Einstein Tensor; What is wrong here?

nobraner

Start with

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]

Insert

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\nabla^{\mu}[/itex][itex]\frac{g_{\mu\nu}g^{\mu\nu}}{4}[/itex]R[itex]_{\mu\nu}[/itex]

Contract the Ricci Tensor

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex] = [itex]\nabla^{\mu}[/itex][itex]\frac{g_{\mu\nu}}{4}[/itex]R

Thus

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\frac{1}{4}[/itex][itex]\nabla^{\mu}[/itex][itex]{g_{\mu\nu}}[/itex]R

But General Relativity says

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\frac{1}{2}[/itex][itex]\nabla^{\mu}[/itex][itex]{g_{\mu\nu}}[/itex]R

What is wrong here?

robphy

Your inserted factor should be
[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\nabla^{\mu}[/itex][itex]\frac{g_{\sigma\tau}g^{\sigma\tau}}{4}[/itex]R[itex]_{\mu\nu}[/itex] (since [itex]\mu[/itex] and [itex]\nu[/itex] are already "taken").


Note that since your proposed proof makes no use of the unique properties of Ricci, it would seem that your result would work for any symmetric tensor. So, you must look at it with suspicion.

nobraner

I don't understand your declaration that [itex]\mu \nu[/itex] are already taken. Does that mean we can never assume that such a metric as

g[itex]^{\mu\nu}[/itex]

exists with out first proving that it is so for the specific case of

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\frac{1}{4}[/itex][itex]\nabla^{\mu}[/itex]g[itex]_{\mu\nu}[/itex]R

elfmotat

You're overloading your indices. An index shouldn't appear more than twice in any term.

juanrga

There is a double sum in the term that you inserted. Therefore you cannot contract the Ricci leaving out the g[itex]_{\mu\nu}[/itex]