Static spacetime as static universe

[johank]

When Einstein conceived GR he added the cosmological constant in an attempt to create a static universe model. The problem with this was that it was unstable.

I've recently read about static spacetime (static spacetime admits a global, non-vanishing, timelike Killing vector field K which is irrotational). Would a static spacetime also be a static universe?

 

[bcrowell]

Hi, johank,

Welcome to PF!

Would a static spacetime also be a static universe?

The static spacetimes that people write down are not normally intended as cosmological models. For example, the Schwarzschild metric is usually applied as a model of the spacetime surrounding a spherically symmetric, gravitating body such as the earth.

The Hawking singularity theorem http://en.wikipedia.org/wiki/Singularity_theorems , combined with observation, proves that there was a big bang singularity. (This is within the context of GR, not other theories, and it assumes certain energy conditions http://en.wikipedia.org/wiki/Energy_condition .) That means that, since about the 1970's, it's been clear that no static spacetime can be a valid cosmological model of our universe.

-Ben

 

[TrickyDicky]

The Hawking singularity theorem http://en.wikipedia.org/wiki/Singularity_theorems , combined with observation, proves that there was a big bang singularity. (This is within the context of GR, not other theories, and it assumes certain energy conditions http://en.wikipedia.org/wiki/Energy_condition .) That means that, since about the 1970's, it's been clear that no static spacetime can be a valid cosmological model of our universe.

I think this statement is misleading without further explanation, for instance one can read in the wiki article:
Wiki quote
"Hawking's singularity theorem is for the whole universe, and works backwards-in-time: in Hawking's original formulation, it guaranteed that the Big Bang has infinite density. Hawking later revised his position in "A Brief History of Time" (1988) where he stated "There was in fact no singularity at the beginning of the universe" (p50). This revision followed from quantum mechanics, in which general relativity must break down at times less than the Planck time. Hence general relativity cannot be used to show a singularity
" End Quote

So what the Hawking theorem proved in the 70's first version is that every FRW solution of General relativity (without cosmological constant) must have an initial singularity with infinite density, so first of all it only applies to a certain type of GR solutions and it claims nothing about other cosmological models, static or otherwise. But the later revision of the theorem doesn't even assert this.
This is reinforced later in the article by remembering that :
Wiki quote
"During inflation, the universe violates the stronger dominant energy condition (but not the weak energy condition), and inflationary cosmologies avoid the initial big-bang singularity, rounding them out to a smooth beginning." End quote

And also when it says:
Wiki quote
"Singularities can be found in all the black-hole spacetimes, the Schwarzschild metric, the Reissner–Nordström metric and the Kerr metric, and in all cosmological solutions which don't have a scalar field energy or a cosmological constant" End quote

I think it is important to clarify this as Hawking theorem is often used to make unwarranted claims about cosmology and it usually is used pretending to prove things it actually doesn't prove at all, I'm not sure if this is done relying on the fact not many people know exactly what the theorem says and how it has been revised based on QM or by posterior observations like accelerated expansion interpreted as CC, but it is obvious the theorem has as "a priori" assumptions not only an expanding universe, but no cosmological constant or no inflationists scenarios so it can hardly say anything about any other model.

 

[bcrowell]

The Hawking singularity theorem http://en.wikipedia.org/wiki/Singularity_theorems , combined with observation, proves that there was a big bang singularity. (This is within the context of GR, not other theories, and it assumes certain energy conditions http://en.wikipedia.org/wiki/Energy_condition .) That means that, since about the 1970's, it's been clear that no static spacetime can be a valid cosmological model of our universe.

I think this statement is misleading without further explanation,

No, the statement is not misleading without further explanation. Please see the text in bold above.

for instance one can read in the wiki article:
Wiki quote
"Hawking's singularity theorem is for the whole universe, and works backwards-in-time: in Hawking's original formulation, it guaranteed that the Big Bang has infinite density. Hawking later revised his position in "A Brief History of Time" (1988) where he stated "There was in fact no singularity at the beginning of the universe" (p50). This revision followed from quantum mechanics, in which general relativity must break down at times less than the Planck time. Hence general relativity cannot be used to show a singularity
" End Quote

So what the Hawking theorem proved in the 70's first version is that every FRW solution of General relativity (without cosmological constant) must have an initial singularity with infinite density, so first of all it only applies to a certain type of GR solutions and it claims nothing about other cosmological models, static or otherwise.

No, this is incorrect. The Hawking singularity theorem is not restricted to FRW cosmologies.

I think it is important to clarify this as Hawking theorem is often used to make unwarranted claims about cosmology and it usually is used pretending to prove things it actually doesn't prove at all, I'm not sure if this is done relying on the fact not many people know exactly what the theorem says and how it has been revised based on QM or by posterior observations like accelerated expansion interpreted as CC, but it is obvious the theorem has as "a priori" assumptions not only an expanding universe, but no cosmological constant or no inflationists scenarios so it can hardly say anything about any other model.

There are various versions of the Hawking singularity theorem, which are based on various energy conditions. The long and the short of it is that the validity of the result is not affected by the cosmological constant.

 

[Passionflower]

When Einstein conceived GR he added the cosmological constant in an attempt to create a static universe model. The problem with this was that it was unstable.

What do you mean by unstable? What do you think is exactly the problem?

 

[TrickyDicky]

No, the statement is not misleading without further explanation. Please see the text in bold above.

I was being polite, not only is misleading but wrong. Perhaps you should check the literature on this subject, there are several papers published in peer reviewed journals -see J.M.M. Senovilla, Phys. Rev. Lett. 64, 2219 (1990), or J.M.M. Senovilla Gen. Rel. Grav.30,701(1998) or "A singularity-free space-time" Chinea et al.Phys.Rev. D45 (1992) 481 available in arxiv-that show there are solutions within GR that are singularity free, in fact this is not just in journals , for instance in the textbook "An introduction to General relativity and cosmology" by Krasinski, page 231 you can read:
Quote
"the singularity theorems are not a general as it was initially claime. Several interesting solutions that do not contain any singularities have been found by Senovilla and coworkers (for an extended review see Senovilla (1998). They have not, so far, been shown to describe any actual astrophysical situations, but teir very existence proves that singularities are an inevitable part not of relativity as such, but of the collection of models of matter defined by the assumptions of the singularity theorems. " End Quote

If you are relatively new to GR, maybe you could inform yourself better about this before making evidently wrong statements.

No, this is incorrect. The Hawking singularity theorem is not restricted to FRW cosmologies.

More accurately is restricted to "the collection of models of matter defined by the assumptions of the singularity theorems" of which FRW cosmologies seem to be the main subset.

There are various versions of the Hawking singularity theorem, which are based on various energy conditions. The long and the short of it is that the validity of the result is not affected by the cosmological constant.

You are disagreeing here with the wiki page, perhaps you'd like to modify the phrase from it I quoted in my last post.

 

[bcrowell]

I was being polite, not only is misleading but wrong. Perhaps you should check the literature on this subject, there are several papers published in peer reviewed journals -see J.M.M. Senovilla, Phys. Rev. Lett. 64, 2219 (1990), or J.M.M. Senovilla Gen. Rel. Grav.30,701(1998) or "A singularity-free space-time" Chinea et al.Phys.Rev. D45 (1992) 481 available in arxiv-that show there are solutions within GR that are singularity free, in fact this is not just in journals , for instance in the textbook "An introduction to General relativity and cosmology" by Krasinski, page 231 you can read:
Quote
"the singularity theorems are not a general as it was initially claime. Several interesting solutions that do not contain any singularities have been found by Senovilla and coworkers (for an extended review see Senovilla (1998). They have not, so far, been shown to describe any actual astrophysical situations, but teir very existence proves that singularities are an inevitable part not of relativity as such, but of the collection of models of matter defined by the assumptions of the singularity theorems. " End Quote

This is why my original post stated "...combined with observation..."

More accurately is restricted to "the collection of models of matter defined by the assumptions of the singularity theorems" of which FRW cosmologies seem to be the main subset.

I don't know what you mean by "main subset," but it looks like you've recognized your mistake.

You are disagreeing here with the wiki page, perhaps you'd like to modify the phrase from it I quoted in my last post.

I don't see any errors in the WP article. The WP article does not contradict anything I've said. You simply failed to read my post carefully enough to notice the assumptions that I explicitly stated.

 

[TrickyDicky]

This is why my original post stated "...combined with observation..."

And once again with that assumption included your statement is either wrong or misleading without specifying what observations exactly together with Hawking theorem prove that there was a big bang singularity. Have you discovered a new theorem? Please share it with us.

I don't know what you mean by "main subset," but it looks like you've recognized your mistake.

what mistake? Since you are so careful with what is explicitly stated, If you read again my post I don't explicitly say that is restricted to FRW models but that the 1970's version proved that the FRW models must have an initial singularity and then go on to say that it only applies to a certain type of GR solutions, which is true.
BTW, have you already recognized your confusion about what the singularity theorems prove or not prove?