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I would like to derive the fact that the flat FRW metric has a time-like conformal Killing vector.

Is there an easy way to do this?

@ValterMoretti showed how one can do this for metrics with a Killing vector by using a metric representation that is independent of a particular co-ordinate :

https://physics.stackexchange.com/a/120089/22307

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A normal Killing vector is obtanied by solving the usual Killing equation

$\nabla_{(\mu} \xi_{\nu)} = 0$

A conformal Killing vector is obtanied by solving a slightly different equation, the conformal Killing equation:

$\nabla_\mu \xi_\nu + \nabla_\nu \xi_\mu = 2 \alpha g_{\mu \nu}$

where $\alpha$ is obtanied by taking the trace of the equation above. Using that particular co-ordinate representation that you alluded these equations probably won't be impossible to solve (if they have a solution).

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Consider the metric in comoving conformal coordinates: $ds^2 = a^2(\eta)[-d\eta^2 + \delta_{ij}dx^i dx^j]$. It is easy to show that $\partial_{\eta}$ solves $\mathcal{L}_{\partial_{\eta}}g_{\mu\nu} = \frac{2}{a(\eta)}g_{\mu\nu}$; furthermore $\partial_{\eta}$ is clearly time-like.

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  • $\begingroup$ Do I have to calculate the expression given by @GabrielCozzella in answer#1 ? Seems like a lot of work calculating all those covariant derivatives with their Christoffel symbols etc! is there an easier way? $\endgroup$ – John Eastmond Jun 9 '15 at 21:57

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