- #1

AstroPhysWhiz

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So I have been following various derivations of the FRW metric and have a bit of confusion due to varying convention...

Would it be correct to say that curvature K can be expressed as both [tex]K = \frac{k}{a(t)^2}[/tex] and [tex]K = \frac{k}{R(t)^2}[/tex] where k is the curvature parameter?

If so, is it correct to say that the spatial line element for the k = 1 (closed) case may be expressed as

[tex]

dl^2=\frac{dr^2}{1-\frac{r^2}{R(t)^2}}+ r^2d\Omega^2

[/tex]

If I then sub

[tex]

r =R(t)\sin(\chi)

[/tex],

using the fact that the full line element is

[tex]

ds^2 = dt^2 - a(t)^2dl^2

[/tex]

I find

[tex]

ds^2 = dt^2 - a(t)^2R(t)^2[d\chi^2 + \sin^2(\chi) d\Omega^2]

[/tex]

but the texts I have read state the metric to be

[tex]

ds^2 = dt^2 - R(t)^2[d\chi^2 + \sin^2(\chi) d\Omega^2]

[/tex]

or

[tex]

ds^2 = dt^2 - a(t)^2[d\chi^2 + \sin^2(\chi) d\Omega^2]

[/tex]

so I am clearly misunderstanding something with my extra factor, anyone able to clear things up for me?

Thanks in advance.

Would it be correct to say that curvature K can be expressed as both [tex]K = \frac{k}{a(t)^2}[/tex] and [tex]K = \frac{k}{R(t)^2}[/tex] where k is the curvature parameter?

If so, is it correct to say that the spatial line element for the k = 1 (closed) case may be expressed as

[tex]

dl^2=\frac{dr^2}{1-\frac{r^2}{R(t)^2}}+ r^2d\Omega^2

[/tex]

If I then sub

[tex]

r =R(t)\sin(\chi)

[/tex],

using the fact that the full line element is

[tex]

ds^2 = dt^2 - a(t)^2dl^2

[/tex]

I find

[tex]

ds^2 = dt^2 - a(t)^2R(t)^2[d\chi^2 + \sin^2(\chi) d\Omega^2]

[/tex]

but the texts I have read state the metric to be

[tex]

ds^2 = dt^2 - R(t)^2[d\chi^2 + \sin^2(\chi) d\Omega^2]

[/tex]

or

[tex]

ds^2 = dt^2 - a(t)^2[d\chi^2 + \sin^2(\chi) d\Omega^2]

[/tex]

so I am clearly misunderstanding something with my extra factor, anyone able to clear things up for me?

Thanks in advance.

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