Tensor #39. The End텐서 #39. 마지막 강의

These are my notes on XylyXylyX’s What is Tensor lectures. This post covers lecture 40 of the playlist, the last lecture, The End. In the previous post we organized all the operations made from two objects, and this time we read the basis of each of those objects as a picture. We stay in three dimensions.

This is the last lecture

The lecture opens by announcing that this is the end. It is lecture 40 and each lecture is about an hour, so it adds up to nearly 40 hours. The later part on differential forms took up a lot of that time. The lecture judges that even without that part this is a sufficient basic course on the mathematical nature of tensors, and enough to start on general relativity, the subject the lecturer really wants to teach.

What we have built and where the gap is

The lecture first looks back at what we have. A connection lets us compare vectors at different points, and on top of it we learned three derivatives: the Lie derivative, the covariant derivative, and the exterior derivative. These derivatives act on tensors and produce tensors again.

It also points out the weak spot. We placed all of this on spacetime but did not treat the concept of a manifold in this course. The story of laying down a coordinate system, plotting points, attaching a vector space to each point, and binding them with a connection works up to elementary general relativity, but studying the subject seriously requires manifolds, and that requires topology too. The lecture says these are covered in a separate set of manifold lectures. For the next general relativity lectures, the lecturer plans to require as prerequisites the first part of this course, up to the two derivatives (the Lie derivative and the covariant derivative), together with the first part of the manifold lectures. All 40 hours are not needed.

The small cube cut out by coordinate planes

The last topic is visualizing the coordinate basis. When a 3-dimensional manifold has coordinates, we can think of coordinate planes, the surfaces where one coordinate value is held fixed. Place two planes for two values of $x^1$, two planes for two values of $x^2$, and two for $x^3$, and one cube is cut out. In Cartesian coordinates it is exactly a cube, and in spherical or cylindrical coordinates too, if we shrink the spacing small enough, the curved edges and faces straighten and the shape becomes a cube in the limit. Any coordinate system worth using fills space with such small cubes.

The bases live in a space attached to the point

There is one thing to be careful about. The cube sits on the manifold, but the arrows and stacks we take as bases do not live on the manifold. They live in a separate vector space attached to the point, a space hanging off the manifold at each point. That is why we must think in terms of a limit to use the cube. We do not worry about the case where we draw an arrow so long that it reaches where the coordinate system curves. The cube is only a tool for fixing directions and sizes.

Q. Why do we need a limit at all to read a basis?
A. Because the basis lives in the vector space attached to the point, while the cube is a piece of the manifold. As the piece shrinks, the curvature of the manifold disappears and the pictures of the two spaces can be read on top of each other. The lecture also notes that this limit is why partial derivatives appear.

The bases of arrow and stack

At the point $P$, the three edges of the cube fix three directions, and fixing the sizes by the lengths gives three basis arrows. In the limit these arrows become the partial derivatives $\partial_\mu$.

The basis of the stack comes from the same cube. Two opposite faces are two parallel sheets, and the gap between the sheets sets the density of the sheets. The front and back, top and bottom, and left and right faces each give one basis stack, so there are three. A stack has a direction, so we must say which way it counts.

The bases of thumbtack, sheaf, and honeycomb

The basis of the sheaf is a bundle of arrows through each corner of the cube. Only the direction and the density of the arrows carry meaning and the length does not, so we read it as a bundle of parallel lines. Earlier the arrow’s meaning was direction and magnitude, while the sheaf’s is direction and density.

The basis of the thumbtack is easy. It is one face of the cube, and it must be a face touching the point $P$. For the stack the density of infinite sheets mattered, but for the thumbtack, conversely, one finite area matters, and it also carries a direction. Three faces touch the point, so there are three basis thumbtacks.

The basis of the honeycomb is the cell between the lines of the sheaf. The lecture explains that the sheaf makes a small honeycomb of its own. The sheaf lives in the same space as the honeycomb, so they are the same story.

Scalar density and scalar capacity

The scalar side is where the numbers do not use three dimensions. An ordinary scalar has no basis, so there is nothing to worry about. The basis of the scalar density is the points marked at the corners of the cube. The number of points per unit volume is the density, and the lecture says Weinreich calls this a swarm of points. The basis of the scalar capacity is the volume of the cube itself. With this we have the bases of every object: stack, honeycomb, arrow, thumbtack, sheaf, scalar density, and scalar capacity.

In the toy below, changing the edge lengths changes the value of each basis. You can see the arrow side grow in proportion to the length while the paired stack side shrinks in inverse proportion.

Multiplying the values of a pair always gives 1.

That this product is 1 is added so we can check it in the toy and was not covered in the lecture. The pairs thumbtack with sheaf and scalar capacity with scalar density also give 1 in the same way. (The calculation of the values $1/(l_j l_k)$ for the sheaf and $1/(l_1 l_2 l_3)$ for the scalar density is also a supplement for this explanation and was not covered in the lecture.)

Q. What does it mean that an ordinary scalar has no basis?
A. The lecture only says that an ordinary scalar has no basis, so there is no need to worry about it. It can be read to mean that the value is just the number and there is no direction or volume basis multiplied in. A density and a capacity have a 3-form basis and a 3-vector basis multiplied in, so they need the cube. (This explanation adds a way of reading it and was not covered in the lecture.)

The bases hidden behind component notation

The lecture links this picture to component notation. Most general relativity textbooks write tensors and vectors only by components such as $a^\mu$ or $T^{\alpha\beta}$, without the unit vectors. Now we know those unit vectors are floating there. The reason the indices have their positions and order is also these bases. As for choosing textbooks, the lecture says to postpone the very advanced ones until after finishing an elementary one, and says there are many good ones, many of them free on the web. It adds that seeing this material again in the books will make it better understood.

Q. If components are enough, do we need to know the bases?
A. The lecture admits that component notation is far more convenient for computing, and plans to use it a lot in the general relativity lectures. But its position is that using it with understanding of why it is written that way differs from using it without. It also says that if we are not yet comfortable, we may have moved through this course too fast.

Summary and closing

What we checked in this post is the following.

• In any coordinate system, looked at small enough, coordinate planes cut out a small cube.
• In this cube the edges are the basis arrows, two opposite faces are the basis stack, the face touching the point is the basis thumbtack, the line bundle through the corners is the basis sheaf, the cell between the lines is the basis honeycomb, the swarm of points at the corners is the basis scalar density, and the volume is the basis scalar capacity.
• These bases live not on the manifold but in the vector space attached to the point.
• Component notation hides these bases, and the positions of the indices are their trace.

The series ends here. The lecture says it has finished the elementary story of tensors and closes by saying we are ready to start on general relativity. The next post is a special lecture on the tangent space and the coordinate basis. If you want to know more about manifolds, the first post of the manifold series continues from here.