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I'm wondering why when we study spherically symmetric objects we neglect the existence of the cosmological constant. His contribution is just to small for "small" scales?

I mean, for a spherically symmetric objects usually we use the schwarzschild and the Einstein equations $$G_{\mu\nu}=\frac{8\pi G}{3} T_{\mu\nu}$$ but not the "full" equations $$G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{3} T_{\mu\nu}$$

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The De Sitter-Schwarzschild metric describes a black hole with no charge and no angular momentum in a spacetime with a cosmological constant.

The observed value of the cosmological constant is so small that it is essentially irrelevant to the geometry near a non-cosmologically-sized black hole.

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  • $\begingroup$ The question was probably interested in whether there might be a phase transition that would introduce a qualitative difference even for the smallest possible $\Lambda$ value? Is there some general way to rule this out? $\endgroup$
    – Kagaratsch
    Apr 27 '20 at 0:48
  • $\begingroup$ @Kagaratsch A phase transition in what? $\endgroup$
    – G. Smith
    Apr 27 '20 at 2:52
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    $\begingroup$ @Kagaratsch The Wikipedia article has the scalar curvature. You can determine whether it has some characteristic that you could convincingly call a phase transition. $\endgroup$
    – G. Smith
    Apr 27 '20 at 2:57
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    $\begingroup$ @Kagaratsch I get $R=4\Lambda$, just like for De Sitter space. It would be hard to argue that a constant has a phase transition. $\endgroup$
    – G. Smith
    Apr 27 '20 at 3:04
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    $\begingroup$ @Kagaratsch: If you are indeed interested in phase transitions that may occur in black hole spacetimes with “varying” cosmological constant I recommend looking at the review Black hole chemistry: thermodynamics with Lambda. (I do not think that this is the viewpoint OP is asking about, so I won't be writing an answer). $\endgroup$
    – A.V.S.
    Apr 27 '20 at 6:48

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