I Adapting Schwarzschild Metric for Nonzero Λ

[Sciencemaster]

TL;DR Summary

The Schwarzschild metric is designed with a cosmological constant in mind. Would it be feasible to make some modifications to it that would make it apply in a spacetime with a nonzero Λ?

So, there are a fair amount of metrics designed with a zero value for the cosmological constant in mind. I was wondering if there was some method to modify metrics to account for a nonzero cosmological constant. Say, for instance, the Schwarzschild metric due to its relative simplicity. A feature of the metric is that it has a stress-energy tensor of 0 due to both the ricci tensor and scalar curvature going to 0. However, this doesn't happen if the cosmological constant is nonzero. So, would there be a way to modify a metric to account for a nonzero cosmological constant while keeping fundamental features intact--in this case, the 0 stress energy tensor and general form of the Schwarzschild metric? I tried multiplying by a constant as well as a few other minor modifications, and none of them gave a SET of 0. So, would it be feasible to make some modification to a metric designed for Λ=0 in order for it to apply when Λ is nonzero while still keeping similar basic properties?

 

[Ibix]

That's Schwarzschild-deSitter spacetime, if I understand what you're trying to do.

I don't know about "keeping similar basic properties". There are some quite significant differences, such as a maximum mass for a black hole.

 

[PeterDonis]

would there be a way to modify a metric to account for a nonzero cosmological constant while keeping fundamental features intact--in this case, the 0 stress energy tensor and general form of the Schwarzschild metric?

Of course you can't keep everything else exactly the same when you add a nonzero cosmological constant. But the Schwarzschild-de Sitter solution, which @Ibix mentioned, is what you get when you assume spherical symmetry, a nonzero cosmological constant, and no other stress-energy present. If you look it up, you will see that its metric is quite similar to the Schwarzschild metric; the only difference (at least in the coordinates that correspond to standard Schwarzschild coordinates) is an extra term in $\Lambda$ in $g_{rr}$ and $g_{tt}$.

 

[Sciencemaster]

First off, I want to add that I made a mistake. I said something along the lines of, 'keeping the Schwarzschild metric's property of a 0 SET. However, I neglected the ambient energy density in the universe that contributes to the nonzero cosmological constant. So...yeah, I was thinking about a metric that gave $$T_{00}=\frac{\rho_{vacuum}}{c^2}$$. Ignoring that, thanks for suggesting the Schwarzschild-de Sitter, that definitely helps and might be close enough to what I'm looking for!

 

[PeterDonis]

Schwarzschild-de Sitter, that definitely helps and might be close enough to what I'm looking for!

It is what you are looking for; it is the unique metric that satisfies the condition you give in post #4 and also contains a Schwarzschild black hole.