Тензори у СТВ. Доведення

Доведення 1
Загальні перетворення Лоренца для тензора другого рангу.

$$\ {T^{00}}' = \gamma^{2}A^{0}B^{0} - \frac{\gamma^{2}}{c}u_{k}(A^{0}B^{k} + A^{k}B^{0}) + \frac{\gamma^{2}}{c^{2}}u_{k}u_{p}A^{k}B^{p} = \gamma^{2}T^{00} - \frac{\gamma^{2}}{c}u_{k}(T^{0k} + T^{k0}) + \frac{\gamma^{2}}{c^{2}}u_{k}u_{p}T^{kp}$$,

$$\ {T^{ij}}' = A^{i}B^{j} + \frac{\Gamma}{c^{2}}u^{j}u_{l}B^{l}A^{i} + \frac{\Gamma}{c^{2}}u^{i}u_{l}A^{l}B^{j} - \frac{\gamma}{c}u_{j}B^{0}A^{i} - \frac{\gamma}{c}u_{i}A^{0}B^{j} - \frac{\gamma \Gamma}{c^{3}}u^{i}u^{j}u_{l}A^{l}B^{0} - \frac{\gamma \Gamma}{c^{3}}u^{i}u^{j}u_{l}A^{0}B^{l} + \frac{\Gamma^{2}}{c^{4}}u^{i}u^{j}u_{l}A^{l}u_{n}B^{n} + \frac{\gamma^{2}}{c^{2}}u_{i}u_{j}A^{0}B^{0} = $$

$$\ = T^{ij} + \frac{\Gamma}{c^{2}}u_{l}(u^{j}T^{il} + u^{i}T^{lj}) - \frac{\gamma}{c}(u^{j}T^{i0} + u^{i}T^{0j}) - \frac{\gamma \Gamma}{c^{3}}u^{i}u^{j}(T^{l0} + T^{0l}) + \frac{\Gamma^{2}}{c^{4}}u^{i}u^{j}u_{l}u_{n}T^{ln} + \frac{\gamma^{2}}{c^{2}}u_{i}u_{j}T^{00}$$,

$$\ {T^{0j}}' = \gamma A^{0}B^{j} + \frac{\gamma \Gamma}{c^{2}}u^{j}u_{l}B^{l}A^{0} - \frac{\gamma^{2}}{c}u^{j}A^{0}B^{0} - \frac{\gamma}{c}u_{k}A^{k}B^{j} - \frac{\gamma \Gamma}{c^{3}}u^{j}u_{k}u_{l}A^{k}B^{l} + \frac{\gamma^{2}}{c^{2}}u_{k}u^{j}A^{k}B^{0} = $$

$$\ = \gamma T^{0j} + \frac{\gamma \Gamma}{c^{2}}u^{j}u_{l}T^{0l} - \frac{\gamma^{2}}{c}u^{j}T^{00} - \frac{\gamma}{c}u_{k}T^{kj} - \frac{\Gamma \gamma}{c^{3}}u^{j}u_{k}u_{l}T^{kl} + \frac{\gamma^{2}}{c^{2}}u_{k}u^{j}T^{k0} = $$

$$\ = \gamma \left( T^{0j} - \frac{1}{c}u_{k}T^{kj}\right) - \frac{\gamma^{2}}{c}u^{j}\left( T^{00} - \frac{1}{c}u_{k}T^{k0}\right) + \frac{\Gamma \gamma}{c^{2}}u^{j}\left( u_{l}T^{0l} - \frac{1}{c}u_{k}u_{l}T^{kl} \right)$$,

$$\ {T^{i0}}' = \gamma T^{i0} + \frac{\gamma \Gamma}{c^{2}}u^{i}u_{l}T^{l0} - \frac{\gamma^{2}}{c}u^{i}T^{00} - \frac{\gamma}{c}u_{k}T^{ik} - \frac{\Gamma \gamma}{c^{3}}u^{i}u_{k}u_{l}T^{lk} + \frac{\gamma^{2}}{c^{2}}u_{k}u^{i}T^{0k} = $$

$$\ = \gamma \left( T^{i0} - \frac{1}{c}u_{k}T^{ik}\right) - \frac{\gamma^{2}}{c}u^{i}\left( T^{00} - \frac{1}{c}u_{k}T^{0k}\right) + \frac{\Gamma \gamma}{c^{2}}u^{i}\left( u_{l}T^{l0} - \frac{1}{c}u_{k}u_{l}T^{lk} \right)$$,

де враховано, що при перемноженні двох величин із індексами, що мають однакові позначення, одне з позначень змінюється:

$$\ (u_{k}A^{k})(u_{k}A^{k}) = u_{k}A^{k}u_{l}A^{l}$$.