Final answer:
The determinant of the metric tensor depends on the coordinate system and is not a property of spacetime itself. A transformation to Cartesian coordinates does not ensure a constant determinant in an arbitrarily curved spacetime, as curvature information is still represented in those coordinates.
Step-by-step explanation:
The determinant of a metric tensor does indeed depend on the choice of coordinates, and it is not a characteristic of the physical spacetime itself. Rather, it reflects the properties of the coordinate system used to describe that spacetime. For example, the Schwarzschild metric in spherical coordinates will have a non-constant determinant because it reflects the radial stretching of space due to the mass concentration. However, a transition to Cartesian coordinates does not guarantee a constant determinant in an arbitrarily curved spacetime; this is because the metric tensor in Cartesian coordinates will still contain information about the curvature induced by mass and energy distributions. Therefore, in general, the determinant of the metric tensor in Cartesian coordinates is not always constant in an arbitrarily curved spacetime.
Einstein's theory of general relativity posits that matter and energy determine the curvature of spacetime. This curvature in turn affects the paths of objects and light passing through it. For black holes in particular, the Schwarzschild solution to the equations of general relativity demonstrates how mass creates such a curvature that beyond the Schwarzschild radius, spacetime is warped to the extent that nothing can escape, not even light.