Two days ago a friend told me about a wonderful auroral display over Greenland and Canada as he flew from Paris to San Francisco. We're used to the notion of great circle paths. The shortest distance between two points on a sphere is a geodesic - the fancy pants generalization of the concept of straight line to curved spaces.^{1}

I didn't think about this until a professor who I corresponded with as a kid sent a letter that told me to draw an equilateral triangle on a sphere, the seat of a saddle (I did have a Montana address and knew what a horse was), and a sheet of paper and measure the angles. Next draw two parallel lines. Finally I was to draw a circle on each and tell him what π meant. I won't offer a drawing. If you haven't thought about this, give it a try. Euclidian geometry falls apart.

People started thinking about this seriously a few hundred years ago. A specific geometry was developed for one class of curved spaces - a certain type of space where the equivalent of a plane would fit hug the surface of a saddle seat. Interesting, but limited and mathematicians lust for generality.^{2}

Bernard Riemann went off to college to study theology - one of the few fields where a college degree offered reasoanble employment prospects. He went to a lecture by Gauss and gave it all up to study math. A few years later, as the Civil War neared in America, Riemann was going for Habilitation - a sort of Ph.D.++ in Germany. He suggested a few dissertation topics and Gauss told him to work on the one Riemann thought boring - namely push the foundation of geometry. After a year and a considerable amount of time the heavens opened. A short paper that allowed you to understand the geometry of a space of any number of dimensions with any curvature at any point in space. My mentor said it was like Euclid had a few lines in the sand for counting, the first non-Euclidian geometry mathematicians moved to a hand crank adding machine and Riemann developed a supercomputer that we'll have in a hundred years.

People still wonder what kind of coffee he drank

He thought about the intrinsic geometry of space .. not how it might look from the outside. For him each point in the space would have information relevant to the space. I won't go into the math, but tensors are perfect containers for the book keeping. But the real breakthrough (actually the first item - looking at space from the inside was a huge breakthrough) was realizing that everything you needed to know about the geometry of a space was encoded in the distance along any curve you might draw in the space and that distance could be written in the form of a tensor he called the metric. So if you tell me any segment of a curve of a space you're fond of, I can use the metric and tell you how long it is. From this you can build up and calculate areas, volumes, geodesics (like flying from Paris to San Francisco), angles and everything you'd like about geometry. You could even talk about the path of a bug walking on an apple. Geometry from the ground up.

It was beautiful math and for about 50 years mathematicians had it to themselves. Then this guy named Einstein comes along with this crazy idea that gravity is really the bending of spacetime by a mass. (This is where all of those lame demos with balls and rubber sheets come from.) He had no idea how to approach the problem mathematically. After all, this space-time would have constantly changing curvatures. Riemann's work was an almost tailor made for the task. He worked with a people more conversant in the math for awhile until the training wheels came off.

If you've played with gravity this is all old hat. I've avoided the math attempting to describe the beauty of Riemann's work - the notion that any kind of space can be understood by looking at it from the inside and using this metric thingamabobby. Some may be too difficult to calculate, but in principal it's always possible. It turns out to be just the tool for a revolution in physics. Physics that is necessary if the GPS in your phone is to work properly and physics that still stands after over 100 years as our deepest understanding of gravity.

There are no lone geniuses in math or science. It's all connected. And the future of technology depends on it. That part scares me given the current anti-science sentiment in government.

Oh - the book. That's the undergrad textbook I had for General Relativity. Beautifully written with a sense of humor. One of the authors was very young at the time. In a few days Kip Thorne gets his Nobel Prize for his work detecting the sound of some of the most violent events in the universe... events that generate ripples in spacetime as predicted by Einstein with a theory made possible by Riemann.

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And one more thing. I've made this point several times, but it bears repeating.

People tend to talk about some theories as replacing others. While that's often true, sometimes the "superior" theory is just more general and reduces to the old one under certain conditions. If you're large - say over a millimeter in size - and moving slowly - say less than a tenth of a percent of the speed of light, Newtonian physics is just fine thank you. It may not be a deep way of thinking about the world, but for some practical matters like engineering, driving a car and playing volleyball it's just dandy and is much easier to use calculate than relativity or quantum mechanics.

The stories we tell are are equally valid to use if they're relevant to the domain we're using them for and if they don't contract physical evidence at the domain level.

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^{1} In physics you'd say it's a locally length minimizing curve ... the path a moving particle follows if it isn't accelerating.

^{2} only surfaces of constant negative curvature.

Steve just plain wonderful!

--Gregg

Posted by: Gregg Vesonder | 11/27/2017 at 06:08 PM

I so enjoy your posts. Thank you.

Posted by: Jean M Russell | 11/28/2017 at 12:08 PM