Derivative of quantities with internal indices

Question

In the context of the 3 + 1 decomposition of spacetime needed for a Hamiltionian formulation of general relativity, quantities with so called internal indices are introduced (in the book I am reading on p.43). For such quatities $G^i$ , some kind of a "covariant derivative" is defined:

$D_aG^i = \partial_a G^i + \Gamma _{aj}^iG^i$

Using this derivative, a corresponding "curvature tensor" $\Omega_{ab}^{ji}$ is then calculated by

$D_aD_b - D_bD_a = \Omega_{ab}^{ji}G^i$

My quastions about this are:

1) Why is $\Gamma _{aj}^i$ called spin connection; it has to do with the spin of what ...?

2) How is the so called curvature of connection $\Omega_{ab}^{ji}$ related to the "conventional" curvature tensor ?

Comments

Which book are you using? (and by any chance does this have to do with the spin networks of loop quantum gravity?)

This post imported from StackExchange Physics at 2014-03-17 03:27 (UCT), posted by SE-user David Z

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Ha ha @DavidZaslavsky, I`m reading this children`s book :-P and they introduce the notation and concepts I`m asking about around p.43.

This post imported from StackExchange Physics at 2014-03-17 03:27 (UCT), posted by SE-user Dilaton

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... the link in my above comment does not work as I expected it; You have to scroll down to the bottom of the article to see why it is a children`s book :-)

This post imported from StackExchange Physics at 2014-03-17 03:27 (UCT), posted by SE-user Dilaton

Answer 1

1) The spin connection allows you to define covariant derivatives of spinors in curved spacetime. For example, to do this, you want a set of gamma matrices which are covariantly constant, so you use the combinations

$\Gamma^i=\gamma^a E_a^i$

where $\gamma^a$ are the usual flat space gamma matrices and $E_a^i$ are the tetrad components, i.e.

$E_a=E_a^i\partial_i$

where $E_a$ is the tetrad basis for the tangent space.

2) The differential geometric relations between the vielbein formalism and the "standard" one is described in detail here. Section IV B describes the curvature relationship I think you're looking for.

This post imported from StackExchange Physics at 2014-03-17 03:27 (UCT), posted by SE-user twistor59

Comments

Thanks @twistor59 this is very helpful; and the whole paper looks interesting. Although I`ll probably have to brush up my knowledge about the tetrade formalism (from my relativity demystified book) a bit first to fully understand it ... :-)

This post imported from StackExchange Physics at 2014-03-17 03:27 (UCT), posted by SE-user Dilaton