Different definitions of exterior derivative

Question

In Sean Carroll's GR book, pg 84, the exterior derivative $d$ is defined as
$$(dA)_{mu_1mu_2...mu_{p+1}} = (p+1) partial_{[mu_1}A_{mu_2...mu_{p+1}]}tag{2.76}$$
where $A$ is a $p$-form and the RHS is the appropiately normalized and antisymmetrized partial derivative.
But on pg. 454, equation E.5,  he replaced the partial derivative with a covariant derivative:
$(dA)_{mu_1mu_2...mu_{p+1}} = (p+1) nabla_{[mu_1}A_{mu_2...mu_{p+1}]}tag{2.76}$
Are these two definitions the same?

Nihar Karve · Accepted Answer

Provided that the connection is torsion-free, $Gamma^{sigma}_{[munu]}=0$, there's a neat cancellation that happens. I'll work out a few examples to get a feel for this:
$$
require{cancel}
newcommandccancel[2][black]{color{#1}{cancel{color{black}{#2}}}}
(dA)_{mu_1mu_2dotsmu_{p+1}}simnabla_{[mu_1}A_{mu_2dotsmu_{p+1}]}
$$
$$
(dA)_{munu}simnabla_{[mu}A_{nu]} tag{p = 1}
simnabla_mu A_nu-nabla_nu A_mu
=partial_mu A_nu-ccancel[red]{Gamma^{sigma}{}_{munu}A_sigma}
-partial_nu A_mu+ccancel[red]{Gamma^{sigma}{}_{numu}A_sigma}
=partial_{[mu}A_{nu]}
$$
$$
(dA)_{munusigma}simnabla_{[mu}A_{nusigma]} tag{p = 2}
simnabla_mu A_{nusigma}-nabla_nu A_{sigmamu}+nabla_sigma A_{munu}
=partial_mu A_{nusigma}-ccancel[red]{Gamma^rho{}_{munu}A_{rhosigma}}-ccancel[blue]{Gamma^rho{}_{musigma}A_{rhonu}}
-partial_nu A_{sigmamu}+ccancel[green]{Gamma^rho{}_{nusigma}A_{rhomu}}+ccancel[red]{Gamma^rho{}_{numu}A_{rhosigma}}
+partial_sigma A_{munu}-ccancel[blue]{Gamma^rho{}_{sigmamu}A_{rhonu}}-ccancel[green]{Gamma^rho{}_{sigmanu}A_{rhomu}}
=partial_{[mu}A_{nusigma]}
$$
since the Christoffel symbols cancel out diagonally due to the absence of torsion. It is not the greatest leap of faith to assume that this result generalises for all $p$-forms. However, we have not yet made use of the crucial point - the antisymmetry of the form! (although we didn't actually need to for the $p=1$ and $p=2$ cases)
There are prima facie three antisymmetric "factors" here: the torsion-free Christoffel symbols, the antisymmetrisation over all the indices, and the antisymmetric nature of the forms.
Let $(a,b,c,...n)$ be a permutation of $(1,2,...p+1)$. For the exterior derivative of a $p$-form, $(dA)_{mu_1mu_2dotsmu_{p+1}}$, compare the terms $Gamma^rho{}_{mu_amu_b}A_{rhomu_c...mu_n}$ and $Gamma^rho{}_{mu_bmu_a}A_{rho,sigma(mu_c...mu_n)}$ where $sigma$ is some permutation (remember, $a$ and $b$ need not be adjacent numbers). Both terms are unique, and are specified up to a sign, which we would like to be opposite in order to get them to cancel out. There are two cases:

$sigma$ is even: the Christoffel symbols give a minus sign, the antisymmetry of the form does nothing and the overall antisymmetrisation does nothing - in total, a relative minus sign

$sigma$ is odd: the Christoffel symbols give us a minus sign, the antisymmetry of the form gives us a minus sign and the overall antisymmetrisation also yields a minus sign - again in total, a relative minus sign

We conclude that
$$Gamma^rho{}_{mu_bmu_a}A_{rho,sigma(mu_c...mu_n)}=-Gamma^rho{}_{mu_amu_b}A_{rhomu_c...mu_n},$$
for the specific permutation $sigma$ of the indices that appears in the exterior derivative. So all the Christoffel symbol terms vanish. Thus, depending on the normalisation conventions of the author, $(dA)_{mu_1mu_2dotsmu_{p+1}}simnabla_{[mu_1}A_{mu_2dotsmu_{p+1}]}$ up to a prefactor.

DanielC · Answer

Nihar Karve provided an explanation why the two formulas which are denoted by $(text{2.76})$ in the original post are identical in a curved spacetime with no torsion, i.e. with a symmetric connection. Here is the missing part to his post (which is useful in GR extensions with torsion):
Let $nabla$ be an operator which generalizes the $d$ from curved spacetime with no torsion. This acts on a p-form on a (cotangent bundle of a) manifold with torsion $T^{mu}_{~~nurho} :=Gamma^{mu}_{~~nurho} - Gamma^{mu}_{~~rhonu}$.
Then:
$$ displaystyle{left(nablaomegaright)_{sigmamu_1 mu_2...mu_p} :=left(domegaright)_{sigmamu_1 mu_2...mu_p} +  (-)^p frac{p(p+1)}{2}T^{nu}_{~~[sigmamu_1} omega_{mu_2...mu_p]nu}}  $$
The first term is given by $(text{2.76})$ in the original post with $nabla_s$ a symmetric connection. Of course, since $d^2 =0$, we have that $nablanablaomega neq 0$.

Different definitions of exterior derivative

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