Place an observer in spacetime such that the feet of the observer touches the Schwarzschild radius and the head is $x$ meters away from the Schwarzschild radius.
I want to calculate the relative acceleration between the head and the feet of the observer in Schwarzschild geometry. Hence with the metric
$$ds^2=-\left(1-\frac{r_s}{r}\right)dt^2+\left(1-\frac{r_s}{r}\right)^{-1}dr^2+r^2d\theta+r^2\sin^2(\theta)d\phi,$$
where $r_S=2M$ the Schwarzschild radius and $M$ is the mass of the black hole.
At first I calculated the acceleration. My result, which when I compare it with different sources in the literature seems to be correct: $$a=\frac{M}{r^2\sqrt{1-\frac{r_S}{r}}}.$$
For $r \gg r_S$ I'm getting the the same acceleration, I'd with Newtons law. But my problem is that I can't evaluate the acceleration at $r=r_S$, i.e. I can't calculate the acceleration from the observers feet.
Is there another way how to approach this exercise? I know I could avoid the singularity of the metric by transforming the Schwarzschild coordinates to Eddington Finkelstein coordinates, but it seems like more work then necessary. Also in the hint for this exercise that I can even use the Newton approximation but if so, I have to argue why I'm allowed to do that. This is not clear to me. As I've written above, the expression of the acceleration matches the Newtonian acceleration for $r\gg r_S$.
Can someone help me with this exercise?
