D'Alembertian | Physics Library

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

2 1 ∂2 □ = ∇ − -2---2 c ∂t

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

Metric: ds2 = dx2 + dy2 + dz2

2 ∂2 ∂2 ∂2 Operator: ∇ = ∂x2- + ∂y2-+ ∂z2-

0.2 D’Alembertian

2 2 2 2 2 Metric: ds = dx + dy + dz − cdt

2 2 2 2 Operator: □ = ∂---+ -∂--+ -∂--− 1-∂--- ∂x2 ∂y2 ∂z2 c2 ∂t2

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:

2 ∇2u = -1 ∂-u- c2 ∂t2

Factorising in terms of operators, we obtain:

2 (∇2 − 1--∂--)u = 0 c2∂t2

□u = 0

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 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:

1 ∂2 □ = 2---2-− ∇2 c ∂t

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