Scale Factor & Space Curvature

[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 traveling 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!:)

 

[Hootenanny]

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!

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

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. We live in an expanding Universe. Does this mean that a meter stick and the atoms in your body will also expand and get bigger?

While it is true that the Universe is expanding as a whole, a meter stick and our own bodies do not expand. These are system which are "gravitationally bound", which means that gravity is stronger than other forces and the expansion of space. Atoms are held together by the nuclear force which is 1038 times stronger than gravity and stronger than the expansion of the Universe.

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 traveling 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?

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.

 

[m2003]

1) The length of the stick at redshift 1.0 can be calculated using the formula: L = L0/a, where L0 is the original length and a is the scale factor. In this case, L0=1 meter and a=1/2. So, the length of the stick at redshift 1.0 would be 2 meters. This is because as the scale factor decreases, the size of objects in the universe increases. This applies to the meter stick as well, as it is a physical object in the expanding universe.

2) The correct statement(s) for a positively curved space are a, d, and e. Parallel lines do converge, if you start traveling in one direction, it is possible to come back to where you started, and as the radius of curvature increases, the space appears flatter. The sum of angles in a triangle is actually greater than 180 degrees in a positively curved space, so b is incorrect. The volume of a sphere in a positively curved space is also larger than (4/3)pi*r^3, so c is incorrect. This is because the curvature of space causes objects to take up more space than they would in a flat space.