d’Alembertian
d’Alembert operator, or wave operator, is the Lapace operator in Minkowski space.
\[\Box\equiv \partial _ \mu\partial^\nu = \eta _{\mu\nu}\partial^\mu \partial^\nu\]
In the usual {t,x,y,z} natural orthonormal basis,
\[\begin{split}\begin{eqnarray}
\Box&=&-\partial_t^2+\partial_x^2+\partial_y^2+\partial_z^2 \\
&=&-\partial_t^2+\Delta^2 \\
&=&-\partial_t^2+\nabla
\end{eqnarray}\end{split}\]
- On wiki , they give some applications to it.
- klein-Gordon equation
\((\Box+m^2)\phi=0\)
- wave equation for electromagnetic field in vacuum:
For the electromagnetic four-potential $Box A^mu=0$footnote{Gauge}
- wave equation for small vibrations
\(\Box_c u(t,x)=0\rightarrow u_{tt}-c^2 u_{xx}=0\)