Surely there must be a better approach than this?
Certainly; it's the Cartan formalism which employs differential forms. Consider your case of a sphere, with a metric tensor,
ds2=r2dr2+r2sin2θdθ2
We can choose an orthonormal basis ea such that ds2=ηabeaeb, and ea=eaμdxμ. For our case:
er=rdr,eθ=rsinθdθ
We now compute the exterior derivative of each, and re-express them in terms of the basis:
der=0,deθ=sinθdr∧dθ=1r2er∧eθ
We can now use the first structure equation,1 namely dea+ωab∧eb=0, to deduce the connection components ωab. With some experience, you'll be able to read them off:
ωrr=ωθθ=0,ωθr=1r2eθ=1rsinθdθ
The second structure equation allows us to directly compute components of the Ricci tensor Rab in the orthonormal basis, namely, Rab=dωab+ωac∧ωcb. We find that,
Rθθ=Rrr=0,Rθr=1r4eθ∧er
Given that Rab=Rabcdec∧ed, we can compute the terms of the full Riemann tensor. To convert back from our orthonormal basis, we can use the relation
Rλμνσ=(e−1)λaRabcdebμecνedσ
By this method, we find for example, Rθrθr=sin2θ as expected for the sphere. Generally, the Cartan formalism is much easier than direct computation, but be careful in particular when,
- Deducing the connections from the first structure equation; it's not always straightforward;
- Converting back to the orthonormal basis.
For an excellent exposition of the method, see the gravitational physics lectures by Professor Ruth Gregory available from the Perimeter Scholars website.
1 In fact, the structure equation states dea+ωab+eb=Ta, where Ta is the torsion, but in general relativity, we may assume Ta=0. See my own question: Why can we assume torsion is zero in GR?
This post imported from StackExchange Physics at 2016-07-21 06:45 (UTC), posted by SE-user JamalS