# Can I write the following eq. (2) with the metric tensor?

I'm a beginner at relativity, I have a question about eq 2 if its true. I know that the interval can be calculated like this in special relativity: $$ds^2 = \eta_{\mu\nu}dx^\mu dx^\nu \tag{1}$$ where $$\eta_{\mu\nu}$$ is the metric tensor (for example the conventional $$\text{diag}(-1,+1,+1,+1)$$ in Cartesian coordinate system). But what about with the upper index version: $$ds^2 = \eta^{\mu\nu}dx_\mu dx_\nu, \tag{2}$$ of course the units are $$c=1$$, $$\hbar=1$$, $$G=1$$. The covariant components we can get from this equation: $$dx_\mu=\eta_{\mu\nu}dx^\nu, \tag{3}$$ also between the upper and down index metric tensors we can write the following: $$\eta_{\mu\alpha}\cdot\eta^{\alpha\nu}=\delta^\nu_\mu, \tag{4}$$ or more simply: $$\eta^{\mu\nu}=(\eta_{\mu\nu})^{-1}. \tag{5}$$ Eq 2 seems a bit weird for me, I'm not sure about it. If we write everything out in eq 2 using eq 3, we get the following: $$ds^2 = \eta^{\mu\nu}\eta_{\mu\alpha}dx^\alpha \eta_{\nu\beta}dx^\beta. \tag{6}$$ Where we can notate $$\eta_{\mu\alpha}dx^\alpha=dx_\mu$$ and $$\eta_{\nu\beta}dx^\beta=dx_\nu$$. Is this right?

• Your eq. 4 is incorrect. The left hand side would sum over both indices leaving 4 on the right hand side. I think you want $\eta_{\mu\alpha} \eta^{\alpha\nu} = {\delta_\mu}^\nu$. – Paul T. May 30 '20 at 19:09
• yes, I'll correct that one, thank you – waerx May 30 '20 at 19:18

• Thank you, so as I understand I can write the following: $$ds^2=\eta^{\mu\nu}\eta_{\mu\alpha}dx^\alpha\eta_{\nu\beta}dx^\beta=\delta^\nu_\alpha dx^\alpha\eta_{\nu\beta}dx^\beta=\eta_{\alpha\beta}dx^\alpha dx^\beta$$ where at last step I used the Kronecker delta index property. – waerx May 30 '20 at 19:38
It is right, but I probably would have written (5) $$\eta^{\mu\nu}={(\eta^{-1}})_{\mu\nu}$$ In practice, you wouldn't need to write this because it is clearly true and you never have to write $$\eta^{-1}$$.
It helps to think of $$\eta^{\mu\nu}$$ and $$\eta_{\mu\nu}$$ as index raising and index lowering operators. This idea carries forward to $$g^{\mu\nu}$$ and $$g_{\mu\nu}$$ in curved spacetimes and makes the relationship between contravariant and covariant tensors quite easy and natural to work with.