Physics 725: Special and General Relativity
Thornton 428, San Francisco State University
Fall 2014, MWF 9:10AM
Homework 3 Due 9:10AM 9/22
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.
Name
Signature:
1. In the last problem set, you were able to embed a 2D non-Euclidean geometry in the 3D Euclidean manifold of real
numbers (also known as R3 ), so that you could visualize it better. You did this by requiring that the path-lengths on
the surface embedded in R3 should equal the path-lengths on the 2D non-Euclidean geometry. For example, the 2D
line element
ds 2 = a2 [dθ2 + f (θ)2 dφ2 ]
could be embedded in R3 space by setting x(θ) = af (θ) and solving for the z(θ) that would give the right pole-to-pole
distance, in this case πa. Rotating the curve [x(θ), z(θ)] about the z axis would then yield the desired surface.
Show that it is impossible to do the same for the 2D Minkowski line element:
ds 2 = −c 2 dt 2 + dx 2
That is, show mathematically why there is no function z(x, t) which defines a surface embedded in R3 such that
path-lengths along this surface equal path-lengths in the 2D non-Euclidean Minkowski space.
It follows that only a subset of all non-Euclidean N-1 dimensional geometries can be embedded in RN .
2. Your amazing spacecraft is at rest in some unprimed frame “A” at t = 0, x = 0. The ship is capable of a Lorentz
boost of any speed up to vmax , where vmax < c. It also has an FTL drive capable of instantaneously teleporting the
ship a distance of up to xmax in any direction, where distance is measured in whatever frame the ship is at rest in at
the moment of the jump. Describe the procedure requiring the fewest steps using which you could time travel to
your own past using this spacecraft. You must list the exact set of steps required, with velocities in units of vmax and
jumps in units of xmax , and show explicitly that you have reached x = 0, t < 0 in frame A.
3. (10 points) Hartle 5.6
4. (10 points) Hartle 5.7
5. (10 points) Hartle 5.12
6. (10 points) You are in a plane at an altitude of 30000 feet above the Earth—still well within the Earth’s atmosphere.
Your cockpit is made of clear glass, so that you can see all around and above you, like Wonder Woman. There’s
not a cloud in the sky—the sky is a perfect, monochromatic blue (the wavelength of the light from all directions is
λ0 = 450nm). You now begin moving with a relativistic velocity v in a direction parallel to the surface.
(a) Describe what you see, i.e. find λ(θ, φ, v ), where θ = 0 points directly above you and φ = 0 points straight ahead.
Neglect air resistance and assume you go from 0 to v instantly.
(b) You look straight up (at θ = 0). The sky is green (λ = 530nm). How fast are you going?
(c) Plot the colors of the entire sky for the velocity you find in (b). See the reverse side of this page for the Mathematica
code to do this.
7. (10 points) Hartle 5.23
H* Below is the only line you need to modify. Insert your solution
of lambdaHp,tL Hwith p=phi and t=thetaL below. The solution
needs to give the color of the sky in nanometers as a function
of direction Hphi, thetaL. I have put in an example here which
clearly is not the solution to the problem;
it makes the
sphere red at the equator and ultraviolet at the poles *L
lambda@p_, t_D := 400 Sin@tD + 250
H* The code below is a single Mathematica command
that does the plotting. Do not change any of it--it will automatically plot the colors of the sky,
cutting off anything below 400nm or above 650nm to black *L

ParametricPlot3D@
8Cos@pDSin@tD,Sin@pD Sin@tD,Cos@tD<,8p,0,2Pi<,8t,0,Pi<,
ColorFunction®Function@8x,y,z,p,t<,[email protected]*H650-lambda@p,tDL250 ,
1,If@lambda@p,tD<400,0,If@lambda@p,tD>650,0,1DDDD,
ColorFunctionScaling®FalseD
1.0
0.5
0.0
-0.5
-1.0
1.0
0.5
Out[226]=
0.0
-0.5
1.0
0.5
0.0
-0.5
-1.0
-1.0