The End 2016 Mathematics A To Z: General Covariance


Today’s term is another request, and another of those that tests my ability to make something understandable. I’ll try anyway. The request comes from Elke Stangl, whose “Research Notes on Energy, Software, Life, the Universe, and Everything” blog I first ran across years ago, when she was explaining some dynamical systems work.

General Covariance

So, tensors. They’re the things mathematicians get into when they figure vectors just aren’t hard enough. Physics majors learn about them too. Electrical engineers really get into them. Some material science types too.

You maybe notice something about those last three groups. They’re interested in subjects that are about space. Like, just, regions of the universe. Material scientists wonder how pressure exerted on something will get transmitted. The structure of what’s in the space matters here. Electrical engineers wonder how electric and magnetic fields send energy in different directions. And physicists — well, everybody who’s ever read a pop science treatment of general relativity knows. There’s something about the shape of space something something gravity something equivalent acceleration.

So this gets us to tensors. Tensors are this mathematical structure. They’re about how stuff that starts in one direction gets transmitted into other directions. You can see how that’s got to have something to do with transmitting pressure through objects. It’s probably not too much work to figure how that’s relevant to energy moving through space. That it has something to do with space as just volume is harder to imagine. But physics types have talked about it quite casually for over a century now. Science fiction writers have been enthusiastic about it almost that long. So it’s kind of like the Roman Empire. It’s an idea we hear about early and often enough we’re never really introduced to it. It’s never a big new idea we’re presented, the way, like, you get specifically told there was (say) a War of 1812. We just soak up a couple bits we overhear about the idea and carry on as best our lives allow.

But to think of space. Start from somewhere. Imagine moving a little bit in one direction. How far have you moved? If you started out in this one direction, did you somehow end up in a different one? Now imagine moving in a different direction. Now how far are you from where you started? How far is your direction from where you might have imagined you’d be? Our intuition is built around a Euclidean space, or one close enough to Euclidean. These directions and distances and combined movements work as they would on a sheet of paper, or in our living room. But there is a difference. Walk a kilometer due east and then one due north and you will not be in exactly the same spot as if you had walked a kilometer due north and then one due east. Tensors are efficient ways to describe those little differences. And they tell us something of the shape of the Earth from knowing these differences. And they do it using much of the form that matrices and vectors do, so they’re not so hard to learn as they might be.

That’s all prelude. Here’s the next piece. We go looking at transformations. We take a perfectly good coordinate system and a point in it. Now let the light of the full Moon shine upon it, so that it shifts to being a coordinate werewolf. Look around you. There’s a tensor that describes how your coordinates look here. What is it?

You might wonder why we care about transformations. What was wrong with the coordinates we started with? But that’s because mathematicians have lumped a lot of stuff into the same name of “transformation”. A transformation might be something as dull as “sliding things over a little bit”. Or “turning things a bit”. It might be “letting a second of time pass”. Or “following the flow of whatever’s moving”. Stuff we’d like to know for physics work.

“General covariance” is a term that comes up when thinking about transformations. Suppose we have a description of some physics problem. By this mostly we mean “something moving in space” or “a bit of light moving in space”. That’s because they’re good building blocks. A lot of what we might want to know can be understood as some mix of those two problems.

Put your description through the same transformation your coordinate system had. This will (most of the time) change the details of how your problem’s represented. But does it change the overall description? Is our old description no longer even meaningful?

I trust at this point you’ve nodded and thought something like “well, that makes sense”. Give it another thought. How could we not have a “generally covariant” description of something? Coordinate systems are our impositions on a problem. We create them to make our lives easier. They’re real things in exactly the same way that lines of longitude and latitude are real. If we increased the number describing the longitude of every point in the world by 14, we wouldn’t change anything real about where stuff was or how to navigate to it. We couldn’t.

Here I admit I’m stumped. I can’t think of a good example of a system that would look good but not be generally covariant. I’m forced to resort to metaphors and analogies that make this essay particularly unsuitable to use for your thesis defense.

So here’s the thing. Longitude is a completely arbitrary thing. Measuring where you are east or west of some prime meridian might be universal, or easy for anyone to tumble onto. But the prime meridian is a cultural choice. It’s changed before. It may change again. Indeed, Geographic Information Services people still work with many different prime meridians. Most of them are for specialized purposes. Stuff like mapping New Jersey in feet north and east of some reference, for which Greenwich would make the numbers too ugly. But if our planet is mapped in an alien’s records, that map has at its center some line almost surely not Greenwich.

But latitude? Latitude is, at least, less arbitrary. That we measure it from zero to ninety degrees, north or south, is a cultural choice. (Or from -90 to 90 degrees. Same thing.) But that there’s a north pole an a south pole? That’s true as long as the planet is rotating. And that’s forced on us. If we tried to describe the Earth as rotating on an axis between Paris and Mexico City, we would … be fighting an uphill struggle, at least. It’s hard to see any problem that might make easier, apart from getting between Paris and Mexico City.

In models of the laws of physics we don’t really care about the north or south pole. A planet might have them or might not. But it has got some privileged stuff that just has to be so. We can’t have stuff that makes the speed of light in a vacuum change. And we have to make sense of a block of space that hasn’t got anything in it, no matter, no light, no energy, no gravity. I think those are the important pieces actually. But I’ll defer, growling angrily, to an expert in general relativity or non-Euclidean coordinates if I’ve misunderstood.

It’s often put that “general covariance” is one of the requirements for a scheme to describe General Relativity. I shall risk sounding like I’m making a joke and say that depends on your perspective. One can use different philosophical bases for describing General Relativity. In some of them you can see general covariance as a result rather than use it as a basic assumption. Here’s a 1993 paper by Dr John D Norton that describes some of the different ways to understand the point of general covariance.

By the way the term “general covariance” comes from two pieces. The “covariance” is because it describes how changes in one coordinate system are reflected in another. It’s “general” because we talk about coordinate transformations without knowing much about them. That is, we’re talking about transformations in general, instead of some specific case that’s easy to work with. This is why the mathematics of this can be frightfully tricky; we don’t know much about the transformations we’re working with. For a parallel, it’s easy to tell someone how to divide 14 into 112. It’s harder to tell them how to divide absolutely any number into absolutely any other number.

Quite a bit of mathematical physics plays into geometry. Gravity physicists mostly see as a problem of geometry. People who like reading up on science take that as given too. But many problems can be understood as a point or a blob of points in some kind of space, and how that point moves or that blob evolves in time. We don’t see “general covariance” in these other fields exactly. But we do see things that resemble it. It’s an idea with considerable reach.


I’m not sure how I feel about this. For most of my essays I’ve kept away from equations, even for the Why Stuff Can Orbit sequence. But this is one of those subjects it’s hard to be exact about without equations. I might revisit this in a special all-symbols, calculus-included, edition. Depends what my schedule looks like.

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