$2.$ As given in class, the time dilation effect in a gravitational field is given by$,$

r $2$GM

$$tg$=$tf $1$ rc$2(1)$

a$)$ Write this equation in terms of the Schwarzschild radius.

b$)$ Assuming that the Earth has collapsed to a black hole where all of its mass is concentrated at a point, at what distance from the center of the black hole is the passage of time $50\%$ that of a clock in free space?

c$)$ A space alien thought it would be fun for the residents of Earth to have a black hole within our solar system and compresses the mass of Jupiter to fit within its Schwarzschild radius. What distance from the center of the black hole is the passage of time $99\mathrm{.}99\%$ that of a clock in free space? Do you think we have to worry if Jupiter turns into a black hole?