Hi as a break from Topology and Waves I have been working on my long term ambition to understand the Big Bang line by line. I'm currently trying to understand the derivation of the Friedmann equations from General relativity. This exercise is somewhat tedious and heartbreaking to say the least. However reading some text books eg Lasenby I have noticed at best a misleading error in the calculations of the connection coefficients and this has led me to all sorts of confusion. A connection coefficient is a three indexed quanitity which can be calculated from the metric tensor g as follows

$$\Gamma^{i}_{jk}= \frac{1}{2}g^{il}(\partial_{k}g_{li}+\partial_{j}g_{lk}-\partial_{l}g_{jk})$$

Where the indices run from 0-3 0 corresponding to t and 1,2,3 x,y,z or whatever coodinate system is required. As there are potentially 128 different terms to caclulate one can see the temptation to take short cuts. However if we interchange the indices we get

$$\Gamma^{i}_{kj}= \frac{1}{2}g^{il}(\partial_{j}g_{li}+\partial_{k}g_{lj}-\partial_{l}g_{kj})$$

Which for cases where g is a diagonal metric ie the only non zero terms are

$$g_{00},g_{11},g_{22},g_{33}$$ or $$g_{ii}$$

as in the case of the Robertson Walker, the first equation reduces to

$$\Gamma^{i}_{jk}= \frac{1}{2}g^{ii}(\partial_{j}g_{ii}+\partial_{k}g_{ii}-\partial_{i}g_{kj})$$

and the second equation reduces to

$$\Gamma^{i}_{kj}= \frac{1}{2}g^{ii}(\partial_{k}g_{ii}+\partial_{j}g_{ii}-\partial_{i}g_{jk})$$

But as addition is commutative this is just the first equation with the first two terms interchanged and the last term in both equations will be zero for the off diagonal terms (ie when j does not equal k) . So for diagonal metrics then we must have

$$\Gamma^{i}_{jk} = \Gamma^{i}_{kj}$$

However this symmetry is often missed by standard textbook accounts (eg Lasenby et al page 377) and the claim is made that the only non zero off diagonal connection coefficients for the Robertson Walker metric are

$$\Gamma^{1}_{01}, \Gamma^{2}_{02},\Gamma^{2}_{12},\Gamma^{3}_{03},

\Gamma^{3}_{13},\Gamma^{3}_{23}$$

When they should also include the other off diagonal terms:

$$\Gamma^{1}_{10}, \Gamma^{2}_{20},\Gamma^{2}_{21},\Gamma^{3}_{30},

\Gamma^{3}_{31},\Gamma^{3}_{32}$$

The reason why this is important is because a key quantity in general relativity is the Ricci Tensor which can be calculated from the connection coeffients as

$$R_{ij}=\partial_{j}\Gamma^{k}_{ik}-\partial_{k}\Gamma^{k}_{ij}+

\Gamma^{l}_{ik}\Gamma^{k}_{l}{j}-\Gamma^{l}_{ij}\Gamma^{k}_{lk}$$

So if some gullible reader like myself neglects the other off diagonal terms then you will not get the correct answer for the Ricci Tensor and will spend many hours of frustration wondering why you can't get the answers quoted in the text books. Of the many books I have which give results for the connection coefficients for the Robertson Walker metric at least three namely

Narlikar " Introduction to Cosmology" page 106

http://www.amazon.co.uk/An-Introduction-Cosmology-J-Narlikar/dp/0521793769/ref=sr_1_1?ie=UTF8&qid=1336329189&sr=8-1

Lasenby "General Relativity An Introduction for physicists" page 377

http://www.amazon.co.uk/General-Relativity-An-Introduction-Physicists/dp/0521829518/ref=sr_1_1?s=books&ie=UTF8&qid=1336329247&sr=1-1

and

Collins, Martin and Squires "Particle physics and Cosmology" page 373

http://www.amazon.co.uk/Particle-Physics-Cosmology-P-Collins/dp/0471600881/ref=sr_1_2?s=books&ie=UTF8&qid=1336329321&sr=1-2

All neglect the other off diagonal connection coefficients for the Robertson Walker metric in their tables of connection coefficients.

All this goes to show is that one should not just naively take on trust results quoted in text books I can only agree with my colleague Duncan in his last post about the importance of pedantry when trying to understand maths or physics books

http://matrices-reloaded.blogspot.co.uk/2012/04/last-book.html

On another note I have decided not to embark on the composition course immediately as my finances couldn't really cope with it and also I want to spend more time on my extra curricular physics.

Very interesting Chris and well done for spotting such an error. If we always take things as gospel then we would never make any progress! I know that I am probably making myself unpopular at the OU by challenging things but if part of our training is to understand proofs, then to really get to grips with them, you have to pick them apart.

ReplyDeleteI do think you're being a little harsh on the authors of the textbooks here. I don't know if you've ever come across the textbook by Hobson, but I was lectured GR by him and whilst he isn't an amazing lecturer, the book was very clear. IMHO it's actually fair to expect you to realise that for a symmetric metric without torsion, this symmetry will exist in the affine connection. I think it would be fair to say that you have had to think harder about what is wrong with the answers you come up with whilst neglecting this symmetry, and that's precisely what textbooks on something like differential geometry are (or should be) aiming for.

ReplyDeleteDan W

Sorry given the hours I had to struggle through whilst checking the Ricci Scalar components are correct, which could have been avoided by a simple statement or reminder that the connection coefficuents are symmetric or including the other components then I feel justified in criticising key omissions. Thats not to say that the textbooks aren;t great it's just they could make life a bit easier for those who just don't want to accept results quoted in textbooks.

ReplyDelete