In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box:
nabla symbol) is the Laplace operator of Minkowski space.
The operator is named after French mathematician and physicist Jean le Rond d'Alembert.
In Minkowski space, in standard coordinates (t, x, y, z), it has the form Here
is the 3-dimensional Laplacian and ημν is the inverse Minkowski metric with Note that the μ and ν summation indices range from 0 to 3: see Einstein notation.
(Some authors alternatively use the negative metric signature of (− + + +), with
Lorentz transformations leave the Minkowski metric invariant, so the d'Alembertian yields a Lorentz scalar.
(Unicode: U+2610 ☐ BALLOT BOX) whose four sides represent the four dimensions of space-time and the box-squared symbol
which emphasizes the scalar property through the squared term (much like the Laplacian).
Another way to write the d'Alembertian in flat standard coordinates is
This notation is used extensively in quantum field theory, where partial derivatives are usually indexed, so the lack of an index with the squared partial derivative signals the presence of the d'Alembertian.
Sometimes the box symbol is used to represent the four-dimensional Levi-Civita covariant derivative.
is then used to represent the space derivatives, but this is coordinate chart dependent.
The wave equation for small vibrations is of the form where u(x, t) is the displacement.
The wave equation for the electromagnetic field in vacuum is where Aμ is the electromagnetic four-potential in Lorenz gauge.
A special solution is given by the retarded Green's function which corresponds to signal propagation only forward in time[2] where