■ - うべの時空代数

$\def\ou#1#2#3{\overset{#2}{\underset{#3}{#1}}}\def\os#1#2{\overset{#2}{#1}}\def\diff{\mathrm{d}}\def\biff{\mathrm{b}}\def\D{\mathrm{D}}\def\p{\mathrm{p}}\def\pu#1{\underset{#1}{\mathrm{p}}}\def\G#1#2{\overset{#1}{\underset{#2}{\Gamma}}}\def\abs#1{|#1|}$ 最近，マクスウェル方程式の三次のほうである
$A\wedge\D\wedge\D=0$
は $A$ を共変成分として計算を行うと自然と共変形式の電磁気におけるビアンキの恒等式が導かれることを確かめ，逆に反変成分で行うと
$A\wedge\D\wedge\D=\frac12\ou{A}{\alpha}{}\ou{R}{}{\rho\alpha\nu\mu}\ou{\gamma}{\rho}{}\wedge\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}=0$
とリーマン曲率テンソルが出現することを知った．また基底を対象にすると
$\ou{\gamma}{}{\alpha}\wedge\D\wedge\D=\frac12\ou{R}{\sigma}{\alpha\nu\mu}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}=\frac12\ou{R}{}{\rho\alpha\nu\mu}\ou{\gamma}{\rho}{}\wedge\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}=0$
とリーマン曲率テンソルを三つ足して0になるというあの恒等式と等価な恒等式を導いたのだった．僕はリーマン曲率テンソルの幾何代数的定義みたいなのを期待したのだが恒等的に0なものなのでかなりつまらないと感じた．そこでせっかく幾何代数では基底が各点で定義されているのだから，あの平行移動の微小変化させてから別の微小変化させたものとその別の微小変化を先にしてから微小変化させたものの差でリーマン曲率テンソルを定義するやつを基底で考えてみよう．
$\ou{\gamma}{}{\mu}(\os{x}{\iota}+\diff\os{x}{\iota})\\=\ou{\gamma}{}{\mu}+\ou{\gamma}{}{\mu}\pu{\nu}\diff\os{x}{\nu}\\=\ou{\gamma}{}{\mu}+\G{\alpha}{\mu\nu}\ou{\gamma}{}{\alpha}\diff\os{x}{\nu}\\=\{\ou{\delta}{\alpha}{\mu}+\G{\alpha}{\mu\nu}\diff\os{x}{\nu}\}\ou{\gamma}{}{\alpha}$
今回はこれを使って計算してく．ただし $(\os{x}{\iota})$ は省略する．
$\ou{\gamma}{}{\mu}(\os{x}{\iota}+\diff\os{x_1}{\iota}+\diff\os{x_2}{\iota})\\=\{\ou{\delta}{\alpha}{\mu}+\G{\alpha}{\mu\nu}(\os{x}{\iota}+\diff\os{x_1}{\iota})\diff\os{x_2}{\nu}\}\ou{\gamma}{}{\alpha}(\os{x}{\iota}+\diff\os{x_1}{\iota})\\=\{\ou{\delta}{\alpha}{\mu}+\{\G{\alpha}{\mu\nu}+\G{\alpha}{\mu\nu}\pu{\rho}\diff\os{x_1}{\rho}\}\diff\os{x_2}{\nu}\}\{\ou{\delta}{\beta}{\alpha}+\G{\beta}{\alpha\sigma}\diff\os{x_1}{\sigma}\}\ou{\gamma}{}{\beta}\\=\{\ou{\delta}{\alpha}{\mu}+\G{\alpha}{\mu\nu}\diff\os{x_2}{\nu}+\G{\alpha}{\mu\nu}\pu{\rho}\diff\os{x_1}{\rho}\diff\os{x_2}{\nu}\}\{\ou{\delta}{\beta}{\alpha}+\G{\beta}{\alpha\sigma}\diff\os{x_1}{\sigma}\}\ou{\gamma}{}{\beta}\\=\{\ou{\delta}{\beta}{\mu}+\G{\beta}{\mu\sigma}\diff\os{x_1}{\sigma}+\G{\beta}{\mu\nu}\diff\os{x_2}{\nu}+\G{\alpha}{\mu\nu}\G{\beta}{\alpha\sigma}\diff\os{x_1}{\sigma}\diff\os{x_2}{\nu}+\G{\beta}{\mu\nu}\pu{\rho}\diff\os{x_1}{\rho}\diff\os{x_2}{\nu}\}\ou{\gamma}{}{\beta}\\=\{\ou{\delta}{\beta}{\mu}+\G{\beta}{\mu\nu}\diff\os{x_1}{\nu}+\G{\beta}{\mu\nu}\diff\os{x_2}{\nu}+\{\G{\alpha}{\mu\nu}\G{\beta}{\alpha\sigma}+\G{\beta}{\mu\nu}\pu{\sigma}\}\diff\os{x_1}{\sigma}\diff\os{x_2}{\nu}\}\ou{\gamma}{}{\beta}$
$\ou{\gamma}{}{\mu}(\os{x}{\iota}+\diff\os{x_2}{\iota}+\diff\os{x_1}{\iota})\\=\{\ou{\delta}{\beta}{\mu}+\G{\beta}{\mu\nu}\diff\os{x_2}{\nu}+\G{\beta}{\mu\nu}\diff\os{x_1}{\nu}+\{\G{\alpha}{\mu\nu}\G{\beta}{\alpha\sigma}+\G{\beta}{\mu\nu}\pu{\sigma}\}\diff\os{x_2}{\sigma}\diff\os{x_1}{\nu}\}\ou{\gamma}{}{\beta}\\=\{\ou{\delta}{\beta}{\mu}+\G{\beta}{\mu\nu}\diff\os{x_2}{\nu}+\G{\beta}{\mu\nu}\diff\os{x_1}{\nu}+\{\G{\alpha}{\mu\sigma}\G{\beta}{\alpha\nu}+\G{\beta}{\mu\sigma}\pu{\nu}\}\diff\os{x_1}{\sigma}\diff\os{x_2}{\nu}\}\ou{\gamma}{}{\beta}$
$\ou{\gamma}{}{\mu}(\os{x}{\iota}+\diff\os{x_1}{\iota}+\diff\os{x_2}{\iota})-\ou{\gamma}{}{\mu}(\os{x}{\iota}+\diff\os{x_2}{\iota}+\diff\os{x_1}{\iota})\\=\{\{\G{\alpha}{\mu\nu}\G{\beta}{\alpha\sigma}+\G{\beta}{\mu\nu}\pu{\sigma}\}-\{\G{\alpha}{\mu\sigma}\G{\beta}{\alpha\nu}+\G{\beta}{\mu\sigma}\pu{\nu}\}\}\diff\os{x_1}{\sigma}\diff\os{x_2}{\nu}\ou{\gamma}{}{\beta}\\=\{\G{\alpha}{\mu\nu}\G{\beta}{\alpha\sigma}-\G{\alpha}{\mu\sigma}\G{\beta}{\alpha\nu}+\G{\beta}{\mu\nu}\pu{\sigma}-\G{\beta}{\mu\sigma}\pu{\nu}\}\diff\os{x_1}{\sigma}\diff\os{x_2}{\nu}\ou{\gamma}{}{\beta}\\=\ou{R}{\beta}{\sigma\nu\mu}\diff\os{x_1}{\sigma}\diff\os{x_2}{\nu}\ou{\gamma}{}{\beta}$
$\therefore\ou{\gamma}{}{\mu}(\os{x}{\iota}+\diff\os{x_1}{\iota}+\diff\os{x_2}{\iota})-\ou{\gamma}{}{\mu}(\os{x}{\iota}+\diff\os{x_2}{\iota}+\diff\os{x_1}{\iota})=\ou{R}{\beta}{\sigma\nu\mu}\diff\os{x_1}{\sigma}\diff\os{x_2}{\nu}\ou{\gamma}{}{\beta}$
ちなみにこれに $\ou{\gamma}{\mu}{}$ との峡積をとったものがリッチテンソルである．
$\ou{R}{\beta}{\sigma\nu\mu}\diff\os{x_1}{\sigma}\diff\os{x_2}{\nu}\ou{\gamma}{}{\beta}\vee\ou{\gamma}{\mu}{}=\ou{R}{}{\sigma\nu}\diff\os{x_1}{\sigma}\diff\os{x_2}{\nu}$