Momentum constraint in GR in ADM formalism is written in the form(adsbygoogle = window.adsbygoogle || []).push({});

$$\mathcal M_i=\gamma_{ij}D_k\pi^{kj},~~~~~~~~~~(1a)$$ or equivalently

$$\mathcal M_i=D_k\pi^{k}_i,~~~~~~~~~~(1b)$$ where

##\pi^{ij}=-\gamma^{1/2}\left(K^{ij}-\gamma^{ij}K\right)~##, ##K=\gamma^{ij}K_{ij}~##, ##\gamma=\det \gamma_{ij}~## and ##D_i~## is covariant derivative. This is from DeWitt1967 parer and original ADM parer.

However, those who deal with numerical relativity uses $$\mathcal

M_i=D_jK^j_i-D_iK.~~~~~~~~~~~~~~~(2)$$

What formula is right? (they coincides only if ##\gamma## does not depend on spatial coordinates, which is evidently not the case.

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# A What is right expression for momentum constraint in GR?

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