Second fundamental form and Mean Curvature

[aCHCa]

Homework Statement

Metric ansatz:
$ds^{2} = e^{\tilde{A}(\tilde{\tau})} d\tilde{t} - d\tilde{r} - e^{\tilde{C}(\tilde{\tau})} dΩ$

where: $d\tilde{r} = e^{\frac{B}{2}} dr$

Homework Equations

How to calculate second fundamental form and mean curvature from this metric?

The Attempt at a Solution

Metric tensor:

$g_{00}= e^{\tilde{A}(\tilde{\tau})}$
$g_{11}= -1$
$g_{22}= - e^{\tilde{C}(\tilde{\tau})}$
$g_{33}- e^{\tilde{C}(\tilde{\tau})} sin^2 θ$

Second fundamental form:

$h_{ij}= g_{kl} \Gamma^{k}_{ij} n^{l}$

where:

$i, j, k, l = 0, 1, 2, 3$

$n^{l} =$normal vector $= (0, 1, 0, 0)$

so:

$n^{0} = 0$
$n^{1} = 1$
$n^{2} = 0$
$n^{3} = 0$

Second fundamental form:

$h_{ij}= diag (\frac{1}{2}e^{\tilde{A}}\tilde{A'}, 0, \frac{1}{2}e^{\tilde{C}}\tilde{C'}, sin {θ} cos {θ})$

Mean curvature:

$h = g^{ij}h_{ij}= \frac{1}{2}\tilde{A'}-\frac{1}{2}\tilde{C'}-e^{-\tilde{C}}cot {θ}$

 

[clamtrox]

Your idea seems correct, but I don't understand what you're doing. Is there any r-dependence in the metric? If not, then all the relevant Christoffel symbols vanish and the extrinsic curvature should be zero (all r=const hyperslices are flat)