NetNews Usenet Archive 1992 #31/NN_1992_31.iso/spool/sci/physics/21610

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From: [e-mail] (John C. Baez)
Subject: Huyghen's principle revisited
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Organization: MIT Department of Mathematics, Cambridge, MA
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Date: Mon, 21 Dec 92 09:38:15 GMT
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I have a certain fondness for Huyghen's principle, which (recall) says
that no light "lingers" in a region after the source is turned off, or
more precisely, the fundamental solution of the wave equation is
supported on the lightcone (the interesting thing being that it's zero
*inside* the lightcone).  The fact that this is true only for
n-dimensional Euclidean space for n odd and > 1, like our 3-d world, but
not for n even or n = 1, is one of those facts that is nice to understand
from many angles.  That God was trying to create the lowest-dimensional
world in which music would sound nice doesn't count - though it's a
pleasant thought.  (Note that sound too satisfies the wave equation, at
least roughly.)

The most brutal way is just to figure out the fundamental solution of
the wave equation, which one can do by Fourier transforms or by the
method of inspired guessing followed by verification.  If space is
n-dimensional it turns out to be simply delta^{(n-3)/2}(t^2 - r^2), by
which I mean: take the (n-3)/2-th derivative of the Dirac delta function
and evaluate it on (t^2 - r^2).  Now, the charming thing is that
fractional derivatives of the delta function make perfectly fine sense
(using Fourier transform technology which I gleefully teach my grad students)
but the (n-3)/2-th derivative of the delta function is only supported at
the origin when (n-3)/2 is a nonnegative integer!  I.e., n odd and > 1.

More sophisticated is the following approach due to Lax.  The wave
equation is conformally invariant so one may work on the conformal
compactification of n+1-dimensional Minkowski space, which is just the
"Einstein universe" R x S^n.  Here the wave equation takes the form
(d^2/dt^2 - Delta + c_n)psi = 0, where t is the "time" coordinate on
R x S^n, Delta is the Laplacian on the sphere S^n, and c_n is a curious
constant, just ((n-1)/2)^2.  The reason for this constant is a bit
tricky to explain, so I won't.  However, one can solve this equation
quite explicitly using spherical harmonics on S^n (in other words,
eigenfunctions of Delta), and one sees that the solutions are all
*periodic* for n odd and > 1.  I.e., what goes around, comes around.
Lax noted that this implies Huyghen's principle, as follows.  Take a
solution of the wave equation that at t = 0 has psi being a delta
function at the north pole.  Then 2pi later psi is back to a delta
function.  If any of the wave had "lingered," there is no way it could
have "caught up" and gotten back to the north pole in time for dinner -
since it can't propagate faster than the speed of light (i.e., unit
speed).  One can easily deduce the periodicity, by the way, by working
out the eigenvalues of the Laplacian on S^n - they're of the form
j(n - 1 + j) - and noting that when you add the curious constant c_n you
get a perfect square!

I will be glad to explain some of the cryptic comments above if anyone
asks.  I used to work on conformally invariant wave equations (such as
"the" wave equation), hence my erudition.  A nice treatment of this
topic - heavy on the group theory - is "Group representations arising
from conformal geometry" by Thomas Branson (another student of my
advisor), in Jour. Funct. Analysis 73.  He works out loads of
conformally invariant differential operators on conformally compactified
Minkowski space and figures out which ones satisfy Huyghen's principle.
Other experts in Huyghen's principle and its group-theoretic roots
include Helgason and Bent Oersted.

I will not go into the relationship of conformal invariance and the
original topic of Michael Weiss -- the blackbody radiation seen by
accelerating observers -- because I don't understand it well enough.
But there *does* seem to be some sort of relationship.