, which since this is a constant is independent of whether we take derivatives or not. The key assumption of perturbation theory after this point is that the equations of motion should hold order by order, meaning we can separate our equations for each order in $\alpha$, using the solutions from the lower order equations to simplify the higher order ones.

This is the formulation of perturbation theory. Whether or not this is a good idea is then a completely different question which would need to be addressed based on your needs and the particular theory you are attempting to solve by perturbation methods.

There are various methods for getting around this highly oscillatory issue you bring up, for example when doing a perturbation around the harmonic oscillator in point particle mechanics, it can sometimes be helpful to use a change in the frequency as a perturbation parameter, aside from just the coupling to whatever higher order perturbing terms are added to the action. But to the best of my knowledge, these things must be considered in a case-by-case fashion.

So I think the real question here is: are you planning to look for solutions which are both nearly Minkowski (or whatever metric you're perturbing about) while simultaneously being very high frequency? And the answer to that depends on what you're looking for in the first place.