Light elements abundance in a static toy universe

[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.