The radius RH of a black hole, also known as the event horizon, marks the location where not even light itself can escape from the black hole. That is, no information about the interior of the black hole may escape to any observer located outside of the black hole. According to general relativity, RH = (2GM / c^2), where M is the mass of the black hole and c is the speed of light. you want to study a black hole by getting near it ith a radial distance of 50 RH. However, you don’t want the difference in gravitational acceleration between your head and your feet to exceed 10 m/s^2.
a) As a multiple of the Sun’s mass, approximate what is the limit to the mass of the black hole you can tolerate at the given radial distance.
b) Is the limit an upper limit(you can tolerate smaller masses) or a lower limit (you can tolerate larger masses)?
Solution:
a) We know acceleration due to gravity, g = GM/r²
Differential change, dg/dr = -2GM/r³
Here, r = 50*Rh = 50*2GM/c² = 100GM/c
²
My height, h=dr = 1.7 m
Difference in gravitational acceleration between my head and my feet, dg = -10 m/s²
or, dg/dr = -10/1.7 = -2GM/(100GM/c²)³
or, 5.9*100³*G²*M² = 2c⁶
or, M = 0.59*c³/(1000G) = 2.39*1032 kg = [(2.39*1032)/(1.99*1030 )]Ms = 120*Ms
Mass of black hole which we can tolerate at the given distance is 120 time the mass of Sun.
b) This limit an upper limit ,we can tolerate smaller masses only.