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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).

Cham
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2 Answers2

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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.

JanG
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Yes, it is possible. What you need are the so-called junction conditions, which to my knowledge were originally derived by Israel. They are reviewed in some standard books in general relativity, such as Poisson's A Relativist's Toolkit, Misner, Thorne, and Wheeler's Gravitation, and Padmanabhan's Gravitation.

The conditions are similar to the matching conditions you get in electrodynamics to patch two solutions across a surface, but they are more intricate due to the need of coordinate independence (as you noted). Shortly, they state the following:

  1. the induced metric on the hypersurface separating the two regions must be well-defined, and hence should be the same from either side.
  2. the discontinuity of the extrinsic curvature (which is essentially the derivative of the induced metric along the normal direction to the hypersurface) is proportional to a surface stress contribution.

The surface stress contribution is analogous to surface charges and currents in electromagnetism. You get a shell of matter at the boundary of the matching.

By glancing at the metrics, I expect the first junction condition to impose a relation between $\Lambda$ and $M$ and the radius at which you perform the matching, and the second junction condition will tell you whether the spacetime needs a shell of matter to exist in order for the Einstein equations to hold everywhere.

I think it could be a little weird to perform this calculations in Schwarzschild coordinates, since they are singular at the horizon (which is your interest), but if you use a different coordinate system it might be fine. I would need to open up the calculations to be sure. I'm pretty sure you can do the matching at $r=2M+\epsilon$, but I'm not sure about what happens when $\epsilon \to 0$.

Níck Aguiar Alves
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