Editing Laplace operator (section)

===D'Alembertian===
The Laplacian can be generalized in certain ways to [[non-Euclidean]] spaces, where it may be [[elliptic operator|elliptic]], [[hyperbolic operator|hyperbolic]], or [[ultrahyperbolic operator|ultrahyperbolic]].

In [[Minkowski space]] the [[Laplace–Beltrami operator]] becomes the [[D'Alembert operator]] <math>\Box</math> or D'Alembertian:
<math display="block">\square = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2}.</math>

It is the generalization of the Laplace operator in the sense that it is the differential operator which is invariant under the [[isometry group]] of the underlying space and it reduces to the Laplace operator if restricted to time-independent functions. The overall sign of the metric here is chosen such that the spatial parts of the operator admit a negative sign, which is the usual convention in high-energy [[particle physics]]. The D'Alembert operator is also known as the wave operator because it is the differential operator appearing in the [[wave equation]]s, and it is also part of the [[Klein–Gordon equation]], which reduces to the wave equation in the massless case.

The additional factor of {{math|''c''}} in the metric is needed in physics if space and time are measured in different units; a similar factor would be required if, for example, the {{mvar|x}} direction were measured in meters while the {{mvar|y}} direction were measured in centimeters. Indeed, theoretical physicists usually work in units such that {{math|1=[[Natural units|''c'' = 1]]}} in order to simplify the equation.

The d'Alembert operator generalizes to a hyperbolic operator on [[pseudo-Riemannian manifold]]s.