Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 19; Problem P19.9.
Using the Riemann tensor, we can get an idea of the force felt by someone falling into a black hole. Recall the original definition of the Riemann tensor was in terms of the equation of geodesic deviation:
where is the four-vector separating two infinitesimally close geodesics, and is the four-velocity of an object in freefall along one of the geodesics.
For an object, such as a person, that has a large enough size that different parts of the object would, if they weren’t connected, follow different geodesics, a tension force is felt as the various geodesics that pass through different parts of the object diverge during the object’s journey. Suppose our unfortunate person is falling feet first into a black hole (we’ll assume that the person started at rest very far away from the black hole). If we set up a locally inertial frame (LIF) at the person’s centre of mass and align the person’s local axis with the radial direction in the Schwarzschild (S) metric, then we’ve seen that we can write the geodesic deviation as
In the case of the falling person, we can look at the direction, since this is where most of the tidal effects will be felt. In that case, we get, using the person’s LIF as the reference frame:
If we neglect separations in the and directions, this becomes
To get the Riemann component, we can use the symmetry of the tensor:
In the LIF, so this equation becomes
and we worked out the RHS in the last post, so we have
The acceleration of the separation of the two geodesics is then
which we can rewrite with as a function of the acceleration felt at that distance from the black hole:
To see how long it takes the person to fall from this distance to , we can use the formula we derived earlier:
So the time measured by the observer is
To put this in practical terms, a typical person can handle up to about 5g (five times the acceleration due to gravity at the Earth’s surface) along their vertical direction before losing consciousness. If we take (about half a person’s height) and then the time from first experiencing this force to annihilation at the singularity at the centre of the black hole is about
Using Moore’s estimate of the speed of pain impulses (around 1 m/sec) any pain resulting from this probably wouldn’t be felt before the person gets annihilated.