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-   -   Quick question to clear up some confusion on Riemann tensor and contraction (http://www.physicsforums.com/showthread.php?t=495402)

Deadstar May2-11 03:43 PM

Quick question to clear up some confusion on Riemann tensor and contraction
 
Let's say I want to calculate the Ricci tensor, [tex]R_{bd}[/tex], in terms of the contractions of the Riemann tensor, [tex]{R^a}_{bcd}[/tex]. There are two definitions of the Riemann tensor I have, one where the [tex]a[/tex] is lowered and one where it is not, as above.

To change between the two all that I have ever seen written is 'we lower the indices' but I don't think I fully understand this. Does this mean...

[tex]R_{abcd} = g_{aa} {R^a}_{bcd}[/tex]

So the answer to my original question of finding the Ricci tensor is...

[tex]R_{bd} = g^{ac} g_{aa} {R^a}_{bcd}[/tex]

Following this, I also have written before me that...

[tex]{R^b}_{bcd} = 0[/tex] since [tex]R_{abcd}[/tex] is symmetric on a and b. Shouldn't this be antisymmetric on a and b?

Sorry if these are basic questions but I'm finding the vagueness of 'lowering the indices' a bit confusing...

Cheers.

DrGreg May2-11 05:25 PM

Re: Quick question to clear up some confusion on Riemann tensor and contraction
 
Quote:

Quote by Deadstar (Post 3279651)
[tex]R_{abcd} = g_{aa} {R^a}_{bcd}[/tex]

You can't have the same index appearing 3 times in an expression, what you mean is
[tex]R_{abcd} = g_{ae} {R^e}_{bcd}[/tex]

jfy4 May2-11 06:33 PM

Re: Quick question to clear up some confusion on Riemann tensor and contraction
 
The Riemann tensor

[tex]R^{\alpha}_{\;\beta\gamma\delta}=g^{\alpha\lambda}R_{\lambda\beta\gamma \delta}[/tex].

The raising and lowing of indices is done using the fundamental metric tensor [itex]g_{\alpha\beta}[/itex] and the inverse metric [itex]g^{\alpha\beta}[/itex].

The Ricci tensor

[tex]R^{\alpha}_{\;\beta\gamma\delta}\rightarrow R^{\alpha}_{\;\beta\alpha\delta}=R_{\beta\delta}[/tex]

is a contraction of the first and third indices.

robphy May2-11 08:57 PM

Re: Quick question to clear up some confusion on Riemann tensor and contraction
 
I would argue that [tex]
R^{\alpha}_{\;\beta\gamma\delta}
[/tex]
is the more fundamental expression for the Riemann tensor
since this is defined from the derivative operator [without reference to a metric].
Its trace yields the Ricci tensor [as jfy4 wrote]... and this too doesn't make use of a metric.

To lower the upper index of Riemann or to get the scalar-curvature from Ricci now requires a metric.


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