Physics 725: Relativity
Thornton 428, San Francisco State University
Fall 2014 MWF 9:10AM
Homework 1 Due 9:10AM 9/8
While I may have consulted with other students in the class regarding this homework, the solutions presented here are my
own work. I understand that to get full credit, I have to show all the steps necessary to arrive at the answer, and unless it
is obvious, explain my reasoning using diagrams and/or complete sentences. Please staple this signed sheet to your
homework.
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1. Hartle 2.2. Just to be clear, Gauss’s experiment is being conducted on Earth in both cases. Repeat the problem, now
considering the Andromeda galaxy instead of the Sun. You may use Wikipedia to find any missing numbers.
2. Hartle 2.3.
3. Hartle 2.4.
4. Hartle 2.5.
5. Hartle 2.7.
6. Consider a cylinder that is infinite in the z direction and has radius R in the x-y plane. Show that the surface of this
cylinder has a flat, two-dimensional, Euclidean geometry.
7. As human beings, we can only comprehend curved 2D surfaces when they are “embedded” in a 3rd dimension, such
as the peanut in Figure 2.7
Imagine line element 2.21 with arbitrary f (θ). A sane requirement for embedding is that distances calculated using
the 2D line element 2.21 should be the same as distances measured along the embedded surface. Use this
requirement to work out arbitrary x(θ, φ), y (θ, φ), and z(θ, φ) for embedding line element 2.21 in terms of generic
f (θ).
Use the above result to calculate x(θ, φ), y (θ, φ), and z(θ, φ) for (a) f (θ) = sin θ (a sphere, sanity check); (b) equation
2.24 (the peanut); (c) f (θ) = θ/2. Does anything strike you as strange about the last case.
Once you are happy with your generic solution, plot (b) and (c) in Mathematica. See the sample Mathematica code
(which plots a sphere) in the back as a starting point.
8. Come up with an f (θ) that generates an egg. The egg should be realistic-looking; i.e., it should not be symmetric
about the z=0 plane. Use Mathematica to plot its embedding.
9. Hartle 3.2.
10. Hartle 3.4.
11. Consider the attractive central force potential V (r ) = −GMme −r /r0 /r0 , where M is much greater than m and r0 is a
constant, characteristic radius.
(a) Which, if any, of Kepler’s three laws are valid for this potential? Back up your answer with mathematical proof
as much as possible.
(b) Write down the relationship between the speed and radius of a circular orbit for the above potential.
(c) Show that for circular orbits around fixed M, the speed-radius relationship reaches a maximum value at a unique
value of r > 0.
(d) Compare with the behavior of the V (r ) above with Newtonian gravity by numerically plotting the speed of a
circular orbit versus its radius for both cases on top of each other on the same plot (use G = M = r0 = 1).
(e) Both V (r ) and the Newtonian gravity potential −GMm/r are monotonic functions of r . Why is it that one of
them has a maximum circular velocity for a given M and the other one is unbounded?
In[20]:=
x@theta_, phi_D := Cos@phiD Sin@thetaD
y@theta_, phi_D := Sin@phiD Sin@thetaD
z@theta_, phi_D := Cos@thetaD
ParametricPlot3D@8x@t, pD, y@t, pD, z@t, pD<, 8t, 0, Pi<, 8p, 0, 2 Pi<D
1.0
0.5
0.0
-0.5
-1.0
1.0
0.5
Out[23]=
0.0
-0.5
-1.0
-1.0
-0.5
0.0
0.5
1.0