In special relativity, electromagnetism and wave theory, the d'Alembert operator (represented by a box:
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In Minkowski space, in standard coordinates (t, x, y, z), it has the form
Here ∇² is the 3-dimensional Laplacian and gμν is the inverse Minkowski metric with
Note that the μ and ν summation indices range from 0 to 3: see Einstein notation. We have assumed units such that the speed of light c = 1.
Some authors also use the negative metric signature of (− + + +), with
Lorentz transformations leave the Minkowski metric invariant, so the d'Alembertian yields a Lorentz scalar. The above coordinate expressions remain valid for the standard coordinates in every inertial frame.
Alternate notations
There are a variety of notations for the d'Alembertian. The most common is the symbol
Another way to write the d'Alembertian in flat standard coordinates is ∂². This notation is used extensively 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.
Sometimes
Applications
The wave equation for small vibrations is of the form
where u(x,t) is the displacement.
The wave equation for the electromagnetic field in vacuum is
where Aμ is the electromagnetic four-potential.
The Klein–Gordon equation has the form
Green's function
The Green's function,
where δ(~x−~x') is the multidimensional Dirac delta function and ~x and ~x' are two points in Minkowski space.
A special solution is given by the retarded Green's function which corresponds to signal propagation only forward in time
where Θ is the Heaviside step function.