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

[wandering.the.cosmos]

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

$$det| g_{\mu \nu} | = g_{00} det | g_{ij} |$$

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 $g_{01}, g_{02}, g_{03}$ when we do the co-factor expansion? Do they somehow vanish by symmetry?

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

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

How could the last four terms cancel out?

 

[pervect]

$g_{\mu\nu}$ can be any symmetric matrix.

If we try for an example

$$\left[ \begin {array}{cccc} m&1&2&3\\\noalign{\medskip}1&4&5&6\\\noalign{\medskip}2&5&7&8\\\noalign{\medskip}3&6&8&9\end {array} \right]$$

we find the determinant is -m-2, which is not a multiple of m.

So I don't think your statement can be correct as written. Start reading the fine print :-).

(Is the ADM paper online anywhere?)

 

[wandering.the.cosmos]

It is equation 3.12 of "The Dynamics of General Relativity", Arnowitt, Deser and Misner:

http://arxiv.org/abs/gr-qc/0405109

Perhaps someone could explain to me if I am mis-reading it.

 

[HallsofIvy]

That is correct assuming that the metric tensor is of the form
$$\left[ \begin {array}{cccc} g_{00}&0&0&0\\\noalign{\medskip}0&g_{11}&g_{12}&g_{13}\\\noalign{\medskip}0&g_{21}&g_{22}&g{23}\\\noalign{\medskip}0&g_{31}&g_{32}&g_{33}\end {array} \right]$$, the "three plus one dimensional decomposition of the Einstein field" referred to in the text.

 

[George Jones]

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

$$det| g_{\mu \nu} | = g_{00} det | g_{ij} |$$

both in Landau's "Classical theory of fields",

When a metric is static, coordinates can be chosen such that the components of the metric take the form given by HallsofIvy.

as well as ADM's paper on the initial value formulation of GR ("Dynamics of General Relativity", Arnowitt et al.)

I don't think equations (3.10) and (3.12) of the ADM paper give this.

 

[wandering.the.cosmos]

I have indeed mis-read the ADM paper. What it says is

$$g^{00} = \frac{\textrm{det} |g_{ij}|}{\textrm{det} |g_{\mu \nu}|}$$

where $\textrm{det} |g_{ij}|$ is the determinant of the space-space part of the metric and $\textrm{det} |g_{\mu \nu}|$ is the determinant of the whole metric tensor.

This follows from Cramer's rule.