# How to understand Friedman equation

1. Nov 12, 2014

### cyc454

Hi, can you help mi understand Friedman equation with cosmological constant:

(\frac{\dot a}{a})^{2} - \frac{8\pi G \rho}{3}=-\frac{k}{a^2}+\frac{\Lambda}{3}

I don't get it: Einstein wanted to get flat ($k=0$) and static ($(\frac{\dot a}{a})^{2}=0$) model with $\rho>0$ so he put $\Lambda$ into his equation of field (Friedman equation should look like above I think). He put $\Lambda$ greater then 0 so how it is possible that it compensates positive curvature caused by $\rho$?

2. Nov 12, 2014

### bapowell

The Einstein static universe is closed, i.e. it has positive spatial curvature.

3. Nov 13, 2014

### cyc454

soo... Did he put k=1 or what? Is $\lambda$ required in that case?

4. Nov 13, 2014

### Chronos

The Friedman equation is merely a conversion of Einstein's field equations into differential equations that have calculable results. I fail to see how that is controversial.

Last edited: Nov 13, 2014
5. Nov 13, 2014

### bapowell

Yes, $k=1$. What is $\lambda$?

6. Nov 13, 2014

### cyc454

sorry, my mistake;) I am asking about $\Lambda$---so Einstein thought that $\rho$ of our galaxy (which he considered our universe) is too small? That's why he put $\Lambda$ into his eq.? And how is it possible that non-empty universe with k=0 is flat? I thought that presence of matter calls curvature of spacetime..

Last edited: Nov 13, 2014
7. Nov 13, 2014

### bapowell

No, he assumed a closed, matter-filled universe. Without $\Lambda$, this universe eventually collapses under the attractive force of gravity. The cosmological constant provides a repulsive component that keeps the universe static.

8. Nov 16, 2014

### ChrisVer

the easiest (at least conceptually) reason to see why someone would add this parameter is by looking at the Lagrangian of matter and adding an extra constant term:
$L_{matter} \rightarrow L_{matter} + \frac{ \Lambda}{8 \pi G}$

This will keep the equations of motion for the matter unchanged. However it will affect the equations for the metric because this term will be (like matter) coupled to the gravitational field ( $S \sim \int d^4 x \sqrt{-g} \Lambda$) (and thus add the cosmological constant). So mathematically it is totally legible to add this term...
I don't know but since this term is actually allowed by the mathematics, even if someone wants to drop it away, they should find a reason to set it equal to zero...So I don't understand how Einstein named it 1 of his biggest mistakes.....

9. Nov 16, 2014

### Staff: Mentor

He said that because, if he hadn't been so fixated on finding a static solution for the universe, he would have realized that his original field equation, without a cosmological constant, predicted a dynamic universe--i.e., the only solutions for a homogeneous, isotropic mass distribution were either expanding or contracting. So he could have told astronomers that his theory predicted that the universe would be expanding (or contracting) and advised them to look for evidence of this. But he didn't.

I agree that, conceptually, the cosmological constant term should be there; there is no a priori reason to expect it to be zero. But since, experimentally, it has turned out to be very small, Einstein could still have reasoned as I described above, based on his original field equation (with no cosmological constant), and his conclusion would have been valid (at least as a good enough approximation for that time).

10. Nov 16, 2014

### George Jones

Staff Emeritus
I am not convinced that he did.

Can anyone find a source for this that doesn't trace back to the tall-tale-teller George Gamow?

Last edited: Nov 16, 2014
11. Nov 16, 2014

### ChrisVer

With a simple google search, for historical maybe reasons:
http://phys.org/news/2011-10-einstein-wrong.html
http://www.theatlantic.com/technolo...-one-of-his-most-oft-quoted-phrases/278508/2/
which targets against Gamow.... of course (in my opinion) it's all a game of words... in this Livio states that Einstein just didn't like the cosmological constant (proved by Einstein's letters to friends) but of course he didn't call it his biggest blunder (thus in the end he tries to change the words to be smoother)...

Apart from those "Einstein's things" even the smallness of the cosmological constant (or better put the vacuum energy) is a big question in cosmology nowadays. I mean, whatever you do, you have to find a mechanism that generates such a small value for the vacuum energy (maybe through a fine-tuning if I understand that term correct)...

Last edited: Nov 16, 2014
12. Nov 16, 2014

### Staff: Mentor

Yes, it is. John Baez gives a good quick summary of the main issue here.