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.
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 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, or , 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:
|Date of creation||2013-03-22 17:55:18|
|Last modified on||2013-03-22 17:55:18|
|Last modified by||invisiblerhino (19637)|