Published on January 21st, 2014 | by Emily Corbett0
De Rham’s theorem
By Benjamin Szczesny, University of Sydney
This student took part in the 2012/13 AMSI Vacation Research Scholarship program. For more information on this years program please click here
My project was on something called De Rham’s theorem which is named after Georges De Rham who first proved it in 1931. What it says is that these two different “cohomologies” of “differentiable manifolds” are really the same. I’m not going to explain what these things are, as they are way too abstract and difficult to explain to even most undergraduates! Instead I’m going to talk about gravity, storms, ducks and rivers.
But first let’s talk about energy. Who or what energy is, isn’t all that important to us. What is important to us is the idea of potential and kinetic energy. When we lift a heavy box, we fight against gravity and raise the boxes gravitational potential energy. That is, a measure of it’s potential to do something; in this case fall and crush our feet (What really happens is that all that potential energy gets converted into kinetic energy, a measure of its speed).
There are other kinds of potentials too. One boring example is that of two strong magnets. We have to push hard to get like poles to touch (raising their potential energy), but if we let go they’ll fly apart (converting this potential into kinetic energy again!). A slightly more interesting example involves a storm. Imagine that this storm has the unusual property that at each point the force and direction the wind is moving doesn’t change over time. Now, imagine you’re a duck in this storm and for some reason known only to yourself you decide to go against the wind. As you move against it, your storm potential energy increases. But after a while you give up and let the storm push you in the direction of the wind. Now your storm potential decreases as your kinetic energy increases.
The storm is an example of a “vector field”. You can imagine at each point in the space there is an arrow that points in the direction of the force and the stronger the force, the bigger the arrow. For vector fields, we can further think of objects (the duck for example) in this vector field as having a potential energy. And as we’ve seen, when it moves against the vector field the potential gets larger, while if it goes in the direction of the arrow it’s potential gets smaller.
When this vector field is gravity, it has the interesting property that no matter what path we take to a point, we will always have the same potential energy at that point. It doesn’t matter if we take a sea voyage around the world, go to the moon and back or do a barrel roll. When we return to that specific point we will always have the same potential energy. We call gravity conservative. But do all vector fields have this property?
Let’s imagine your the duck again and now you’ve found yourself caught in a tornado. The wind is too strong to fly against so you let it push you in circles. Now, your storm potential is decreasing as you go around and eventually you come back to the same spot but with a lower potential! Thus, this storm mustn’t be conservative. Why is this? Well the vector field has a vortex in it, so we can think of the eye of the tornado as having a ‘twist’ in it. So vector fields with twists can’t possibly be conservative. Now the question becomes: Are all twistless vector fields conservative? After a long night of being thrown around by the tornado you decide to go relax in a river. Except this river is special, it flows all in the same direction with exactly the same force, it also never ends and stretches all the way to infinity in a perfectly straight line. This river is also a vector field and is twistless, but is it also conservative?
It’s probably not too surprising that this will be conservative as well. However, because you’re a smart duck with a phd in astronomy, after a bit of stargazing you eventually realise that this river doesn’t go on for infinity, it only looks that way because it lives on a really really big sphere! (The earth). So now you map the river and come to realise it does a loop around the earth. But hold on, if you let the river just push you around, you would go around the earth and eventually reach the same point again. Your potential energy would have been decreasing the entire time so you would have arrived back with a different potential energy. The river isn’t conservative after all! So what changed? why, the shape of the river of course! De Rham’s theorem in essence relates the failure of these twistless vector fields to be conservative to how “holey” the space they live in is. The infinite straight line isn’t holey so it always works. But a river that goes around the world? That has a hole through the earth and things start to go wrong.