Effect of diffeomorphisms on the <PERSON> metric
This question was prompted by the following passage in <PERSON> (1984)'s chapter on spinors:
Consider, now, a diffeomorphism $\phi:M\to M$ and allow $\phi$ to map the basis fields $(e_\alpha)^a$ into $\phi^(e_\alpha)^a$ in the manner described in appendix C. In general, the basis $\phi^(e_\alpha)^a$ will not be orthonormal at each point and thus will not correspond to a physically realizable family of observers, However, when (and only when) $\phi$ is an isometry, $\phi^(e_\alpha)^a$ will be orthonormal, and we can use $\phi$ to map our original family, $O$, of physical observers associated with the basis field $(e_\alpha)^a$ into a new family $\tilde{O}$ of physical observers associated with $\phi^(e_\alpha)^a$.
I don't understand how the basis vectors would cease to be orthonormal after the diffeomorphism. Surely the diffeo must also induce a pullback $(\phi^{-1})$ (described in the apprendix <PERSON> refers to) such that $(\phi^{-1})_g(\phi^ e_\alpha,\phi^ e\beta)=g(e_\alpha,e_\beta)=\eta_{\alpha\beta}$?
This is an issue that has been troubling me lately. 346
I see something similar in the "derivation" of the Lorentz transformation condition from invariance of the line element. It goes: postulate $ds^2$ is invariant under <PERSON> transformations, then $$\eta_{\mu\nu}dx^\mu dx^\nu=\eta_{\alpha\beta}(dx^\prime)^\alpha (dx^\prime)^\beta=\eta_{\alpha\beta}\Lambda^\alpha_{\,\,\mu}dx^\mu\Lambda^\beta_{\,\,\nu}dx^\nu=\Lambda^\alpha_{\,\,\mu}\eta_{\alpha\beta}\Lambda^\beta_{\,\,\nu}dx^\mu dx^\nu$$So one concludes that the transformations must satisfy $\Lambda^\alpha_{\,\,\mu}\eta_{\alpha\beta}\Lambda^\beta_{\,\,\nu}=\eta_{\mu\nu}$.
But doesn't $\eta_{\mu\nu}dx^\mu dx^\nu=\eta_{\alpha\beta}(dx^\prime)^\alpha (dx^\prime)^\beta$ already assume that the transformations are isometries of $\eta_{\mu\nu}$? Otherwise the metric components would also change and the invariance of the line element would be a trivial fact. After all, it is a scalar and so is invariant under arbitrary coordinate transformations.
I guess my question is, why do we sometimes treat transformations such as <PERSON> transformations as purely acting on vectors, while other times we treat them as diffeomorphisms (or coordinate transformations in the passive viewpoint) that also affect the metric through a pullback? Surely these two cases are very distinct? 818