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(1) He says

*In our discussion of path lengths in special relativity we (somewhat handwavingly) introduced the line element as [itex] ds^2=\eta_{\mu\nu}dx^{\mu}dx^{\nu}[/itex], which was used to get the length of the path. Of course now that we know that [itex]\text{d} x^{\mu} [/itex] is really a basis dual vector, it becomes natural to use the terms "metric" and "line element" interchangeably, and write [tex] ds^2=g_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}[/tex]*

My question here is: isn't what he's doing right now just as handwavy as before? It's very hard to see a concrete reason justifying what he's doing, unless he's

*defining*[itex] ds^2 [/itex] by the above expression, which I'd have no problem with. But just come out and say it man!

(2) Why does he talk about [itex] ds^2 [/itex] in terms of basis dual vectors, instead of just differentials? What's the point of doing this? I realize that this is not just Carroll's idea; that's the way it is in GR. But in treating [itex] ds^2 [/itex] as a (0,2) tensor, instead of the square of a differential length, don't we lose the physical meaning inherent in [itex] ds^2 [/itex]? How do we go back to interpreting [itex] ds^2 [/itex] as a length so quickly when we changed its meaning so drastically?

Thanks in advance!