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.
Answer
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...
No comments:
Post a Comment