Can we glue the Schwarzschild and the de Sitter metrics at their event horizon?

Question

Is there a way to glue the de Sitter metric inside the event horizon of the Schwarzschild metric, without an explicit reference to a particular coordinates system?

Using the standard radial coordinates $r$ of both metrics, we have \begin{align} ds_{\text{Sch}}^2 &= \Bigl( 1 - \frac{2GM}{r} \Bigr) \, dt^2 - \frac{1}{1 - \frac{2GM}{r}} \, dr^2 - r^2 \, d\Omega^2, \tag{1} \\[2ex] ds_{\text{deS}}^2 &= \Bigl( 1 - \frac{\Lambda}{3} \, r^2 \Bigr) \, dt^2 - \frac{1}{1 - \frac{\Lambda}{3} \, r^2} \, dr^2 - r^2 \, d\Omega^2. \tag{2} \end{align} So, naively gluing both metrics at $r = 2 G M = \sqrt{\frac{3}{\Lambda}}$ imposes a specific relation between the mass $M$ and the constant $\Lambda$ inside the horizon. This is coordinate dependent, and the metric is not smooth at the event horizon. We could also use the "isotropic" radial coordinate of the Schwarzschild metric instead of (1), so the relation would be different (and the metric derivatives still be discontinuous).

So is it possible to define a smooth spacetime metric from the Schwarzschild metric with a de Sitter spacetime inside the event horizon? I suspect it's not possible, since the cosmological constant $\Lambda$ is supposed to be a constant over the whole of spacetime. If it is possible to introduce a discontinuous $\Lambda$, I would like to see an explicit example (from an explicit coordinates system).

Answer 1

@Cham, it is not possible because the derivative of $\sqrt{g_{00}}$ would be discontinuous on your event horizon. Einstein's field equations for static spherically symmetric spheres are first-order for $g^{-1}_{rr}$ and second-order for $\sqrt{g_{00}}$. Therefore, there are three boundary conditions which ensure continuity of $g^{-1}_{rr}$, $\sqrt{g_{00}}$, and $d(\sqrt{g_{00}})/dr$. The last one would be is $-\infty$ on the inner side, and $+\infty$ on the outer side of event horizon.