Adapting Schwarzschild Metric for Nonzero Λ

In summary, the conversation discusses the possibility of modifying a metric to account for a nonzero cosmological constant while still retaining its fundamental properties, specifically the 0 stress-energy tensor and general form of the Schwarzschild metric. The solution is the Schwarzschild-de Sitter spacetime, which combines the Schwarzschild metric with a nonzero cosmological constant and maintains similar basic properties.
  • #1
Sciencemaster
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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?
 
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  • #2
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.
 
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  • #3
Sciencemaster said:
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}##.
 
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  • #4
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!
 
  • #5
Sciencemaster said:
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.
 

Related to Adapting Schwarzschild Metric for Nonzero Λ

1. What is the Schwarzschild metric?

The Schwarzschild metric is a mathematical solution to Einstein's field equations in general relativity. It describes the curvature of spacetime around a non-rotating, spherically symmetric mass, such as a black hole.

2. What does nonzero Λ mean in the context of the Schwarzschild metric?

In general relativity, Λ (Lambda) represents the cosmological constant, which is a measure of the energy density of the vacuum of space. A nonzero Λ in the Schwarzschild metric indicates the presence of a cosmological constant, which can affect the curvature of spacetime.

3. Why is it important to adapt the Schwarzschild metric for nonzero Λ?

Adapting the Schwarzschild metric for nonzero Λ allows us to study the effects of a cosmological constant on the curvature of spacetime around a spherically symmetric mass. This is important for understanding the behavior of gravity in the presence of dark energy, which is thought to be responsible for the expansion of the universe.

4. How is the Schwarzschild metric adapted for nonzero Λ?

The Schwarzschild metric is modified by adding a term involving Λ to the equation. This term represents the contribution of the cosmological constant to the curvature of spacetime. The resulting metric is known as the Schwarzschild-de Sitter metric.

5. What are some applications of the adapted Schwarzschild metric for nonzero Λ?

The adapted Schwarzschild metric has been used in various studies, such as analyzing the effects of dark energy on the motion of particles around a black hole, investigating the properties of black holes in a universe with a nonzero cosmological constant, and testing the validity of general relativity in the presence of dark energy.

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