Metric in SR: \eta^{\alpha \beta}=\eta_{\alpha \beta}?

[dEdt]

Does \eta^{\alpha \beta}=\eta_{\alpha \beta} in all coordinate systems or just inertial coordinate systems?

 

[stevendaryl]

Does \eta^{\alpha \beta}=\eta_{\alpha \beta} in all coordinate systems or just inertial coordinate systems?

For non-inertial coordinate systems, the symbol g_{\alpha \beta} is used instead of \eta_{\alpha \beta}. And in general, g_{\alpha \beta} is unequal to g^{\alpha \beta}. g^{\alpha \beta} is the inverse of g_{\alpha \beta}.

Here's an example: In polar coordinates t, \rho, \phi, z,

g_{tt} = 1
g_{zz} = -1
g_{\rho \rho} = -1
g_{\phi \phi} = -\rho^2

g^{tt} = 1
g^{zz} = -1
g^{\rho \rho} = -1
g^{\phi \phi} = -\frac{1}{\rho^2}

 

[jcsd]

g^{\alpha \beta}=g_{\alpha \beta} means that the coordinate basis is orthonormal, which only corresponds to 'inertial coordinates' (Minkowski coordinates) in flat spacetime.

 

[dEdt]

Thanks guys.