dAlembert operator
E505995
The d'Alembert operator is a second-order differential operator used in relativistic wave equations to describe how fields propagate through spacetime.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
differential operator
ⓘ
linear operator ⓘ second-order differential operator ⓘ wave operator ⓘ |
| actsOn |
scalar fields
ⓘ
tensor fields ⓘ vector fields ⓘ |
| alsoKnownAs |
box operator
ⓘ
d'Alembertian NERFINISHED ⓘ |
| appearsIn |
Klein–Gordon–Fock equation
NERFINISHED
ⓘ
Maxwell's equations in covariant form ⓘ classical field theory NERFINISHED ⓘ linearized gravity equations ⓘ |
| commutesWith | Poincaré transformations in Minkowski spacetime ⓘ |
| coordinateSystem | Minkowski spacetime NERFINISHED ⓘ |
| covariantForm | uses covariant derivatives ⓘ |
| definitionInMinkowskiCoordinates |
□ = -\partial_t^2 + \partial_x^2 + \partial_y^2 + \partial_z^2
ⓘ
□ = \eta^{\mu\nu} \partial_\mu \partial_\nu ⓘ |
| describes | propagation of fields in spacetime ⓘ |
| domain | spacetime manifolds ⓘ |
| eigenfunctions | plane waves in flat spacetime ⓘ |
| equationType | wave equation ⓘ |
| field |
differential geometry
ⓘ
mathematical physics ⓘ partial differential equations ⓘ theoretical physics ⓘ |
| generalizationOf | Laplace operator to spacetime ⓘ |
| generalRelativisticForm | \Box = g^{\mu\nu} \nabla_\mu \nabla_\nu ⓘ |
| linearity | linear ⓘ |
| LorentzInvariant | yes ⓘ |
| metricSignatureDependent | yes ⓘ |
| namedAfter | Jean le Rond d'Alembert NERFINISHED ⓘ |
| order | second order ⓘ |
| relatedConcept |
Green's function
ⓘ
advanced propagator ⓘ retarded propagator ⓘ |
| relatedTo | Laplace operator NERFINISHED ⓘ |
| role | governs propagation of waves at finite speed ⓘ |
| spacetimeDimension | four-dimensional Minkowski spacetime ⓘ |
| symbol |
abla^2
ⓘ
□ ⓘ |
| type | hyperbolic differential operator ⓘ |
| usedIn |
Klein–Gordon equation
NERFINISHED
ⓘ
electromagnetism ⓘ general relativity NERFINISHED ⓘ quantum field theory ⓘ relativistic wave equations ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.