Hello there, suppose we take ##M## to denote the spacetime manifold. Suppose also that ## ds^2 = g_{\mu \nu} dx^\mu dx^\nu##. I have some confusions with regards to the metric and the line elements.(adsbygoogle = window.adsbygoogle || []).push({});

My main confusion is at which points in the manifold are ## ds^2## defined? Is it correct that if ## p \in M##, then it only makes sense to fix ## ds^2## for each tangent space ## T_pM##, or is it fixed for each ## v \in T_pM ##? More specifically:

(1) If there is some position dependent line element, such as the Schwartzschild line element, then ##g_{\mu \nu}= g_{\mu \nu}(t,r, \theta, \phi) ##. Are these ##(r, \theta, \phi)## the coordinates of some ##(t,r, \theta, \phi) \in M##, or are they the coordinates of some ## (t,r, \theta, \phi) \in T_pM ##?

(2) Are the differentials ## dx^\mu(x)## evaluated at each ## x \in M## or ##x \in T_pM##?

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# A General relativity -- Who reports the components of ds^2?

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