Question

Given the Schwarzschild radius defined by
R_s = (2GM)/ c²

we can also define a “surface gravity” κ, which is the acceleration (from gravity) an imaginary
mass would feel if it were held at the Schwarzschild radius. Calculate κ using classical
Newtonian gravity.
(a) How does κ depend on the total black hole mass M?
(b) What is the value of κ for a black hole with a mass of 40 x 109 M_@ (this is the most
massive black hole measured, where M@ is the mass of our sun)

Answer 1

According to the classical Newtonian gravity, the surface gravity at the Schwarzschild radius, is given by

where, M is the mass of the black hole and G is the Newton constant.
And the Schwrazschild radius is given by
  
And so, we get the surface gravity as

  
Thus, we see that the surface gravity depends upon the mass M of the black hole inversely, i.e.,

(b)
Given the mass of the black hole and using the values of the standard constants
  
we get the magnitude of the surface gravity as