Up 一般座標の計量テンソル 作成: 2017-12-30
更新: 2017-01-01


    直線座標の上では,つぎの形式が計量テンソルとなった ( 「直線座標の計量テンソル」):
      \[ s^2 = \sum_{i,j=1}^n d_{ij}\, {x^i}{x^j} \]
    これから一般座標 \( u^i \) での計量テンソルを導いてみる。

    直線座標 \( x^i \) と一般座標 \( u^i \) 間の変換は:
      \[ \left( \begin{array}{c} x^1 \\ \vdots \\ x^n \\ \end{array} \right) \begin{array}{c} \left(\,\frac{\partial x^i}{\partial u^j}\,\right) \\ \longleftarrow \\ \longrightarrow \\ \left(\,\frac{\partial u^i}{\partial x^j}\,\right) \\ \end{array} \left( \begin{array}{c} u^1 \\ \vdots \\ u^n \\ \end{array} \right) \\ x^i = \sum_{j=1}^n \frac{\partial x^i}{\partial u^j} u^j \ \ \ \ (\, i = 1, \cdots, n \,) \]
    よって,
      \[ ds^2 = \sum_{k,l=1}^n d_{kl}\, {dx^k}{dx^l} \\ = \sum_{k,l=1}^n d_{kl} \left( \sum_{i=1}^n \frac{\partial x^k}{\partial u^i} du^i \right) \left( \sum_{j=1}^n \frac{\partial x^l}{\partial u^j} du^j\right) \\ = \sum_{i,j=1}^n \left( \sum_{k,l=1}^n d_{kl}\, \frac{\partial x^k}{\partial u^i} \frac{\partial x^l}{\partial u^j} \right) du^i \, du^j \\ \]
    形式
      \[ \sum_{i,j=1}^n \left( \sum_{k,l=1}^n d_{kl}\, \frac{\partial x^k}{\partial u^i} \frac{\partial x^l}{\partial u^j} \right) du^i \, du^j \]
    は,\( x^i,\, u^i \) の座標変換 \[ \left( \begin{array}{c} x^1 \\ \vdots \\ x^n \\ \end{array} \right) \begin{array}{c} \left(\,a^i_j\,\right) \\ \longleftarrow \\ \longrightarrow \\ \left(\,b^i_j\,\right) \\ \end{array} \left( \begin{array}{c} {x^{'}}^1 \\ \vdots \\ {x^{'}}^n \\ \end{array} \right) \ \ \ \ \ \ \ \ \ \left( \begin{array}{c} u^1 \\ \vdots \\ u^n \\ \end{array} \right) \begin{array}{c} \left(\,\frac{\partial u^i}{\partial {u^{'}}^j}\,\right) \\ \longleftarrow \\ \longrightarrow \\ \left(\,\frac{\partial {u^{'}}^i}{\partial u^j}\,\right) \\ \end{array} \left( \begin{array}{c} {u^{'}}^1 \\ \vdots \\ {u^{'}}^n \\ \end{array} \right) \] で保たれる: \[ \sum_{i,j=1}^n \left( \sum_{k,l=1}^n d_{kl}\, \frac{\partial x^k}{\partial u^i} \frac{\partial x^l}{\partial u^j} \right) du^i \, du^j \\ = \sum_{i,j=1}^n \left( \sum_{k,l=1}^n d_{kl} \left( \frac{\partial x^k}{\partial {x^{'}}^k} \frac{\partial {x^{'}}^k}{\partial {u^{'}}^i} \frac{\partial {u^{'}}^i}{\partial u^i} \right) \left( \frac{\partial x^l}{\partial {x^{'}}^l} \frac{\partial {x^{'}}^l}{\partial {u^{'}}^j} \frac{\partial {u^{'}}^j}{\partial u^j} \right) \right) du^i \, du^j \\ = \sum_{i,j=1}^n \left( \sum_{k,l=1}^n d_{kl} \left( \frac{\partial x^k}{\partial {x^{'}}^k} \frac{\partial x^l}{\partial {x^{'}}^l} \right) \left( \frac{\partial {x^{'}}^l}{\partial {u^{'}}^i} \frac{\partial {x^{'}}^l}{\partial {u^{'}}^j} \right) \right) \frac{\partial {u^{'}}^i}{\partial u^i} du^i \frac{\partial {u^{'}}^j}{\partial u^j} du^j \\ = \sum_{i,j=1}^n \left( \sum_{k,l=1}^n d_{kl} \left( a^k_k\, a^l_l \right) \frac{\partial {x^{'}}^k}{\partial {u^{'}}^i} \frac{\partial {x^{'}}^l}{\partial {u^{'}}^j} \right) d{u^{'}}^i d{u^{'}}^j \\ = \sum_{i,j=1}^n \left( \sum_{k,l=1}^n d^{'}_{kl} \frac{\partial {x^{'}}^k}{\partial {u^{'}}^i} \frac{\partial {x^{'}}^l}{\partial {u^{'}}^j} \right) d{u^{'}}^i d{u^{'}}^j \\ \ \ \ \ \ \ \ \ \ d^{'}_{kl} = d_{kl} \, a^k_k \, a^l_l \] そこで,形式
      \[ ds^2 = \sum_{i,j=1}^n g_{ij}\, du^i \, du^j \\ g_{ij} = \sum_{k,l=1}^n d_{kl} \frac{\partial x^k}{\partial u^i} \frac{\partial x^l}{\partial u^j} \]
    を「直線座標 \( x^i \) に対する座標 \( u^i \) の上の計量テンソル」と定める。

    \( g_{ij} \) は, (0, 1) テンソル \( \frac{\partial x^k}{\partial u^i},\, \frac{\partial x^l}{\partial u^j}\) の積の和ということで, (0, 2) テンソルである。