Seeking thorough derivation of Friedman equations

[andrewkirk]

I am looking for a derivation of Friedman's equations from the Cosmological Principle and Einstein's field equations.

The text from which I am teaching myself (Schutz) does it all in two massive leaps, whose reasoning are respectively "it is easy to show that ..." (it isn't) and "the Einstein equations [in the FLRW metric, assuming Cosmo Principle] are easy to write down" (they are in fact incredibly time consuming, and hence very easy to make the odd error in, which stops things from cancelling out as needed).

Other sources I have seen on the internet just skip the messy manipulation of Riemann and Ricci tensors and Christoffel symbols and just present the resul, or key steps in it, as a fait accompli.

I'm sure there must be sources out there that work through this properly, but I just haven't been able to find them.

Thanks.

 

[WannabeNewton]

See section 2.11 of the following notes for a coordinate independent, purely geometric derivation: http://www.socsci.uci.edu/~dmalamen/bio/GR.pdf

 

[andrewkirk]

Thanks WBN but when I try to open that file I get a message from Adobe Reader saying it is not a supported file type, or has been damaged.

 

[WannabeNewton]

Try this link then: https://www.dropbox.com/s/4df0y5ujt...on Theory-Chicago University Press (2012).pdf

 

[Chalnoth]

I am looking for a derivation of Friedman's equations from the Cosmological Principle and Einstein's field equations.

The text from which I am teaching myself (Schutz) does it all in two massive leaps, whose reasoning are respectively "it is easy to show that ..." (it isn't) and "the Einstein equations [in the FLRW metric, assuming Cosmo Principle] are easy to write down" (they are in fact incredibly time consuming, and hence very easy to make the odd error in, which stops things from cancelling out as needed).

Other sources I have seen on the internet just skip the messy manipulation of Riemann and Ricci tensors and Christoffel symbols and just present the resul, or key steps in it, as a fait accompli.

I'm sure there must be sources out there that work through this properly, but I just haven't been able to find them.

Thanks.

It's not horribly difficult. I remember doing this for my qualifying exam way back when. I can't tell you any resources because I didn't use any, beyond the definitions of the field equations and the various GR operations you have to do.

The primary trick is recognizing that most of the components of the various tensors are zero, and using that fact so that you don't have to write down so many components of the various required sums. Still, it did take me a good weekend and a number of pages of notebook paper to do the full calculation from scratch myself and verify that it was correct.

 

[WannabeNewton]

Andrew let me know if those notes still refuse to open for you and I can just post the derivation here.

 

[andrewkirk]

Thanks WBN. I managed to download the file when I got home. I suspect there was something in my workplace firewall that was preventing the download from both of the links. I haven't worked through it yet, but will.

 

[George Jones]

Are you asking about:

1) the mathematical analysis that leads from spatial homogeneity and isotropy to the Robertson-Walker metric

2) the mathematical analysis that leads via Einstein's equations from the Roberson-Walker metric to Friedmann's equations

or both?

1) isn't treated that often in detail in textbooks. WannabeNewton gave an interesting reference that specifically avoids Killing vectors. A couple of textbook treatments of the somewhat involved Killing vector treatment of 1) are section 10.2 "Robertson-Walker metric" from "Introduction to General Relativity" by Ryder, and section 10.7 "The isotropic Bianchi-type (Robertson-Walker) spacetimes" from "An Introduction to General Relativity and Cosmology" by Plebanski and Kransinski.

 

[andrewkirk]

It's the second one that is currently stumping me. I feel I understand the first well enough, with a couple of gaps whose existence I attribute to my own laziness in not having yet made the effort to fully comprehend Killing vector fields (which is in turn attributable to a reluctance to confront and conquer Lie derivatives).

There are two steps that are eluding me. I am transferring my hand-written calculations to latex, partly in order to try to detect my mistakes, and partly so I can seek help. I have finished latex-ing one of the two steps and the link below is to the version as a PF blog post. The post shows my problem, which is that the Ricci scalar I get depends on $\theta$ and hence does not comply with the homogeneity requirement.

https://www.physicsforums.com/blog.php?b=4697

 

[Chalnoth]

It's the second one that is currently stumping me. I feel I understand the first well enough, with a couple of gaps whose existence I attribute to my own laziness in not having yet made the effort to fully comprehend Killing vector fields (which is in turn attributable to a reluctance to confront and conquer Lie derivatives).

There are two steps that are eluding me. I am transferring my hand-written calculations to latex, partly in order to try to detect my mistakes, and partly so I can seek help. I have finished latex-ing one of the two steps and the link below is to the version as a PF blog post. The post shows my problem, which is that the Ricci scalar I get depends on $\theta$ and hence does not comply with the homogeneity requirement.

https://www.physicsforums.com/blog.php?b=4697

Link doesn't work.

 

[andrewkirk]

I'll try posting the link again. I've never used the PF blog for anything other than my own purposes before, so I'm not sure how to link it.

THe following link is obtained by viewing the blog post via My PF > Blog > Your blog, and then copying the URL from the address bar and pasting it in the link:

https://www.physicsforums.com/blog.php?b=4697

Hmm, it looks the same as the one above. Any suggestions as to how I can properly link to a PF blog page?

[I'm a mess aren't I? Here I am trying to do some fancy tensor manipulation, and I can't even post a useable web-link!]

 

[andrewkirk]

I've solved the blog link problem. It was that I hadn't 'published' the blog entry so it wasn't visible to other users. I've now done that, so the link should now work. Please let me know if it doesn't.

 

[Chalnoth]

I think that jumping straight to the 4th rank Ricci curvature tensor may be causing issues here. I think it's less error-prone to start by computing the connection, and then going from the connection to the curvature. If I recall, a lot of things simplify for this metric when doing it this way as well.

 

[WannabeNewton]

Why not just calculate everything in a coordinate free manner? It is much more tractable, much faster, and definitely more elegant (not to mention more insightful).

 

[andrewkirk]

Using the connection doesn't sound like it would be any faster, as each Riemann tensor component is expressed in terms of six first derivatives of Christoffel symbols, and each of those symbols needs to be calculated from three first derivatives of the metric and one inverse metric component.

My calculation directly calculates each Riemann component from four second derivatives of the metric and one inverse metric component.

So the two calculations are essentially the same. Sure, many items turn out to be zero in the connection approach but the same thing happens in the direct approach, and there are less terms to possibly be mis-transcribed.

WBN, I am interested in the coordinate-free approach and look forward to working through it but nevertheless the coordinate approach must work, so I'm aiming to discover why the derivation I've written doesn't (ie what error I have made) before moving on to that one. Some things have to be done using coordinates so if there's a flaw in the way I manipulate coordinates it's better for me to discover and correct it now, so I don't get repeatedly stuck in later calculations.

Thanks

 

[Chalnoth]

Using the connection doesn't sound like it would be any faster, as each Riemann tensor component is expressed in terms of six first derivatives of Christoffel symbols, and each of those symbols needs to be calculated from three first derivatives of the metric and one inverse metric component.

My calculation directly calculates each Riemann component from four second derivatives of the metric and one inverse metric component.

So the two calculations are essentially the same. Sure, many items turn out to be zero in the connection approach but the same thing happens in the direct approach, and there are less terms to possibly be mis-transcribed.

There are some simple symmetries to dramatically reduce that, though. Anyway, here was a good thread that goes into some detail on the general process for diagonal metrics:
http://physics.stackexchange.com/questions/14136/ricci-scalar-for-a-diagonal-metric-tensor

Edit: And while it's been a number of years since I did this calculation, this is definitely the way I did it, and I found it to be fairly straightforward at the time.

 

[Chalnoth]

Why not just calculate everything in a coordinate free manner? It is much more tractable, much faster, and definitely more elegant (not to mention more insightful).

I never did see how to do the coordinate-free calculations myself. Sounds interesting.

 

[WannabeNewton]

I never did see how to do the coordinate-free calculations myself. Sounds interesting.

It's in the notes I linked above, in particular section 2.11, if you're interested in seeing it !

Also, while we're on the topic of coordinate-free derivations, check out section 25 of Geroch's GR notes (2nd link in the following site) for a coordinate-free derivation of the Schwarzschild metric because it's also really awesome: http://home.uchicago.edu/~geroch/Links_to_Notes.html

Have fun!

 

[Chalnoth]

It's in the notes I linked above, in particular section 2.11, if you're interested in seeing it !

Also, while we're on the topic of coordinate-free derivations, check out section 25 of Geroch's GR notes (2nd link in the following site) for a coordinate-free derivation of the Schwarzschild metric because it's also really awesome: http://home.uchicago.edu/~geroch/Links_to_Notes.html

Have fun!

After a little bit of looking, it is definitely simpler, at least up until the point of transitioning to FRW coordinates, but it may have some significant overhead in understanding due to use of coordinate-free techniques.

 

[andrewkirk]

There's a tiny bit of light at the end of the tunnel. I've discovered the formula from Schutz (equation 6.67 on p159) that I've been using to express Riemann tensor components in terms of second partial derivatives of metric tensor components is only valid in the local inertial frame, and hence not applicable to the FLRW frame I'm using.

So there's more work to do, but at least I can see a way forward, which is using the Christoffel symbols as Chalnoth suggested.

Yes, WBN, I know that if I were doing it the coordinate-independent way I wouldn't have these problems. You're right. But I want to do it the coordinate way first before I do it the other way. For a start, I need the practice at tensor manipulation. Reading the coordinate-free approach will be a treat to reward me when I solve this problem.

 

[Chalnoth]

There's a tiny bit of light at the end of the tunnel. I've discovered the formula from Schutz (equation 6.67 on p159) that I've been using to express Riemann tensor components in terms of second partial derivatives of metric tensor components is only valid in the local inertial frame, and hence not applicable to the FLRW frame I'm using.

So there's more work to do, but at least I can see a way forward, which is using the Christoffel symbols as Chalnoth suggested.

Ah, yes. That makes sense to me. Those derivatives should have all come along with factors of the connection to account for the curvature, which would have significantly complicated the whole calculation. I'm kicking myself a bit for not catching that that might be a problem.

 

[andrewkirk]

I have finally managed to complete the proof. Once I spotted that I was using an invalid formula, it only took a few hours to get the derivation worked out.

Still, at several hours and about 10 pages of calculations, I think it's a bit cheeky of Schutz to say, as he does in his chapter on Cosmology "The Einstein equations are also easy to write down".

The goal of the derivation is to prove that, in the FLRW coordinate system:

$$G^{00}=\frac{3}{R^2}(\dot{R}^2 +k)$$

The derivation is split over two blog posts because it is too long to be accepted as a single one:

First post (calculates and summarises all Christoffel symbols in a FLRW system): https://www.physicsforums.com/blog.php?b=4697

Second post (uses those symbols to calculate the Einstein tensor component): https://www.physicsforums.com/blog.php?b=4701

Once this result is proved then, together with a result that expresses the energy conservation rule in tensor coordinate terms, one can rapidly generate the Friedman equations and reach interesting conclusions about how the universe will behave in future and how it behaved in the past (after the inflationary era).

 

[WannabeNewton]

For a start, I need the practice at tensor manipulation.

I've always personally been against doing particularly arduous coordinate based (tensor) calculations by hand when things like Mathematica exist but just like with integration, doing things by hand a few times is always good practice I suppose. I'm glad you got it all to work out in the end though! It's always a satisfying feeling when a long calculation works out

Regardless, when I think of tensor manipulations I think of something else, specifically of abstract index manipulations i.e. "index gymnastics" (something like this: https://www.physicsforums.com/showpost.php?p=4424809&postcount=2) but that's probably because I've mostly learned GR from texts that try to be as coordinate-free as possible when it comes to calculations, derivations, and definitions. I just think such an approach would save you a considerable amount of finger pain from writing down so many calculations !

As a side note, you'd be surprised at just how many textbook authors brush off a calculation or proof as "trivial" or "easy" when in reality it takes a considerable amount of effort. I see it more in real analysis texts wherein the proof involves a slew of annoying epsilon delta approximations, than in physics texts.