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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).

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    $\begingroup$ That's Wheeler's bag of gold spacetime. $\endgroup$ Nov 30, 2021 at 15:56
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    $\begingroup$ Also a chap called Markov used this idea and called it a Friedmon geometry. See here for more details. $\endgroup$ Nov 30, 2021 at 16:00
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    $\begingroup$ I've never looked into how the bag of gold geometry is constructed. I just know you patch an FLRW geometry into the inside of a Schwarzschild geometry and you get an expanding universe inside the black hole. It shouldn't be hard to Google for the details. $\endgroup$ Nov 30, 2021 at 16:01
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    $\begingroup$ There's a procedure for patching spacetimes together, that's covered in detail in a relativist's toolkit. It is similar to pasting E&M solutions together. Generally, you have to enforce junction conditions and the like, and you might end up with a surface mass charge. $\endgroup$ Nov 30, 2021 at 16:31
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    $\begingroup$ @benrg: it's discussing a cosmological constant that is constant on a patch of spacetime, which is definitely something that many different models discuss. $\endgroup$ Nov 30, 2021 at 18:13

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@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.

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  • $\begingroup$ This is what I was expecting. But I also expect a non-static metric (collapse or expansion). I think that the glue could be made with the help of some scalar field. $\endgroup$
    – Cham
    Dec 2, 2021 at 16:25
  • $\begingroup$ No objections. My answer is limited to static spherically symmetric solutions with isotropic pressure. $\endgroup$ Dec 2, 2021 at 17:02

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