D'Alembert operator in ADM 3+1 formalism

Question

I am trying to write the D'Alembert operator after 3+1 splitting. We have, 

\[\Box=g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}\]

I can decompose as 

\[\Box=(h^{ab}e^{\alpha}_{a}e^{\beta}_{b}-n^{\alpha}n^{\beta})\nabla_\alpha\nabla_\beta\]

where the Greek indices are 4-dim and latin indices are 3-dim. After a bit of algebra I get, 

\(\Box=h^{ab}D_a D_b-h^{ab}(D_a e^\beta_b)\nabla_\beta-n^\alpha n^\beta\nabla_\alpha\nabla_\beta\)

where D is covariant derivative in 3-dim. I wonder if the last two terms are vanishing somehow. Or generally what's the form of D'Alembert operator in ADM formalism. 

Comments

you cannot expect them to vanish since the result one would obtain ostensibly depends on the chosen coordinate system.