D’Alembertian

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

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

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

$$\text{Metric: }d{s}^2=d{x}^2+d{y}^2+d{z}^2$$

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

0.2 D’Alembertian

$$\text{Metric: }d{s}^2=d{x}^2+d{y}^2+d{z}^2-cd{t}^2$$

$$\text{Operator: }\mathrm{\square }=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}-\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}$$

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

$$\mathrm{\square }u=0$$

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

2 Alternative notation

The symbols $\mathrm{\square }$ and ${\mathrm{\square }}^2$ 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, $\mathrm{\Delta }$ 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:

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

Title	D’Alembertian
Canonical name	DAlembertian
Date of creation	2013-03-22 17:55:18
Last modified on	2013-03-22 17:55:18
Owner	invisiblerhino (19637)
Last modified by	invisiblerhino (19637)
Numerical id	8
Author	invisiblerhino (19637)
Entry type	Definition
Classification	msc 31B15
Classification	msc 31B05
Classification	msc 26B12
Synonym	wave operator
Synonym	D’Alembert operator
Related topic	Laplacian