(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!