The Schwarzschild Spacetime metric is is given by :
$\text ds^2=-\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$
I am currently studying doing an extra project for an introductory GR course, I was reading the book by Eric Poisson "A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics".
And I am trying to solve exercise 1 of Chapter 3.
It asks us to consider a hypersurface $T = constant$ in Schwarzschild Spacetime, where $$T = t + 4M \Bigg[ \sqrt{\frac{r}{2M}} + {\frac{1}{2}\ln\Bigg( \frac{\sqrt{\frac{r}{2M}}-1}{\sqrt{\frac{r}{2M}}+1} \Bigg) } \Bigg] .$$ Use as (r,$\theta$,$\phi$) as coordinates on the hypersurface.
a) Calculate the unit normal $n_{\alpha}$ and find the parametric equations describing the hypersurface.
I am having problem starting as I want to use the equation defined in the book as 3.1.4, which is
$n_{\alpha} = \frac{\xi\Phi_{;\alpha}}{\mid g^{\mu\nu}\Phi_{;\mu} \Phi_{;\nu} \mid^{\frac{1}{2}} }$
but I am not sure if my hypersurface, here denoted by $\xi$ is timelike or spacelike. I have read the introduction but I am not understanding how could I compute it.
Please note $\xi$ is given in eq. 3.1.3 as
$n^{\alpha}n_{\alpha} =\xi = \Bigg\{ \pm $ 1 if $\sum$ (the hypersurface is timelike (+1) or spacelike (-1))
EDIT : I have determine the unit normal and determined the hypersurface to be spacelike.
I am struggling now to determine the parametric equations. Can I use $x = r\sin\theta\cos\phi$, $y = r\sin\theta\sin\phi$ and $z =r\cos\theta$ ? I am not sure how can I use the normal vector though to calculate the tangent vectors