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A spin-off from another thread. I consulted a couple of my GR textbooks on Huygens' principle, and found little. Wiki had a little information, which said that it could be regarded as a consequence of the homogeneity of space-time, and "In 1900, Jacques Hadamard observed that Huygens' principle was broken when the number of spatial dimensions is even.".
So some of my questions are:
1) Does Huygens principle work in general space-times? (Wiki states that "Huygens' principle can be seen as a consequence of the homogeneity of space", but I wouldn't think a general space time would necessarily be homogenous).
2) How would one write the principle in an explicitly covariant manner?
3) In the limit of geometric optics, can Huygen's principle be logically be derived from Fermat's principle of "stationary optical path length" or Hamilton's principle principle of "stationary action"? And what about the reverse? (I ask about the geometric limit because I don't see how one would handle diffraction with Hamilton's principle).
On a related note, I am reminded of Feynman's "sum of all possible paths" approach to QED, I suspect there might be a relation. But I don't want to drag the thread too far afield.
4) Why does Huygen's principle only work for an odd number of spatial dimensions?
So some of my questions are:
1) Does Huygens principle work in general space-times? (Wiki states that "Huygens' principle can be seen as a consequence of the homogeneity of space", but I wouldn't think a general space time would necessarily be homogenous).
2) How would one write the principle in an explicitly covariant manner?
3) In the limit of geometric optics, can Huygen's principle be logically be derived from Fermat's principle of "stationary optical path length" or Hamilton's principle principle of "stationary action"? And what about the reverse? (I ask about the geometric limit because I don't see how one would handle diffraction with Hamilton's principle).
On a related note, I am reminded of Feynman's "sum of all possible paths" approach to QED, I suspect there might be a relation. But I don't want to drag the thread too far afield.
4) Why does Huygen's principle only work for an odd number of spatial dimensions?