D'Alembert operator

In special relativity, electromagnetism and wave theory, the d'Alembert operator $\Box ^{2}$ , also called the d'Alembertian or the wave operator, is the Laplace operator of Minkowski space. The operator is named for French mathematician and physicist Jean le Rond d'Alembert. In Minkowski space in standard coordinates (t, x, y, z) it has the form:

$\Box ^{2}=\partial ^{\mu }\partial _{\mu }=g^{\mu \nu }\partial _{\nu }\partial _{\mu }=\left({\frac {\partial ^{2}}{\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}-{\frac {\partial ^{2}}{\partial y^{2}}}-{\frac {\partial ^{2}}{\partial z^{2}}}\right)={\partial ^{2} \over \partial t^{2}}-\left(\nabla _{\mathbf {R} ^{3}}^{2}\right)$

Here

$\nabla _{\mathbf {R} ^{3}}^{2}$ is the three dimensional Laplacian.
$g^{\mu \nu }$ is the Minkowski metric with $g^{00}=1$ , $g^{11}=g^{22}=g^{33}=-1$ , $g^{\mu \nu }=0$ for $\mu \not =\nu$ .

Note that the μ and ν summation indices range from 0 to 3: see Einstein notation. We have assumed units such that the speed of light $c=1$ . Few authors also use the negative metric signature of [-+++] with $\eta ^{00}=-1,\eta ^{11}=\eta ^{22}=\eta ^{33}=1$ .

Lorentz transformations leave the Minkowski metric invariant, so the d'Alembertian is a Lorentz scalar. The above coordinate expressions remain valid for the standard coordinates in every inertial frame.

Alternate notations

There is a variety of notations for the d'Alembertian. The most common is the symbol $\Box ^{2}$ : the four sides of the box representing the four dimensions of space-time. In keeping with the triangular notation for the Laplacian sometimes $\Delta _{M}$ is used.

Another way to write the d'Alembertian in flat standard coordinates is $\partial ^{2}$ . 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 $\Box ^{2}$ is used to represent the four-dimensional Levi-Civita covariant derivative. The symbol $\nabla$ is then used to represent the space derivatives, but this is coordinate chart dependent. In such case, the three sides of the triangular nabla may be taken to represent the three dimensions of space.