Physics 725: Special and General Relativity
Thornton 335, San Francisco State University
(c) 2014 Andisheh Mahdavi
Fall 2014, MWF 9:10AM
Homework 5 Due 10/20
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.
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1. (80 points) Suppose an Alcubierre warp ship departs from space station A at a time t = 0 xA = 0 and
arrives at space station B at a time t = 1, xB = 2. The ship’s warp attenuation field is
(
f (rs ) =
1−
0
rs2
R2
rs2 < R 2
rs2 > R 2
where rs2 = (x − xs )2 + y 2 + z 2 is the distance from the ship and R = 0.2. Let us assume y = z = 0
throughtout this problem. Email me your Mathematica notebooks for this problem.
a) (10 points) Find a smooth function Vs (t) such that both the acceleration and the velocity of the ship
are zero at the beginning and at the end of the trip (i.e. Vs (0) = Vs (1) = Vs0 (0) = Vs0 (1) = 0, and
such that the average trip speed is twice the speed of light as viewed by the space stations. Remember
that Vs (t) = dxs (t)/dt.
b) (10 points) In Mathematica, use the principle of extremal proper time to set up the equations of motion
for a particle anywhere in the spacetime, assuming y (t) = z(t) = 0. Use the If statement, e.g.:
f[rs_] := If[rs^2<R^2,(1-rs^2/R^2),0]
The above defines a function that is evaluated for variable rs . Be sure to use If in your xs (t) and Vs (t)
as well to make the ship stand still at x = 2 for t > 1.
xs[t_] := If[t<1, .... , 2]
Vs[t_] := D[xs[t],t]
c) (15 points) Is it possible for a non-warp capable “scout” ship to make use of the field of the warp ship
to get from A to B? To answer this question, use Mathematica to solve the equations in part (b) using
my suggested f (rs ) as well as the results of (a). For simplicity, put the piggybacking ship at y (t) = 0,
z(t) = 0, x(0) = ds , and assume zero initial velocity for the scouts. Try both ds = 0.02 and ds = −0.02.
Plot1 t(τ ) vs. x(τ ) for each ship on top of xs (t). Comment on why the motion looks like this. You will
need to pass the MaxStepFraction->0.01 option to NDSolve for sufficient accuracy.
d) (15 points) Now consider a piece of debris sitting at t = 0, x = 1, i.e., halfway across the journey. Give
the piece of debris an initial velocity of dx/dt = −0.2, i.e., it’s heading towards the ship. Calculate the
worldline for this piece of debris and plot it on the t −x spacetime diagram as you did in (c). Specifically,
carefully compare the initial and final γ of the piece of debris. You will need to evaluate t 0 (τ ) to do
this. (Assume the debris narrowly misses the ship, darting very close to it but not actually hitting it).
e) (15 points) Now repeat (d) for a positive initial velocity. Start at x = 1 with low inital velocity
dx/dt = 0.01. The result is quite surprising—can you explain it and reconcile it with the results of the
previous section? What about initial dx/dt = 0.2?
f) (15 points) At t = 0, space station B sends a photon with energy h̄ω0 (in the space station frame)
towards the warp ship. At what coordinate time t does the ship receive the signal, and what is the
observed photon energy? Use E = −p · uobs and a spacetime diagram to answer this question.
2. (10 points) Hartle 8.2
3. (10 points) Hartle 8.5
1
In HW4 we showed that t = τ for the ship—but that is not true for anything else outside the position of the ship.