In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: ), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator^{[1]} (cf. nabla symbol) is the Laplace operator of Minkowski space. The operator is named after French mathematician and physicist Jean le Rond d'Alembert.
In Minkowski space, in standard coordinates (t, x, y, z), it has the form
Here is the 3dimensional Laplacian and η^{μν} is the inverse Minkowski metric with
 , , for .
Note that the μ and ν summation indices range from 0 to 3: see Einstein notation.
(Some authors alternatively 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.
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INVARIANCE OF D'ALEMBERTIAN OPERATOR UNDER LORENTZ TRANSFORMATION  RELATIVISTIC ELECTRODYNAMICS 
Transcription
The box symbol and alternate notations
There are a variety of notations for the d'Alembertian. The most common are the box symbol (Unicode: U+2610 ☐ BALLOT BOX) whose four sides represent the four dimensions of spacetime and the boxsquared symbol which emphasizes the scalar property through the squared term (much like the Laplacian). In keeping with the triangular notation for the Laplacian, sometimes is used.
Another way to write the d'Alembertian in flat standard coordinates is . 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 the box symbol is used to represent the fourdimensional LeviCivita covariant derivative. The symbol is then used to represent the space derivatives, but this is coordinate chart dependent.
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 A^{μ} is the electromagnetic fourpotential in Lorenz gauge.
The Klein–Gordon equation has the form
Green's function
The Green's function, , for the d'Alembertian is defined by the equation
where is the multidimensional Dirac delta function and and 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^{[2]}
where is the Heaviside step function.
See also
 Fourgradient
 d'Alembert's formula
 Klein–Gordon equation
 Relativistic heat conduction
 Ricci calculus
 Wave equation
 Oneway wave equation
References
 ^ Bartelmann, Matthias; Feuerbacher, Björn; Krüger, Timm; Lüst, Dieter; Rebhan, Anton; Wipf, Andreas (2015). Theoretische Physik (Aufl. 2015 ed.). Berlin, Heidelberg. ISBN 9783642546181. OCLC 899608232.
{{cite book}}
: CS1 maint: location missing publisher (link)  ^ S. Siklos. "The causal Green's function for the wave equation" (PDF). Archived from the original (PDF) on 30 November 2016. Retrieved 2 January 2013.
External links
 "D'Alembert operator", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 Poincaré, Henri (1906). Wikisource., originally printed in Rendiconti del Circolo Matematico di Palermo. – via
 Weisstein, Eric W. "d'Alembertian". MathWorld.