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The postulate that the velocity of light in a vacuum is the same in all inertial frames, can be exploited to establish that the spacetime interval between two events $x^\mu=(ct,{\bf r})$ and $x^\mu+dx^\mu=(ct+cdt,{\bf r}+d{\bf r})$ defined as $$ds^2=\eta_{\mu\nu}dx^\mu dx^\nu=c^2dt^2-d{\bf r}^2\tag{1}$$ remains invariant.

In general relativity, we require $$ds^2=g_{\mu\nu}(x)dx^\mu dx^\nu \tag{2}$$ to be invariant. Which postulate of general relativity gives rise to this requirement?

Qmechanic
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SRS
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  • This is not a postulate, this is $g_{\mu \nu}$ definition. – warlock Dec 22 '20 at 17:17
  • In SR we work with Lorentz transformations, under which the line element (1) is manifestly invariant. In GR we work with general coordinate transformations, under which (2), which is just the metric tensor ($g_{ab}(x)$ are the components), is manifestly invariant. – Eletie Dec 22 '20 at 17:24

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