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).
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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.
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