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I am learning about harmonic gauge for weak field limit of Einstein's equation and have some problems with the tensor calculus invovlved.

Consider the weak field approximation

$$g_{\mu\nu}(x)=\eta_{\mu\nu} + h_{\mu\nu}(x)$$ where $g_{\mu\nu}(x)$ is the metric, $\eta_{\mu\nu}$ is the Minkowski metric and $h_{\mu\nu}(x)$ is a small perturbation.

When there is a small coordinate transform $$x^\mu \rightarrow x'^{\mu}=x^\mu+\epsilon^\mu(x),$$ the transformation matrix is $$\frac{\partial x^\mu}{\partial x'^\nu}=\delta^\mu_\nu-\partial_\nu\epsilon^\mu.$$

I am trying to use this transformation matrix to show that in the primed coordinate system, the perturbation term transforms approximately as $$h'_{\mu\nu}\approx h_{\mu\nu}-\partial_\mu\epsilon_\nu-\partial_\nu\epsilon_\mu.$$

My attempt is

$$h'_{\mu\nu}=\frac{\partial x^\alpha}{\partial x'^\mu}\frac{\partial x^\beta}{\partial x'^\nu}h_{\alpha\beta} = (\delta^\alpha_\mu-\partial_\mu\epsilon^\alpha)(\delta^\beta_\nu-\partial_\nu\epsilon^\beta)h_{\alpha\beta}$$$$\approx\delta^\alpha_\mu\delta^\beta_\nu h_{\alpha\beta}-\delta^\beta_\nu (\partial_\mu\epsilon^\alpha) h_{\alpha\beta} - \delta^\alpha_\mu ( \partial_\nu\epsilon^\beta ) h_{\alpha\beta} $$ $$=h_{\mu\nu}- (\partial_\mu\epsilon^\alpha) h_{\alpha\nu}-(\partial_\nu\epsilon^\beta) h_{\mu\beta}$$

So it seems that to arrive at the solution I want, $$\epsilon^\alpha h_{\alpha\nu}=\epsilon_\nu,$$$$\epsilon^\beta h_{\mu\beta}=\epsilon_\mu$$ needs to be true. Are these two statements true? I have only learned that$$\epsilon^\nu g_{\mu\nu}=\epsilon_\nu,$$ i.e. the metric can be used to the lower index.

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You need to transform the entire tensor $g^{\mu\nu} = \eta^{\mu \nu} + h^{\mu\nu}$. Then you 1) neglect cross-terms $\sim \epsilon h$, 2) require that the transformed metric $g_{\mu\nu}$ is still equal to $\eta_{\mu\nu}$ plus some new $h'_{\mu\nu}$. Good luck!

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