Does the steady-state model resolve Olbers' paradox?

[Jaime Rudas]

[Moderator's note: Thread spun off due to topic change.]

No. Things far away are seen as they were in the past, and redshifted. Far enough away and you see the universe as it was before star formation and ultimately you see the last scattering surface, so you do not see stars in every direction. This is the resolution to Olber's paradox.

In the steady-state model, is Olbers' paradox resolved by redshift?

 

[Ibix]

In the steady-state model, is Olers' paradox resolved by redshift?

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

A point about the Einstein static universe - AFAIK it's unstable to small perturbations, so it can't have stars anyway without becoming an expanding or contracting FLRW universe. In fact, it can only be filled with an ideal FLRW medium - i.e. an ideal fluid everywhere in eternal thermal equilibrium with itself. So I think that the resolution of Olber's paradox in that spacetime is that the sky is equally bright everywhere.

 

[Jaime Rudas]

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.

 

[PeterDonis]

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).

 

[Ibix]

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.

 

[Jaime Rudas]

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.

 

[PeterDonis]

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.

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.

 

[Jaime Rudas]

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.

 

[PeterDonis]

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.

 

[Jaime Rudas]

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 $a(t)=a_0 e^{Ht}$?

 

[renormalize]

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.

 

[Jaime Rudas]

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.

 

[Ibix]

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

Unfortunately, the paper then shows that the equations of motion of the cosmological fluid in such a system are completely undetermined. At the end of section 3 (about half way down p 477) it points out that formulae for redshift and other such phenomena in this universe therefore include an unknown function $dr/ds$ that would have to be supplied by some other physical theory. (And much of the rest of the paper is dedicated to pointing out how implausible any such theory would have to be.)

 

[PeterDonis]

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

Not if your theory violates the Einstein Field Equation.

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

The Bonnor paper that @renormalize cited draws this conclusion, yes--basically that the spacetime must be de Sitter spacetime (pure exponential expansion) in order to satisfy the requirement of constant "density" (why that's in scare quotes will become apparent below) and rate of expansion.

Unfortunately, there is no "continuous creation of matter" in de Sitter spacetime. There can't be, because that spacetime is a solution of the EFE--the Bonnor paper uses the EFE to derive it, though IIRC de Sitter's original derivation was somewhat different--and "continuous creation of matter" is inconsistent with the EFE, as I've already pointed out. But we know it anyway because we know all of the properties of de Sitter spacetime, and that "continuous creation of matter" is not one of them.

Indeed, there is no matter at all in de Sitter spacetime; the only "stress-energy" present is the cosmological constant. The Bonnor paper doesn't call it that because it starts with the EFE with no "cosmological term", but in fact what it ends up with for what it calls the "stress-energy tensor" is the cosmological term--in terms of the Friedmann equation, $p = - \rho$. Of course this is well known, that you can just as well move the cosmological term to the RHS of the EFE and call it "stress energy" ("dark energy"). But that doesn't change the fact that it's not "matter"--you can't make stars and planets out of it.

The Bonnor paper glosses over this at first as well, concluding instead that what it calls the "motion of matter" in the model is "indeterminate". What it is really saying, in more standard GR language, is that de Sitter spacetime is maximally symmetric--it has a 10-parameter group of Killing vector fields, just like "empty" Minkowski spacetime. So there is no preferred congruence of timelike worldlines picked out by the geometry, as there is in, for example, a matter-dominated ($p = 0$ in the Friedmann equation instead of $p = - \rho$) FLRW universe. Describing de Sitter spacetime as an FLRW spacetime obscures this fact because it requires picking out some particular slicing, corresponding to picking some particular inertial frame in Minkowski spacetime, and calling that the "comoving" slicing--but unlike in a matter-dominated FLRW universe, there are an infinite number of such slicings you could pick that all look the same, just as in Minkowski spacetime there are an infinite number of inertial frames you could pick that all look the same.

Later on, the Bonnor paper does conclude that "both the inertial and passive gravitational mass densities" are zero in the model. This, of course, is just another way of saying that de Sitter spacetime has no matter in it--the stress-energy tensor has only "dark energy" ($p = - \rho$) and nothing else.

In short, the Bonnor paper does not support the claim that the model it studies is a viable model within GR of the steady-state cosmology--rather the opposite, it illustrates why you can't have a viable model within GR of the steady-state cosmology.