How Do You Derive the FRW Metric for a Closed Universe?

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SUMMARY

The discussion focuses on deriving the Friedmann-Robertson-Walker (FRW) metric for a closed universe. The user transitions from the equation ds² = dt² - a(t)²[(dr²/(1-Kr²))+r²dΩ] to ds² = dt² - a(t)²[dχ² + Skχ²dΩ], utilizing the substitution r = Skχ = (1/√K)sin(√Kχ). The user successfully resolves the problem, indicating a solid understanding of the FRW metric despite a hiatus from physics studies.

PREREQUISITES
  • Understanding of the Friedmann-Robertson-Walker (FRW) metric
  • Familiarity with differential geometry concepts
  • Knowledge of cosmological parameters, specifically curvature (K)
  • Basic proficiency in mathematical physics, particularly in manipulating equations
NEXT STEPS
  • Study the derivation of the FRW metric in detail
  • Explore the implications of different curvature values (K) on cosmological models
  • Learn about the role of scale factors in cosmology
  • Investigate applications of the FRW metric in modern cosmological theories
USEFUL FOR

Students of physics, cosmologists, and anyone interested in the mathematical foundations of cosmological models, particularly those focusing on the FRW metric and its applications in closed universe scenarios.

Skye
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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\Omega]

to

ds2 = dt2 - a(t)2[d\chi2 + Sk\chi2d\Omega]

using r = Sk\chi = (1/\sqrt{}Ksin(\sqrt{}K\chi)

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 :)
 
Physics news on Phys.org
Never mind. I get it. Guess I'm not as rusty as I thought.
 

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