Maths & Physics

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It is said that every other math kid at Goettingen knew all about matrices but it was left to Heisenberg to "invent" his Matrix Mechanics. And that many math stalwarts including Hilbert were hunting keenly for a curved space version of Gravity, but it was left to Einstein to come up with what we now know as the 'correct' field equations. And Whittaker was so pissed off with the upstart physicist Einstein that he gave all the credit of SR to his math guru Poincare and to Lorentz by the way. And that even Gauss was scared to publish his discovery of what is now known as the Riemannian Geometry of 2-dimensions...much like the astronomers of the Dark Ages.

And in a small way my Guru SDM was scared a bit...the story is like this:

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It required a good 25 years for the world to recognize Cherenkov Effect as important (in particle detectors) and give him and Frank and Tamm their Nobels in 1958. And then physicists started taking interest in the Cherenkov Radiation. Most of them worked, naturally, in the lab frame where the medium is at rest.

But SDM wanted to be different, as always. So, he preferred to jump to the rest frame of the moving charge, where the fields become 'static'. But the 'number' of Cherenkov photons emitted must be the same in every inertial frame. So, how does a charge at rest radiate? That is because the moving medium becomes peculiarly 'magneto-electric'.

See G P Sastry:

http://ajp.aapt.org/resource/1/ajpias/v46/i5/p554_s1?isAuthorized=no

And SDM wanted to work out Cherenkov Radiation in crystals...as usual not knowing that experimental work in this field is very much alive at the Proton Synchrotron at Dubna, USSR, by V P Zrelov.

And straightaway SDM attacked the problem in crystals which are simultaneously anisotropic in their electrical permittivities (ep) and magnetic permeabilities (mp). And discovered that he landed up in a fourth order differential equation which could be factorized under certain conditions.

Now, an isotropic dielectric medium like glass has all its three principal (ep)s equal. Its wave surface is a sphere. If two of them are equal and the third is different we call it a uniaxial crystal, like calcite. This has its wave surface split into a sphere and an ellipsoid. If all the three (ep)s are different we call it a biaxial crystal, like TGS (Tri-Glycine Sulphate). Its wave surface is a single self-intersecting monster, which leads to very interesting effects like conical refraction.

Similar with magnetic crystals, with (ep) replaced by (mp).

Now, SDM and his RS (R Pal) discovered that their 4th order differential equation factorizes into two under the condition that two of the ratios (ep / mp) are equal. This was new and strange to them but SDM had the courage to call it a 'generalized uniaxilaity condition'. But he didn't have the courage to say that such a crystal is a 'generalized uniaxial crystal'. Note the difference: SDM thought that his condition is a mathematical curiosity arising out of his choice of the rest frame of the particle.

R Pal left after his Ph D to take up manufacturing electronic components and selling them and doing original work on Alexander the Great and Lord Buddha.

That was when I joined SDM. I could see that his condition is very physical, because I was working on a more impressive monster medium in which the dielectric and magnetic principal axes are not parallel but are arbitrarily inclined. So, I had the advantage of generalizing (even more) his 'generalized uniaxiality condition'.

See: G P Sastry:

http://iopscience.iop.org/0301-0015/7/5/010

Lesson: Always start with the most general case and fail...and then descend to workable special cases...not the other way round.

So, within a year I could see that the most general crystals with arbitrarily oriented principal axes of electric and magnetic properties can behave like isotropic media like glass, or uniaxial crystals like calcite, or biaxial crystals like TGS if appropriate conditions are imposed on the ratios of their (ep)s and (mp)s and the angles between the two sets of axes.

So, one afternoon I went to his Office at 4.30 and told him that his 'generalized unixaililty condition' is a crystal property and not a mathematical ogre. That is: he is doing new Physics and not just new Math. He put me on the blackboard and listened to my talk for an hour and was as pleased as Punch. So, I returned to my Faculty Hostel Room (E-13) for a peaceful smoke. And within half an hour there was a knock and SDM entered my room and put down his heavy trademark bag on my table and started shaking and said:

"What all you said and what all I agreed to is WRONG!"

That is what happens when you don't do the math yourself.

And I asked him to sit down and showed all my calculations. And within five minutes he was as happy as ever and asked:

"Does it mean I made a minor 'discovery' in Crystal Optics?"

And I replied emphatically: "Undoubtedly!"

And then he went home. At that time he was doing his 'independent' Paper on Cherenkov Effect in Biaxial Crystals and had sent his manuscript to Annals of Physics (NY) which at that time was top-class with Hamermesh as Consulting Editor. And, he found that, timid as he was, he talked about his 'generalized uniaxialtiy condition' in an obscure sentence in an introductory paragraph. And some equally timid subtitle for that para.

He then boldly changed the subtitle to: "Generalized Uniaxiality Condition".

And added an Acknowledgment to his 'independent' Paper saying something like:

"I thank GPS for showing that the 'Generalized Uniaxiality Condition' remains unaltered (i.e. a crystal property) even when we take the most general doubly anisotropic medium; and for certain other conclusions."

And next day, before sending his Addendum to Ann Phy, he called me and showed me the changes and asked me:

"Is this OK or should I make the Acknowledgment stronger?"

And I laughed.

Later on, while I was going through the Literature on the subject for writing my thesis, I discovered as usual that the most genreal GUC was embedded in an even more abstruse text by a math wizard in the Courant Institute (NY)...I forget his name.

As they say:

"There is nothing truly original in this world":

"Dhata yatha purvam akalpayat"

See:

http://gpsastry.blogspot.in/2012/02/noriginality.html

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French Leave

My blog-posts may become a bit sporadic in the next few days...not to worry.

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