## Sunday, 6 May 2012

### The Importance of Pedantry

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