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Observables have to be gauge invariant, but clearly the spacetime interval:

$$ds = g_{ij}dx^idx^j = (\eta_{ij}+h_{ij})dx^idx^j\tag{1}$$

is not invariant under a transformation:

$$ h_{ij} \rightarrow h_{ij}+\partial_i\xi_j+\partial_j\xi_i\tag{2}$$

Since:

$$g_{ij}dx^idx^j \rightarrow (g_{ij}+\partial_i\xi_j+\partial_j\xi_i)dx^idx^j\neq g_{ij}dx^idx^j\tag{3}$$

So how can (2) be a valid gauge transformation?

Quanta
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1 Answers1

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The transformation you're talking about is generated by the infinitesimal coordinate transformation $$ x^{\mu} \rightarrow \hat{x}^{\mu} = x^{\mu} + \xi^{\mu}(x) $$ where we assume $\xi^{\mu}$ is small (i.e., working up to first order in $\xi$). Applying the appropriate transformations on the coordinate basis one forms $dx^{\mu}$ shows the interval $ds^2$ is indeed invariant.

Eletie
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