How to understand Friedman equation

[cyc454]

Hi, can you help mi understand Friedman equation with cosmological constant:
\begin{equation}
(\frac{\dot a}{a})^{2} - \frac{8\pi G \rho}{3}=-\frac{k}{a^2}+\frac{\Lambda}{3}
\end{equation}
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 $\rho$?

 

[bapowell]

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

 

[cyc454]

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

 

[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.

 

[bapowell]

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

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

 

[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..

 

[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.

 

[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:
$L_{matter} \rightarrow L_{matter} + \frac{ \Lambda}{8 \pi G}$
(http://ned.ipac.caltech.edu/level5/Sept02/Padmanabhan/Pad1_1.html)

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...

 

[PeterDonis]

I don't understand how Einstein named it 1 of his biggest mistakes...

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).

 

[George Jones]

So I don't understand how Einstein named it 1 of his biggest mistakes...

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?

 

[ChrisVer]

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

With a simple google search, for historical maybe reasons:
http://phys.org/news/2011-10-einstein-wrong.html

however, Einstein was more concerned with its consequences for general relativity. If the universe was expanding, the cosmological constant wasn't needed. His beautiful equations had been right to begin with. In 1931, Einstein came to Mount Wilson to shake Hubble's hand and thank him for saving relativity from the cosmological constant, whose invention Einstein denounced as "the greatest blunder of my life."

unfortunately many moe articles write about this... In contrast I give also this:
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)...

But since, experimentally, it has turned out to be very small.

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)...

 

[PeterDonis]

the smallness of the cosmological constant (or better put the vacuum energy) is a big question in cosmology nowadays

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