Bert reads Einstein
I spent two years, on and off, studying relativity theory. Today I finished my summary of the basics:
Relativity Theory: Lorentz Transformation Made Easy October 2017: how light shapes space and time. Comprehensive, only basic math, 28 pages, 20 hours proper studying download pdf
Contents: After reading some handfuls of introductions to relativity theory, and, frankly, being more puzzled than enlightened by quite some of them, these 28 pages form, in my perception, for readers in the position as I was before I started, the best, simplest, straightest, complete explanation of the measurement of time and place or Lorentz transformation of relativity theory, in which the reader is not asked to take anything for granted. Most authors in the field are physics talents, simply too smart and too far ahead of most of us, and skip many steps we need. Excellent physics students need only half a word and thus have no problems with this. This short statement, and I write this because I see it as a recommendation and even source of some modest pride, is the logbook kept and refined by a curious ignorant outsider who needed every single step while struggling hard to grasp the subject.
There are two ways to log a study like this and I did both. One results in what is linked to above: to keep and update your latest account of what you learned. This means you not only extend, but also remove mistakes, change order etc. This makes errors and clumsy reasoning of earlier versions disappear. Important as this is, it obscures another matter of interest: the track the student followed, for better and for worse, to take him where he got. Here is that track.
45 years ago I was a student in mathematical economics. I embarked, as a second study, on philosophy of science, the text-books of which dealt with relativity theory with more awe than detail. This prompted my ambition to study relativity theory. But there was much to learn and it failed to reach the top of the priority list. Once retired, a rereading of Robert Musils Der Mann Ohne Eigenschaften rekindled my eagerness. [details]
Autumn 2015 I started to collect and study literature and browse web pages (I tend to restart mij studies when the harvest is done, just as in old days that was the moment the wars were continued). Then came 2 years occupied, on and off, with relativity theory. And though I did more than that, only the basics (Lorentz transformation) got totally clear to me. 2 years, that is about as long as it took Einstein (largely on his own) to conceive it. But then, I seem to suffer unusually strict criteria for understanding: surprisingly often my reading contained, to my standards, gaps - often even consciously buried stealth gaps that delayed me for days and weeks, that made true understanding impossible, and even occasionally seemed to reveal misunderstandings by its authors. Such errors, especially in the verbal "elucidations" between the formulas make soft landings nowadays, for now, after over a century, everybody knows what are the correct formulas that should result. I threw a lot of it aside (sometimes even in the river on which I live) and ended up close reading a remarkable little book by ... Einstein, A., Uber die spezielle und die allgemeine Relativitätstheorie, Braunschweig: Vieweg 1956, orig. 1916). Here was a great mind who could explain - and clearly - to simple minds too. But I got totally stuck at a place where some unhappy formulations, seemingly harmless to any expert but lethal to who knows nothing, made me get hopelessly stuck off the road.
Looking for help is not my favourite occupation. But after afer a few weeks of dangling like a fly in a web, in September, I decided to find myself a course at the nearest university, Utrecht. [my experiences there].
I am really not a type for lectures. You can't pauze them to check something, and the guys have the habit to go extra fast at tricky places they 'd rather not have questions about. You come home with bits and pieces and do not succeed to get them all together. You are in continuous danger to be put on the wrong leg. But something had to happen. After the course was done (these bachelors have three more of those courses parallel!) I could, quietly on my boat, try to make order. Though I would probably have passed the exam, I still did not understand it at all.
Neither would I. I got headaches. I never have headaches. Long after the course had ended I gave up on the course syllabus and in desperation returned to the Einstein-book - hoping, against the odds for a cerebral flash helping me back on the road - to discover that on the very first day of the Utrecht course I was - without realizing! - told things that gave me the clue to what I had misunderstood at the place where I got stuck.
That same afternoon I understood the Lorentz transformation. I logged my understanding in My Einstein math booklet.pdf, and My Einstein exercise booklet.pdf.
To me this was the sign to get into general relativity. There the Lorentz
transformation, working only under very ideal conditions, gets deformed in
difficult ways by gravitation and elctromagnetic fields, which is the rule in
the universe big and small. First the math: matrix
calculus, handling complex numbers, and the first principles of non-eucidian
geometry: affine connections, where you do not first walk around giving
coordinate numbers to what's there, and then prove things about it, but, starting at
some place get things fitted one by one around what is there, en from the
system of fitting you have to calculate to what kind of thing the construction is
going to build up. Terrible. I blogged it in
despair and perseverance.html.
There were no hopeless obstacles but progress was hard and very slow. Do a piece of math, do the exercizes, next piece etc. Every now and then I could not stop myself trying whether it would already launch me in some relativity text. But no. Meanwhile I developed mistrust concerning my understanding of Lorentz transformation. I got confronted with suspicious loose ends and absurdities in the take I had acquired on it. When the sailing season 2016 started I took my 12 foot dinghy to the races, and became the summer's most frequent starter. At the end of the sailing season I got listed 6 in the Dutch year ranking. Anything better then relativity!
But with autumn in sight the ghosts of relativity started to haunt me again. And the doubts whether I had really understood Lorentz. Soon I worked with simple Minkovski graphs that I later turned into an animated graph of Lorentz rotations.
To my relief I got a much clearer view by finding that all the points in a Minkovski diagram where all movers read the same time on their clocks form a hyperbola, and that a similar hyperbola for distance can be obtained by a mirroring over the (45o) x=t axis, and that the Lorentz equations could be derived from those hyperbolas. I wrote this down in Lorentz and the hyperbola.
While now I thought things had become simple, physics professors I know thought it was unnecessarily cumbersome. Prof. Gerard 't Hooft, whom I of course needed to send nothing but one graph, with the formulas written in, reacted, thought it correct but useless: Lorentz transformation is a rotation, he wrote. Write it in matrix form right away and you will easily generalize to 3 space dimensions and insert the modifications of general relativity (gravity, electromagnetic fields) by replacing the matrix elements with the appropriate functions.
He should of course be right and after some more study in his awesome and concise Introduction to General Relativity [download pdf], I got some understanding why. Plus the hard conclusion: my brain is too slow. To creep into general relativity, smart physics talents need a year, for me it would mean quite some years of my retirement and more stamina than is fun to invest - and the worst is that I now know what I'm talking about. I quit.
But meanwhile I got the Lorentz transformation straight: I found (mostly by harvesting clarifying mathematical moves left and right in the literature) that you can actually draw, without invoking any formulas, clear pictures of Lorentz rotations, then from there mathematically derive the Lorentz equations, then get the time and distance frontier hyperbolas out of the Lorentz equations and have everything so clear and straight as to make me ask myself: why didn't they just tell me this? I ended up writing this down in Lorentz Transformation Made Easy.
Up to here, where I quit, I loved the journey. It was hard occasionally but I can appreciate such episodes. And it somehow feels I have a new virtual friend: the spirit of a totally sympathetic man, void of any perkiness, capable to agreeably communicate with anyone about anything: Albert Einstein.
Bert Hamminga [about me]