The metric for de Sitter spacetime in static coordinates is $$ds^2 = \left(1-\frac{r^2}{\ell^2}\right)dt^2 - \frac {1}{1 - \frac{r^2}{\ell^2}}dr^2 - r^2\,d\Omega_2^2.$$ It is evident that there exists a singularity at $r = \ell$. My question is, in what way can we remove this singularity? I am studying a complex scalar field theory in this particular spacetime, and I would like to make an appropriate coordinate transformation so that the singularity disappears from the equations of motion.
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You can remove the singularity by the same trick that removes the singularity of Schwarzschild coordinates: substitute $t = u + f(r)$ for some $f$. Taking $f(r) = \ell \tanh^{-1} (r/\ell) - r$ gives the metric
$$ds^2 = (1 - r^2/\ell^2)\,du^2 + 2\ (r^2/\ell^2)\,du\,dr - (1 + r^2/\ell^2)\,dr^2 - r^2 dΩ^2$$
in which the radial outward speed of light is $1$ and the radial inward speed is $\dfrac{1-r^2/\ell^2}{1+r^2/\ell^2}$. Whether that's suitable for your application I don't know. It covers only half of the de Sitter spacetime (again, like the analogous metric for the Schwarzschild manifold).
benrg
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