Metric on Lorentzian manifold

Question

Since the signature of a Lorentzian manifold (M,g) is (-,+,+,+), am I right to assume that the determinant of $g_{ij}$ with EACH coordinate system is $-1$?

Answer 1

Yes, you are.

If you change the coordinate system the metric in the new coordinates will be the conjugate of $g_{ij}$ by the coordinate change’s differential map. Since the differential belongs to the general linear group, this will not change neither the metric’s signature nor eigenvalues. Thus, the determinant shall remain the same.