The concept of a black hole, that is, an object whose gravity is so strong that not even light, traveling at a speed of 3 x 10⁸ m/s, can escape from it, dates back to at least the 18th century and the work of Pierre-Simon Laplace, who referred to them as “dark stars.” From Newtonian mechanics, the escape velocity (see Section 3.5) from a spherical mass is given by the equation 

v_e = (2GM/R)^1/2

where M and R are the mass and radius of the body, respectively. G is the gravitational constant, 6.67 x 10^-11 N-m²/kg².

(a) Compute the escape velocity from the surface of the Earth, and confirm the result quoted at the end of Section 3.5, viz., v_e ∼ 11,200 m/s.

(b) Ignoring for the moment that the physics of strong gravity in the vicinity of a black hole requires general relativity and not Newtonian mechanics for its proper understanding, use this relationship to show that the radius of a spherical object of mass M whose escape velocity is the speed of light must be equal to

R = 2GM/c²

Notice that this is precisely the same equation as that for Schwarzschild radius, d_S, introduced in Section 12.2 in connection with the time dilation formula. Indeed, the Schwarzschild radius is the distance from a nonrotating spherical mass at which the escape velocity reaches the speed of light and thus may be taken to be the boundary of Laplace’s dark star and our black hole. This is one example where the predictions of Newtonian physics happen to match those of general relativity.

(c) Find the Schwarzschild radius of Earth and then calculate how large the density Earth would have to be for it to become a black hole. Compare your result to the current density of Earth, approximately 5514 kg/m³.