Huygens Principle

The dispute of 1689 between Newton and Huygens was won by Newton (he was by far the more important person) but in reality, Newton was wrong (*). With all the information available at that time, the “guess” that light is made of corpuscular objects should not have been tenable at the timeenIt failed to explain refraction phenomena. Of course, one can be “wrong” even if one should later be right. Democritus was dead wrong with his wild guess about the discreteness of matter. It was a wild guess due to observations that very tiny particles could no more be separated. It does not matter that one has later seen atoms. In the case of Newton’s guess, photon picture was only justified once the photoelectric effect was found. Also, later, de Broglie went full circle and suggested that also point particles like an electron are waves. Today, we keep the matter in a schizophrenic state and call both pictures “correct”. The particle-wave duality is at the heart of fundamental physics. The particle picture, where point particles move on geodesics is closer to GR, while the wave picture where we look at solutions of wave equations is closer to QM. It is interesting that still today, it is fashionable to claim that QM is “hard”. In reality, it is just fancy linear algebra and linear stuff is in principle much, much easier than classical mechanics or relativity with all these non-linear ordinary differential equations. One can imagine the “architect of our world” during the design process see what sort of mess nonlinear ordinary differential equations can lead into and say “lets do it linear, meaning lets let things be quantum and not classical”. What is “beautiful” and what is “difficult” is always very relative and part of the fashion of time, and as usual, also depends of what the VIP’s claim to be the case. It would have been nice to see the meeting of Newton and Huygens in June of 1689 as they were discussing such fundamental stuff. Huygens was a an admirer of Newton but had concerns about the laws of gravity, concerns which only much later were resolved with Einstein. Of course, Huygens was no match to the heavyweight Newton.

[(*) Update of January 25: in more philosophical texts, Newton’s arguments are still today favored and Huygens is pictured as a “mechanist” who could not image forces to happen without any medium transmitting it. Maybe that was an other reason for his arguments. More favorable is to see the objection as an early relativistic point of view: it should take time for signals. Only Einstein was essentially revived a “mechanical view”. Events must be connected in a causal way (even so it also killed ideas like the Aether which are a mechanical part but which is also not meaningless today as particle-anti-particle pairs can be created anywhere, also in a complete vaccuum. Space can not be completely empty.). Betelgeuse is 700 light years away and is expected to explode soon (in the next 100’000 years maybe), but the event could already have happened at the time, when Newton and Huygens met in the summer of 1689. The signal of the event just might not have reached us. Newton’s gravity assumes that the gravitational effect is immediate. His particle picture was motivated by his theory of color and it appeared natural that such quantities are attached to matter. It does not explain for example that in even dimensions with the weak Huygens picture only, a light source at the center the infinite plane emitting photons would see photons bounce back. It would suggest that like in one or three dimensions, the photons at time t are in distance t from the source P only. ]

Maybe the two were discussing scenarios which would be puzzling even today: assume you would have a way to make the sun disappear instantaneously (Scotty from Startrek, please beam our sun to Andromeda ….!) The earth would still be pulled to the sun for 8 minutes and 20 seconds, until we would learn about this event and until then feel the gravitational pull become absent. The Newton law of gravity does not take such things into account. This came only with general relativity. But Huygens seemed have worried about it already. In GR, one would describe this as a problem to solve the Einstein equations in a situation where one has a mass point up time. This can be described with a suitable matter tensor. The Einstein equations would come up with a metric g in space time. It would be a hard problem to get an explicit solution but it would have the effect the space metric would flatten out and a non-flat wave move out with the speed of light. One sees similar events in the case of a black hole merger. There are gravitational waves involved too and the experimental evidence is strong that such waves indeed exist. The statement that gravitons exist is at the moment still just pure speculation. None have been observed. But it belongs into the “particle-wave” debate.

In this presentation I talk a bit about solutions of the wave equations. Usually this is done for scalar wave equations in flat Euclidean space, where one can use spherical coordinates. But all works nicely in any Riemannian manifold (M,g) once one can see that everything can be done coordinate free using differential forms and operators which define the solutions through “Taylor type expansions” where the derivative is replaced by D or L. The Kirchhoff solution for example is completely determined by the Laplacian L and pushing the solution from the Euclidean case to a manifold does not even require to change notation.

Update January 25, 206: The Fourier and Operator method solution are taught in intro linear algebra courses: here is a 1 minute video about the wave equation or this 1 minute video about the operator method (in that course D is used for the derivative operator and not for the Dirac operator but it is almost equivalent: in one dimension i d/dx is self adjoint and can be seen (if written using 2×2 matrices) as which maps 0-forms to 1-forms and 1-forms to 0-forms. For the self-adjoint operator D=id/dx, the Schroedinger equation is the transport equation which has the solution g(x+t) if u(0,x)=g(x) is the initial condition. A Taylor expansion of the solution is the same than the solution of the Schroedinger equation This is a reason for calling as a Taylor solution of the wave equation. What is cool is that works on any Riemannian manifold and (without any change of notation) verbatim also in the discrete or even in any delta set (and in particular in any simplicial set and even more particular in a simplicial complex and even more particular in the Whitney complex of a graph and even more particular in a one dimensional simplicial complex which some authors equate with “graph”. The remark about quantum dynamics which was written 30 years ago but appeared in 1998) which writes the solution of a modified wave equation as discrete infinite dimensional symplectic map renders this “Taylor expansion” into a discrete type CA type evolution similar as a random walk, but it is an actual interpolation of a wave equation. It solves the finite speed of propagation problem one has when looking at space discretizations. With a discrete time there is a weak Huygens principle. If we let a wave start at a simplex x and an other simplex y is 1000 steps away, and if start the wave equation with an initial condition localized at x, then it will take 1000 time steps to change the value at y. Any semi-discrete equation like the wave equation immediately affects the amplitude of the wave at and the reason is that in a finite dimensional situation the solution is given by explicit entire functions that can not be zero for some time and then suddenly change once the signal has arrived (an entire function in t can not be locally constant on some part of an interval). At the time when writing the 1998 paper, I had not been concerned with philosophical causal questions but worried about “numerical integrity”. When you make an experiment in a finite laboratory, there are natural boundary effects (which can be periodic boundary conditions). The presence of a boundary or global topology of the space should not be part of the consideration when looking at a wave for a short time. The method described in the paper allowed to compute thousands of Fourier coefficients exactly. But it is a weak Huygens principle restoration.