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  D'Alembert operator in ADM 3+1 formalism

+ 2 like - 0 dislike

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


I can decompose as 


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. 

asked Jan 10, 2016 in Theoretical Physics by William [ no revision ]

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

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