Understand Tensors Like a Physicist! (The Easy Way)（原声文稿）

Okay, so according to our brother's mathematicians, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Does it make sense?

Let's find something more easy, tensor product.

A type puq tensor is defined in the context of an element of the tensor product of vector spaces.

something else, a definition via tensor products of vector spaces.

Given a finite set v1 up to vn of vector spaces over a common field f, one might form their tensor product v1 up to vn, an element of which is termed tensor.

I hate you mathematicians.

But seriously, when I first time encountered tensors in us on a math class at a mathematical institute and I got scared of the word tensor ever since.

I never learned what it is and I just passed the exam because the professor didn't ask me about it.

But I was happy because I thought that I won't have to deal with tensors ever again.

Until this word reappeared at physics class. And I got scared even more.

I was scared so badly that I didn't even try to understand what it is.

I kinda just carried along by accepting this is tensor, this is not okay. Until it wasn't enough.

One day I told myself Enough is enough and I'm going to finally learn what this tensor is.

At least why we talk about them in physics.

And what I figured out is that I was scared the whole time for no reason.

Getting a physical intuition is actually very straightforward and the reason why we use them is actually very natural.

So let Let me explain what a tensor is and why we use them in physics.

So for a long time, I understood tensors as being some sort of generalization of vectors.

You don't have to limit yourself to just one row of numbers, that you can have also a matrix of numbers, or even a cube of numbers, and so on.

To refer to a specific component of this object, you can use index notation.

For vector, you only need one index to find the corresponding element.

For matrix you need two, and for cube you need three.

So we can write it in this generalized form, where the number of indices represents the rank of the tensor.

So, this is rank 1, rank 2 and rank 3 tensor.

You can also add a scalar with zero indices and call it a rank zero tensor.

Then by the range these indices run through, we determine the dimension of the tensor quantity.

In physics we use Latin letters for three -dimensional quantities, which is the xyz component, and Greek letters for four -dimensional quantities. in special or general relativity for T x y z.

And finally, how to kind of imagine all of these objects? Well, the scalar is easy.

It can be a temperature at a point, for example, or mass of an object, and so on. So it can be represented by a single number.

A vector can be, for example, velocity, electric and magnetic field, and so on.

And these are represented by an arrow, which has one direction and length.

Rank 2 tensor is kinda more tricky, but it is usually explained by the stress tensor where on each independent plane you can have 3 force components.

So I imagined that it is just more and more complicated versions of the same.

These indices don't have to be always up but also on the bottom.

If they are up we call them contra variant and if they are down we call them co -variant and they are connected by the metric which is also a tensor.

And I was still kind of confused why all of it is defined this way. Why we need upper and downster indices and and so on.

And that is how I understood tensors for a long time.

But then when I was taking the general relativity course, I heard about these Christopher symbols, which are used to define the derivative on a curved manifold, which you can understand as something that describes the gravitational force.

And then I heard that it's not a tensor.

And I was like what why it has three indices one up two down so it should be rank three tensor one time contra variant and two times covariant so where is the problem and i also remembered that angular momentum which we know from classical mechanics is not a vector it can be represented by an arrow but arrow is just representation of something.

It is not the quantity itself, so while we can represent the angular momentum by an arrow, it is not a vector. But why?

It has 3 components, so we could say it's rank 1 3 -dimensional tensor.

And I was very confused about why we call some quantities tensors and other not.

But trying to understand this using Christoffel symbols is difficult.

Just too many indices and dimensions.

So now I'm going to tell what tensor is.

So that is sensible for physicists. So mathematicians cover your ears.

A tensor under a certain set of coordinate transformations is a quantity that has invariant magnitude under such transformations. So what does it mean?

In physics, we usually work with several types of coordinate transformations.

For example, in including 3d space, we can rotate the coordinate system, and we can also move its origin.

Therefore, we can perform rotations and translations.

If we have a velocity vector, for example, it doesn't matter how we rotate the coordinate system and where we put the origin.

The components of the vector might change, but the magnitude of this velocity vector remains unchanged.

So velocity is a rank 1 tensor under transformations of rotations and translations. But what about angular momentum?

The mathematical expression for it is a vector product of a distance from the origin times momentum.

So the magnitude surely depends on where I put the origin.

If you put it close to the moving body, it is small, and if I put it further away, it's bigger.

So angular momentum is not a vector, it's also called pseudo -vector.

Okay, now let's talk about something more physically relevant. What about Galilean transformations?

In Galilean transformations, you can perform a boost, which means you transform the coordinates into a moving frame.

So, while one observer would say this spaceship is moving with a certain velocity, the co -moving observer would say the velocity is zero.

So these two observers disagree on the magnitude of the velocity vector.

And therefore, velocity is no longer a tensor under the Galilean transformations.

The same is true for angular momentum, of course.

So, what does it physically mean when a quantity is not a tensor? And why it is so important?

And it means that the quantity is not real in some sense.

In Galilean transformations, the velocity not being a tensor reflects the Galilean principle of relativity, that there is no absolute velocity, there is only relative velocity between objects.

So it's a coordinate dependent phenomena.

But for example acceleration vector is invariant and therefore we can call it a tensor under Galilean transformations.

So it is an absolute quantity and therefore real in some sense. And this is why we look for tensor quantities.

Because those are the quantities that we can consider to be the physical ones.

From special relativity onwards, it starts to be more tricky, since we look for quantities that are tensors under Lorentz transformations.

Now, the quantities that we consider real under Galilean transformations fall apart.

For example, if you take a difference between two points, this is a tensor under 3D rotations, translations, and also Galilean transformations.

And this represents a length. But this is not a tensor under Lorentz transformations.

So when we jump into the special relativity, we need to build a zoo of tensor quantities again, since those are the real ones.

So we have the space -time interval, for velocity, for momentum, electromagnetic for potential and so on.

So it is not the length interval, nor it is the time interval, but it's the spacetime interval that is the real one.

And I say that we have 4 velocity as tensor, so it has an invariant magnitude, which is the speed of light.

So we all move with the speed of light through Minkowski spacetime.

The components of the 4 velocity might change, you can move faster through space and slower through time or the other way around, but the total velocity is still the same.

And what about these Christoffel symbols in general relativity?

Well, they kinda represent the gravitational force.

But if you are freely falling with the object, they are zero.

But globally, they are not zero everywhere, if the spacetime is curved.

So this tells us that the gravitational force is also not real, since it's not a tensor.

It is just the effect of the curvature of the spacetime.

A side note, in general relativity we look for quantities that are tensors under general coordinate transformations, not just Lorentz transformations, but Lorentz transformations are a subset of these general coordinate transformations.

But before I end this video, be aware of one important thing.

The way you calculate the magnitude of a vector, or in general a tensor, depends on the space time you are working with.

In Euclidean space it's easy, you just take the scalar product of the vector with itself. But in more complicated spaces this is no longer true.

You need to take a scalar product of the tensor with its covariant counterpart.

And the way you calculate this covariant counterpart is by using metric tensor.

In Euclidean space, this metric tensor is just identity matrix, but in Minkowski spacetime, it is different, and you have to keep that in mind.

In general relativity, the metric tensor can be sometimes a bunch of very complicated functions and calculating a magnitude suddenly isn't that easy.

So not very mathematical explanation of tensors, I get it.

But if you would like to see this video in a more mathematical manner, let me know in the comments.