The Fascinating perspective of Geometric Algebra #SoMEpi

Transcript

This is a video about gravity, electricity, and complex numbers. Those are things we usually learn in high school, but not altogether. They seem to be three unrelated branches of math and physics, but aren't they? Let me show you one curiosity about the mathematics of classical physics. This is Kulum's inverse square law. Right? With this equation, we calculate the electric force that two charges exert on each other at a given distance.

Now, you might think this is just for electricity, but look a little bit closer because by applying some elementary steps, especially this one, which introduces two imaginary units at the cost of a negative sign, we can get to a different but well-known expression that I'm sure you can recognize because this is Newton's law of universal gravitation. The laws of gravity and electricity are so similar that simply by considering complex numbers, you can make them look just identical. This is interesting because think about it this way. Is it even possible to consider a complex number to describe the amount of mass and electric charge of an object? We would end up with just one law. Sounds optimal, right? Okay.

But wait, because if we compute a force now, its components wouldn't be just real numbers anymore. And we describe all sorts of things with the mathematical concept of a real valued vector. So what about complex numbers? What if the components of a vector themselves are complex numbers? Does it describe something at all? We can always decompose a complex valued vector into two parts. A pure real and a pure imaginary vector. Then when it comes to representation, well surely we can represent them as two vectors in a coordinate system and no problem with the real part.

But but we find a much more intuitive and satisfactory description of a pure imaginary vector when we consider a concept taken from geometric algebra. The concept of by vectors. Have you ever heard about geometric algebra? Here we're going to take a look at its most basic concepts and in the process we will find that two things we teach separately. They can actually be described in terms of single entities called multiv vectors leaving us with this spectacularly compact representation of the electromagnetic and gravidomic field. And then we're going to take the source of this field which is that particular multiv vector we considered before and we're going to see how it actually relates to the mathematics of magnetic monopoles through something we call duality transformation.

So this is a three-dimensional multi vector. Do you recognize some of its elements? There are some coefficients, the three basis vectors, a combination of two of them, and finally all of the basis vectors multiplied together. For those that are unfamiliar to this subject, let me first show you how this data structure ends up being related to some more familiar structures. Choose the right coefficients and a scaler is a multiv vector. For reasons that will become clear in a moment, a complex number is a multiv vector as well.

A three-dimensional vector is also a multi vector as well as a quatian and other important structures like a scalar plus a vector and a vector plus a b vector which is what the force equation resulted in at the beginning are also multiv vectors. Addition is defined component-wise just as we do with vectors and much of the same when it comes to multiplication. Although this is not generally a commutative operation. Now let's write all the terms. Let's focus on one of them and let's distribute the product.

There are many good introductions about this topic on the internet. So from here I'm going to list some of the properties about the multiplication of the basis vectors as if they were the rules of a game. Wherever you find two basis vectors multiplied together, you can swap them at the cost of a negative sign and wherever you find a basis vector multiplied by itself, the result would be one. So I'm going to leave these two untouched. Here I'm going to swap them and change its corresponding sign.

Convert this to one. Here swap once and convert to one. Convert to one. Swap twice. And swap twice again and convert to one.

Reorganize so that we get a clean multiv vector. And remember we were performing a multiplication. So do the same with the remaining terms and add them up to get a new multi vector. Now notice this. Take what we call the tri vector part and square it.

Swap twice, convert to one, swap once, and convert to one twice. And this is how we relate this part of a multiv vector with the imaginary unit. With this relationship, we can now take a pair of basis vectors and write them as the remaining basis vector time [Music] I. Lastly, we will find with the geometric product of two vectors in the next section which results in a scalar plus a by vector which can be conveniently written as the dotproduct between the two vectors plus i * the crossroduct. And this will become useful because the equations of electromagnetism are often expressed in terms of these operations which brings us to part two classical electronamics.

These are Maxwell's equations. If you are unfamiliar with them, but you know something about vectors, these are the basics. Everything with an arrow on top is a vector and this one is a number. We call them electric charge density, electric field, magnetic field and electric current density. Epsilon 0 and C are numerical constants.

The left hand side of the first equation is a dotproduct between these two and it equals a scaler. The second is also a dot product but it equals zero. This is a way to say we are not accounting for magnetic charges. Third one is a crossroduct and it equals the derivative in time of a vector. Here we just take the derivative component-wise.

And the last one is a crossroduct again and it equals a vector plus a derivative of a vector. This set of equations describe things related to electricity, magnetism and electromagnetic waves like light, radio waves, microwaves, x-rays and so on. they actually underpin modern technology. So our first step is to multiply by C both sides of these two equations and by one right here so that the electric field and C * the magnetic field have the same units. And when we consider not only electric charges but magnetic charges as well, the equations would look like this.

We call them the generalized Maxwell's equations. More about that in a moment. Let me now bring what we call gem equations. They are like Maxwell's equations but from the perspective of gravity. It's a similar story.

Vectors, scalers, dotproducts, crossroducts, derivatives and so on. This time we call them mass density, gravitational field, gravitational magnetic field and mass current density. They were first written by Oliver Heide as an extension of Newtonian mechanics. They were later derived from the general theory of relativity as an approximation and experimentally confirmed around 2008. So let me show you how these two sets of equations are actually two perspectives on the same matter.

Let's make the same consideration we made at the beginning and write mass density correspondingly. Next step is to consider mass to be zero. Bring the Newtonian gravitational field and write it in terms of our new values. Multiply by one. Rearrange and cancel.

Here's the definition of Kulom's electric field. So rewrite and take its divergence. Negative i's cancel, rearrange and cancel. Play the same game everywhere. And what we get are Maxwell's equations.

This is a way to say that both sets of equations are telling the same story, which is not a surprise once you consider what we call a dual transformation. But I'm getting ahead of myself. Now multiply by I both sides here and here. Multiply by1 here. Write it as I squared and distribute.

Write every term related to fields in the left hand side and let everything related to the sources on the right hand side. Next step is to write the full set of equations as a geometric sum so that we get a multi vector at each side of the identity. A vector plus a scalar plus a b vector and so on. These are the geometric product between vectors. Let's simplify that.

And one last simplification takes us here. This is the geometric representation of the full set of equations which we can simplify further to get to this super compact identity. Let's bring back the original gem equations. And remember we were considering complex values for mass and current density. Just let me write them like this to save some space and write everything else accordingly.

We know how to go from here to here. But let me show you a much more interesting place to start because we can get right there also from here. And these are the generalized gem equations. We can actually account for gravitational magnetic charges and get to the same place. It's just that this time this wouldn't be zero anymore.

which sounds odd, but it actually has to do with something we call duality transformation. And it's easy to explain from a geometric perspective. These fields are related to material properties and since we are considering the equations in vacuum, the transformations become the same. Let's bring Maxwell's equations in its geometric representation and identify these two fields with these two. Everything else is inside J.

From this perspective, it's easy to see why the transformations will keep consistency since it's actually the same thing as to multiply both sides by the same amount. So that given an angle we are always going to find a new field with a vector part and a bi vector part which we can just re label. And equally important, it also works the opposite way because given an angle, we can always decompose the fields, whatever they are, in such a way that we can split the identity and conveniently reabel things to get both sets of equations back again. It's just that this time these two values are forced to be zero since there is no imaginary part on the right hand side of any of these two equations. And whatever value you may put there, there will be always a transformation that will make them end up in their counterpart set of equations.

So in the search for magnetic monopoles, what do we expect to find? Lastly, we pick gravity's perspective because this way we are just one step away from writing this elegant equation which looks just like energy under a dual transformation. That was it. I hope you enjoyed it and I really hope you leave this video with something interesting to think about. If you do, consider to hit the like button and subscribe. Thanks for watching.

See you next time. [Music]