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.

* __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.

**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].

@ __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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