D'Alembert Operator
   HOME

TheInfoList



OR:

In
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
,
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''.
nabla symbol The nabla symbol The nabla is a triangular symbol resembling an inverted Greek delta:Indeed, it is called ( ανάδελτα) in Modern Greek. \nabla or ∇. The name comes, by reason of the symbol's shape, from the Hellenistic Greek word ...
) is the Laplace operator of
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
. The operator is named after French mathematician and physicist
Jean le Rond d'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Encyclopédie ...
. In Minkowski space, in standard coordinates , it has the form : \begin \Box & = \partial^\mu \partial_\mu = \eta^ \partial_\nu \partial_\mu = \frac \frac - \frac - \frac - \frac \\ & = \frac - \nabla^2 = \frac - \Delta ~~. \end Here \nabla^2 := \Delta is the 3-dimensional
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
and is the inverse
Minkowski metric In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
with :\eta_ = 1, \eta_ = \eta_ = \eta_ = -1, \eta_ = 0 for \mu \neq \nu. Note that the and summation indices range from 0 to 3: see
Einstein notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
. We have assumed units such that the speed of light = 1. (Some authors alternatively use the negative
metric signature In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...
of , with \eta_ = -1,\; \eta_ = \eta_ = \eta_ = 1.)
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
s leave the
Minkowski metric In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
invariant, so the d'Alembertian yields a
Lorentz scalar In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
. The above coordinate expressions remain valid for the standard coordinates in every inertial frame.


The box symbol and alternate notations

There are a variety of notations for the d'Alembertian. The most common are the ''box'' symbol \Box (
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expre ...
: ) whose four sides represent the four dimensions of space-time and the ''box-squared'' symbol \Box^2 which emphasizes the scalar property through the squared term (much like the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
). In keeping with the triangular notation for the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
, sometimes \Delta_M is used. Another way to write the d'Alembertian in flat standard coordinates is \partial^2. This notation is used extensively in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, 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 four-dimensional Levi-Civita covariant derivative. The symbol \nabla is then used to represent the space derivatives, but this is
coordinate chart In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
dependent.


Applications

The
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
for small vibrations is of the form : \Box_ u\left(x,t\right) \equiv u_ - c^2u_ = 0~, where is the displacement. The
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
for the electromagnetic field in vacuum is : \Box A^ = 0 where is the
electromagnetic four-potential An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. W ...
in
Lorenz gauge In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ha ...
. The
Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. ...
has the form :\left(\Box + \frac\right) \psi = 0~.


Green's function

The
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
, G\left(\tilde - \tilde'\right), for the d'Alembertian is defined by the equation : \Box G\left(\tilde - \tilde'\right) = \delta\left(\tilde - \tilde'\right) where \delta\left(\tilde - \tilde'\right) is the multidimensional
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
and \tilde and \tilde' are two points in Minkowski space. A special solution is given by the ''retarded Green's function'' which corresponds to signal
propagation Propagation can refer to: * Chain propagation in a chemical reaction mechanism *Crack propagation, the growth of a crack during the fracture of materials * Propaganda, non-objective information used to further an agenda * Reproduction, and other fo ...
only forward in time :G\left(\vec, t\right) = \frac \Theta(t) \delta\left(t - \frac\right) where \Theta is the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
.


See also

*
Four-gradient In differential geometry, the four-gradient (or 4-gradient) \boldsymbol is the four-vector analogue of the gradient \vec from vector calculus. In special relativity and in quantum mechanics, the four-gradient is used to define the properties and r ...
*
d'Alembert's formula In mathematics, and specifically partial differential equations (PDEs), d'Alembert's formula is the general solution to the one-dimensional wave equation u_(x,t) = c^2 u_(x,t) (where subscript indices indicate partial differentiation, using the d' ...
*
Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. ...
*
Relativistic heat conduction Relativistic heat conduction refers to the modelling of heat conduction (and similar diffusion processes) in a way compatible with special relativity. In special (and general) relativity, the usual heat equation for non-relativistic heat conduct ...
* Ricci calculus *
Wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
*
One-way wave equation A one-way wave equation is a first-order partial differential equation describing one wave traveling in a direction defined by the vector wave velocity. It contrasts with the second-order two-way wave equation describing a standing wavefield resu ...


References


External links

* * , originally printed in
Rendiconti del Circolo Matematico di Palermo The Circolo Matematico di Palermo (Mathematical Circle of Palermo) is an Italian mathematical society, founded in Palermo by Sicilian geometer Giovanni B. Guccia in 1884.
. * {{physics operators Differential operators Hyperbolic partial differential equations