Welcome to Cadabra Q&A, where you can ask questions and receive answers from other members of the community.
+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.



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

g_{a b}::Metric.

            # see https://cadabra.science/qa/473/is-this-legal-syntax
            # 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};


 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}});


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?

in General questions by (130 points)
edited by

Please log in or register to answer this question.