How Does General Relativity Predict an Expanding Universe

[117]

Hello Everyone,
Back when Einstein was formulating General Relativity his equations just could not predict a static universe. I have read that they actually predicted an expanding Universe. Later Friedmann derived an equation from GR that would explain how an Expanding Universe would evolve. This was a very unpopular opinion at the time, I think they didn't even think there was more than one galaxy in the universe, this caused Einstein to add a Cosmological Constant such that it would result in a static universe. Now my understanding of General Relativity is quite elementary so I am curious as to these equations would predict an expanding universe without the need for some sort of 'Big Bang' to set the universe into motion

I am also curious as to the role of radiation density in an expanding universe.
Thanks in advance,

 

[PeterDonis]

Back when Einstein was formulating General Relativity his equations just could not predict a static universe.

More precisely, they could not predict a static universe without a cosmological constant in them. Einstein's original derivation did not include one, but that was because his original derivation, as we now know, was actually not the most general one possible.

I have read that they actually predicted an expanding Universe.

The original equations, without a cosmological constant, allow solutions that describe both and expanding and a contracting universe. There is nothing in the equations that privileges one over the other.

I am curious as to these equations would predict an expanding universe without the need for some sort of 'Big Bang' to set the universe into motion

They don't. More precisely, they don't tell you what originally "set the universe into motion". They tell you that at some time in the past, the universe was very hot, very dense, and expanding very rapidly, and that since then, it has gotten much cooler, much less dense, and is expanding much more slowly. But they don't tell you what originally caused the early hot, dense, rapidly expanding state.

I am also curious as to the role of radiation density in an expanding universe.

In the early universe, radiation density was very significant; it was actually larger than the density of matter, so the radiation density dominated the dynamics, i.e., it was the primary determiner of how the rate of expansion evolved. Over time, however, the radiation density decreased faster than the matter density, and so it gradually became negligible. Today it is several orders of magnitude smaller than the matter density and has no significant effect on the dynamics.

 

[117]

What did Einstein think his equations predicted, did he think it would predict expansion, or contraction. Would this impact which side of the equation Einstein would want to put a cosmological constant in?

 

[Smattering]

What did Einstein think his equations predicted, did he think it would predict expansion, or contraction.

When Einstein worked on GR, people thought that the universe was static. But without a cosmological constant, a current state of equilibrium will pass over into contraction. So I guess that without the cosmological constant, Einstein would have predicted contraction.

Would this impact which side of the equation Einstein would want to put a cosmological constant in?

I do not understand your question.

 

[117]

As in Left Hand Side or Right Hand Side, a cosmological constant would supposedly have opposite effects based on which side of the equation you put the term on.

 

[Smattering]

As in Left Hand Side or Right Hand Side, a cosmological constant would supposedly have opposite effects based on which side of the equation you put the term on.

But in principle, you can move any constant from the lhs of an equation to the rhs by simple addition resp. subtraction. So the difference would merely be the sign of the constant. Thus, I still do not get the question.

 

[117]

Yeah that is true when you're rearranging an equation, but I'm referring to just randomly adding a term in. Like if you added a positive cosmological constant to one side it might have a repulsive effect, but on another side an attractive effect? Anyway I understand my question was a bit vague so sorry.

 

[PeterDonis]

What did Einstein think his equations predicted, did he think it would predict expansion, or contraction.

It predicts both. As I said in post #2, without a cosmological constant there are expanding solutions and contracting solutions, but no static solutions. AFAIK Einstein didn't really care which one, expanding or contracting; the main thing he cared about was that there were no static solutions. That was why he added the cosmological constant.

 

[PeterDonis]

a cosmological constant would supposedly have opposite effects based on which side of the equation you put the term on.

Yes. The solution with a positive cosmological constant on the LHS and zero on the RHS (i.e., no other stress-energy present) is called "de Sitter spacetime", and the solution with a negative cosmological constant on the LHS and zero on the RHS is called "anti-de Sitter spacetime". These are certainly different solutions.

In the Einstein static universe, there is a positive cosmological constant on the LHS of the field equation, and a nonzero stress-energy tensor describing ordinary matter and energy on the RHS, and the effects of the two balance in just the right way to make the solution as a whole static. There is no way to do this with a negative cosmological constant on the LHS.

 

[Chalnoth]

When Einstein worked on GR, people thought that the universe was static. But without a cosmological constant, a current state of equilibrium will pass over into contraction. So I guess that without the cosmological constant, Einstein would have predicted contraction.

Even with a cosmological constant, it's not possible to have a static (non-expanding) universe.

The problem is that the static universe solution with a cosmological constant is unstable: if there is any motion of matter, then some parts of the universe will get a little bit more or less dense than others. The more dense regions will collapse on themselves, while the less dense regions will expand forever. It's just not possible to have a universe which neither expands nor contracts.

 

[PeterDonis]

Even with a cosmological constant, it's not possible to have a static (non-expanding) universe.

Not in practical terms, yes, for the reason you give. But the idealized model still exists. AFAIK Einstein didn't realize the stability issue with the model when he proposed it; it was only realized later.