# Static Point in de Sitter-Schwarzschild Spacetime

• I
• timmdeeg
In summary, in the conversation, it is discussed that in de Sitter-Schwarzschild spacetime, objects near the black hole are falling towards it while objects further away are receding. There is a specific (unstable) r-coordinate where objects are stationary. The de Sitter-Schwarzschild metric has two solutions, and by equating them, a value for r can be derived. However, this is not the correct method for finding the static r-coordinate. Instead, it is found by looking for the value of r where the proper acceleration of a worldline with constant r is zero. This gives a value that is close but not exactly the same as the one derived by equating the solutions. The correct formula
timmdeeg
Gold Member
In de Sitter-Schwarzschild spacetime things close to the black hole are falling towards it whereas in greater distance they are receding. So there should be a certain (unstable) ##r##-coordinate, where things are static. The de Sitter-Schwarzschild metric has according to Wikipedia
https://en.wikipedia.org/wiki/De_Sitter–Schwarzschild_metric
2 solutions:

##f(r)=1-2M/r##

##f(r)=1-\Lambda*r^2##

Equating yields ##r=(2M/\Lambda)^{1/3}##

Is this the wanted static r-coordinate? Heuristically it seems to make sense, r increases with increasing ##M## and decreases with increasing ##\Lambda##.

timmdeeg said:
2 solutions

No, one solution with two extra terms in ##f(r)##, i.e., the superposition of the two individual solutions. The correct formula is ##f(r) = 1 - 2M / r - \Lambda r^2 / 3##. (Note the factor of ##3## in the ##\Lambda## term; the Wikipedia article is a little misleading since it fails to include that.)

timmdeeg said:
Equating

That's not how you find the static ##r## coordinate. You find it by looking for the value of ##r## for which the proper acceleration of a worldline with constant ##r## is zero. That gives a value that is close to, but not the same as, the one you derived. See here:

https://www.physicsforums.com/threa...ween-two-spherical-shells.989533/post-6347791

Note that the ##A## in that post is ##\Lambda / 3##.

timmdeeg
PeterDonis said:
No, one solution with two extra terms in ##f(r)##, i.e., the superposition of the two individual solutions. The correct formula is ##f(r) = 1 - 2M / r - \Lambda r^2 / 3##. (Note the factor of ##3## in the ##\Lambda## term; the Wikipedia article is a little misleading since it fails to include that.)

That's not how you find the static ##r## coordinate. You find it by looking for the value of ##r## for which the proper acceleration of a worldline with constant ##r## is zero. That gives a value that is close to, but not the same as, the one you derived. See here:

https://www.physicsforums.com/threa...ween-two-spherical-shells.989533/post-6347791

Note that the ##A## in that post is ##\Lambda / 3##.
Ok, got it, so zero proper acceleration of this timelike geodesic is the correct criterion.
Then with ##A=\Lambda/3## the correct result is

##r=(3M/\Lambda)^\frac{1}{3}##

Its by a factor 2 different from what results by equating the two extra terms.

As always I appreciate your explanation, thanks.

Last edited:
timmdeeg said:
the correct result is

With the exponent fixed (it should be ##\frac{1}{3}##, not ##\frac{1}{1/3}##), yes.

timmdeeg said:
Its by a factor 2 different from what results by equating the two extra terms.

A factor of ##2^\frac{1}{3}##.

timmdeeg
PeterDonis said:
With the exponent fixed (it should be ##\frac{1}{3}##, not ##\frac{1}{1/3}##), yes.
Indeed, fixed.

PeterDonis said:
A factor of ##2^\frac{1}{3}##.
Yes, my fault.

## 1. What is a static point in de Sitter-Schwarzschild spacetime?

A static point in de Sitter-Schwarzschild spacetime refers to a point in the universe where the gravitational effects of both a black hole and dark energy are present. It is a hypothetical concept used in theoretical physics to study the effects of combining these two phenomena.

## 2. How is a static point different from a black hole or a de Sitter spacetime?

A static point is different from a black hole or a de Sitter spacetime because it combines the properties of both. Unlike a black hole, which has a singularity at its center, a static point does not have a singularity and is not infinitely dense. It also differs from a de Sitter spacetime, which is characterized by a positive cosmological constant, by having the additional gravitational effects of a black hole.

## 3. What are the implications of a static point in de Sitter-Schwarzschild spacetime?

The existence of a static point in de Sitter-Schwarzschild spacetime has significant implications for our understanding of the universe. It suggests that the effects of dark energy and black holes can coexist and potentially interact with each other. This could have implications for the expansion of the universe and the formation of galaxies.

## 4. Is there any evidence for the existence of a static point in de Sitter-Schwarzschild spacetime?

Currently, there is no direct evidence for the existence of a static point in de Sitter-Schwarzschild spacetime. It is a theoretical concept that has not yet been observed or confirmed through observational data. However, some studies have suggested that the effects of dark energy and black holes may be connected, which supports the idea of a static point.

## 5. How does the concept of a static point impact our understanding of the universe?

The concept of a static point in de Sitter-Schwarzschild spacetime challenges our current understanding of the universe and its evolution. It suggests that the effects of dark energy and black holes may be intertwined, and could potentially lead to new insights into the nature of gravity and the expansion of the universe. Further research and observations are needed to fully understand the implications of a static point in de Sitter-Schwarzschild spacetime.

• Special and General Relativity
Replies
5
Views
1K
• Special and General Relativity
Replies
35
Views
3K
• Special and General Relativity
Replies
8
Views
1K
• Special and General Relativity
Replies
7
Views
1K
• Special and General Relativity
Replies
1
Views
1K
• Special and General Relativity
Replies
8
Views
3K
• Special and General Relativity
Replies
58
Views
4K
• Special and General Relativity
Replies
3
Views
1K
• Special and General Relativity
Replies
14
Views
6K
• Special and General Relativity
Replies
8
Views
2K