In a thread a while back, Mentz114 posted a Painleve chart for FRW spacetime; here's the link to the post:(adsbygoogle = window.adsbygoogle || []).push({});

https://www.physicsforums.com/showpost.php?p=2985307&postcount=60

He posted the metric in Cartesian coordinates, and I've derived a corresponding metric for polar coordinates. (I'm doing this so I can then see what the worldlines of "comoving" observers look like in this chart.) [Edit: originally I thought my answer looked different than what I expected based on Mentz114's post, but I made an error in deriving it; the error is now corrected below.]

Here's what I'm getting: I start with the FRW metric for k = 0 (i.e., flat spatial slices) and with a matter-dominated equation of state, so the scale factor is proportional to [itex]t^{\frac{2}{3}}[/itex]. (Mentz114 didn't say so, but it looks to me like that's the equation of state for the metric he wrote down.) I pick units so that the constant of proportionality for the scale factor is 1 (i.e., a(t) = 1 at t = 1), so

[tex]ds^{2} = - dt^{2} + t^{\frac{4}{3}} \left( dr'^{2} + r'^{2} d\Omega^{2} \right)[/tex]

We want a coordinate transformation that will make the purely spatial part of the metric static (i.e., independent of t). I try this:

[tex]r' = t^{- \frac{2}{3}} r[/tex]

(leaving all other coordinates the same), which gives

[tex]dr' = t^{- \frac{2}{3}} dr - \frac{2 r}{3} t^{- \frac{5}{3}} dt[/tex]

Substituting into the metric gives, after some algebra,

[tex]ds^{2} = - dt^{2} \left( 1 - \frac{4 r^{2}}{9 t^{2}} \right) - \frac{4 r}{3 t} dt dr + dr^{2} + r^{2} d\Omega^{2}[/tex]

If the above is correct, then the worldlines of comoving observers are easy. We haven't changed the t coordinate so for comoving observers we want [itex]ds^{2} = - dt^{2}[/itex]. That gives:

[tex]\frac{dr}{dt} = \frac{2 r}{3t}[/tex]

Does all this look correct?

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# Painleve chart for FRW spacetime

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