Second fundamental form and Mean Curvature

[aCHCa]

Homework Statement

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

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&#039;}, 0, \frac{1}{2}e^{\tilde{C}}\tilde{C&#039;}, sin {θ} cos {θ})

Mean curvature:

h = g^{ij}h_{ij}= \frac{1}{2}\tilde{A&#039;}-\frac{1}{2}\tilde{C&#039;}-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)