PlanetPhysics/D'Alembertian

The D'Alembertian is the equivalent of the [[../LaplaceOperator/|Laplacian]] in Minkowskian geometry. It is given by:

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

Here we assume a Minkowskian [[../MetricTensor/|metric]] of the form $(+,+,+,-)$ as typically seen in [[../SR/|special relativity]]. The connection between the Laplacian in Euclidean space and the D'Alembertian is clearer if we write both [[../QuantumOperatorAlgebra4/|operators]] and their corresponding metric.

Laplacian

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

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

D'Alembertian

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

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

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 [[../WaveEquation/|wave equation]] is given by:

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

Factorising in terms of operators, we obtain:

$({\mathrm{\nabla }}^2-\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2})u=0$

or

$◻u=0$

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

Alternative notation

The symbols $◻$ and ${◻}^2$ are both used for the D'Alembertian. Since it is unheard of to [[../PiecewiseLinear/|square]] the D'Alembertian, this is not as confusing as it may appear. The symbol for the Laplacian, $\mathrm{\Delta }$ or ${\mathrm{\nabla }}^2$, 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:

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