Register to reply

Einstein Tensor; What is wrong here?

by nobraner
Tags: einstein, tensor
Share this thread:
nobraner
#1
Feb18-12, 08:11 AM
P: 14
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?
Phys.Org News Partner Science news on Phys.org
Scientists develop 'electronic nose' for rapid detection of C. diff infection
Why plants in the office make us more productive
Tesla Motors dealing as states play factory poker
robphy
#2
Feb18-12, 09:02 AM
Sci Advisor
HW Helper
PF Gold
robphy's Avatar
P: 4,136
Quote Quote by nobraner View Post
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]

[snip]

What is wrong here?
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
#3
Feb18-12, 10:58 AM
P: 14
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
#4
Feb18-12, 01:21 PM
elfmotat's Avatar
P: 260
Einstein Tensor; What is wrong here?

Quote Quote by nobraner View Post
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
You're overloading your indices. An index shouldn't appear more than twice in any term.
juanrga
#5
Feb18-12, 02:46 PM
P: 476
Quote Quote by nobraner View Post
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
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]


Register to reply

Related Discussions
Einstein Tensor; super simple derivation; where did I go wrong? Special & General Relativity 8
General Tensor contraction: Trace of Energy-Momentum Tensor (Einstein metric) Special & General Relativity 5
Einstein Tensor... Special & General Relativity 15
Einstein Tensor Special & General Relativity 5
Einstein Tensor Calculus & Beyond Homework 12