General metric and flat metric

[Tony Stark]

What is the difference between General metric g_αβ and flat metric η_βα in GR?
Elaborate answers are appreciated.

 

[Nugatory]

The metric tensor describes the geometry of spacetime in a coordinate-independent way. One very important special case is the metric tensor that describes a flat (no significant gravitational effects) spacetime; the $\nu_{\alpha\beta}$ that you're calling the "flat metric" are the components of that metric tensor written in Minkowski coordinates. You could use a different set of coordinates (Rindler or spherical or...) and the components would come out looking completely different, but it would still be the same flat spacetime.

 

[Tony Stark]

The metric tensor describes the geometry of spacetime in a coordinate-independent way. One very important special case is the metric tensor that describes a flat (no significant gravitational effects) spacetime; the $\nu_{\alpha\beta}$ that you're calling the "flat metric" are the components of that metric tensor written in Minkowski coordinates. You could use a different set of coordinates (Rindler or spherical or...) and the components would come out looking completely different, but it would still be the same flat spacetime.

I do understand that the metric tensor used in provides independence from a particular coordinate system. But this still doesn't answer my question about the difference between general and flat metric clearly.Sorry sir.

 

[Nugatory]

Your question is a bit like asking for the difference between an elephant and animal: all elephants are animals but not all animals are elephants.

If there exists a coordinate system in which the metric for a given spacetime takes the form $diag(-1,1,1,1)$ then the spacetime is flat; if not, then it's not.

 

[Ibix]

It's kind of like F=ma and F=mg. g is just an acceleration, but we give it a special symbol because it's a particular case of an acceleration that we use a lot.

Similarly, $\eta_{\mu\nu}$ is just a metric tensor. It does everything any other metric tensor does. It's just that it describes a case of particular interest, the one where there is no gravity, or where gravity can be neglected.

That analogy is a bit strained. but the best I can think of.

 

[Dale]

What is the difference between General metric g_αβ and flat metric η_βα in GR?
Elaborate answers are appreciated.

The flat metric has 0 curvature. The general metric may not have 0 curvature.

Sorry that is not very elaborate, but the difference doesn't seem to require much elaboration.