Uncovering the Metric Sign in Einstein's Equations

[blumfeld0]

hi.
1. Why does the sign of the metric have to be (+,-,-,-) or (-,+,+,+)?
meaning why does the sign in front of the time part of the metric have to be different than the signs in front of the spatial part of the metric?
is there a good answer for this question other than it gives the right answer.

2. are the signs something that just pop out after solving the Einstein Equations for various scenarios and doing the math or is it put there ad hoc?

thank you

 

[smallphi]

Einstein derived from a few postulates (most notably that the speed of light is constant in any inertial coordinate system) and logical reasoning the coordinate transformations between two inertial coordinate systems (also known as Lorentz transformations).

Minkowski was the one that realized those transformations conserve the following combination called spacetime interval:
ds^2 = dt^2 - (dx^2 + dy^2 + dz^2) (units of c=1)
That fixed the metric in SR to (+, -, -, -) or (-, +, +, +) depending on the sign convention. SR is GR of the flat Minkowski spacetime.

GR models spacetime mathematically with a Riemannian manifold. Every such manifold has the property of local flatness, i.e. the metric at any point in spacetime can be diagonalized with a suitable coordinate transformation. The resulting local diagonal metric is used by an observer at that spacetime point to measure events around him/her. GR postulates that the diagonalized metric at any point is the SR metric thus the signature of the metric in GR is imported from SR.

The bottom line is that the sign difference between dt^2 and dr^2 originates in the Einstein's SR postulate that light speed is the same in all inertial coordinate systems. If you have a light signal connecting two events separated by dt and dr then according to the SR postulate c=1=dr/dt in any innertial coordinate system. That leads to ds^2 = dt^2 - dr^2 = 0 =invariant (light travels along null paths) which is generalized to nonzero ds^2 for other possible paths not followed by light.