I have a metric g on spacetime and a spatial metric ##\gamma## such that the components of g can be written in matrix form as(adsbygoogle = window.adsbygoogle || []).push({});

$$ g_ {\alpha, \beta} = \begin{pmatrix} g_{00} & g_{0 j} \\

g_{i 0} & \gamma_{ij} \end{pmatrix} $$

where ##i,j = 1,2,3## and ##\alpha = 0,1,2,3##. Now I want to find a relation between the determinant of ##g## and the determinant of ##\gamma## expressed in terms of the components of ##g##. Using Cramer's rule I get

$$ det(g_{\alpha, \beta}) \equiv g = \frac{C_{00}}{g^{00}}$$

where ##C_{00}## is the cofactor of the (0,0)-element of the matrix above; i.e. it is ##det(\gamma_{ij}) = \gamma##. But in order to find the full relation I will still need to find the (0,0) element of the inverse. What is the best way to go about doing this? Do I really have to solve the full set of equations ##g^{\alpha \lambda}g_{\lambda \beta} = \delta^{\alpha}_{\ \ \ \beta}##? Or is there a better way?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Relation between det(spacetime metric) and det(spatial metric)

Tags:

Loading...

Similar Threads for Relation between spacetime |
---|

I Diffeomorphism invariance and contracted Bianchi identity |

I Conformal Related metrics |

I Difference between vectors and one-forms |

I Difference between diffeomorphism and homeomorphism |

I About extrinsic curvature |

**Physics Forums | Science Articles, Homework Help, Discussion**