then $A(r)<0$ for all $r>0$, so that $ \xi^\mu$ is spacelike. In this case, the Kottler spacetime contains no static region but it is spatially homogeneous.

2.  If $9(GM)^2\Lambda<1$, then $A(r)>0$ for $r$ between $r{_{\text{b}}}$ and $r{_{\text{c}}}>r{_{\text{b}}}$ which are the two positive roots of the polynomial $rA(r)$, and correspond respectively to the black hole and cosmological horizons. We have $$\begin{aligned}
    r{_{\text{c}}} &= \frac{2}{\sqrt{\Lambda}}\cos\left(\frac{\psi}{3} + \frac{\pi}{3} \right),\\
    r{_{\text{b}}} &= \frac{2}{\sqrt{\Lambda}}\cos\left(\frac{\psi}{3} - \frac{\pi}{3} \right),\end{aligned}$$ with $\cos\psi = 3\,GM\sqrt{\Lambda}$, so that $$r{_{\text{S}}} < r{_{\text{b}}} < \frac{3}{2}\,r{_{\text{S}}} < \frac{1}{\sqrt{\Lambda}} < r{_{\text{c}}} < \frac{3}{\sqrt{\Lambda}} .$$ In the region $r{_{\text{b}}}<r<r{_{\text{c}}}$, the Kottler spacetime is static. Note also that $r=r{_{\text{b}}}$ and $r=r{_{\text{c}}}$ are Killing horizons, since $\xi$ vanishes on these hypersurfaces.

In practice, we use the Kottler solution to describe the vicinity of a gravitationally bound object, such as a galaxy, or a cluster of galaxies. In this context, we have typically $9(GM)^2\Lambda < 10^{-14}$ (see Subsec. \[subsec:numerical\_values\]), so we are in the second case. Moreover, this solution only describes the exterior region of the central object; it is thus valid only for $r>r{_{\text{phys}}}$, where $r{_{\text{phys}}}$ is the physical size of the object. For the cases we are interested in, $r{_{\text{phys}}} \gg r{_{\text{b}}}$, so that there is 