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.