D'Alembertian

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

$\displaystyle \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.

Laplacian

   Metric: $\displaystyle ds^2 = dx^2 + dy^2 + dz^2
$

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

D'Alembertian

   Metric: $\displaystyle ds^2 = dx^2 + dy^2 + dz^2 -cdt^2
$

   Operator: $\displaystyle \Box = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\parti...
...\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 wave equation is given by:

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

Factorising in terms of operators, we obtain:

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

or

$\displaystyle \Box u = 0
$

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

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.

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:

$\displaystyle \Box = \frac{1}{c^2} \frac{\partial^2}{\partial t^2} -\nabla^2
$



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