In special relativity, electromagnetism and wave theory, the d'Alembert operator Δ, also called the d'Alembertian or the Wave operator, is the Laplace operator of Minkowski space and other solutions of the Einstein equation. In Minkowski space in standard coordinates (t, x, y, z) it has the form Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ... Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, which exerts a force on those particles that possess the property of electric charge, and is in turn affected by the presence and motion of such particles. ... A wave is a disturbance that propagates through space, often transferring energy. ... In mathematics and physics, the Laplace operator or Laplacian, denoted by Δ, is a differential operator, specifically an important case of an elliptic operator, with many applications in mathematics and physics. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or the Einstein equation is a tensor equation in the theory of gravitation. ...

where is the three dimensional Laplacian, η⁰⁰ = 1, η⁰ⁱ = 0 and η^ij = − δ^ij for i,j = 1 to 3; η being the Minkowski metric, and δ being the Kronecker delta. Note that μ and ν range from 0 to 3, whereas i and j range from 1 to 3: see Einstein notation. The sign of these expressions depends on the sign convention used for the Minkowski metric. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae. ... In some physics textbooks and articles, certain quantities are defined with the opposite sign from that which is used in other publications. ...

Lorentz transformations leave the metric invariant, thus the above coordinate expressions remain valid for the standard coordinates in every inertial frame. A Lorentz transformation (LT) is a linear transformation that preserves the spacetime interval between any two events in Minkowski space, while leaving the origin fixed. ...

Alternate notations

In physics the symbol or is usually used for the d'Alembertian: the four sides of the box representing the four dimensions of space-time. Sometimes is used to represent the four-dimensional Levi-Civita covariant derivative. The symbol is then used to represent the space derivatives, but this is coordinate chart dependent. In such case, the three sides of the triangular nabla may be taken to represent the three dimensions of space. In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... In vector calculus, del is a vector differential operator represented by the nabla symbol, ∇. In the three-dimensional Cartesian coordinate system R3 with coordinates (x, y, z), del can be defined as or alternatively, where is the standard basis in R3. ...

Another way to write the d'Alembertian in flat standard coordinates is . The notation is useful in quantum field theory where partial derivatives are usually indexed: so the lack of an index with the squared partial derivative signals the presence of the D'Alembertian. Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

Applications

The continuity equation for the four-current J = (ρc, j) All the examples of continuity equations below express the same idea; they are all really examples of the same concept. ... In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density where c is the speed of light, ρ the charge density, and j the conventional current density. ...

can be written

The Klein-Gordon equation would look like The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is the relativistic version of the Schrödinger equation. ...

.

A wave equation for the electromagnetic field is The wave equation is an important partial differential equation which generally describes all kinds of waves, such as sound waves, light waves and water waves. ...

where A is the vector potential. In vector calculus, a vector potential is a vector field whose curl is a given vector field. ...