It’s a clear winter night as I write these last words. I’ve stepped out to look at the sky. With the stars up above and the blackness of space, I can’t avoid feeling awe.
How could we, Homo sapiens, an insignificant species on an insignificant planet adrift in a middleweight galaxy, have managed to predict how space and time would tremble after two black holes collided in the vastness of the universe a billion light-years away? We knew what that wave should sound like before it got here. And, courtesy of calculus, computers, and Einstein, we were right.
That gravitational wave was the faintest whisper ever heard. That soft little wave had been headed our way from before we were primates, before we were mammals, from a time in our microbial past. When it arrived that day in 2015, because we were listening—and because we knew calculus—we understood what the soft whisper meant.
Steve Strogatz
Chatting with a friend the subject of gravity came up. It isn't a conventional force, but we perceive it as one. Rather it's connected with the curvature of a four-dimensional object called space-time that we happen to live in. The fundamental concepts aren't intuitive and the math that describes it is beyond what you might get in an engineering degree. So how to describe it? There are very high level books and videos that speak in terms of balls on rubber sheets and then jump to oddities like clocks running faster on mountain tops and blackholes. I usually find them a bit fluffy and disappointing. They're bound by assumptions about their audience, so it's not a major criticism. It's just that I was talking with one person and something a bit deeper seemed appropriate. Just how do you find the right level and where do you go from there?
A few years ago Wired produced the 5 Levels series. An expert would try to explain something about a complex subject to five people: a child, a teenager, an undergrad majoring in the same subject, a grad student and, finally, a colleague. My favorite is Donna Strickland on lasers - give it a watch, she's excellent:
I went through something like this just before my Ph.D. thesis defense. I was to explain my thesis work to a group of high school students. At first I thought it would be easy, but that notion quickly evaporated. It turned out to be one of the most difficult and embarrassing things I've done. My advisor stepped in near the end of the talk and summed everything up beautifully. Afterwards he told me if you understand something deeply, you can explain the gist of it to a high school student. That's high on the list of the most important things I've learned.
My friend has a BA in biochemistry and knows about differential equations so I figured I should aim for something between the vague videos and what a senior level general relativity course offers. The more I thought about it I realized it would be best to talk about the history and use a bit of calculus and geometry. I had the advantage of knowing her well and she can stop and ask questions or give a blank look along the way. And afterwards she was going to talk about something where she has serious expertise.
Steve Strogatz is an applied math professor at Cornell. He's written several books including the one the opening quote is taken from: Infinite Powers: How Calculus Reveals the Secrets of the Universe. It's an excellent read that makes no assumption on your mathematical background.1 Thinking about it gave me some hints of how I might proceed.
First why do I have to use math? Why won't words work? The laws of nature obey logic that we can make predictions from. Math, calculus in particular, is a logical calculating tool. In a way it's a prothesis. Math lets us take a bit of logic, write down and perform logical manipulations that far exceed what we can do in our head, and then interpret the results. The logic can be crafted to represent something about the science. Every once and awhile the results can make predictions that can lead to new discoveries, but more often they're used to to solve an enormous range of problems. And why calculus? Much of the underlying structure of nature has been successfully expressed with the corner of calculus known as differential equations. "Why?' is a deeper question. If one encountered an intelligent alien who understood some aspects of how the Universe worked, I suspect they'd use math. I suspect, but it's only my suspicion, that it's deeper than an artifact of how we think about things.
I spent a couple of hours writing. How general relativity came about, a bit on the structure of space-time, what goes into the main equation and how a simple prediction could be calculated. Then about an hour of chatting that left me with two delightful philosophical questions that will probably lead to more discussions as well as her turn to teach me something.
Whatever your expertise, it can be fun to try and explain the gist and maybe even the beauty of it (those can be he same thing) to someone with a very different background. If you're like me you'll fail at first, but eventually you'll get to a point where you can find the right grounding, the right words and perhaps the right drawings or even music (some of you are artists and musicians).
__________
1 By all accounts he's a wonderful teacher and has become a popularizer of math. Among other things he's created a college course at Cornell for people who think they're afraid of math and have generally put it off until their senior year.
an almost lost word from the dictionary
The warmth of a bright Sun in on a clear cold Winter day. That's something that captures most of us know - the radiant heat of the Sun isn't isn't perturbed by the cold air and can be quite comfortable. Then it hit me I heard the word in a John McPhee lecture in Princeton.2 A few decades had passed since he wrote Coming into the Country, but it was about three Alaskas including a look at the Winter. A perfect word for some of the days.
A day of apricity can make snow melt in subzero air temperatures on Southern-facing roofs only to freeze into beautifully clear long icicles at the edge. This type of ice formation is often free of bubbles producing beautifully clear ice. If you have wide enough icicles you might try making an ice lens and starting a fire.
Such days are made for outdoor recreation if there isn't any wind. Perhaps more importantly the basic principle - heat transfer by radiation - can keep heating expenses down. Heating and moving air to fill rooms is enormously inefficient. If you aren't moving around much you might try a small radiant heater and point it at the area where you are. Heat what you need to heat with infrared light. Direct sunlight is the same... sit in the Sun. These measures, plus wearing warm indoor clothing and being somewhat active lets you get away with low thermostat settings. And be on the lookout for aprocity as that can give a big psychological boost!
Another Winter word I like is tingilinde. It's a constructed word based on J.R.R. Tolkien's elfin language Quenya meaning the sparkle of the starlight reflecting on the snow on a dark moonless Winter night. Such sights could be spectacular in Montana away from town when it was really cold and a bit of new snow had fallen as tiny ice crystals. Another great spot is Yellowstone .. super cold air forms the right kind of ice crystal snow near the hot springs and geysers. Once it was magical - the cold starlight reflecting from millions of tiny diamonds along with the greens and reds of a bright aurora dancing overhead. On such nights you don't notice that it's really cold.
__________
1 The Compact Edition of the Oxford English Dictionary is the 1928 edition. Devilishly small print that requires a magnifying glass, but the real OED and only about $80 (still a lot back then!) rather than the fortune the full dictionary went for. There's a lovely bit of historical fiction on the dictionary and the people who made it happen: The Dictionary of Lost Words by Pip Williams. It's set in the time and place with many of the characters, but focuses on what was left out.. those who weren't white, male and of a certain class. Recommended!
2 McPhee is a great writer - a master of the creative non-fiction genre.
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