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


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

0.1 Laplacian

Metric: ds2=dx2+dy2+dz2
Operator: 2=2x2+2y2+2z2

0.2 D’Alembertian

Metric: ds2=dx2+dy2+dz2-cdt2
Operator: =2x2+2y2+2z2-1c22t2

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:


Factorising in terms of operators, we obtain:




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

2 Alternative notation

The symbols and 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, Δ or 2, is often used when it is clear that a Minkowski spaceMathworldPlanetmath 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:

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