- #1
Airsteve0
- 83
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I understand that in order for a space to be conformally flat that the vanishing of the Weyl tensor is a necessary condition. I was wondering if the following arguments are also (if not the same) in arguing that Robertson-Walker spacetime is conformally flat.
Starting with the Robertson-Walker metric:
[itex]ds^{2}[/itex]=[itex]a^{2}(t)\left(\frac{dr^{2}}{1-Kr^{2}}+r^{2}(d\theta^{2}+sin(\theta)^{2}d\phi^{2})\right)-dt^{2}[/itex]
Could it be argued that there exists coordinates [itex]x^{\gamma}[/itex] such that:
[itex]ds^{2}[/itex]=[itex]f(x^{\gamma})\eta_{\alpha\beta}dx^{\alpha}dx^{β}[/itex]
where [itex]\eta_{\alpha\beta}[/itex] is flat. As such, the 4 dimensional space would possesses signature of (+ or -) 2 and be describable as locally Minkowskian.
Starting with the Robertson-Walker metric:
[itex]ds^{2}[/itex]=[itex]a^{2}(t)\left(\frac{dr^{2}}{1-Kr^{2}}+r^{2}(d\theta^{2}+sin(\theta)^{2}d\phi^{2})\right)-dt^{2}[/itex]
Could it be argued that there exists coordinates [itex]x^{\gamma}[/itex] such that:
[itex]ds^{2}[/itex]=[itex]f(x^{\gamma})\eta_{\alpha\beta}dx^{\alpha}dx^{β}[/itex]
where [itex]\eta_{\alpha\beta}[/itex] is flat. As such, the 4 dimensional space would possesses signature of (+ or -) 2 and be describable as locally Minkowskian.