DAlembertOperator
The DAlembertOperator, commonly called the d'Alembertian, is a differential operator that generalizes the Laplacian to Lorentzian (spacetime) manifolds. It is denoted by □ and can be written as □ = ∇^μ ∇_μ, the covariant d'Alembertian. In flat Minkowski space it reduces to a simple second-order differential operator, and it is form-invariant under Lorentz transformations. Different sign conventions exist in the literature, reflecting choices of metric signature.
In four-dimensional flat spacetime with metric signature (-,+,+,+), the operator takes the form □ = -∂^2/∂t^2 + ∇^2. Some authors
The d'Alembertian appears in relativistic wave equations. For a scalar field φ, the Klein-Gordon equation is □φ + m^2
In curved spacetime, the operator generalizes to the covariant d'Alembertian □ ≡ ∇^μ ∇_μ acting on scalar fields. It plays
The operator is named after Jean le Rond D'Alembert, reflecting its origins in the study of wave