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D'Alembertian (Definition)

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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See Also: Laplacian in Cartesian Coordinates

Other names:  wave operator, D'Alembert operator

Cross-references: square, wave equation, operators, special relativity, metric, Laplacian
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This is version 2 of D'Alembertian, born on 2008-03-20, modified 2008-03-20.
Object id is 271, canonical name is DAlembertian.
Accessed 1545 times total.

Classification:
Physics Classification: 40. (ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID MECHANICS)
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