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Even though I have done substantial tensor calculus, I still don't get one thing. Probably I am being naive or even stupid here, but consider

$$R_{\mu\nu} = 0$$.

If I expand the Ricci tensor, I get

$$g^{\sigma\rho} R_{\sigma\mu\rho\nu} = 0$$.

Which, in normal algebra, should imply,

$$ g^{\sigma\rho} = 0$$ (which is meaningless) or $$R_{\sigma\mu\rho\nu} = 0$$ ( which isn't always true).

So, Why can't we do normal algebra here?( it is perfectly valid step in algebra)

Also, consider a simple case

$$dS^2 = g_{\mu\nu}dx^{\mu}dx^{\nu}$$.

Here, why can't we simply transpose(or divide both sides by) the differentials on RHS, i.e.,

$$\frac{dS^2}{dx^{\mu}dx^{\nu}} = g_{\mu\nu}$$ ???

Why is this expression not valid? Or, another example, Why can't

$$R_{\mu\nu} = g^{\sigma\rho} R_{\sigma\mu\rho\nu}$$ imply that

$$g^{\sigma\rho} = \frac{R_{\mu\nu}}{R_{\sigma\mu\rho\nu}}$$ ??

Is there a reason why this is wrong? Or is there a different way to transpose tensors from one side of the equation to the other side? Can you do this to vacuum field equations(as an example)?

Thanks in advance!!

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# Why can we not do algebraic methods like transposing with tensors

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