While reading through Spacetime and Geometry by Sean Carroll, I came across the following passage:
"Don't get tricked into thinking that the timelike component of the four velocity of a particle at rest will always equal unity: we need to satisfy the normalisation condition, $g_{\mu\nu}U^{\mu}U^{\nu}=-1$ which in the rest frame ($U^{i}=0$) implies $U^0=\sqrt{-g_{00}}$."
However, shouldn't $U^0=\frac{1}{\sqrt{-g_{00}}}$? Since $g_{\mu\nu}U^{\mu}U^{\nu}=-1$, and since spatial components are zero, this would translate to $g_{00}(U^{0})^2=-1$, and thus $U^0=\frac{1}{\sqrt{-g_{00}}}$? I couldn't find anything about this in Sean Carroll's Errata as well, could someone explain this?
