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Does Huygens' principle hold in even dimensional (2m+1,1) curved spacetimes, or are there certain necessary conditions for it to hold? In other words, if I have Cauchy data for a field satisfying the wave equation on curved space, does the field value at a point only depend on the intersection of the past light cone with the Cauchy surface?

In addition, what are the physical implications in cases when Huygens' principle fails, both in odd dimensional flat space and curved spacetimes? Are there complications with the Cauchy problem or notable physical phenomena other than wave tails? I would be interested in implications for electromagnetic and gravitational radiation.

Qmechanic
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Dan
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  • It is my understanding that Hyugen's principle only holds up via flat metric. It may be possible for the principle to function in a very slightly curved space-time, however, in terms of General relativity, I would think that gravitational scattering might occur. I added this as a comment because I would like you to correct me or fill me in if I am misinterpreting your question. – Gödel Aug 02 '14 at 21:00
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    Isn't Huygens' principle just an approximation even in flat spacetime? –  Aug 02 '14 at 22:07
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    related: http://mathoverflow.net/a/5396/21349 –  Aug 03 '14 at 18:34
  • @BenCrowell No I believe my statement of the principle is valid in flat space in odd spatial dimensions, as your link points out, but correct me if I'm wrong. NaturalPhilosopher, the reason for the question is essentially this: In 4 dimensions there is a very effective description of asymptotically flat spaces using conformal compactification. As described in several places (see http://arxiv.org/abs/gr-qc/0407014) this method does not generalize in the case of even spatial dimension spacetimes. So I am looking for other heuristic differences between radiation in even and odd dim. spacetimes – Dan Aug 04 '14 at 04:17

2 Answers2

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It generally does not work in curved spacetime. There is a quite thick book almost completely devoted to study this issue by P. Günther: Huygens' Principle and Hyperbolic Equations. Some discussions can be found in Friedlander's book about the wave equation in curved spacetime. A necessary condition for the validity of the Huygens principle is that the spacetime be an Einstein space. For Ricci-flat spacetimes there are only two cases, one is Minkowski spacetime the other is a space containing plane gravitational waves.

There are also implications regarding the characteristic Cauchy problem...

Valter Moretti
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  • That is what I thought, but I would like him to specify his reasoning for posing the question. – Gödel Aug 02 '14 at 21:44
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It is possible for spacetime curvature to scatter and reflect light. The most obvious case of this is gravitational lensing. It's probably best to just solve the wave equation for the underlying light against the correct metric than to appeal to a simplifying principle like Huygen's principle.

Zo the Relativist
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