Tensor #10. Metric Spaces텐서 #10. 메트릭 스페이스

These are my notes on XylyXylyX’s What is Tensor lectures. This post covers lecture 10 of the playlist, Metric spaces. To what we built through the previous post, we now add one new ingredient for the first time.

What we add

Starting from a single 4-dimensional real vector space $\V$, we built the dual space, many tensor product spaces and changes of basis. All of that arose from $\V$ and logic alone, and we have been strict about not adding new ingredients. This post is where we break that promise for the first time.

We look for a function that takes two arbitrary vectors of $\V$ in two slots and returns one real number. Many functions are possible, and we will pick one we like and put it on top of $\V$. A vector space originally had only the vector sum and scalar multiplication as its parts, and now this function joins them as one more part. Mathematicians like to say that a space with extra parts has been promoted.

Properties the function must have

We write the chosen function as $\ip{v}{w}$ and require the properties below. First, in either slot a scalar $\lambda \in \mathbb{R}$ comes out.

Vector sums split too, in both slots.

Together the two equations say that the function is linear in each slot. And, as general relativity needs, it must be symmetric.

Seen across all of mathematics, symmetry is not required. Antisymmetric functions do a similar job. We only require symmetry for the purpose of this series.

Properties we do not use

Many books attach the following properties to the definition of an inner product.

This property is called positive definite, and many books define the inner product without it. Which one is used depends on where the book is heading, and general relativity does not use this property. An inner product with the property gives a Riemannian metric, and one without it is indefinite. A space with an indefinite inner product is a pseudo-Riemannian space, and general relativity uses this kind.

The function we pick and put on top is called an inner product or a metric, and a vector space that has one is called an inner product space or a metric space. Some books define a metric space as one that is positive definite as well, but this series does not. (This differs from the use of metric space in mathematics for a set with a distance function. This explanation is added to tell the terms apart and was not covered in the lecture.)

Notation

Several notations do the same job. This series uses parentheses with a comma, $\ip{v}{w}$. It overlaps with ordered pairs, but we no longer use ordered pairs, so confusion is unlikely. The bra-ket of quantum mechanics, $\langle v \vert w \rangle$, means the same, but it suits complex vector spaces better, so we do not use it. The dot product of elementary physics, $v \cdot w$, is an inner product too. But with the dot product we have always seen that $v \cdot v$ cannot be negative, so positive definiteness comes to mind automatically, and we do not use it.

Q. Can a vector of another vector space $\mathcal{W}$ be put into the inner product of $\V$?
A. It cannot. An inner product takes only two vectors from the space it is attached to. $\mathcal{W}$ has its own inner product, and both vectors must come from that space.

Q. Is $\ip{v}{w}$ the same kind of map as the $\map{\beta}{v}$ we have used so far?
A. No. $\map{\beta}{v}$ is a map of the dual space that takes one dual vector and one vector. $\ip{v}{w}$ is a special map that takes two vectors, and the shape of the brackets shows the difference.

Defining it through a basis

Having an inner product means knowing the inner products of the basis vectors. Put in two vectors $a^\mu e_\mu$ and $b^\nu e_\nu$, and the constants come out and the sums split, giving this.

Expanded, it is 16 terms: $a^0 b^0 \ip{e_0}{e_0} + a^0 b^1 \ip{e_0}{e_1} + \dots + a^3 b^3 \ip{e_3}{e_3}$. The coefficients in front are components we already know, so all we need are the 16 values $\ip{e_\mu}{e_\nu}$. It is exactly the way we fixed a map of the dual space by its values on the basis. So to define an inner product we only have to give these 16 values. Any real numbers will do. In the lecture the values are deliberately whimsical, such as $\ip{e_0}{e_0} = 3$ and $\ip{e_0}{e_1} = \pi/2$. Fill in the table this way, and any two vectors we put in give a real number through the expansion above, and at that moment $\V$ is an inner product space. (If we require symmetry, then $\ip{e_\mu}{e_\nu} = \ip{e_\nu}{e_\mu}$, so only 10 distinct values need to be fixed. This explanation follows from the properties above and was not covered in the lecture.)

If even one negative term is large enough, the whole $\ip{v}{v}$ can drop below zero. That is fine, because we decided to use an indefinite inner product. Let us play with it. I reduced it to 2 dimensions.

Q. Do we choose the inner product freely?
A. As long as the properties above hold, the values in the table can be chosen freely. In that sense it is arbitrary. The problem general relativity tries to solve is not choosing one at random but finding out which inner product nature actually picked.

An example from special relativity

In special relativity, with a suitable coordinate system and basis, the table of values keeps only its diagonal. The diagonal values are $-1, 1, 1, 1$ and all the rest are 0.

We call $\eta$ the Minkowski metric. Depending on the book or paper the signs are reversed, with $\eta_{00}$ equal to $+1$ and the rest $-1$, so check first which one a text uses. This is the story of special relativity, and in general relativity the table of values becomes much more complicated.

Reading it as a tensor

This map is not unfamiliar. A map that takes two vectors and returns a real number already exists in the (0, 2) tensor product space $\V^* \otimes \V^*$. Tensors of the form $g_{\mu\nu}\, e^\mu \otimes e^\nu$ live in that space, and they also take two vectors and return a real number. Then can we find a map inside that space that imitates the inner product exactly? The answer is yes, and that tensor is the metric tensor. We look at it more in the next post.