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I am a little bit confused about whether the inner product between the Riemannian metric tensor and Minkowski metric tensor is equal to the Kronecker delta function:

$${\eta}^{{\mu}{\nu}}{g}_{{\mu}{\alpha}}={\delta}^{{\nu}}_{ {\alpha}}.$$

Is the above inner product correct and valid?

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    $\begingroup$ The Minkowski metric is just a special case of the metric tensor for flat space... so it's not clear why you would want to write something like this. The metric tensor is in general a tensor field which means it also varies from point to point in spacetime, but the Minkowski metric is defined as constant... so I don't see the sense in combining them both like that. Also, this is not an inner product. $\endgroup$
    – Amit
    Apr 1, 2023 at 19:02

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Short Answer:

If you work in Minkowski space then $g_{\mu\alpha}=\eta_{\mu\alpha}$ and the identity holds, otherwise it does not hold.

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