Equation regarding the Riemann tensor in the Cartan formalism

+ 2 like - 0 dislike
115 views

I have a problem verifying the following equation (in three dimensions)

$$\epsilon_{abc} e^a\wedge R^{bc}=\sqrt{|g|}Rd^3 x$$

where $R$ is the Ricci scalar and $R^{bc}$ is the Ricci curvature

Attempt at a solution:

$$\epsilon_{abc} e^a\wedge R^{bc}=\epsilon_{abc} e_\mu^ae_\alpha^be_\beta^c R^{\alpha\beta}_{\nu\rho} dx^\mu\wedge dx^\nu\wedge dx^\rho$$

Now the idea is that the number of dimensions and the Levi-Civita tensor and the antisymmetry of the three-form forces the set $\{\alpha,\beta\}=\{\nu,\rho\}$. This will give the expression

\begin{align}\epsilon_{abc} e^a\wedge R^{bc}&=\epsilon_{abc} e_0^ae_1^be_2^c R^{12}_{12} dx^0\wedge dx^1\wedge dx^2+\\&\epsilon_{abc} e_0^ae_1^be_2^c R^{12}_{21} dx^0\wedge dx^2\wedge dx^1+\\&\epsilon_{abc} e_0^ae_2^be_1^c R^{21}_{21} dx^0\wedge dx^2\wedge dx^1+\\&\epsilon_{abc} e_0^ae_2^be_1^c R^{21}_{12} dx^0\wedge dx^1\wedge dx^2+({\rm cyclic\,permutations})\end{align}

The problem now is that the Ricci scalar is $R^{12}_{12}+R^{21}_{21}+({\rm cyclic\,permutations})$, so when counting the number of terms I obtain $2\sqrt{|g|}Rd^3 x$ which is wrong by a factor of 2. Can anyone see where I made a mistake?

This post imported from StackExchange Physics at 2015-10-11 18:32 (UTC), posted by SE-user user2133437
I don't know if you are intentionally overstacking the indices on the Riemann tensor, but in case not: $R^{\alpha\beta}{}_{\nu\rho}$ can be hacked R^{\alpha\beta}{}_{\nu\rho} or {R^{\alpha\beta}}_{\nu\rho}. The former risks breaking across line wraps, and the latter is semantically wrong, but they work in a pinch.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysi$\varnothing$sOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.