# (wald) method for calculating curvature

1. Nov 21, 2012

### nulliusinverb

R$_{a}$$_{b}$$_{c}$$^{d}$ω$_{d}$=((-2)$\partial$$_{[a}$$\Gamma$$^{d}$$_{b] }$$_{c}$+2$\Gamma$$^{e}$$_{[a]}$$_{c}$$\Gamma$$^{d}$$_{}$$_{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!

2. Nov 21, 2012

### Bill_K

Brackets around a pair of indices means antisymmetrize. So

[aΓdb]c = ½(∂aΓdbc - ∂bΓdac)

3. Nov 21, 2012

### pervect

Staff Emeritus
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?

4. Nov 22, 2012

### 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!