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In "Quantum Gravity in 2+1 dimension" by S Carlip, Sec 3.1 (where the metric of a spacetime with a point source is derived, using the ADM formalism), equation 3.8 states that (this is the momentum constraint for a metric on a constant surface),

$$ \nabla_{j} \Pi^{ij} = 0 = g^{il}\partial_{k} \Pi^{k}_{l} - (1/2)g^{il}(\partial_{l}g_{jk})\Pi^{jk}. $$

There should be two terms with Christoffel connections from the covariant derivative, however this has only one. I am getting another term which I find to be non-zero. Can somebody please tell me how the other term vanishes?

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1 Answer 1

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Recall that momentum $\pi^{ij}$ is a tensor density in 2-space, cf. e.g. Ref. 2. In other words, $$\frac{\pi^{ij}}{\sqrt{\det g}}$$ is a symmetric (2,0) tensor. Therefore the covariant derivative is

$$(\nabla_{\ell}\pi)^{ij}~=~ \partial_{\ell} \pi^{ij} + \Gamma^i_{\ell k} \pi^{kj} + \pi^{ik}\Gamma^j_{\ell k} -\pi^{ij} \partial_{\ell}\ln\sqrt{\det g}.$$

The contraction becomes

$$(\nabla_j\pi)^{ij}~=~ \partial_j \pi^{ij} + \Gamma^i_{jk} \pi^{kj} + \pi^{ik}\underbrace{\left(\Gamma^j_{jk} - \partial_k\ln\sqrt{\det g}\right)}_{=0}$$ $$~=~\partial_j \pi^{ij} + g^{i\ell} \left(\partial_{j} g_{\ell k}-\frac{1}{2}\partial_{\ell} g_{jk}\right)\pi^{kj} $$ $$~=~g^{i\ell} \left( \partial_{j} \left(g_{\ell k}\pi^{kj}\right) -\frac{1}{2}\pi^{kj}\partial_{\ell} g_{jk}\right), \tag{3.8} $$ in agreement with eq. (3.8) in Ref. 1.

References:

  1. S. Carlip, Quantum Gravity in 2+1 Dimensions, Cambridge University Press, 1998, Section 3.1, p. 39.

  2. C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation, 1973, Section 21.7, p. 521.

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  • $\begingroup$ Thanks a lot! I was treating $\pi^{ij}$ as a tensor instead of a tensor density and so I was getting wrong results. $\endgroup$
    – Sourav
    Commented Mar 23, 2014 at 19:25

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