The approach taken in linearized gravity seems to be to 'perturb' the 'Minkowski metric' such that(adsbygoogle = window.adsbygoogle || []).push({});

$$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$$

where ##|h_{\mu \nu}| <<1##. As I've understood it, the goal is to get an approximate theory for gravity, i.e. for weak gravitational fields and thus for small curvature. However the perturbation above seems more to be about choice of basis. After all if we had ##g_{\mu \nu} = \eta_{\mu \nu}## everywhere, that would say nothing about the curvature of spacetime, but rather that we had chosen to use an orthonormal basis everywhere. In this light the perturbation above seems more like the demand that we are only allowing 'almost orthonormal bases' than a demand on the weakness of gravity.

Would it not be more sensible to perturb about a ##g_{\mu \nu}## with vanishing second derivatives and demand that the second derivatives of the perturbation be small, since it is the second derivatives of the metric that contribute to the Riemann tensor?

Has approaches like this been tried somewhere?

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# Linearized Gravity and the smalness of the pertubation

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