(wald) method for calculating curvature

[nulliusinverb]

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

good, me question is about of:

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

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

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

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

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

[tex] 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] [/tex]

i.e [itex]\frac{1}{n!} [/itex] (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[itex]_{a}[/itex][itex]_{[b}[/itex][itex]_{c}[/itex][itex]_{d]}[/itex]=0

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

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


thank very much!

 

[sardar]

1. The coefficient (-2) and (2) appear in the second term of the equation, which is the product of the partial derivatives of the Christoffel symbols. This coefficient comes from the fact that the Christoffel symbols are defined as the second derivatives of the metric tensor, and when taking the derivative of the derivative, a factor of (-2) appears. The (2) comes from the fact that the Christoffel symbols are symmetric in their lower indices, so when taking the partial derivative, we need to account for both terms.

2. Yes, it is assumed that the partial derivatives of the Christoffel symbols follow the general form you have mentioned. This is a common notation in differential geometry and is used to simplify the expressions. The notation with the brackets [a] and indicates that we are taking the antisymmetric combination of the indices a and b. This is necessary because the Christoffel symbols are not symmetric in their upper indices, so we need to take the antisymmetric combination to get the correct result.