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.

 

[sardar]

In Special Relativity, the metric tensor is defined as \eta^{\alpha \beta}=\eta_{\alpha \beta}, where \eta^{\alpha \beta} is the contravariant form and \eta_{\alpha \beta} is the covariant form. This equation holds true in all coordinate systems, not just inertial coordinate systems.

The metric tensor is a fundamental concept in the theory of General Relativity and is used to describe the geometry of spacetime. It represents the relationship between space and time and is responsible for the curvature of spacetime. In Special Relativity, the metric tensor takes on a simpler form due to the absence of gravitational effects.

The equation \eta^{\alpha \beta}=\eta_{\alpha \beta} is a manifestation of the principle of covariance, which states that physical laws should have the same form in all coordinate systems. This principle is crucial for the development of a consistent and universal theory of physics.

Therefore, in conclusion, \eta^{\alpha \beta}=\eta_{\alpha \beta} holds true in all coordinate systems, not just inertial coordinate systems, and is a fundamental concept in the theory of Special Relativity.