I've been challenged by coordinate transformations lately, as most of you know during solving any problem in GR we have to go through lots of transformations, my question is how we decide the best coordinate at each stage, as an example let's consider de Sitter space-time which has the following metric:

$$ds^2=-(1-Λ/3 r^2 )dt^2+(1-Λ/3 r^2 )^{-1} dr^2+r^2 dΩ^2 .$$

how do we determine the best coordinate transformation to write the metric above in a comoving synchronous coordinate system?


First, allow the cosmological constant to be expressed via the Hubble constant as $$\frac{\Lambda}{3c^2}=H_0^2 \tag{1}$$ Then the metric is called the static form of the de Sitter metric. The coordinate transform that turns a static metric to a metric in comoving frame is $$d\tau=dt-\frac{v}{c^2} \left(1-\frac{v^2}{c^2}\right)^{-1}dr\tag{2}$$ where $\tau$ is time in comoving frame. The velocity of comoving frame in this particular case is $$v=H_0 r\tag{3}$$ The substitution of (2) with the use of (1)&(3) into the given metric leads to the Gulstrand-Painleve (GP) form $$ds^2=-c^2\left(1-\frac{v^2}{c^2} \right)d\tau^2- 2 v d\tau dr+dr^2+r^2 dΩ^2 \tag{4}$$ Basically this is the metric in comoving frame. $v$ is given by (3) and $\tau$ is time in comoving frame. However, $r$ is still coordinate distance. Therefore, in order to to use comoving distance one may resolve (3) for $r(\tau)$ which yields $$r=R \: e^{H_0\tau} \tag{5} \quad \mathrm{and} \quad v=R \:H_0 \:e^{H_0\tau}$$ where the arbitraty constant of integration was set to $R$ which is the comoving distance. Using $$dr= dR \: e^{H_0\tau} +v \: d\tau \tag{6}.$$

The substitution of (6) into (4) results in $$ds^2=-c^2 d\tau^2+ \left(e^{H_0\tau}\right)^2\left(dR^2+R^2 dΩ^2\right)\tag{7}$$ which is the FLRW form for the spatially flat metric with the cosmological scale factor $a(\tau)=e^{H_0 \tau}$.

You may check Tolman for the de Sitter case or this paper https://www.nature.com/articles/srep00923 for general correspondence of static and comoving metrics.

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