Huygens' Principle  

In General
$ Def: (Hadamard's point of view) A wave equation satisfies Huygens' principle if the support of its Green function with no incoming waves, or fundamental solution, the one with source δ(x) δ(t), is entirely contained on the future or past light cone of that point.
* Idea: Roughly, data concentrated in a small region at time t' will only produce a disturbance on a sphere of radius c(tt') (without leaving tails behind); The value of a massless field at a point in spacetime is determined by data on the intersection of its past light cone with some spacelike surface.
* Scalar fields, flat spacetime: It holds for massless Klein-Gordon particles in even d > 2.
* Other fields, flat spacetime: A spin-1/2 field in 4D satisfies the Huygens principle if and only if m = 0.

Related Concepts > see Hadamard's Conjecture; perturbations in general relativity; velocity [speedup in random media]; wave equations and wave tails.

@ General: Künzle PCPS(68); Günther 88; Bombelli, Couch & Torrence JMP(91); Soodak & Tiersten AJP(93)may; Bombelli & Sonego JPA(94)mp/00; Liu mp/03 ["proof"]; Dai & Stojković EPJP-a1309 [physical origin of the tail in odd dimensionalities].
@ Electromagnetism: Kaiser AACA-a0906 [distributional approach].
@ Other fields: Noonan CQG(95) [higher-rank]; Baum JGP(97) [Dirac]; Berest & Loutsenko CMP(97) [KdV solitons].
@ Variations: Luís EJP(07) [complementary version, in terms of Wigner functions and rays].

In Curved Spacetime > s.a. phenomenology of gravity.
* General results: A necessary condition for the validity of the Huygens principle is that (M, g) be an Einstein space.
* Ricci-flat spacetime: For ∇2φ = 0, the only cases are Minkowski space and plane waves.
* de Sitter spacetime: The Klein-Gordon equation obeys Huygens' principle if and only if the mass m and the spatial dimension n are related by m2 = (n2 – 1)/4.
@ General references: McLenaghan PCPS(68); Goldoni JMP(77); Sonego & Faraoni JMP(92); Noonan CQG(96) [necessary condition, Bach tensor = 0].
@ Specific types of spacetimes: Noonan CQG(95) [conformally flat], CQG(01) [scalar, Einstein spacetime]; McLenaghan & Sasse AIHP(96)mp/05 [Petrov III, electromagnetism and Dirac]; Anderson et al AIHP(99)mp/05 [Petrov III, scalar]; Czapor et al AIHP(99)mp/05 [Petrov III, conformally invariant scalar]; Yagdjian JMP(13)-a1206 [de Sitter spacetime].
@ Other types of fields: Wünsch GRG(85) [higher-spin fields]; Yagdjian a2009 [Dirac fields in de Sitter spacetime].
> Specific theories: see metric perturbations in general relativity; wave phenomena.

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