- #1
cr7einstein
- 87
- 2
Hello everyone!
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!
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!