ASTR 498 Problem Set 3 Due Thursday, March 13 1. 4 points Suppose that matter in a circular orbit at the ISCO around a nonrotating black hole spirals into the black hole without any further loss of energy or angular momentum. Derive the angular velocity > of the matter as seen at inflnity, as a function of the radius r of the matter, where r < 6M as the matter falls. Check in particular what the angular velocity is when r = 2M, i.e., when the matter is just about to enter the event horizon. Discuss whether the prediction of your expression make sense in this case. 2. 4 points Is the surface of a cylinder at or curved, in a geometric sense? To determine this, consider the outside of the cylinder (not the at caps), and think about two initially parallel geodesics. Either prove that they always remain parallel regardless of their initial angle to the axis of the cylinder, or exhibit a speciflc case in which they converge or diverge. Hint: set up a coordinate system on the surface (only the two spa...