It is well known that if a black hole fulfills certain conditions, it's tidal forces will rip incoming objects apart even before they cross the event horizon. So at some point, the curvature of space seems to be so intense that your feet are attracted by orders of magnitude higher than your head, leading to your painful death.
I was curious about the analogous effect in the temporal direction, that is how high will the gravitational time dilation between your head and feet get at some point before reaching the event horizon. Using the Schwarzschild-metric and the geodesic equation, I can try to answer this question and ultimately want to plug in some numerical values for a person who is 1.90m tall just for fun. However, I doubt that I interpret the radius coordinate in the right way, and would like to know your opinion on how to best approach this problem. I will show you my approach first:
From the geodesic equation of a radial free fall into a schwarzschild black hole follow the two equations:
- $ \frac{dt}{d\tau}(1-\frac{r_s}r)=F$
- $(\frac{dr}{d\tau})^2+(1-\frac{r_s}r)=F²$
Where I set c=1, F is a constant of integration and other equations were eliminated because of $d\phi=d\theta=0$ as well as $\theta=\frac{\pi}2$
$dt$ is the time interval measured by an observer who is "infinitely" far away. $d\tau$ is the time interval measured by the person falling inwards. I could now write down equation 1) two times for my head and feet. Where r is the radial coordinate of my feet, and $r_{head}$ the one for my head So:
$ \frac{dt}{d\tau_{head}}(1-\frac{r_s}{r_{head}})=F$
$ \frac{dt}{d\tau_{feet}}(1-\frac{r_s}{r})=F$
I could now eliminate $dt$ and F, thus obtaining:
$\frac{d\tau_{head}}{d\tau_{feet}}=\frac{1-\frac{r_s}{r_{head}}}{1-\frac{r_s}{r}}$
Now comes my question:
In flat spacetime, I could say that $r_{head}=r+1.90$. But as has been discussed in many threads, because of the curved spacetime the radial coordinate in the Schwarzschild-metric cannot be interpreted as the physical distance from the center/singularity, which means I shouldn't be able to use this simple expression. And because I'm particularly interested in the extreme regime of a black hole, it should deviate extremly.
When trying to obtain some actual numerical values from problems regarding black holes, how can I properly deal with the radius coordinate, specifically in this example?
Of course it might be that my approach written above is just wrong. I could maybe simply look at the line element, set $dr=d\phi=d\theta=0$ and then write down the resulting relationship between $dt$ and $d\tau$ two times, for head and feet in the same way as above. In any case the question on how to deal with the radial coordinate is still relevant.