SOLVED Physics 396 Homework Set 8

1. In geometrized units, the Schwarzschild line element takes the form ds2 = ✓ 1 2M r ◆ dt2 + ✓ 1 2M r ◆1 dr2 + r2 (d✓2 + sin2 ✓d2 ), (1) where M is the total mass of the central object. Consider two massless shells concentric with the central object. The inner shell has a circumference of 6⇡M and the outer shell has a circumference of 20⇡M. a) Calculate the physical radial distance between these two shells. b) Calculate the spatial volume between the two spherical shells. 2. In class, we arrived at an equation of motion describing the radial coordinate of a freelyfalling particle about a spherically-symmetric massive object of the form E = 1 2 ✓dr d⌧ ◆2 + Vef f (r), (2) where M is the mass of the central object, E ⌘ (e2 1)/2, and Vef f (r) = M r + `2 2r2 M`2 r3 . (3) a) Setting r ⌘ Mx and ` ⌘ M ˜`, show that Eq. (3) can be written in the form Vef f (x) = 1 x + ˜`2 2x2 ˜`2 x3 . (4) b) Using your favorite plotting software (i.e. Maple, Matlab, Wolframalpha, etc.), plot Vef f (x) vs x for ˜` = 3, ˜` = p12, ˜` = 4, and ˜` = 4.5. Your x-axis should have a range of 2