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What is physical meaning of $\kappa$ and $R$ in curved space?

$$dl^2 = \frac{dr^2}{1 - \kappa\frac{r^2}{R^2}} + r^2d\theta^2 + r^2\sin^2\theta d\phi^2$$

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This is not a direct answer to your question, but relevant and I found it useful: In this and this paper, the author discusses the reasonable coordinate systems for some spacetimes of physical interest. Specifically, check page 21 and 10 respectively. – NikolajK Dec 17 '12 at 11:00
See… – John Rennie Dec 17 '12 at 11:23
up vote 3 down vote accepted

That's a metric for a space of constant curvature in 3 dimensions. Typically you would parameterise $\kappa$ as -1, 0, or 1 and $R$ as a non zero real number. So $\kappa$ just tells you if you have positive, negative or no curvature and $R$ tells you the magnitude of the curvature. $R$ can be thought of as a radius of curvature if you choose to embed your 3-space in a higher dimensional space.

For example, if you take $\kappa=1$ and subsitute $$r=R \sin\psi$$ you'll get a 3 sphere metric (in spherical polars) with an $R^2$ overall factor.

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