FRW metric for closed universe

In summary, the speaker is a new member of the Physics Forum and is unsure of where to post their question, which involves the FRW metric. They found a problem in their old lecture notes and are trying to solve it, using a specific equation. They mention being rusty in physics due to taking a break from it after their degree, but then realize they understand the problem.
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Hi, I'm new to Physics Forum and wasn't really sure where to post this since its not strictly speaking a homwork question. So if it happens to be in the wrong place I apologise.

I was looking through some lecture notes from when I did my Physics degree years ago and come across a problem involving the FRW metric.

Basically I'm trying to get from

ds2 = dt2 - a(t)2[(dr2/(1-Kr2))+r2d[tex]\Omega[/tex]]

to

ds2 = dt2 - a(t)2[d[tex]\chi[/tex]2 + Sk[tex]\chi[/tex]2d[tex]\Omega[/tex]]

using r = Sk[tex]\chi[/tex] = (1/[tex]\sqrt{}K[/tex]sin([tex]\sqrt{}K[/tex][tex]\chi[/tex])

After I competed my degree I took a bit of a break from Physics so I'm quite rusty at this stuff. Would appreciate any help i can get :)
 
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  • #2
Never mind. I get it. Guess I'm not as rusty as I thought.
 

1. What is the FRW metric for a closed universe?

The FRW metric, also known as the Friedmann-Lemaitre-Robertson-Walker metric, is a mathematical description of the expanding universe in the framework of general relativity. It is used to describe the evolution of a closed universe, which is one where the geometry of space is positively curved, similar to the surface of a sphere.

2. How does the FRW metric describe the expansion of a closed universe?

The FRW metric takes into account the effects of both matter and energy on the expansion of the universe. It describes how the distances between objects in the universe change over time as the universe expands. In a closed universe, this expansion is eventually slowed down and reversed by the gravitational pull of matter and energy, leading to a "big crunch" scenario.

3. What is the role of dark energy in the FRW metric for a closed universe?

Dark energy, a mysterious force that is thought to be responsible for the accelerating expansion of the universe, is also included in the FRW metric for a closed universe. Its presence is needed to explain the observed expansion rate of the universe, and it is represented by the cosmological constant in the metric.

4. Can the FRW metric be used to explain the cosmic microwave background radiation in a closed universe?

Yes, the FRW metric can be used to explain the cosmic microwave background radiation, which is the leftover radiation from the early stages of the universe. In a closed universe, the radiation is expected to be slightly hotter on one side and colder on the other, due to the curvature of space.

5. Is the FRW metric for a closed universe consistent with other observations and measurements?

Yes, the FRW metric has been extensively tested and is consistent with a wide range of observations and measurements, such as the expansion of the universe, the cosmic microwave background radiation, and the distribution of galaxies. It is currently the most widely accepted model for the evolution of the universe.

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