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I'm considering a metric of the following form, $$ds^2 = \left[F(r,t)-G(r,t)\right]dt^2 - \left[F(r,t)+G(r,t)\right]dr^2 - r^2d\Omega^2$$

with signature $(1,3)$, where $F(r,t)$ and $G(r,t)$ are arbitrary scalar functions and $d\Omega^2$ the line element on the unit sphere.

I am trying to find a coordinate and/or conformal transformation to one of the 'standard' Robertson-Walker forms, e.g.

$$ds^2 = dT^2 - \frac{a(T)^2}{1-kR^2}dR^2 - R^2 d\Omega^2$$

for any $k=0,-1,+1$, or show that there isn't one.

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  • $\begingroup$ (The conformal field theory tag is inaccurate, I have removed it; this is a standard GR problem...) $\endgroup$ – Alex Nelson Apr 16 '13 at 14:18
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So what happens if you try to solve directly, writing:

$$ F(r,t) - G(r,t) = 1 $$

and

$$ F(r,t) + G(r,t) = \frac{a(t)^{2}}{1-kr^{2}}$$

Hint: you can solve this since it's a system of linear equations in $F$ and $G$...

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Start with the FRLW metric. Make a redefinition $r = f(\tau, R)$ and $t = g(\tau,R)$. Then, do the coordinate transformation. You will have a system of three PDEs:

$g_{\tau \tau} = F - G$

$g_{\tau R} = 0$

and

$g_{R R} = F + G$

that will depend on $f$ and $g$ and their derivatives. The set of the $F$ and $G$ where you can find a $f$ and $g$ that solve these equations is the set of answers you have. You'll have to do the rest yourself.

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