# Change of variables in an interval expression

This question is a continuation of How to calculate a scalar curvature fast? .

Let's have Lorentz-Fock spacetime with an interval $$d \hat {s}^{2} = \frac{t_{0}^{2}R^{2}}{\hat {t}^{4}}\left( d \hat {t}^{2} - \frac{\hat {t}^{2}}{R^{2}}d\hat {l}^{2}\right), \qquad (1)$$ and $$d\hat {l}^{2} = \frac{d \hat {r}^{2}}{1 + \frac{\hat {r}^{2}}{R^{2}}} + \hat {r}^{2}\left( d\theta^{2} + \sin^{2}(\theta )d\varphi^{2}\right) , \quad t_{0} = \frac{R}{c} .$$ Recently, I realized that expression $(1)$ is looks like interval for Frieedman-Robertson-Walker model. Let's have $$r = R \sinh(\psi ).$$ So $(1)$ can be transformed as $$d \hat {s}^{2} = \frac{t_{0}^{2}R^{2}}{\hat {t}^{4}}\left( d \hat {t}^{2} - \hat {t}^{2}\left[d \psi^{2} + \sinh^{2}(\psi)\left( d\theta^{2} + \sin^{2}(\theta )d\varphi^{2}\right) \right]\right) \qquad (2).$$ This expression is almost corresponding Frieedman-Robertson-Walker metric. To fully comply I need to change variables: $$\tau = \frac{t_{0}^{2}}{\hat {t}}, d \tau = -\frac{t_{0}^{2}d \hat {t}}{\hat t^{2}},$$ and $(2)$ can be rewrite as $$d\hat {s}^{2} = c^{2}d \tau^{2} - c^{2}\tau^{2}\left[d \psi^{2} + \sinh^{2}(\psi)\left( d\theta^{2} + \sin^{2}(\theta )d\varphi^{2}\right) \right],$$ which is equal to FRW metric.

So, my questions are following:

1. Is it possible to make the change of variables specified?

2. What can I do with singularity of $\tau$ for $\hat {t} -> 0$?

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Also, where did your factor of $R^{2}$ go to? It doesn't affect anything, and can be gotten rid of with just a rescaling, but you did seem to just drop it. – Jerry Schirmer Apr 25 '13 at 14:48
Jerry Schirmer , $t_{0} = \frac{R}{c}, so R = ct_{0}$. – user8817 Apr 25 '13 at 14:56

2) Now, look at the function that you have for ${\hat t}$ in terms of $\tau$. You know, physically what $\tau$ means, thanks to the correspondance with the FRLW spacetime. What is special about the values of $\tau =0$ and $\tau = \infty$? What are the corresponding values of $\hat t$? Where should the spacetime be regular?
"...What is special about the values of $\tau = 0$ and $\tau = \infty$? What are the corresponding values of $\hat {t}$?.." This corresponds to the final and initial time $\tau$, if I understood your question correctly. But I don't understand the physical meaning of time $t_{0}$. – user8817 Apr 25 '13 at 14:58
@PhysiXxx: $t_{0}$ is just a time scale. You can take it to be a unit quantity. You asked what you do with the singularity. Use your intiution about the FRLW spacetime to tell you what the singularity in $\hat t$ means. – Jerry Schirmer Apr 25 '13 at 15:04
And anyone that I know would use units where $c=1$, so that they would only have to carry around one of $R,c$ and $t_{0}$ – Jerry Schirmer Apr 25 '13 at 15:12