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Phys 4147

Homework Set 9

Problem 1 (10 points)

Use your favorite software package to reproduce the Figure 9.3 (page 196) of the textbook. Specifically, plot the effective potential Veffprq for different values of ℓ{M, as shown in the Figure. Your graph should show the following (each worth 3 points):

• Correct shapes of the curves.

• The range of axes should be chosen so that the key features of the curves are enclosed in the graph window.

• Axes should be marked with the appropriate titles.

• Axes should be marked with the appropriate units.

• Axes should be marked with the correct numerical values.

Include the PDF printout of your script (please do not sent Mathematica and similar notebooks) with the problem (5 points).

Problem 2 (10 points)

An advanced civilization living outside a spherical neutron star of mass M constructs a massless shell concentric with the star such that the area of the inner surface is 144πM2 and the area of the outer surface is 400πM2 . What is the physical thickness of the shell?

Problem 3 (10 points)

An observer is stationed at fixed radius R in the Schwarzschild geometry produced by a spherical star of mass M. A proton moving radially outward from the star traverses the observer’s laboratory. Its energy E and momentum |P| are measured. What are the components of the four-momentum of the proton in the Schwarzschild coordinate basis in terms of E and P?

Problem 4 (10 points)

Two particles fall radially in from infinity in the Schwarzschild geometry. One starts with e = 1, and the other starts with e = 2. A stationary observer at r “ 6M measures the speed of each when they pass by. How much faster is the second particle moving at that point?