L Problem 4. (25 points) In this problem, the spacetime manifold under consider- ation is the same as in problem 3. a) Let A> 0 be constant and assume that r> A. Compute and simplify the derivative of the right hand side to verify the following equation: √r dr = √√r A+ Aln (√- A+ √√r) + constant b) A stick of negligible mass and constant angular and time coordinates is ori- ented radially in the empty space surrounding a star of mass M. The stick has radial coordinates at its endpoints r₁ and r₂ with r₂>r> 2GM/c². Show that the proper length of the stick is L(r2) - L(ri), where, L(r) = p³/2 ( 2GM) 1/2 {p³/² + (r− 2GM) 1/²} 2GM + In 1/2 2 c) Find the limit of L(r2) - L(ri) as GM/c² 0, and give a physical interpre- tation of the result. d) Find the proper circumference of the circle in Schwarzschild spacetime given by t=constant, r=ro and 0=/2, where ro > 2GM/c². e) Find the proper surface area of the sphere S consisting of all spacetime points with fixed radial coordinate ro and fixed time coordinate to. By "proper surface the proper length of the stick is L(r2) L(ri), where, L(r) = r³½² (r 1/2 2GM 2GM 2GM 1/2 ()()() c) Find the limit of L(r2) -L(r) as GM/c0, and give a physical interpre- tation of the result. d) Find the proper circumference of the circle in Schwarzschild spacetime given by t-constant, r = ro and 0 = /2, where ro > 2GM/c². e) Find the proper surface area of the sphere S consisting of all spacetime points with fixed radial coordinate ro and fixed time coordinate to. By "proper surface area" of S, I mean the (two dimensional) hypervolume of the submanifold S as calculated from metric induced on S by the metric determined by the line element -ds² given by, -ds²=-c² 1 2GM c²r 2GM dt²+ cer dr²+r²(do² + sin² Odo²). Display the induced metric on S and the integral expression that you use to calculate the surface area of S. Problem 5. (25 points) Solve problem 8.11 on page 195 of the textbook.