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*(*t*–*t'*) (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.

* __Flat space__: It holds
for massless Klein-Gordon particles in even *d* > 2.

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

**References**

@ __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 ∇^{2}*φ* = 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 *m*^{2} = (*n*^{2} – 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].

> __Specific theories__:
see metric perturbations in general relativity; wave
phenomena.

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