If we let latin alphabets {i,j,k,...} denote the spatial indices, and the greek ones run from 0 to 3, then I've seen the following(adsbygoogle = window.adsbygoogle || []).push({});

[tex]det| g_{\mu \nu} | = g_{00} det | g_{ij} |[/tex]

both in Landau's "Classical theory of fields", as well as ADM's paper on the initial value formulation of GR ("Dynamics of General Relativity", Arnowitt et al.) I don't quite understand it though, or am I reading it right?

Specifically, what about the terms that involve [itex]g_{01}, g_{02}, g_{03}[/itex] when we do the co-factor expansion? Do they somehow vanish by symmetry?

In (2+1) dimensions I got the determinant to be

[tex]det| g_{\mu \nu} | = g_{00} det | g_{ij} | - g_{10}(g_{01} g_{22} - g_{02} g_{21}) + g_{20} (g_{01} g_{12} - g_{11} g_{02})[/tex]

How could the last four terms cancel out?

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# Determinant g = g_{00} det |g_{ij}|

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