PlanetPhysics/D'Alembertian

The D'Alembertian is the equivalent of the Laplacian in Minkowskian geometry. It is given by:

$\Box =\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}$

Here we assume a Minkowskian metric of the form $(+,+,+,-)$ as typically seen in special relativity. The connection between the Laplacian in Euclidean space and the D'Alembertian is clearer if we write both operators and their corresponding metric.

Laplacian

${\mbox{Metric: }}ds^{2}=dx^{2}+dy^{2}+dz^{2}$

${\mbox{Operator: }}\nabla ^{2}={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}$

D'Alembertian

${\mbox{Metric: }}ds^{2}=dx^{2}+dy^{2}+dz^{2}-cdt^{2}$

${\mbox{Operator: }}\Box ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}$

In both cases we simply differentiate twice with respect to each coordinate in the metric. The D'Alembertian is hence a special case of the generalised Laplacian.

Connection with the wave equation

The wave equation is given by:

$\nabla ^{2}u={\frac {1}{c^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}$

Factorising in terms of operators, we obtain:

$(\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}})u=0$

or

$\Box u=0$

Hence the frequent appearance of the D'Alembertian in special relativity and electromagnetic theory.

Alternative notation

The symbols $\Box$ and $\Box ^{2}$ are both used for the D'Alembertian. Since it is unheard of to square the D'Alembertian, this is not as confusing as it may appear. The symbol for the Laplacian, $\Delta$ or $\nabla ^{2}$, is often used when it is clear that a Minkowski space is being referred to.

Alternative definition

It is common to define Minkowski space to have the metric $(-,+,+,+)$, in which case the D'Alembertian is simply the negative of that defined above:

$\Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}$