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A Component of a Riemann Tensor

Calculation: long. Result: messy!

While studying general relativity, I learned about the Riemann curvature tensor Rⁱ_jkl, which is defined in terms of Christoffel symbols $Γ_{jk}^{i}$ and their partial derivatives $[R_{jkl}^{i} = \frac{\partial Γ_{jl}^{i}}{\partial x^{k}} - \frac{\partial Γ_{jk}^{i}}{\partial x^{l}} + Γ_{mk}^{i} Γ_{jl}^{m} - Γ_{ml}^{i} Γ_{jk}^{m}]$ . The Christoffel symbols in turn are defined in terms of partial derivatives of the metric tensor g $[Γ_{jk}^{i} = \frac{1}{2} g^{im} (\frac{\partial g_{mj}}{\partial x^{k}} + \frac{\partial g_{mi}}{\partial x^{j}} - \frac{\partial g_{mj}}{\partial x^{m}})]$ . 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 R^j_iik in a 4-space with coördinates i, j, k, l. I’ll let you judge whether it was worth the effort.

R_{iik}^{j} = \frac{\partial \frac{1}{2} [g^{ji} (\frac{\partial g_{ii}}{\partial x^{k}} + \frac{\partial g_{ij}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{i}}) + g^{jj} (\frac{\partial g_{ji}}{\partial x^{k}} + \frac{\partial g_{jj}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{j}}) + g^{jk} (\frac{\partial g_{ki}}{\partial x^{k}} + \frac{\partial g_{kj}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{k}}) + g^{jl} (\frac{\partial g_{li}}{\partial x^{k}} + \frac{\partial g_{lj}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{l}})]}{\partial x^{i}} - \frac{\partial \frac{1}{2} [g^{ji} (\frac{\partial g_{ii}}{\partial x^{i}} + \frac{\partial g_{ij}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{i}}) + g^{jj} (\frac{\partial g_{ji}}{\partial x^{i}} + \frac{\partial g_{jj}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{j}}) + g^{jk} (\frac{\partial g_{ki}}{\partial x^{i}} + \frac{\partial g_{kj}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{k}}) + g^{jl} (\frac{\partial g_{li}}{\partial x^{i}} + \frac{\partial g_{lj}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{l}})]}{\partial x^{k}} + (\frac{1}{2} [g^{ji} (\frac{\partial g_{ii}}{\partial x^{i}} + \frac{\partial g_{ij}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{i}}) + g^{jj} (\frac{\partial g_{ji}}{\partial x^{i}} + \frac{\partial g_{jj}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{j}}) + g^{jk} (\frac{\partial g_{ki}}{\partial x^{i}} + \frac{\partial g_{kj}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{k}}) + g^{jl} (\frac{\partial g_{li}}{\partial x^{i}} + \frac{\partial g_{lj}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{l}})] \times \frac{1}{2} [g^{ii} (\frac{\partial g_{ii}}{\partial x^{k}} + \frac{\partial g_{ii}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{i}}) + g^{ij} (\frac{\partial g_{ji}}{\partial x^{k}} + \frac{\partial g_{ji}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{j}}) + g^{ik} (\frac{\partial g_{ki}}{\partial x^{k}} + \frac{\partial g_{ki}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{k}}) + g^{il} (\frac{\partial g_{li}}{\partial x^{k}} + \frac{\partial g_{li}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{l}})]) - (\frac{1}{2} [g^{ji} (\frac{\partial g_{ii}}{\partial x^{k}} + \frac{\partial g_{ij}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{i}}) + g^{jj} (\frac{\partial g_{ji}}{\partial x^{k}} + \frac{\partial g_{jj}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{j}}) + g^{jk} (\frac{\partial g_{ki}}{\partial x^{k}} + \frac{\partial g_{kj}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{k}}) + g^{jl} (\frac{\partial g_{li}}{\partial x^{k}} + \frac{\partial g_{lj}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{l}})] \times \frac{1}{2} [g^{ii} (\frac{\partial g_{ii}}{\partial x^{i}} + \frac{\partial g_{ii}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{i}}) + g^{ij} (\frac{\partial g_{ji}}{\partial x^{i}} + \frac{\partial g_{ji}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{j}}) + g^{ik} (\frac{\partial g_{ki}}{\partial x^{i}} + \frac{\partial g_{ki}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{k}}) + g^{il} (\frac{\partial g_{li}}{\partial x^{i}} + \frac{\partial g_{li}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{l}})]) + (\frac{1}{2} [g^{ji} (\frac{\partial g_{ij}}{\partial x^{i}} + \frac{\partial g_{ij}}{\partial x^{j}} - \frac{\partial g_{ji}}{\partial x^{i}}) + g^{jj} (\frac{\partial g_{jj}}{\partial x^{i}} + \frac{\partial g_{jj}}{\partial x^{j}} - \frac{\partial g_{ji}}{\partial x^{j}}) + g^{jk} (\frac{\partial g_{kj}}{\partial x^{i}} + \frac{\partial g_{kj}}{\partial x^{j}} - \frac{\partial g_{ji}}{\partial x^{k}}) + g^{jl} (\frac{\partial g_{lj}}{\partial x^{i}} + \frac{\partial g_{lj}}{\partial x^{j}} - \frac{\partial g_{ji}}{\partial x^{l}})] \times \frac{1}{2} [g^{ji} (\frac{\partial g_{ii}}{\partial x^{k}} + \frac{\partial g_{ij}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{i}}) + g^{jj} (\frac{\partial g_{ji}}{\partial x^{k}} + \frac{\partial g_{jj}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{j}}) + g^{jk} (\frac{\partial g_{ki}}{\partial x^{k}} + \frac{\partial g_{kj}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{k}}) + g^{jl} (\frac{\partial g_{li}}{\partial x^{k}} + \frac{\partial g_{lj}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{l}})]) - (\frac{1}{2} [g^{ji} (\frac{\partial g_{ij}}{\partial x^{k}} + \frac{\partial g_{ij}}{\partial x^{j}} - \frac{\partial g_{jk}}{\partial x^{i}}) + g^{jj} (\frac{\partial g_{jj}}{\partial x^{k}} + \frac{\partial g_{jj}}{\partial x^{j}} - \frac{\partial g_{jk}}{\partial x^{j}}) + g^{jk} (\frac{\partial g_{kj}}{\partial x^{k}} + \frac{\partial g_{kj}}{\partial x^{j}} - \frac{\partial g_{jk}}{\partial x^{k}}) + g^{jl} (\frac{\partial g_{lj}}{\partial x^{k}} + \frac{\partial g_{lj}}{\partial x^{j}} - \frac{\partial g_{jk}}{\partial x^{l}})] \times \frac{1}{2} [g^{ji} (\frac{\partial g_{ii}}{\partial x^{i}} + \frac{\partial g_{ij}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{i}}) + g^{jj} (\frac{\partial g_{ji}}{\partial x^{i}} + \frac{\partial g_{jj}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{j}}) + g^{jk} (\frac{\partial g_{ki}}{\partial x^{i}} + \frac{\partial g_{kj}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{k}}) + g^{jl} (\frac{\partial g_{li}}{\partial x^{i}} + \frac{\partial g_{lj}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{l}})]) + (\frac{1}{2} [g^{ji} (\frac{\partial g_{ik}}{\partial x^{i}} + \frac{\partial g_{ij}}{\partial x^{k}} - \frac{\partial g_{ki}}{\partial x^{i}}) + g^{jj} (\frac{\partial g_{jk}}{\partial x^{i}} + \frac{\partial g_{jj}}{\partial x^{k}} - \frac{\partial g_{ki}}{\partial x^{j}}) + g^{jk} (\frac{\partial g_{kk}}{\partial x^{i}} + \frac{\partial g_{kj}}{\partial x^{k}} - \frac{\partial g_{ki}}{\partial x^{k}}) + g^{jl} (\frac{\partial g_{lk}}{\partial x^{i}} + \frac{\partial g_{lj}}{\partial x^{k}} - \frac{\partial g_{ki}}{\partial x^{l}})] \times \frac{1}{2} [g^{ki} (\frac{\partial g_{ii}}{\partial x^{k}} + \frac{\partial g_{ik}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{i}}) + g^{kj} (\frac{\partial g_{ji}}{\partial x^{k}} + \frac{\partial g_{jk}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{j}}) + g^{kk} (\frac{\partial g_{ki}}{\partial x^{k}} + \frac{\partial g_{kk}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{k}}) + g^{kl} (\frac{\partial g_{li}}{\partial x^{k}} + \frac{\partial g_{lk}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{l}})]) - (\frac{1}{2} [g^{ji} (\frac{\partial g_{ik}}{\partial x^{k}} + \frac{\partial g_{ij}}{\partial x^{k}} - \frac{\partial g_{kk}}{\partial x^{i}}) + g^{jj} (\frac{\partial g_{jk}}{\partial x^{k}} + \frac{\partial g_{jj}}{\partial x^{k}} - \frac{\partial g_{kk}}{\partial x^{j}}) + g^{jk} (\frac{\partial g_{kk}}{\partial x^{k}} + \frac{\partial g_{kj}}{\partial x^{k}} - \frac{\partial g_{kk}}{\partial x^{k}}) + g^{jl} (\frac{\partial g_{lk}}{\partial x^{k}} + \frac{\partial g_{lj}}{\partial x^{k}} - \frac{\partial g_{kk}}{\partial x^{l}})] \times \frac{1}{2} [g^{ki} (\frac{\partial g_{ii}}{\partial x^{i}} + \frac{\partial g_{ik}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{i}}) + g^{kj} (\frac{\partial g_{ji}}{\partial x^{i}} + \frac{\partial g_{jk}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{j}}) + g^{kk} (\frac{\partial g_{ki}}{\partial x^{i}} + \frac{\partial g_{kk}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{k}}) + g^{kl} (\frac{\partial g_{li}}{\partial x^{i}} + \frac{\partial g_{lk}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{l}})]) + (\frac{1}{2} [g^{ji} (\frac{\partial g_{il}}{\partial x^{i}} + \frac{\partial g_{ij}}{\partial x^{l}} - \frac{\partial g_{li}}{\partial x^{i}}) + g^{jj} (\frac{\partial g_{jl}}{\partial x^{i}} + \frac{\partial g_{jj}}{\partial x^{l}} - \frac{\partial g_{li}}{\partial x^{j}}) + g^{jk} (\frac{\partial g_{kl}}{\partial x^{i}} + \frac{\partial g_{kj}}{\partial x^{l}} - \frac{\partial g_{li}}{\partial x^{k}}) + g^{jl} (\frac{\partial g_{ll}}{\partial x^{i}} + \frac{\partial g_{lj}}{\partial x^{l}} - \frac{\partial g_{li}}{\partial x^{l}})] \times \frac{1}{2} [g^{li} (\frac{\partial g_{ii}}{\partial x^{k}} + \frac{\partial g_{il}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{i}}) + g^{lj} (\frac{\partial g_{ji}}{\partial x^{k}} + \frac{\partial g_{jl}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{j}}) + g^{lk} (\frac{\partial g_{ki}}{\partial x^{k}} + \frac{\partial g_{kl}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{k}}) + g^{ll} (\frac{\partial g_{li}}{\partial x^{k}} + \frac{\partial g_{ll}}{\partial x^{i}} - \frac{\partial g_{ik}}{\partial x^{l}})]) - (\frac{1}{2} [g^{ji} (\frac{\partial g_{il}}{\partial x^{k}} + \frac{\partial g_{ij}}{\partial x^{l}} - \frac{\partial g_{lk}}{\partial x^{i}}) + g^{jj} (\frac{\partial g_{jl}}{\partial x^{k}} + \frac{\partial g_{jj}}{\partial x^{l}} - \frac{\partial g_{lk}}{\partial x^{j}}) + g^{jk} (\frac{\partial g_{kl}}{\partial x^{k}} + \frac{\partial g_{kj}}{\partial x^{l}} - \frac{\partial g_{lk}}{\partial x^{k}}) + g^{jl} (\frac{\partial g_{ll}}{\partial x^{k}} + \frac{\partial g_{lj}}{\partial x^{l}} - \frac{\partial g_{lk}}{\partial x^{l}})] \times \frac{1}{2} [g^{li} (\frac{\partial g_{ii}}{\partial x^{i}} + \frac{\partial g_{il}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{i}}) + g^{lj} (\frac{\partial g_{ji}}{\partial x^{i}} + \frac{\partial g_{jl}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{j}}) + g^{lk} (\frac{\partial g_{ki}}{\partial x^{i}} + \frac{\partial g_{kl}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{k}}) + g^{ll} (\frac{\partial g_{li}}{\partial x^{i}} + \frac{\partial g_{ll}}{\partial x^{i}} - \frac{\partial g_{ii}}{\partial x^{l}})])

Lessons (re)learned

This exercise reinforced a few things I already knew:

XML isn’t meant to be written or manipulated by humans.
XML isn’t meant for data storage.
LaTeX is much easier to write than MathML, and is much more compact (in this case, 400%).
No one writes components of the Riemann tensor in terms of the metric tensor because it’s a bloody mess and not particularly enlightening.
I need to study more differential geometry.

Last updated 1 May 2006
http://www.rdrop.com/~half/math/riemann.xhtml
All contents released into the public domain by Mark L. Irons

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