# Partial derviative indices issues

+1 vote

As a new user of Cadabra I am a bit confused what is going on with the following code. I am trying to repeat the steps done in Schutz's GR book for equations 6.63 - 6.67 as shown below.

Note I have edited my original post and added print outputs for when I substitute Gammpd into the equation for R. Cut and paste of my outputs produced garbage.

{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u#}::Indices(name=fixed,position=fixed).
{\alpha,\beta,\nu,\mu,\sigma}::Indices(name=fixed,position=fixed);
{\rho,\delta,\kappa,\theta}::Indices(name=fixed,position=fixed);

\partial{#}::PartialDerivative.

\Gamma^{a}_{b c}::TableauSymmetry(shape={2}, indices={1,2});

g_{a b}::Metric.

;::Symbol;
# this code works with and without this trick
R := R^{\alpha}_{\beta \mu \nu} -> \partial_{\mu}{\Gamma^{\alpha}_{\beta \nu}}
-\partial_{\nu}{\Gamma^{\alpha}_{\beta \mu}}
+\Gamma^{\alpha}_{\sigma \mu} \Gamma^{\sigma}_{\beta \nu}
-\Gamma^{\alpha}_{\sigma \nu} \Gamma^{\sigma}_{\beta \mu};

exp := R^{\alpha}_{\beta \mu \nu};

substitute(exp,R);

Gammapd := \partial_{\sigma}{\Gamma^{\alpha}_{\mu \nu}} -> (1/2) g^{\alpha \beta}
(\partial_{\nu \sigma}{g_{\beta \mu}} + \partial_{\mu \sigma}{g_{\beta \nu}}
-\partial_{\beta \sigma}{g_{\mu \nu}});

substitute(exp,Gammapd);


Doing the above results in :

1/2 g^{α β}(\partial{ν μ}(g{β β})) - 1/2 g^{α β}(\partial{μ ν}(g{β β})) + Γ^{α}{σ μ} Γ^{σ}{β ν}-Γ^{α}{σ ν} Γ^{σ}{β μ}

Using the following for Gammapd I get what I think is correct.

Gammapd := \partial_{\sigma}{\Gamma^{\alpha}_{\mu \nu}} -> (1/2) g^{\alpha a}
(\partial_{\nu \sigma}{g_{a \mu}} + \partial_{\mu \sigma}{g_{a \nu}}
-\partial_{a \sigma}{g_{\mu \nu}});


1/2 g^{α a}(\partial{ν μ}(g{a β}) + \partial{β μ}(g{a ν})-\partial{a μ}(g{β ν})) - 1/2 g^{α a}(\partial{μ ν}(g{a β}) + \partial{β ν}(g{a μ})-\partial{a ν}(g{β μ})) + Γ^{α}{σ μ} Γ^{σ}{β ν}-Γ^{α}{σ ν} Γ^{σ}{β μ}

Note here that that \partial{ν μ}(g{a β}) should cancel out with \partial{μ ν}(g{a β}. In Shutz we are considering the inertial frame so all the Gammas = 0.

From Schutz: (the alpha on R is contravariant as are the metric's alpha sigma indices)

Rα βμν = 12 gασ (gσβ,νμ + gσ ν,βμ − gβν,σ μ − gσβ,μν − gσ μ,βν + gβμ,σ ν ). (6.65)

I would paste pictures of the Schutz equation and of my output but I can't seem to do that here and I'm not sure why?

I have also used:

Gammapd := \partial_{\sigma}{\Gamma^{\alpha}_{\mu \nu}} -> (1/2) g^{a b}
(\partial_{c d}{g_{b e}} + \partial_{e d}{g_{b c}}
-\partial_{b d}{g_{e c}});

for my substitution and for that I get: Γ^{α}{σ μ} Γ^{σ}{β ν}-Γ^{α}{σ ν} Γ^{σ}{β μ}

only the Gamma terms?

Do I somehow have to use Depends or something to get this to work. Why are the terms with the double partial derivatives not canceling when the indices are symmetric?

edited