Ibix said:

It can't be, because it doesn't expand.

A point about the Einstein static universe

Well, I don't know if Einstein's original model can be considered as steady-state model; I would say it's static. The one I'm referring to is this steady-state model that does expand.

Jaime Rudas said:

The one I'm referring to is this steady-state model that does expand.

That model obviously violates General Relativity because of its "continuous creation of matter", which is impossible by the Einstein Field Equation. AFAIK this criticism has never been fully addressed by proponents of the steady-state model (which has morphed several times as new observations have been made).

Jaime Rudas said:

Well, I don't know if Einstein's original model can be considered as steady-state model; I would say it's static. The one I'm referring to is this steady-state model that does expand.

Oh, I see.

I don't know. I suspect you'd need a complete theory of such a universe to answer that authoritatively, and I'm not sure there is one. I do believe that extinction is an important part of how steady state proponents propose to get redshift, so they probably propose that same mechanism to dim distant stars. Faster-than-inverse-square dimming would resolve Olber's paradox in favour of a dark sky.

PeterDonis said:

That model obviously violates General Relativity because of its "continuous creation of matter", which is impossible by the Einstein Field Equation. AFAIK this criticism has never been fully addressed by proponents of the steady-state model (which has morphed several times as new observations have been made).

The steady-state cosmological model was widely accepted by a large part of the scientific community in the 1950s, especially by those who considered any model implying a beginning for the universe unacceptable, and even more so if that model was formulated by a Catholic priest.

Observations made from the mid-1960s to the present have consistently supported the Big Bang model and contradicted the steady-state model.

But my question wasn't about whether that model fit reality, but rather whether or not it could explain Olbers' paradox through redshift.

Jaime Rudas said:

The steady-state cosmological model was widely accepted by a large part of the scientific community in the 1950s

I wouldn't say "a large part", but it was a signficant number, yes.

Jaime Rudas said:

whether or not it could explain Olbers' paradox through redshift.

I don't know if the mathematical development of the model was sufficiently detailed to know whether it would make a prediction for this, and if so, what it would be.

PeterDonis said:

I don't know if the mathematical development of the model was sufficiently detailed to know whether it would make a prediction for this, and if so, what it would be.

The steady-state model implies that the density of the universe and the rate of expansion are constant in space and time. The question, then, is whether, under these conditions, Olbers' paradox is resolved by the effect of redshift. In other words, my question is whether redshift alone could explain Olbers' paradox.

Jaime Rudas said:

The steady-state model implies that the density of the universe and the rate of expansion are constant in space and time.

But that alone is not sufficient to evaluate the effects of redshift. To do that you also need a spacetime geometry, and, as I've said, the steady-state model is inconsistent with the Einstein Field Equation (because of the continuous creation of matter), so it's not clear what spacetime geometry we would use. Again, I'm not aware of any actual math for this that's been proposed by any steady state model proponents. And without that, the question is simply not answerable.

PeterDonis said:

But that alone is not sufficient to evaluate the effects of redshift. To do that you also need a spacetime geometry, and, as I've said, the steady-state model is inconsistent with the Einstein Field Equation (because of the continuous creation of matter), so it's not clear what spacetime geometry we would use. Again, I'm not aware of any actual math for this that's been proposed by any steady state model proponents. And without that, the question is simply not answerable.

But if spacetime is homogeneous and isotropic, isn't the metric necessarily FLRW?
And if the density is constant, isn't the Hubble parameter H necessarily also constant?
And if H is constant, doesn't that imply that ##a(t)=a_0 e^{Ht}##?

PeterDonis said:

Again, I'm not aware of any actual math for this that's been proposed by any steady state model proponents. And without that, the question is simply not answerable.

Actually, a few different mathematical models have been proposed for the steady-state universe. For example, take a look at this 1960 paper by Bonnor and the references therein: The relativistic model of the steady-state universe.

renormalize said:

Actually, a few different mathematical models have been proposed for the steady-state universe. For example, take a look at this 1960 paper by Bonnor and the references therein: The relativistic model of the steady-state universe.

Equation 1.1 of the cited paper indicates that it is the FLRW metric with constant H.

Jaime Rudas said:

But if spacetime is homogeneous and isotropic, isn't the metric necessarily FLRW?

PeterDonis said:

Not if your theory violates the Einstein Field Equation.

I disagree. A homogeneous and isotropic spacetime is necessarily described by the FLRW metric, regardless of whether or not it satisfies Einstein's field equations.

Jaime Rudas said:

And if the density is constant, isn't the Hubble parameter H necessarily also constant?
And if H is constant, doesn't that imply that ##a(t)=a_0 e^{Ht}##?

PeterDonis said:

The Bonnor paper that @renormalize cited draws this conclusion, yes--basically that the spacetime must be de Sitter spacetime (pure exponential expansion).

Bonnor's paper doesn't draws this conclusion. Bonnor's paper STARTS from the premise that the steady-state model metric is:
$$ds^2=-e^{2kt}(dr^2+r^2 d\theta+ r sin^2 \theta d\phi) +dt$$
where a(t) is of the form ##a_0 e^{Ht}##.

That is, it suggests that the steady-state model has an FLRW metric despite not satisfying Einstein's field equations.

Jaime Rudas said:

A homogeneous and isotropic spacetime is necessarily described by the FLRW metric, regardless of whether or not it satisfies Einstein's field equations.

If it doesn't satisfy the EFEs it won't satisfy the Friedmann equations either. It feels like a bit of a stretch to classify a spacetime as FLRW if it doesn't follow Friedmann.

Jaime Rudas said:

That is, it suggests that the steady-state model has an FLRW metric despite not satisfying Einstein's field equations.

I think what Bonnor is doing is pointing out that the quoted metric isn't a steady state universe in the sense McCrae wants. The metric produces a constant energy density, yes, but it's everywhere exotic matter and yields dynamics inconsistent with observation in the Newtonian limit.

Ibix said:

If it doesn't satisfy the EFEs it won't satisfy the Friedmann equations either. It feels like a bit of a stretch to classify a spacetime as FLRW if it doesn't follow Friedmann.

There is no contradiction; they are simply two different things. One is the FLRW metric, which is general for all homogeneous and isotropic spacetimes, and the other is the Friedman equations, which deal with the behavior of the scale factor for the particular case of spacetimes that satisfy Einstein's field equations.

Jaime Rudas said:

the FLRW metric, which is general for all homogeneous and isotropic spacetimes

What is your basis for this claim?

PeterDonis said:

What is your basis for this claim?

The principal contribution of Robertson and Walker was to establish that the metric previously introduced by Friedman and Lemaître is the most general metric compatible with a homogeneous and isotropic spacetime, as demonstrated in the original papers by Robertson (1935) and Walker (1937).

It turns out that Ben Crowell wrote an Insights article on the steady-state model:

https://www.physicsforums.com/insights/steady-state-model-no-longer-viable/

It discusses Fred Hoyle's way of addressing the "continuous creation of matter" problem, by adding what he called the "C-field" to the model, a sort of scalar field with zero energy density and negative pressure, so the condition ##p = -\rho## only applies to the combination of ordinary matter plus the C-field. That in itself would just be an "interpretation", so to speak, of the de Sitter metric, but Hoyle added to the model the continuous creation of hydrogen atoms by the C-field, which breaks local Lorentz invariance and violates the equivalence principle, as the Insights article discusses.

Crowell notes that the C-field mechanism, on its face, doesn't appear to violate local stress-energy conservation (zero covariant divergence of the stress-energy tensor), which was the issue I raised earlier in this thread with "continuous creation of matter". However, his description of how that works appears to me to leave something out: the "continuous creation" seems to be converting C-field to ordinary matter, and I'm not sure how that can be done while keeping the total energy density ##\rho## constant and continuing to satisfy ##p = - \rho##. I'll have to see if I can find a copy of Hoyle's paper online to look at the details of the math.

Jaime Rudas said:

Bonnor's paper STARTS from the premise that the steady-state model metric is:
$$ds^2=-e^{2kt}(dr^2+r^2 d\theta+ r sin^2 \theta d\phi) +dt$$
where a(t) is of the form ##a_0 e^{Ht}##.

That is, it suggests that the steady-state model has an FLRW metric despite not satisfying Einstein's field equations.

PeterDonis said:

It suggests no such thing. As just noted (and as I already noted in my earlier post), this metric does satisfy the EFE.

Bonnor's paper says:

In the steady-state model of the universe space-time is described by a four-dimensional Riemann space with metric
$$ds^2=-e^{2kt}(dr^2+r^2 d\theta+ r sin^2 \theta d\phi) +dt$$

That metric is the FLRW metric when ##a(t)=-e^{2kt}##

PeterDonis said:

the steady-state model is inconsistent with the Einstein Field Equation

From the above, I conclude that the steady-state model is inconsistent with the Einstein's field equations and has an FLRW metric.

Jaime Rudas said:

From the above, I conclude that the steady-state model is inconsistent with the Einstein's field equations and has an FLRW metric.

You are correct that the metric you quoted describes an FLRW metric, since de Sitter spacetime, which is what that metric describes, is in the class of FLRW spacetimes.

You are wrong that that metric is inconsistent with the EFE. I have already explained why it is not, more than once. I have also explained how the Bonnor paper itself shows that it is consistent with the EFE.

As for the post of mine that you quoted, it has nothing whatever to do with the metric you quoted, nor can it be combined with any of the other things you quoted in your post to reach any useful conclusion. The post of mine that you quoted gave a completely different reason for believing that the steady-state model is inconsistent with the EFE. I have also explained in previous posts how that reason is not present in the metric you quoted, which, as I have also stated (and as the Bonnor paper states), is just de Sitter spacetime. That is a reason for believing, as I've also said in a previous post, that the metric you quoted cannot be a description of the steady-state model--since it lacks the crucial feature I described.

In short, you are cherry picking things that have nothing to do with each other, and then trying to tie them together into some kind of argument. That's not valid. Nor is it contributing anything useful to this discussion.

PeterDonis said:

You are correct that the metric you quoted describes an FLRW metric, since de Sitter spacetime, which is what that metric describes, is in the class of FLRW spacetimes.

Yes, and I also said that Bonnor presents it as the metric that describes the steady-state model.

PeterDonis said:

You are wrong that that metric is inconsistent with the EFE.

What I have said about this is the following:

Jaime Rudas said:

A homogeneous and isotropic spacetime is necessarily described by the FLRW metric, regardless of whether or not it satisfies Einstein's field equations.
[...]
That is, it suggests that the steady-state model has an FLRW metric despite not satisfying Einstein's field equations.

I apologize for my poor writing. What I mean to say is that the FLRW metric describes homogeneous and isotropic spacetimes even when these spacetimes don't satisfy Einstein's field equations, and that the steady-state model, despite not satisfying Einstein's field equations, is described by the FLRW metric.

PeterDonis said:

I have already explained why it is not, more than once. I have also explained how the Bonnor paper itself shows that it is consistent with the EFE.

As for the post of mine that you quoted, it has nothing whatever to do with the metric you quoted, nor can it be combined with any of the other things you quoted in your post to reach any useful conclusion.

No, no, no. I didn't just quote you; I also quoted Bonnor. From your quote, I concluded that the steady-state model is inconsistent with Einstein's field equations, and from Bonnor's quote, I concluded that the steady-state model has an FLRW metric.

PeterDonis said:

The post of mine that you quoted gave a completely different reason for believing that the steady-state model is inconsistent with the EFE. I have also explained in previous posts how that reason is not present in the metric you quoted, which, as I have also stated (and as the Bonnor paper states), is just de Sitter spacetime. That is a reason for believing, as I've also said in a previous post, that the metric you quoted cannot be a description of the steady-state model--since it lacks the crucial feature I described.

So, is Bonnor wrong to say that this is the metric of the steady-state model?

PeterDonis said:

In short, you are cherry picking things that have nothing to do with each other, and then trying to tie them together into some kind of argument. That's not valid. Nor is it contributing anything useful to this discussion.

I consider that a very serious and totally unjustified accusation, so I ask that you explain in detail where the alleged cherry picking takes place.

Jaime Rudas said:

is Bonnor wrong to say that this is the metric of the steady-state model?

Bonnor himself does not make the claim that that metric is the metric of the steady-state model. He references a paper by McCrea that uses this metric and claims that it is the metric of the steady state model. However, Bonnor's analysis, as we've discussed, shows that it can't be. There is no "continuous creation of matter" in the spacetime, and indeed there is no matter in it at all. Nor is there any definite state of motion that the matter (stars and galaxies) could have--but matter (stars and galaxies) has to have some definite state of motion. So that metric simply does not describe what the steady-state model is trying to describe.

Jaime Rudas said:

I ask that you explain in detail where the alleged cherry picking takes place.

I already did. I'm not going to repeat myself.

Jaime Rudas said:

the steady-state model, despite not satisfying Einstein's field equations

So far in this thread we have not seen any mathematical model that does not satisfy the EFE. If you are claiming that the metric referenced in the Bonnor paper is the metric of the steady-state model, then, as I've already pointed out, it is obviously wrong to say that the steady-state model does not satisfy the EFE, since that metric does.

Also, I strongly suggest that you read what I said in post #19 about satisfying the EFE (if you already have, read it again). I don't think you have fully thought through what Bonnor is actually doing in his paper.

PeterDonis said:

Bonnor himself does not make the claim that that metric is the metric of the steady-state model.

He references a paper by McCrea that uses this metric and claims that it is the metric of the steady state model.

Bonnor does make the claim that that metric is the metric of the steady-state model, and at no point does he link this statement to McCrea's paper.

PeterDonis said:

If you are claiming that the metric referenced in the Bonnor paper is the metric of the steady-state model,

I'm not claiming that, Bonnor is claiming that that metric is the metric of the steady-state model

Jaime Rudas said:

There is no contradiction; they are simply two different things. One is the FLRW metric, which is general for all homogeneous and isotropic spacetimes, and the other is the Friedman equations, which deal with the behavior of the scale factor for the particular case of spacetimes that satisfy Einstein's field equations.

My cosmology notes refer to the metric with ##a(t)## unconstrained by the Friedmann equations as the RW metric, and with the additional constraint of the Friedmann equations/EFEs as the FRW metric. Lemaitre isn't referenced. However, reading the Wiki article, I note that this particular distinction may reflect the personal preferences of my lecturer and/or whoever taught him.

Jaime Rudas said:

So, is Bonnor wrong to say that this is the metric of the steady-state model?

I think we need to be careful with what we mean by "the steady state model", especially in light of what Bonnor shows in that article.

He certainly states that the metric is the metric of the steady state model, and McCrea clearly believed this too. However, Bonnor promptly shows that it is not a realistic model of our universe, being what we would now call pure dark energy. So I think Bonnor is arguing along the lines that there is a self-contradiction here: the metric is the metric of a EFE-consistent steady state cosmological model, but it doesn't describe anything remotely like our universe so it can't be the steady state model that Hoyle et al had in mind. Hence I think some care about terminology is needed because Bonnor is deliberately contradicting himself (at least in some senses) to make his point.