I'm studying GR with Schutz' First Course in General Relativity and I have some trouble.

When field is weak enough, we can take such coordinate system that our metric is written as $$ g_{\alpha\beta} = \eta_{\alpha\beta} + h_{\alpha\beta}, \ \ \ |h_{\alpha\beta}| \ll 1 $$ where $\eta_{\mu\nu}$ is Minkowski metric whose components are $\rm{diag(-1, 1,1,1)}$ and $h_{\mu\nu}$ is perturbation field.

In the book, Schutz several times regards products $h_{\alpha\beta}h_{\mu\nu,\gamma}$ as of second order and drops. Why we can do such calculations? I think I cannot say that $h_{\mu\nu,\gamma}$ is small just because $h_{\mu\nu}$ is small enough.... :(

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    $\begingroup$ Those are two independent criterions. i.e. take the case where $h = \sin(1/x)$ for some component. $|h| \leq 1$ but $|h'|$ is unbounded. $\endgroup$ – Slereah Jun 23 at 12:16

You are correct, you cannot assume that $h_{\mu\nu,\gamma} \ll 1$ based only on the fact that $h_{\mu\nu} \ll 1$. The fact that $h_{\mu\nu,\gamma} \ll 1/L$, and that $h_{\mu\nu,\gamma\delta} \ll 1/L^2$, where $L$ is a physical length of interest, are standalone assumptions that together yield the usual weak-field limit.

There is another way the weak-field limit can be characterized and that is that in a region of linear size $L$ there exists a smooth orthonormal frame $e^A_\mu, e^A_\mu e^B_\nu g^{\mu\nu} = \eta^{AB}$ such that all the eigenvalues of the Riemann tensor in the tetrad frame $R_{ABCD}$ are $\ll 1/L^2$. Based on this assumption you can construct a coordinate system such that the coordinate components of the metric fulfill $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$, $h_{\mu\nu}\ll 1, h_{\mu\nu,\gamma} \ll 1/L, h_{\mu\nu,\gamma \kappa} \ll 1/L^2,...$ Of course, the simpler way to get there is to jump into this situation without a broader explanation of its physical meaning.

There is an alternative expansion, the so-called high-frequency limit, where one does not assume the metric perturbation $h_{\mu\nu,\gamma} \ll 1/L$ while assuming $h_{\mu\nu}\ll1$, but there one cannot assume the background (unperturbed) metric to be flat. Just a word of warning: the high-frequency limit should be understood more as a split of the metric into a slowly-varying (the "background") and a quickly varying part (the "perturbation"), which simplifies computation. For more see Isaacson (1968): Gravitational Radiation in the Limit of High Frequency.

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It is definitely true that they are two different conditions as a counter example already mentioned in the comments proves. However, the approximation remains valid within a proper physical context. When one deals with weak field approximation, one usually also assumes that velocities of bodies involved in the problem are very small compared to that of the propagation speed of gravity (light speed) or the typical rate of change of the gravitational field, put in slightly different wording, that within the region of interest the field doesn't vary too much. Hence derivatives are counted as also being way smaller than 1.

P.D. when in need of more mathematical details or more rigor look for "Gravitation" by Misner, Thorne and Wheeler or if you really are inclined into the mathematical physics look for Wald's book on GR.

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  • $\begingroup$ Your discussion implies that you can't meaningfully derive a linearized equation for gravitational waves, since the velocities involved are not small compared to that of light. $\endgroup$ – Michael Seifert Jun 23 at 21:17
  • $\begingroup$ I meant the velocities of any particles traveling in this background, not the propagation speed of the gravitational field, I will make an edit to be clearer on that point, thank you for the comment $\endgroup$ – ohneVal Jun 23 at 21:49

The short answer is that in introductory textbooks in GR, the only perturbations considered are those for which $h_{\mu \nu}$ and its derivatives are "small". In other words, not only do we have $|h_{\mu \nu}| \ll 1$ but also $|h_{\mu \nu,\rho}| \ll 1$, $|h_{\mu \nu,\rho \sigma}| \ll 1$, etc. This is not always stated explicitly.

The mathematically rigorous way to do this sort of perturbation calculation is to make the assumption that there is a one-parameter family of metrics $g_{\alpha \beta}(\lambda)$ with the following properties:

  • $g_{\alpha \beta}(\lambda)$ is a smooth function of the spacetime coordinates and of $\lambda$.
  • For all $\lambda$, $g_{\alpha \beta}$ satisfies the vacuum Einstein equation $G_{\alpha \beta} = 0$.
  • $g_{\alpha \beta}(0) = \eta_{\alpha \beta}$.

Under this assumption, the quantity $h_{\alpha \beta}$ is then defined to be $$ h_{\alpha \beta} \equiv \left.\frac{d g_{\alpha \beta}(\lambda)}{d\lambda} \right|_{\lambda = 0}, $$ or in other words $$ g_{\alpha \beta}(\lambda) = \eta_{\alpha \beta} + \lambda h_{\alpha \beta} + \mathcal{O}(\lambda^2). $$

The linearized equations are then found by expanding the Einstein tensor $G_{\alpha \beta}$ in a power series in $\lambda$: Since $G_{\alpha \beta} = 0$ for all values of $\lambda$, all of the coefficients of the resulting power series must vanish. The linear-order term in this power series will only contain terms that are linear in $h_{\alpha \beta}$, since any term that is (for example) quadratic in $h$ will be of order $\lambda^2$.

Note that pathological cases such as those discussed in the comments are not allowed under the smoothness requirements of this formalism. As a toy example, suppose that $g_{tt}(\lambda) = \eta_{tt} + \delta g_{tt}$, where $\delta g_{tt} = \lambda \sin (x/\lambda)$. This function has the property that $g_{tt} \to \eta_{tt} = -1$ as $\lambda \to 0$, but $\partial_x g_{tt} \not\to 0$. The formalism above cannot treat perturbations of this kind, since the derivative of $g_{tt}$ with respect to $\lambda$ does not exist at $\lambda = 0$: $$ \frac{d g_{tt}(\lambda)}{d\lambda} = \sin \left( \frac{x}{\lambda} \right) - \frac{x}{\lambda^2} \cos \left( \frac{x}{\lambda} \right). $$ If your situation involves calculations such as this, non-standard techniques are required; see Void's answer for a brief description on how to do this. However, such techniques are usually not part of a first course in GR.

For more details, see Section 7.5 of Wald's General Relativity, from which this exposition is drawn. In that section, Wald also shows how this technique can be generalized to perturbations off of a curved background, such as Schwarzschild or FRW.

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  • $\begingroup$ This however doesn't solve the issue of the OP, the focus point is the reasoning behind neglecting derivative terms, $h_{\mu\nu},\alpha$, which only with a series expansion of $g$ is not justified, see the counter example in the comments to the question. As explained briefly in my answer and with a more detail in @Void 's answer you need an extra scale in the problem together with an assumption on the variation of $h_{\mu\nu}$ with it. $\endgroup$ – ohneVal Jun 24 at 7:10
  • $\begingroup$ @ohneVal: pathological cases such as the one in the comments are ruled out by the assumption that you can actually differentiate $g_{\mu \nu}(\lambda)$ with respect to $\lambda$. I've edited by answer to make this clearer. $\endgroup$ – Michael Seifert Jun 24 at 11:58

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