Friedmann Equations connection to Einstein’s GR Field Equations...explain

[storyteller]

Can someone explain to me the connection between equations of Einstein and Friedmann, please?

And how they are relevant to the cosmology?

[joigus]

Einstein's equations in general are complicated. They involve second derivatives of the metric arranged in an object with many (10) components (Einstein tensor; LHS of EE). And they are non-linear. On the RHS of Einstein's eqs. you have the distribution of matter, radiation, etc., in the universe. Schematically, they are:

Geometry = matter

Under assumptions of symmetry at large scale (isotropy=space is the same in every direction; homogeneity=space is the same everywhere) you get to a simple form of EE that's FLRW (Friedmann, etc.) that only involves the scale factor, which codifies the expansion of the universe.

Very briefly, the Friedmann equations are Einstein's equations when you plug in several distributions of matter in the universe. On the RHS you plug in different distributions of matter dependent on the scale factor (radiation-dominated, matter-dominated, vacuum-energy dominated).

And you solve, and get a rough picture of the different phases of the universe.

I hope that was helpful.

[storyteller]

Thank you, joigus.

When I was in college, a whole boatload of physics weren’t needed and weren’t studied in civil engineering course, including Relativity, Quantum Mechanics, Particle Physics, Nuclear Physics, etc. 

So the course only focused on Newtonian theory on motions & on gravitation, and nothing on SR & GR.

So everything relating to astrophysics and cosmology are only recent interests that I have been reading about, trying to understand the explanatory side of the theory (BB) without delving into the mathematical aspects.

 

[joigus]

You may find it interesting to know that the Friedmann equation can be derived from Newtonian physics alone. The spatial curvature term of the Einstein tensor happens to coincide with the energy term in the corresponding Newtonian equations. 

This derivation you can find in Steven Weinberg's excellent book --although somewhat outdated today-- The First Three Minutes. Also in any of Leonard Susskind's lectures on cosmology (Youtube).

[beecee]

9 hours ago, joigus said:

Einstein's equations in general are complicated. They involve second derivatives of the metric arranged in an object with many (10) components (Einstein tensor; LHS of EE). And they are non-linear. On the RHS of Einstein's eqs. you have the distribution of matter, radiation, etc., in the universe. Schematically, they are:

Geometry = matter

Under assumptions of symmetry at large scale (isotropy=space is the same in every direction; homogeneity=space is the same everywhere) you get to a simple form of EE that's FLRW (Friedmann, etc.) that only involves the scale factor, which codifies the expansion of the universe.

Very briefly, the Friedmann equations are Einstein's equations when you plug in several distributions of matter in the universe. On the RHS you plug in different distributions of matter dependent on the scale factor (radiation-dominated, matter-dominated, vacuum-energy dominated).

And you solve, and get a rough picture of the different phases of the universe.

I hope that was helpful.

Nice rundown, just one querie from an old fuddy duddy...."Geometry = matter", should that be "Geometry = Gravity"? [you got a like from me anyway! 😉]

2 hours ago, joigus said:

This derivation you can find in Steven Weinberg's excellent book --although somewhat outdated today-- The First Three Minutes. Also in any of Leonard Susskind's lectures on cosmology (Youtube).

Excellent certainly, while the book is outdated, I'm not sure the explanation of those first three minutes is.

[Eise]

7 hours ago, beecee said:

."Geometry = matter", should that be "Geometry = Gravity"?

I think Joigus made an excellent job here. Not giving the absolute, technical correct explanation, but trying to pickup from the estimated level where the questioner stands.

Joigus presented the Einstein Equation, which really has (kind of) the form he presents. On one side stands a mathematical description of the curvature of spacetime ("Geometry"), and on the other side the possible sources of that curvature ("Matter"). Maybe one could say that "Geometry = Gravity" is a postulate of general relativity, but is definitely not the Einstein Equation itself.

Edited July 12, 2021 by Eise
Typo

[beecee]

3 hours ago, Eise said:

I think Joigus made an excellent job here. Not giving the absolute, technical correct explanation, but trying to pickup from the estimated level where the questioner stands.

Joigus presented the Einstein Equation, which really has (kind of) the form he presents. On one side stands a mathematical description of the curvature of spacetime ("Geometry"), and on the other side the possible sources of that curvature ("Matter"). Maybe one could say that "Geometry = Gravity" is a postulate of general relativity, but is definitely not the Einstein Equation itself.

OK, point  taken.  If I could give him two likes I would! 

[joigus]

It is fair to say that this LHS of Einstein's eqs. is not all of the geometry, as @Eise justly said.

All of the geometry is captured by an object called the Riemann tensor, which in dimension 4=1(time)+3(space) has 20 independent components. In a \( D \)-dimensional space-time, the Riemann would have \( \frac{1}{12}D^2\left( D^2-1 \right) \) independent components.

The "geometry" on the LHS is only part of the geometry. The rest is the degrees of freedom contained in the so-called Weyl tensor. Those are the degrees of freedom carried by gravitational waves.

Only in dimension 3=1(time)+2(space) specifying the Einstein tensor would be tantamount to specifying the Ricci tensor, which would be tantamount to specifying the Riemann tensor, because all of them would have 6 components. Gravity in 1+2 dimensions would have no gravitational waves.

[studiot]

On 7/11/2021 at 3:30 PM, storyteller said:

Can someone explain to me the connection between equations of Einstein and Friedmann, please?

And how they are relevant to the cosmology?

Here are a couple of explanations, the first in plain English by Oxford Professor of AstoPhysics, P Ferreira

 

The second is an excellent pdf from Baez and Bunn (the 2006 version) which is ( a bit) more mathematical

https://www.researchgate.net/publication/238984245_The_Meaning_of_Einstein's_Equation