Homework Help: Scale Factor & Space Curvature

1. May 6, 2007

kingwinner

I will be writing my final exam tomorrow evening, and I am currently terribly stuck on the following practice problems. I have posted my thoughts below each problem. They look tricky to me. It would be very nice if someone could help me out and I will remain eternally grateful for your help!

1) You have a stick 1 meter long (physicsal size) today. What was the length of the stick at redshift 1.0? The scale factor at redshift 1.0 is 1/2 if we assume that the current scale factor is 1.
This looks very very tricky to me. The distance between galaxies is certainly smaller at redshift 1.0 because the scale factor is smaller and "space" itself stretches out as time goes on. But does the same apply to things like the meter stick? I am so confused...

2) (Multiple Choice) Which of the following statement(s) is (are) incorrect for a positively curved space (think about the surface of the earth):
a) Parallel lines converge
b) The sume of angles in a triangle is less than 180 degrees.
c) The volume of a sphere is larger than (4/3)pi*r^3, where r is the radius
d) If you strart travelling in one direction, it is possible that you come back to where you started
e) IF the radius of the curvature become larger and larger, the space will look increasingly flatter to you

I believe that a, d, and e are correct, and b is incorrect, but I am totally unsure about c, is c true or not and why?

Thank you!:)

2. May 7, 2007

Hootenanny

Staff Emeritus
I fear that my reply may come to late for your examination, but looking at the answers you've given I wouldn't be two worried
You are correct in all your answers. (c) is incorrect, the [inside] volume of a sphere in positively curved space is always less than (4/3)pi*r^3, which can be proved mathematically from the generalisation of the sphere into metric spaces, but is something I'm not going to do here.