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