4. In the movie Interstellar, the crew visits planets in orbit around a supermassive black hole named Gargantua. Gargantua has a mass of 1 billion Me. At the rst planet, the time dilation is so strong that only one hour passes on the planet compared to 7 years for an observer far away. Hint: Be careful with precision here. Throughout this problem you will nd cases where you need to use a Taylor Expansion and compare how your results differ from 1 in order to not lose small numbers. [10 pts] (a) What is the Schwarzschild radius ( event horizon) of Gargantua? (1)) Given the gravitational time dilation at the rst planet, how far away must it be from the event horizon of Gargantuaf2 (c) What velocity would be required to leave the planet near Gargantua? How much smaller than the speed of light is this? Hint: Consider the fact that the time dilation is a measurement of the depth of the gravitational potential well