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I'm looking at the wikipedia article about four-momentum and I can't seem to get things right. It says

Calculating the Minkowski norm of the four-momentum gives a Lorentz invariant quantity equal (up to factors of the ''c'') to the square of the particle's proper mass:

[tex]-||\mathbf{P}||^2 = - P^\mu P_\mu = - \eta_{\mu\nu} P^\mu P^\nu = {E^2 \over c^2} - |\vec p|^2 = m^2c^2 [/tex]

where we use the convention that

[tex]\eta^{\mu\nu} = \begin{pmatrix}

-1 & 0 & 0 & 0\\

0 & 1 & 0 & 0\\

0 & 0 & 1 & 0\\

0 & 0 & 0 & 1

\end{pmatrix}[/tex]

is the reciprocal of the metric tensor of special relativity.

So I get that I can do that straightforward by taking the dot product [tex]P^\mu P_\mu[/tex] (I don't know why there is a minus sign, but that's wikipedia after all). But how to calculate it from:

[tex] \eta_{\mu\nu} P^\mu P^\nu[/tex]?

Should I contract the given [tex]\eta^{\mu\nu}[/tex] using [tex]g^{\mu\nu}[/tex]? ([tex]eta^{\mu\nu}=g^{\mu \alpha}g^{\nu \beta}\eta_{\alpha \beta}[/tex])?

And how does that act on [tex]P^\mu P^\nu[/tex]? Since the indices should stand for certain [tex]\eta[/tex] and [tex]\nu[/tex] in those tensors, right?

I'm kinda confused as to how did they manage to get the result using that notation :\

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# Tensor notation issue

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