Riemann Curvature Tensor in 2D

  • #1
Woolyabyss
143
1
Homework Statement:
Show that the Riemann curvature tensor in 2d is given by ##R_{abcd} =\frac{R}{2}(g_{ac}g_{bd} - g_{ad}g_{bc}) ##
Relevant Equations:
## R = R_{ab} g^{ab} ## and ## R_{ab} = R_{acb}^{c}##
Since in 2D the riemman curvature tensor has only one independent component, ## R = R_{ab} g^{ab} ## can be reversed to get the riemmann curvature tensor.

Write
## R_{ab} = R g_{ab} ##

Now
## R g_{ab} = R_{acbd} g^{cd}##
Rewrite this as
## R_{acbd} = Rg_{ab} g_{cd} ##
My issue is I'm not sure how they caught a second term? Any help would be appreciated.
 

Answers and Replies

  • #2
robphy
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Now
## R g_{ab} = R_{acbd} g^{cd}##
Rewrite this as
## R_{acbd} = Rg_{ab} g_{cd} ##
My issue is I'm not sure how they caught a second term? Any help would be appreciated.

Your candidate Riemann tensor doesn't have the expected index-symmetries.
Generally, you can't undo a contraction to arrive at your last line.
There may be other expressions that lead to the same contracted term.
 

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