(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A space-time is said to be conformally flat if there is a frame in which the metric is [itex]g_{ab} = \Omega^2 \eta _{ab} \text{ with } \eta_{ab}[/itex] the metric in Minkowski.

Is the general Friedmann-Robertson-Walkers space-time with line element

[tex] ds^2 = -dt^2 +a^2(t)(dx_1^2 + \ldots + dx_n^2)[/tex]

conformally flat?

2. Relevant equations

[tex] ds^2 = g_{ij} dx_i dx_j [/tex]

3. The attempt at a solution

We can easily use the symmetry of the metric and the line element to find that the metric in FRW spacetime is

[tex] \begin{bmatrix}

-1 & 0 & \ldots & 0\\

0 & a^2(t) & \ldots & 0\\

\vdots & \vdots &\ddots & \vdots\\

0 & 0 & \ldots & a^2(t) \end{bmatrix} [/tex]

Now we weren't told what [itex] \Omega^2 [/itex] was in the question, but when I asked my TA he said that it was a real-valued function from spacetime coordinates. He also told me that I should make a change of coordinates. I've tried using the definition of the line element, and taking derivatives with respect to time, but to no avail. I attempted using

[tex] dt^2 = \Omega^2 d\tau^2[/tex]

[tex] a^2(t)dx_i^2 = \Omega^2 d\xi_i^2 \quad \forall i = 1\ldots n [/tex]

to try and solve for a coordinate transformation, but again I just ended up getting integrals of [itex]\Omega[/itex] which were at best, complex valued. A few of my class-mates and I have been pondering this question for a bit, and we just can't seem to find the trick. Any help would be appreciated.

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# Homework Help: Conformally Flat Space Time

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