]> A Component of a Riemann Tensor

A Component of a Riemann Tensor

Calculation: long. Result: messy!

While studying general relativity, I learned about the Riemann curvature tensor Rijkl, which is defined in terms of Christoffel symbols Γ jk i and their partial derivatives R jkl i = Γ jl i x k Γ jk i x l + Γ mk i Γ jl m Γ ml i Γ jk m . The Christoffel symbols in turn are defined in terms of partial derivatives of the metric tensor g Γ jk i = 1 2 g im g mj x k + g mi x j g mj x m . This raised an obvious question: what does the Riemann tensor look like when the Christoffel symbols are expanded—in other words, when the Riemann tensor is defined directly in terms of the metric tensor? The result was sure to be large and messy, but could be enlightening.

How could I resist a challenge like that?

Since I don’t have access to symbolic math software, I wrote a Perl script to crank through the expansions. Here’s the result for the component Rjiik in a 4-space with coördinates i, j, k, l. I’ll let you judge whether it was worth the effort.

R iik j = 1 2 g ji g ii x k + g ij x i - g ik x i + g jj g ji x k + g jj x i - g ik x j + g jk g ki x k + g kj x i - g ik x k + g jl g li x k + g lj x i - g ik x l x i - 1 2 g ji g ii x i + g ij x i - g ii x i + g jj g ji x i + g jj x i - g ii x j + g jk g ki x i + g kj x i - g ii x k + g jl g li x i + g lj x i - g ii x l x k + 1 2 g ji g ii x i + g ij x i - g ii x i + g jj g ji x i + g jj x i - g ii x j + g jk g ki x i + g kj x i - g ii x k + g jl g li x i + g lj x i - g ii x l × 1 2 g ii g ii x k + g ii x i - g ik x i + g ij g ji x k + g ji x i - g ik x j + g ik g ki x k + g ki x i - g ik x k + g il g li x k + g li x i - g ik x l - 1 2 g ji g ii x k + g ij x i - g ik x i + g jj g ji x k + g jj x i - g ik x j + g jk g ki x k + g kj x i - g ik x k + g jl g li x k + g lj x i - g ik x l × 1 2 g ii g ii x i + g ii x i - g ii x i + g ij g ji x i + g ji x i - g ii x j + g ik g ki x i + g ki x i - g ii x k + g il g li x i + g li x i - g ii x l + 1 2 g ji g ij x i + g ij x j - g ji x i + g jj g jj x i + g jj x j - g ji x j + g jk g kj x i + g kj x j - g ji x k + g jl g lj x i + g lj x j - g ji x l × 1 2 g ji g ii x k + g ij x i - g ik x i + g jj g ji x k + g jj x i - g ik x j + g jk g ki x k + g kj x i - g ik x k + g jl g li x k + g lj x i - g ik x l - 1 2 g ji g ij x k + g ij x j - g jk x i + g jj g jj x k + g jj x j - g jk x j + g jk g kj x k + g kj x j - g jk x k + g jl g lj x k + g lj x j - g jk x l × 1 2 g ji g ii x i + g ij x i - g ii x i + g jj g ji x i + g jj x i - g ii x j + g jk g ki x i + g kj x i - g ii x k + g jl g li x i + g lj x i - g ii x l + 1 2 g ji g ik x i + g ij x k - g ki x i + g jj g jk x i + g jj x k - g ki x j + g jk g kk x i + g kj x k - g ki x k + g jl g lk x i + g lj x k - g ki x l × 1 2 g ki g ii x k + g ik x i - g ik x i + g kj g ji x k + g jk x i - g ik x j + g kk g ki x k + g kk x i - g ik x k + g kl g li x k + g lk x i - g ik x l - 1 2 g ji g ik x k + g ij x k - g kk x i + g jj g jk x k + g jj x k - g kk x j + g jk g kk x k + g kj x k - g kk x k + g jl g lk x k + g lj x k - g kk x l × 1 2 g ki g ii x i + g ik x i - g ii x i + g kj g ji x i + g jk x i - g ii x j + g kk g ki x i + g kk x i - g ii x k + g kl g li x i + g lk x i - g ii x l + 1 2 g ji g il x i + g ij x l - g li x i + g jj g jl x i + g jj x l - g li x j + g jk g kl x i + g kj x l - g li x k + g jl g ll x i + g lj x l - g li x l × 1 2 g li g ii x k + g il x i - g ik x i + g lj g ji x k + g jl x i - g ik x j + g lk g ki x k + g kl x i - g ik x k + g ll g li x k + g ll x i - g ik x l - 1 2 g ji g il x k + g ij x l - g lk x i + g jj g jl x k + g jj x l - g lk x j + g jk g kl x k + g kj x l - g lk x k + g jl g ll x k + g lj x l - g lk x l × 1 2 g li g ii x i + g il x i - g ii x i + g lj g ji x i + g jl x i - g ii x j + g lk g ki x i + g kl x i - g ii x k + g ll g li x i + g ll x i - g ii x l

Lessons (re)learned

This exercise reinforced a few things I already knew:


Last updated 1 May 2006
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