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.

 

[Entropia]

for your question. The sign convention in Einstein's equations is not arbitrary and has a significant physical meaning. It is related to the geometry of spacetime and the way we measure distances and intervals in this space. The metric tensor is used to describe the curvature of spacetime and it contains information about the relationship between time and space.

1. The reason for the specific sign convention (+,-,-,-) or (-,+,+,+) is based on the Minkowski metric, which is the flat spacetime metric used in special relativity. In this metric, the time component has a positive sign while the spatial components have negative signs. This convention was chosen because it reflects the observed nature of spacetime, where time moves forward and space is three-dimensional. Using this convention in general relativity ensures that the equations are consistent with special relativity and that the curvature of spacetime is correctly described.

2. The signs in the metric are not put there ad hoc, but they are derived from the Einstein field equations. These equations describe the relationship between the curvature of spacetime and the energy and matter content of the universe. When solving these equations for different scenarios, the signs in the metric emerge naturally and are not chosen arbitrarily. This is because the equations are based on physical principles and must be consistent with experimental observations.

In summary, the sign convention in Einstein's equations is not just a mathematical choice, but it has physical significance and is derived from the principles of special and general relativity. It is crucial for accurately describing the curvature of spacetime and understanding the behavior of matter and energy in the universe.