Light elements abundance in a static toy universe

[Ken G]

The only explanation I can find to what you are saying is that you might be using the term "steady state" with a different meaning than I am. In fact in wikipedia at least two different meanings can be found: steady state as a kind of equilibrium of a system as used in many disciplines like thermodynamics and economics, and "steady state theory" or cosmology which is the specific model of universe that Hoyle et al. came up with in 1948 and that was seriously considered as alternative to BB universe until the 60's.

The term "steady state" is used in a very wide array of physics models, and it always means one thing: no explicit dependence on time or age. Including, no time dependence of H/He. That's quite a bit more than just a "static spacetime."

A a spacetime is said to be static if it admits a global, non-vanishing, timelike Killing vector field K which is irrotational, this is the standard definition and the one I'm following in my thought experiment as scenario for a putative plausible imaginary equilibrium distribution of chemical elements abundance.

Did you specify an age in your question? Then you don't just mean a static spacetime, you mean a static everything (including a non-varying H/He). Indeed, you said:

Yes, just the stellar nuclear reactions, only in a static universe makes little sense to say what one begins with, since time is invariant.

(my bold). If you didn't actually mean that time was invariant, only that the spacetime didn't depend on it, then ask your question again, but this time specify the age of the universe, rather than referring to a "steady-state" H/He ratio. It sounds like what you meant was, "what would the H/He ratio be, at age 13.7 billion years, in a static spacetime." The answer to that is the same as I said: stellar nucleosynthesis has not had a significant impact on H/He in 13.7 billion years, so H/He is whatever value you assume "at the beginning." The static spacetime, unlike the Big Bang, gives us no constraint on H/He at all. So yes, put like that, it is an interesting point to make-- but it was already made.

 

[TrickyDicky]

The term "steady state" is used in a very wide array of physics models, and it always means one thing: no explicit dependence on time or age. Including, no time dependence of H/He. That's quite a bit more than just a "static spacetime."
Did you specify an age in your question? Then you don't just mean a static spacetime, you mean a static everything (including a non-varying H/He). Indeed, you said:
(my bold). If you didn't actually mean that time was invariant, only that the spacetime didn't depend on it, then ask your question again, but this time specify the age of the universe, rather than referring to a "steady-state" H/He ratio. It sounds like what you meant was, "what would the H/He ratio be, at age 13.7 billion years, in a static spacetime." The answer to that is the same as I said: stellar nucleosynthesis has not had a significant impact on H/He in 13.7 billion years, so H/He is whatever value you assume "at the beginning." The static spacetime, unlike the Big Bang, gives us no constraint on H/He at all. So yes, put like that, it is an interesting point to make-- but it was already made.

In a static spacetime there is no age of the universe concept.

 

[Ken G]

In a static spacetime there is no age of the universe concept.

No, that is wrong. Of course there is still an age of the universe concept, it would just have to do with how old the matter is, not anything about the spacetime. For one thing, it would eventually all be iron, as was pointed out.

 

[TrickyDicky]

No, that is wrong. Of course there is still an age of the universe concept, it would just have to do with how old the matter is, not anything about the spacetime. For one thing, it would eventually all be iron, as was pointed out.

What is the age of a universe that has no beginning in time?

 

[Chronos]

TD, it appears you are implying the universe is infinitely old and all the evidence accumulated to date strongly suggests we do not reside in such a universe.

 

[TrickyDicky]

TD, it appears you are implying the universe is infinitely old and all the evidence accumulated to date strongly suggests we do not reside in such a universe.

We all know for sure we do not live in such universe, I thought words and expressions such as "imaginary","hypothetical", "cosmology-fiction", "thought experiment" in my posts would make that clear enough.
Also note that the words infinitely and old can't be logically put together.

 

[Ken G]

What is the age of a universe that has no beginning in time?

Infinite. So what? This doesn't tell us what H/He will be. For that, you need an age, or a timestamp of some kind (perhaps time since the last periodic event). Or, if you don't, then you have a steady-state value of H/He (which is just what we said you will not get). That exhausts the possibilities, so there is no sense in a question that asks for a static H/He but not a steady-state H/He, and gives no age or time stamp of any kind. The question has no meaning.

 

[TrickyDicky]

then you have a steady-state value of H/He (which is just what we said you will not get).

why?
This is the condition of the exercise.

 

[Ken G]

And this is the answer to the exercise: if you do not give an age, then it makes no difference what the spacetime is (static or expanding), you can never get an H/He unless the latter has reached a steady-state value. I'm sorry, that's just perfectly obvious. So you have two choices, even within a static spacetime:
1) specify the age of the universe, and derive H/He from that. If the age is short (along the lines of our current age), you cannot answer it because it depends on the initial value assumed, since stellar nucleosynthesis hasn't had enough time to do much. If the age is very long, you'll have all iron. If the age is somewhere in between, stellar nucleosynthesis rates, and the age given, will determine H/He.
2) use an effectively infinite age, which is tantamount to the last possibility of #1.
That is the answer to your exercise, and it's all been given above. I'm afraid I don't know what else you are looking for.

 

[TrickyDicky]

And this is the answer to the exercise: if you do not give an age, then it makes no difference what the spacetime is (static or expanding), you can never get an H/He unless the latter has reached a steady-state value. I'm sorry, that's just perfectly obvious. So you have two choices, even within a static spacetime:
1) specify the age of the universe, and derive H/He from that. If the age is short (along the lines of our current age), you cannot answer it because it depends on the initial value assumed, since stellar nucleosynthesis hasn't had enough time to do much. If the age is very long, you'll have all iron. If the age is somewhere in between, stellar nucleosynthesis rates, and the age given, will determine H/He.
2) use an effectively infinite age, which is tantamount to the last possibility of #1.
That is the answer to your exercise, and it's all been given above. I'm afraid I don't know what else you are looking for.

when you say 2) is equivalent to an age somewhere in between (last possibility of 1)) I cannot see how you reach that conclusion:infinite age=age somewhere in between?

 

[Ken G]

Ah, typo-- I meant the second case in #1, not the last case. If the age is long enough to reach a steady state, then age doesn't matter, and that is equivalent to an infinite age, in regard to the question you are asking. The bottom line is, if a question is posed that does not specify the age, one must assume the age doesn't matter, which is always equivalent to assuming a steady state, which is always equivalent to an infinite age, which means the answer is "all iron."

 

[TrickyDicky]

Ah, typo-- I meant the second case in #1, not the last case. If the age is long enough to reach a steady state, then age doesn't matter, and that is equivalent to an infinite age, in regard to the question you are asking. The bottom line is, if a question is posed that does not specify the age, one must assume the age doesn't matter, which is always equivalent to assuming a steady state, which is always equivalent to an infinite age, which means the answer is "all iron."

Ok, so the answer is "all iron", how come we get the same answer for a static "infinite age" universe and a for expanding "arbitrarily old (very old) age" universe?

 

[Ken G]

We don't necessarily-- the "all iron" is not guaranteed in an expanding scenario, the density might eventually drop too low to make stars, and the H/He at that point would be "frozen in" for all time following, much as the H/He ratio was "frozen in" in the original Big Bang nucleosynthesis. So "all iron" is only the static no-age-given answer, whereas "maybe all iron, maybe some frozen-in value of H/He" is the expanding answer. Some even think expansion might get so severe as to rip matter apart. So it's not clear what the asymptotic behavior of the expanding scenario actually is, because of the changes in the background spacetime.

 

[TrickyDicky]

It turns out the answer "all iron" is wrong for a static spacetime because that would require time evolution of the universe which is a feature static spacetimes don't have globally. Thanks Ken G anyway, at least you tried.

 

[Ken G]

It turns out the answer "all iron" is wrong for a static spacetime because that would require time evolution of the universe which is a feature static spacetimes don't have globally. Thanks Ken G anyway, at least you tried.

Simple logic indicates there are only two possibilities here:
1) You are wrong. You are saying that because the spacetime is static, no time evolution in any physical variable is possible. Which theory does that come from?
2) Your original question is meaningless. You asked for the static H/He ratio, and now you are saying that no evolution of that ratio is possible. If you believe that, then obviously the H/He ratio in a static universe is set by the initial condition, which you did not specify.
So take your pick-- your question has no answer, or has a simple answer that you don't believe. What a waste of time.

 

[TrickyDicky]

Simple logic indicates there are only two possibilities here:
1) You are wrong. You are saying that because the spacetime is static, no time evolution in any physical variable is possible. Which theory does that come from?

Not exactly, no time evolution of the H/He ratio would be possible, because it is considered a global time-dependent feature of the static spacetime.

2) Your original question is meaningless. You asked for the static H/He ratio, and now you are saying that no evolution of that ratio is possible. If you believe that, then obviously the H/He ratio in a static universe is set by the initial condition, which you did not specify.

I tried to specify it by considering the nuclear reactions in reversible form.

 

[Ken G]

Not exactly, no time evolution of the H/He ratio would be possible, because it is considered a global time-dependent feature of the static spacetime.

I have no idea why you think it is considered that. It certainly isn't considered that by cosmologists.

I tried to specify it by considering the nuclear reactions in reversible form.

As someone else said, if you change the physics, you can get any answer you want. But in this universe, H-->He is only reversible in the early minutes of the Big Bang, conditions that did not exist in your question. That's why the Big Bang model answers the H/He ratio-- it represents exactly the ratio of neutrons to protons one would expect to be "frozen in" from the reversible process p<-->n in the early minutes of the Big Bang, assuming expansion. The cores of stars tend to only result in p-->n.

 

[TrickyDicky]

It certainly isn't considered that by cosmologists.

You are right, that is because cosmologists generally deal with physically realistic scenarios, I'm having problems getting you people into the "thought experiment mode" here.

As someone else said, if you change the physics, you can get any answer you want. But in this universe, H-->He is only reversible in the early minutes of the Big Bang, conditions that did not exist in your question. That's why the Big Bang model answers the H/He ratio-- it represents exactly the ratio of neutrons to protons one would expect to be "frozen in" from the reversible process p<-->n in the early minutes of the Big Bang, assuming expansion. The cores of stars tend to only result in p-->n.

Again, "this" universe (ours) is not the one I'm talking about.
Yes, the cores of stars as isolated systems tend to p-->n, so in the hypothetical static spacetime some mechanism should be compensating this, I guess.

 

[Ken G]

You are right, that is because cosmologists generally deal with physically realistic scenarios, I'm having problems getting you people into the "thought experiment mode" here.

Not true, we have no issue with thought experiments. We like thought experiments, we think they are a nice way to learn real physics. Not make believe physics, though. You just didn't like the correct answer for some reason.

Again, "this" universe (ours) is not the one I'm talking about.
Yes, the cores of stars as isolated systems tend to p-->n, so in the hypothetical static spacetime some mechanism should be compensating this, I guess.

That's not a thought experiment, that's make believe. There's a difference.

 

[TrickyDicky]

What I cannot understand is why you won't concede that in a static spacetime there is time symmetry and therefore nuclear reactions would be reversible, so the "all iron" answer can never be the correct answer.

 

[Ken G]

I can't concede it because it's wrong, the physics of that claim is confused. The static character of the spacetime has nothing at all to do with the nuclear reactions possible. The latter depends, not on the spacetime (which simply defines the inertial paths, and asserts that they are always the same), but on the conditions of the matter (temperature, density, and so on), and the physical processes allowed in those conditions. The model would have reached a steady state if the age is effectively infinite, so all processes that can occur must balance their inverse process. That doesn't mean you have some known H/He ratio, it might just mean you don't have any of either H or He. I'm saying that is what you would indeed have, because the conditions one can assume for your static spacetime (given that they are unspecified, yet you asked your question anyway, we can assume you intended conditions of T and density like we find in the universe today), do not have a process for turning He back into H, so we are on a one-way street leading to iron. Hence the answer that you don't like. Now, obviously if you are allowed to invent imaginary physics, you can get any H/He you are more happy with, but then there is also no reason to pose your question here.

 

[TrickyDicky]

I can't concede it because it's wrong, the physics of that claim is confused. The static character of the spacetime has nothing at all to do with the nuclear reactions possible. The latter depends, not on the spacetime (which simply defines the inertial paths, and asserts that they are always the same), but on the conditions of the matter (temperature, density, and so on), and the physical processes allowed in those conditions.

Ok, let's imagine this spacetime was a solution of the EFE, in that case the matter conditions would also be fixed by the RHS of the EFE.

 

[Ken G]

Ok, let's imagine this spacetime was a solution of the EFE, in that case the matter conditions would also be fixed by the RHS of the EFE.

You think the H/He ratio, and the nucleosynthesis physics, shows up on the RHS of the EFE? What is actually there?

 

[twofish-quant]

You are right, that is because cosmologists generally deal with physically realistic scenarios, I'm having problems getting you people into the "thought experiment mode" here.

That's because it's not clear what rules you are imposing. If you can state the rules of the game, we can figure out what goes on.

Yes, the cores of stars as isolated systems tend to p-->n, so in the hypothetical static spacetime some mechanism should be compensating this, I guess.

And once you specify that mechanism then you get whatever answer you want.

 

[TrickyDicky]

You think the H/He ratio, and the nucleosynthesis physics, shows up on the RHS of the EFE? What is actually there?

Stress-energy tensor.

 

[Ken G]

Yes... no physics about nucleosynthesis at all. You can kind of tell this, actually-- Einstein did have a cosmological model with a static spacetime. So why didn't he go ahead and try to answer the question from your OP? Because he knew it would not be possible to do, there's not enough information without additional assumptions. Now, of course Einstein didn't know squat about nucleosynthesis, but what we do know about it now is what gives the answer "all iron", so Einstein would have then known his static solution was wrong in the absence of some new physics (which is what we are telling you, also). So the bottom line is, as has often been repeated, there are only two possible answers to your question:
1) if no new physics: all iron
2) if new physics: anything you want
I wish I had just said that from the start, but then again, I think I basically did.

 

[TrickyDicky]

And in the simplified case of a static universe with a fluid in thermodynamical equilibrium the stress-energy tensor is proportional to the hydrostatic pressure and the inverse of the metric tensor.

 

[TrickyDicky]

Yes... no physics about nucleosynthesis at all.

What?? So in your opinion, what physics is the matter tensor related to?

 

[Ken G]

The stress-energy tensor is what it is-- there are many different things that can lead to the same stress-energy tensor. You seem to imagine that tensor completely describes everything that is happening, but this is incorrect. Consider this analogy. As I write this, everything happening in my head can be influencing in some way the words that appear, yet you cannot take those words and infer everything happening in my head. So it is for the stress-energy tensor, and so it was for Einstein and his static spacetime cosmology, and that is also why he knew he could not use that cosmology to infer H/He. Why else do you think Einstein could design a theory around the stress-energy tensor without even knowing that nucleosynthesis existed?

To repeat: Einstein could make a static cosmology. He could not infer H/He from that cosmology, because he did not know the physics of nucleosynthesis. We do, so we can get H/He, and it's all iron, unless you want to put in some additional unknown physics, in which case you can get any answer you like.

 

[TrickyDicky]

The stress-energy tensor is what it is-- there are many different things that can lead to the same stress-energy tensor. You seem to imagine that tensor completely describes everything that is happening, but this is incorrect. Consider this analogy. As I write this, everything happening in my head can be influencing in some way the words that appear, yet you cannot take those words and infer everything happening in my head. So it is for the stress-energy tensor, and so it was for Einstein and his static spacetime cosmology, and that is also why he knew he could not use that cosmology to infer H/He. Why else do you think Einstein could design a theory around the stress-energy tensor without even knowing that nucleosynthesis existed?

To repeat: Einstein could make a static cosmology. He could not infer H/He from that cosmology, because he did not know the physics of nucleosynthesis. We do, so we can get H/He, and it's all iron, unless you want to put in some additional unknown physics, in which case you can get any answer you like.

You are weirdly hung upon that Einstein thing, that has nothing to do with my questions.
No, I don't think the stress tensor describes what you are thinking.
Thanks for your valuable help.

 

[Ken G]

You are weirdly hung upon that Einstein thing, that has nothing to do with my questions.

Yeah, why would the fact that your OP stipulated a static spacetime, which is just what Einstein had in his static spacetime cosmology, why would that be relevant? All you added was nucleosynthesis, as if knowledge of that would suddenly let H/He be calculated in Einstein's cosmology simply because our H/He "formed in equilibrium." So all equilibrium are exactly the same then, in your mind? No, they're not. But you can't get this so, this must conclude our conversation. There isn't much point in repeating further-- your question is answered: "all iron if you add nothing to your OP, or anything you want if you add some made up physics you did not stipulate."

 

[TrickyDicky]

That's because it's not clear what rules you are imposing. If you can state the rules of the game, we can figure out what goes on.

And once you specify that mechanism then you get whatever answer you want.

So all equilibrium are exactly the same then, in your mind? No, they're not.

Certainly. But equilibrium is certainly a good condition to start with. And a static fluid in equilibrium - therefore thermodynamic and hydrostatic equilibrium- is a particular equilibrium that simplifies the problem.
So that the ratio of protons and neutrons when they are allowed to freely and reversibly transform into each other (this particular equilibrium), I understand, is determined just by their relative masses. This seems to be the only stipulating that is needed to calculate a H/He ratio under the postulated conditions. But please correct me if this is not so.
Sorry for not making this stipulations clear in the OP.

 

[Ken G]

Certainly. But equilibrium is certainly a good condition to start with. And a static fluid in equilibrium - therefore thermodynamic and hydrostatic equilibrium- is a particular equilibrium that simplifies the problem.
So that the ratio of protons and neutrons when they are allowed to freely and reversibly transform into each other (this particular equilibrium), I understand, is determined just by their relative masses.

Not just that, also the temperature. Just saying you have an equilibrium is only the beginning, it means you have a temperature, but it doesn't tell you what the temperature is. You need that to get H/He in equilibrium between the processes that make He and those that make H. But as I said, simply having a static spacetime doesn't mean you have equilibrium everywhere, you can have stars forming and exploding and so on, and those are the processes that will turn everything into iron regardless of what is the average temperature. It sounds to me like you wanted not only a static spacetime, but also a homogeneous density, but that's generally not stable to gravity even locally (never mind the global instability that dooms static spacetimes). If we could have a stable static spacetime, that was also locally stable, so you have equilibrium everywhere at the same T, even then, you still need to know what that T is before you can know H/He. That's the crucial input from the Big Bang model-- it tells you what happens to T, and that is what inevitably gives you H/He ~ 4 (by mass).

 

[TrickyDicky]

Not just that, also the temperature. Just saying you have an equilibrium is only the beginning, it means you have a temperature, but it doesn't tell you what the temperature is. You need that to get H/He in equilibrium between the processes that make He and those that make H. But as I said, simply having a static spacetime doesn't mean you have equilibrium everywhere, you can have stars forming and exploding and so on, and those are the processes that will turn everything into iron regardless of what is the average temperature. It sounds to me like you wanted not only a static spacetime, but also a homogeneous density, but that's generally not stable to gravity even locally (never mind the global instability that dooms static spacetimes). If we could have a stable static spacetime, that was also locally stable, so you have equilibrium everywhere at the same T, even then, you still need to know what that T is before you can know H/He. That's the crucial input from the Big Bang model-- it tells you what happens to T, and that is what inevitably gives you H/He ~ 4 (by mass).

Thanks, this helps a lot.
You are correct also that I should have specified that it should be a stable homogeneous universe, which is as you very well point out an impossibility as there are no static and homogeneous cosmologies that are stable. In fact the only static homogeneous model is Einstein's universe and it is a well known fact that it is not stable .
I realize I left out a lot of important data in my OP and I apologyze again for it (I see now that could be frustrating from the answering POV).

Regarding temperature, I realize that it is a key component to compute a freeze-out neutron/proton ratio (and from that a H/He) with the Boltzmann statistics formula that includes the temperature and the mass of protons and neutrons in the BBN model. But I'm wondering if the concept of temperature would even make any sense in such a bizarre scenario as the one I'm imagining. It would seem temperature is very related to time asymmetry, and here we would have time symmetry. So I guess by pure logic a H/He ratio could be simply obtained in this imaginary setting from the fact that He-4 has four nucleons and hydrogen has one, and by chance it is also 4. But this leads nowhere so at this point I'm ready to wrap this up unless someone has any further comment to make.

 

[Ken G]

I think a lot has been cleared up. I don't think the temperature concept requires time asymmetry, because it has meaning in equilibrium, but I agree that a static cosmology has a lot of paradoxes associated with it, and I'm a little surprised neither Newton nor Einstein recognized that. Perhaps it was simply that their imaginations didn't grasp the alternative, and needed a little nudge from observations.