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At what distance would it become physically painful to be near a black hole?
Reading about the effects of black holes, its clear one would become stretched, compressed, or just torn to pieces when entering the singularity. But on approach, assuming the transportation could sustain the forces, at what distance would a human start to feel the pain of the force from the black hole?
Ans By pfsico
The answer depends on the mass of the black hole. The larger the black hole, the lower the “tidal force” (that causes the spaghettification) is before you get to the Schwartzschild radius.
The formula for the tensile force felt by a uniform rod (which is not a terrible approximation for a human body, if you just want a rough number) is given in the second sentence here (https://en.wikipedia.org/wiki/Spaghettification#Examples_of_weak_and_strong_tidal_forces), and is
F = (G*M*m*L)/(4*r^3),
where M is the mass of the black hole, m is your mass, L is the length of the rod (your height), and r is the distance from you to the black hole. Solving for r given a known force F we have
r_pain = ( (G*M*m*L)/(4*F) ) ^1/3.
Let’s look at two cases, and figure out what radius corresponds to a painful (I’m going to say) 1000 Newton (about 250 lbs) force, and see where that is relative to the Schwarzschild radius, R_s = (2*G*M/c^2).
Using SI units, and rounding to make things easy, we can use
G = 7×10^-11 kg/m/s^2
c = 3 * 10^8 m/s
F = 1000 N,
L = 2 m,
m = 100kg.
If we plug in all those, and M = 10^6 * 2*10^30 kg for a million solar mass black hole at the center of a galaxy, we find the “radius of pain” is
r_pain = 200 million meters or so.
Meanwhile, the Schwarzschild radius is
R_s = 3 billion meters or so.
That is, the Schwarzschild radius is about 15 times bigger than the “radius of pain”.
Now, let’s consider a one solar-mass black hole, the sort that might be formed from a neutron star, or detected in a merger by LIGO. In that case,
r_pain = 2 million meters = 2000 km or so, and
R_s = 3000 meters or so.
That is, the “radius of pain” almost 1000 times as large as the Schwarzschild radius in this case, so you’ll feel it, and more as you fall further in.
Keep this in mind, and be careful when you’re visiting small black holes in the future.