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\begin{document}

Lorentz covariance means that all parts of the equation transform
the same.

A rotationally invariant Lagrangian is 

\( L=\sum _{i}\frac{m_{i}(\vec r _{i}'\vec r _{i}')}{2}-\sum _{i\neq j}V_{ij}(|\vec r _{I}-\vec r _{j}|) \).

examples of scalars:

speed of light: \( n_{uv}x^{u}x^{v}=0 \)

\( A^{u} \)is a 4-vector \( n_{uv}A^{u}B^{v} \)is a lorentz scalar.

proof:\( n_{uv}A'^{u}B'^{v}=n_{uv}L^{u}_{a}A^{a}L^{v}_{b}B^{b}=n_{ab}A^{a}A^{b} \)

\( A_{u}=\frac{\partial s}{\partial x^{u}} \)is a 1 form.\( A_{u}'=\frac{\partial s'}{\partial x'^{u}} \)\( =\frac{\partial s}{\partial x'^{u}} \)\( = \)\( A_{v}\frac{\partial x^{v}}{\partial x'^{u}} \)

\( L^{a}_{b}n_{au}x'^{u}=L^{a}_{b}n_{au}L^{u}_{v}x^{v}=n_{bv}x^{v} \)\( \Rightarrow  \)\( n^{va}L^{b}_{a}n_{bu}x'^{u}=x^{v} \)

This implies that the inverse transformation: \( L_{u}^{v}=n_{uv}n^{vb}L^{a}_{b} \) so \( A_{u}'=L_{u}^{v}A_{v} \).(covariant vectors)

contravariant vectors follow \( A'^{u}=L^{u}_{v}A^{v} \).

\( A^{u}B_{u} \)is a lorentz scalar.

\end{document}
