(wald) method for calculating curvature

[nulliusinverb]

R_{a}_{b}_{c}^{d}ω_{d}=((-2)\partial_{[a}\Gamma^{d}_{b] }_{c}+2\Gamma^{e}_{[a]}_{c}\Gamma^{d}_{<b>}</b>_{e})ω_{d}

good, me question is about of:

1.- as appear the coefficient (-2) und the (2)?

2.- it is assumed that:
\partial_{[a}\Gamma^{d}_{b]}_{c}=\partial_{a}\Gamma^{d}_{b}_{c}+\partial_{b}\Gamma^{d}_{a}_{c}

also the general form is: (maybe my problem is with the notation)

R_{a}_{b}_{c}^{d}ω_{d}=(\partial_{a}\Gamma^{d}_{b}_{c}-\partial_{b}\Gamma^{d}_{a}_{c}+\Gamma^{e}_{a}_{c}\Gamma^{d}_{b}_{e}-\Gamma^{e}_{b}_{c}\Gamma^{d}_{a}_{e})ω_{d}

thank very much!

 

[Bill_K]

Brackets around a pair of indices means antisymmetrize. So

∂_[aΓ^d_b]c = ½(∂_aΓ^d_bc - ∂_bΓ^d_ac)

 

[pervect]

Brackets around a pair of indices means antisymmetrize. So

∂_[aΓ^d_b]c = ½(∂_aΓ^d_bc - ∂_bΓ^d_ac)

Trying to do the latex is giving me fits. But since we have three variables inside the brackets, shouldn't we write

f([a,d,b],c) = \frac{1}{6} \left[ f(a,d,b,c) + f(d,b,a,c) + f(b,a,d,c) - f(a,b,d,c) - f(b,d,a,c) - f(d,a,b,c) \right]

i.e \frac{1}{n!} (even permutations - odd permutations), where n=3?

 

[nulliusinverb]

pervect, we have two variables inside the brackets, a and b.

ok... but see the curvature tensor:

R_{a}_{[b}_{c}_{d]}=0

it is definition equal of the tensor antisymmetric in the brackets?

(where it origines ∂[aΓdb]c = ½(∂aΓdbc - ∂bΓdac) ? )


thank very much!