Concrete computation of the extrinsic curvature and its trace Consider the following metric:
$$\mathrm{d} s^{2}=-\left(1-\frac{2 m}{r}\right) \mathrm{d} t^{2}+\left(1+\frac{2 m}{r}\right)\left(\mathrm{d} r^{2}+r^{2} \mathrm{~d} \Omega^{2}\right)-\frac{4 j \sin ^{2} \theta}{r} \mathrm{~d} t \mathrm{~d} \phi.$$
I am trying to compute the extrinsic curvature of hypersurfaces of constant $t$. The coordinates on the spacetime manifold are $x^{\alpha} = (t,r, \theta, \phi)$, and take the coordinates on the hypersurface to be $y^{a} = (r, \theta, \phi)$.
The normal to the hypersurfaces is
$n_{\alpha} = -(1-m / r) \partial_{\alpha} t$. The extrinsic curvature is give by (after the pull-back):
$K_{ab} = n_{\alpha ; \beta} e_{a}^{\alpha} e_{b}^{\beta}$, where $e_{a}^{\alpha} = \frac{\partial x^{\alpha}}{\partial x^{a}}$.
When I try and go through the computation I keep getting zero, since the only non-zero component of $n_{\alpha}$ is $n_{t}$ but $e_{a}^{t} = 0$. I am not sure where I am going wrong, any help or insight is very much appreciated.
 A: The problem here is with assuming that because $n_{\alpha}\sim \delta_{t\alpha}$, then the $n_{\alpha;\beta}$ will also vanish for $\alpha \neq t$. This is not true. Even though a field might have only some component, it does not mean that its covariant derivative does not have any other components.
Let us see, what hides behind the $n_{\alpha;\beta}$.
In semicolon notation, we have to be careful and remember that we first take the covariant derivative of the tensor field in the $\frac{\partial}{\partial x^{\beta}}$ direction, and only then evaluate its $\alpha$-th component.
Therefore:
$$n_{\alpha;\beta}\equiv(\nabla_{\beta}n)_{\alpha}$$
where the parentheses serve to indicate the order of operations. It's analogous to how we first differentiate a function and then take its value at a point. In the opposite order, we'd always get zero, like you're getting here!
I encourage you to try and calculate the covariant derivative for
$\beta=(t,r,\theta,\varphi)$ like this:
$$\nabla_{\beta}n = \Big{[} \frac{\partial n_{\alpha}}{\partial x^{\beta}} - \Gamma_{\beta \alpha}^{\lambda}n_{\lambda}  \Big{]}dx^{\alpha}.$$
And then read off the relevant entries $(\nabla_{\beta}n)_{\alpha}$ of the one-form field $\nabla_{\beta}n$ .
Alternatively, since you already have $n$, we can do it like this:
$$ \nabla_{\beta}n = \nabla_{\beta}(n_{t}dt) = \nabla_{\beta}(n_{t})dt+ n_{t}\nabla_{\beta}dt = \partial_{\beta}n_{t}dt - n_{t}\Gamma_{\beta \lambda}^{t}dx^{\lambda} $$
where we used, accordingly, the Leibniz rule and the fact that on functions, the covariant derivative reduces to a partial derivative, and we know its actions on coordinate one-forms and coordinate basis vector fields.
Second problem I see is that in the question's body, you normalized incorrectly the one-form $n$. You should use the $g^{tt}$ that's given below.
For $n_{\alpha}$, I am getting:
$$ n = -\frac{\sqrt{2m+r}}{(2mr+r^{2})\sqrt{r}}\sqrt{4j^{2}\sin(\theta)^{2} +r^{4} -4m^{2}r^{2}} \;dt$$
After an edit, the answer below can be now treated as illustrating an alternative way of calculating the extrinsic curvature tensor:
I allowed myself the liberty to run calculations partly through SageMath.
$$g_{\mu\nu} = \begin{bmatrix}
    -(1-\frac{2M}{r})       & 0 & 0 & \frac{-2j\sin^{2}\theta}{r} \\
    0       & 1+\frac{2M}{r} & 0  & 0 \\
    0       & 0 & r^{2}(1+\frac{2M}{r}) & 0\\
 \frac{-2j\sin^{2}\theta}{r}       & 0& 0& r^{2}(1+\frac{2M}{r})\sin^{2}\theta
\end{bmatrix}
$$
Inverting:
$$g^{\mu\nu} = \begin{bmatrix}
    \frac{r^{3}(2M+r)}{4M^{2}r^{2}-r^{4}-4j^{2}\sin^{2}\theta}   & 0 & 0 & \frac{2jr}{4M^{2}r^{2}-r^{4}-4j^{2}\sin^{2}\theta} \\
    0       & \frac{r}{2M+r} & 0  & 0 \\
    0       & 0 & \frac{1}{2Mr+r^{2}} & 0\\
\frac{2jr}{4M^{2}r^{2}-r^{4}-4j^{2}\sin^{2}\theta}       & 0& 0& - \frac{r(2M-r)\,\sin\theta^{-2}}{4M^{2}r^{2}-r^{4} -4j^{2}\sin^{2}\theta}
\end{bmatrix}
$$
You can calculate the inverse easily if you treat the $t-\varphi$ block of the metric as 2-by-2 and calculate its inverse separately, and the other block to be $r-\theta$ and obtain the corresponding components of the inverse metric simply by raising the entries of that block to the power of $-1$.
Now you should calculate $n_{\mu}= At_{,\mu}$, normalize it: $g^{\mu\nu}n_{\mu}n_{\nu}=-1$ to get the scaling factor $A$ and finally, raise the index to get the normal vector field instead of the normal form.
You should get that the normal vector field has two nonvanishing components - in the $t$ and $\varphi$ coordinates.
$$ n = n^{t}\partial_{t} + n^{\varphi}\partial_{\varphi}$$
Then you can use the formula:
$$ K_{ij}= -\frac{1}{2}(\mathcal{L}_{n}\gamma)_{ij},$$
where $\mathcal{L}$ is the Lie derivative and $\gamma_{ij}$ is the induced metric on a $t=$const slice.
