light elements abundance in a static toy universe

TrickyDicky

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. This latter is the sense I have been giving to the term "steady state universe". It is well known that this model is that of an expanding universe. It is not possible therefore for static universes to be a subset of an expanding universe as I hope you will agree.
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.
Now, as was pointed out before, in abstract terms every distribution is compatible with such a universe. My question is, is there a way to constrain this with the known nuclear reactions (in reversible form) and the physical conditions of stellar's cores?

I thought this was an interesting exercise, I'm not so sure now.

Ken G

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.

TrickyDicky

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

Ken G

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

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

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

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

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

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

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

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

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