# 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.

### 0.1 Laplacian

 $\mbox{Metric: }ds^{2}=dx^{2}+dy^{2}+dz^{2}$
 $\mbox{Operator: }\nabla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial% ^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}$

### 0.2 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.

## 1 Connection with the wave equation

The wave equation is given by:

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

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.

## 2 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.

## 3 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:

 $\Box=\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2}$
Title D’Alembertian DAlembertian 2013-03-22 17:55:18 2013-03-22 17:55:18 invisiblerhino (19637) invisiblerhino (19637) 8 invisiblerhino (19637) Definition msc 31B15 msc 31B05 msc 26B12 wave operator D’Alembert operator Laplacian