Tensor #40. Special Lecture: Tangent Spaces and Coordinate Basis텐서 #40. 특별 강의: 탄젠트 스페이스와 좌표 기저

These are my notes on XylyXylyX’s What is Tensor lectures. This post covers lecture 40 of the playlist, Special Lecture: Tangent Spaces and Coordinate Basis. The previous post ended the main line of the tensor lectures, and this one is a special lecture added separately. The lecturer sees the biggest sticking point for students in using the partial derivative operators $\partial_i$ as a basis of a vector space, and redoes more slowly an explanation given earlier in the manifold lectures. This post condenses that flow into nine sections.

A manifold with coordinates

The starting point is an $n$-dimensional manifold $M$. In general relativity, spacetime is modeled as a manifold with $n = 4$. The word manifold should immediately bring up the following picture. Around a point $p$ there is an open region $U$, and a one-to-one correspondence sends this region to $\R^n$.

The pair of $U$ and $x$ is called a chart. The map $x$ is a homeomorphism, continuous with a continuous inverse. The coordinates of the point $p$ are $x(p) = (x^0(p), \dots, x^{n-1}(p))$, so there are $n$ coordinate functions, one for each component. A single chart often cannot cover the whole manifold. Even a sphere needs two charts. This post stays inside one chart. The change $x \circ y^{-1}$ between two overlapping charts $x, y$ is taken to be differentiable.

Defining a manifold already means that it has coordinates, so every construction below may use them. We begin, however, by stacking up what can be built without relying on coordinates.

The tangent space comes with the manifold

In earlier lectures the lecturer said “we erect a vector space at each point.” The lecture corrects this phrasing because it misleads. Choosing a manifold as the model of spacetime is a decision made by the physicist. But once the manifold is chosen, the collection of all smooth functions on it and the collection of all curves on it are already there without our making them. Only taking one function or one curve out of a collection is our choice. The collections themselves depend on no decision, so they exist as soon as the manifold does.

This post follows that order of existence. There is a collection of functions and a collection of curves. Pairing them gives directional derivatives. We can give addition and scalar multiplication to the collection of directional derivatives, so it becomes a vector space. Choosing coordinates then gives its basis. The point is that nothing in this chain was newly made up and attached.

Functions and curves

A function on the manifold is $f : M \to \R$. We write the collection of all smooth functions as $\Cf$. Gathering the points where the function has the same value gives a level set, like a contour line on a map. A single point can also be the level set of a maximum value.

Care is needed in saying that $f$ is differentiable. $M$ is just a set of points, so we cannot differentiate with respect to its points. So we read it as saying that the function moved to the coordinate side by the chart is differentiable.

$F$ is a function of $n$ coordinates, so partial derivatives apply. From now on we often write $f$ as if it were a function of coordinates, but strictly we mean this $F$.

Q. $f$ is a function on the manifold. How is differentiation defined on it?
A. The points of a manifold have neither addition nor distance, so we cannot define it directly. We turn points into coordinates with the chart and differentiate $F = f \circ x^{-1}$ by ordinary multivariable calculus. That result is the derivative of $f$.

A curve is a map from a real interval into the manifold, $\alpha : \R \to M$. We write the parameter as $\tau$. In coordinates it is $x \circ \alpha : \R \to \R^n$, and each component $x^i \circ \alpha$ is a function from the reals to the reals, so we can take an ordinary derivative. Even if two curves pass through the same points in the same order, they count as different curves when the parameters differ. For example, changing $\tau$ to $2\tau$ gives a different curve that passes along the same road twice as fast. We call the collection of all curves on the manifold $A(M)$, and the collection of those that pass through $p$ with the parameter set so that $\alpha(0) = p$ we call $A_p(M)$. This normalization loses no generality.

The directional derivative

Choosing one function $f$ and one curve $\alpha \in A_p(M)$ gives $f \circ \alpha : \R \to \R$, an ordinary function of one variable. We write its derivative, taken at the moment of passing through $p$, that is at $\tau = 0$, as $V_{\alpha,p}$.

Fixing this number needs both $f$ and $\alpha$. In one dimension the function alone is enough, but with more than one dimension the rate of change at the same point depends on which way we go. Once a curve is fixed, $V_{\alpha,p}$ is a map $\Cf \to \R$ that takes a function and returns a real number, and we call it the directional derivative. The function $f$ is not part of the definition, and its place is left empty in the operator. The lecturer says that just as a hammer can sit in a toolbox without a nail, this operator is an object in its own right even when no function is fed to it.

Each time we pick a curve from the collection $A_p(M)$ of curves through $p$, one directional derivative comes out. The set of directional derivatives gathered this way is the object we will turn into a vector space. Each point has its own such set. Different curves can also give the same operator. For example, two curves passing through $p$ at the same velocity and differing only in how much they bend give the same values. The lecture says curves and directional derivatives correspond one to one, but the later argument in this post holds as it is without distinguishing curves that give the same operator. (This explanation is added for understanding and was not covered in the lecture.)

Scalar multiplication

For a real number $a$ and a directional derivative $V_{\alpha,p}$, we must find a curve so that the product is again a directional derivative. Consider the curve $\alpha_a(\tau) = \alpha(a\tau)$ with the parameter changed to $\lambda = a\tau$. This curve also passes through $p$ at $\tau = 0$. By the chain rule the factor $d\lambda/d\tau = a$ comes out front.

So $a V_{\alpha,p} = V_{\alpha_a,p}$, and multiplying by a real number means switching to the curve that passes the same road $a$ times as fast. If $a$ is negative, it passes backwards. The collection is closed under scalar multiples.

Addition

This is the harder side. Given curves $\gamma, \beta \in A_p(M)$, we must find $\psi$ so that $V_{\gamma,p} + V_{\beta,p}$ is the directional derivative of some curve $\psi \in A_p(M)$. Curves cannot be added, so the job is to construct a curve whose directional derivative is that sum. Here the chart is needed for the first time. Coordinates are ordered tuples in $\R^n$ and can be added.

The first attempt that comes to mind is $x \circ \psi = x \circ \gamma + x \circ \beta$. But at $\tau = 0$ its value is $x(p) + x(p) = 2x(p)$, so it misses $p$ and leaves $A_p(M)$. We only need to remove this excess.

Now the value at $\tau = 0$ is $x(p)$, so $\psi$ passes through $p$. Let us see whether the directional derivatives really add. Since $f \circ \psi = F \circ (x \circ \psi)$, the chain rule multiplies the derivative of $F$ by the derivative of the curve’s coordinates, and $\xp$ is a constant, so it disappears on differentiating. All three curves pass through $p$ at $\tau = 0$, so they are evaluated at the same moment.

Running the chain rule backwards, the two terms on the right are $V_{\gamma,p}(f)$ and $V_{\beta,p}(f)$. So for every $f$ we get this.

Both the left and the right are additions of real numbers, so addition of real numbers took over the job of addition of operators. The directional derivative of the constant curve $\alpha(\tau) = p$ gives 0 on any $f$, so it is the zero vector, and the reversed curve $\alpha(-\tau)$ is the inverse with the opposite sign. Since addition and scalar multiplication are closed and there are a zero and inverses, the collection of directional derivatives is a vector space over $\R$. This is the tangent space $T_pM$ at the point $p$. Directional derivatives are rates of change along directions tangent to curves, which is why it is called tangent.

The coordinate basis

Now we look for a basis of this vector space. Choose a chart, expand $f$ in $V_{\alpha,p}(f)$ as $F \circ x$, put in the curve’s coordinates $x^0 \circ \alpha, \dots, x^{n-1} \circ \alpha$, and apply the multivariable chain rule. $F$ is a function of $n$ variables and every variable is a function of $\tau$, so we multiply the partial derivative in each variable by the $\tau$-derivative of that variable and add.

The two factors in the sum play separate roles. The latter is a real number fixed by the curve alone, and the former is fixed by the function $f$ and the point $p$ alone, with no curve in it. We call the latter number $v^i$, and define the former as the operator $\partial_i$ that takes a function and returns a real number.

Then the directional derivative is written like this.

The $n$ operators $\partial_i$ are the basis and $v^i$ are the components. We call this basis the coordinate basis. This is why an operator $\partial_i$ gets called a vector. Directional derivatives form a vector space, and $\partial_i$ sits in the place of a basis of that space. There is no need to draw arrows. After this we can forget about curves and use only $v = v^i \partial_i$ with the Einstein summation convention.

To confirm that $\partial_i$ is a basis we must check linear independence. Inside the chart, applying this operator to a coordinate function $x^j$ gives $\partial_i (x^j) = \delta^j_i$. If some sum $\sum c^i \partial_i$ were the zero operator, applying it to $x^j$ gives $c^j = 0$, so they are linearly independent. (This explanation is added for understanding, and the lecture only asserts linear independence.) Also, for any reals $v^0, \dots, v^{n-1}$, the straight-line curve with coordinates $x(p) + \tau (v^0, \dots, v^{n-1})$ gives the directional derivative $v^i \partial_i$, so every combination of components actually occurs. (This explanation is added for understanding and was not covered in the lecture.)

Q. If choosing a basis needs a chart, what changes when the chart changes?
A. The variables we differentiate by change, so $\partial_i$ changes, and so the basis in the tangent space at each point changes. The vector $V_{\alpha,p}$ itself is fixed by the curve and the point, so it stays the same, and its components change by the rule seen in the earlier posts. The definition of the directional derivative has no chart, and the chart enters only when we choose a basis.

The toy below lets you handle this procedure by hand. Drag the velocities of two curves through one point $p$ and you see two directional derivatives, the directional derivative of the sum curve, and $\sum v^i \partial_i f$ written with components.

Point-dependent bases and the dual basis

The tag “evaluate at $x(p)$” behind $\partial_i$ is usually dropped, but it actually matters. The basis at a point $p$ and the basis at a point $q$ are the same partial derivative $\partial/\partial x^i$, but the coordinates where they are evaluated, $x(p)$ and $x(q)$, differ. Applied to a function they give different numbers. So the tangent spaces at two points are separate vector spaces.

Q. The basis vector is named $\partial_0$ at both points. Is it the same vector?
A. It has the same name but is a different vector. The partial derivative has the same form and the point of evaluation differs. The value on a function $f$ depends on whether $\partial_0 f$ is read at $x(p)$ or at $x(q)$.

The dual basis is easy. The cotangent space is the space of linear maps that send vectors of the tangent space to real numbers, and the dual basis matching the coordinate basis is the map that gives 1 when it receives $\partial_j$ with $i = j$ and 0 otherwise. We simply name this map $dx^i$.

The lecturer says that even if the name looks like a differential or the exterior derivative of a coordinate function, we should not worry and should read it only as the name of the map that gives 1 on its basis vector. These symbols also exist separately at each point.

Summary

What we checked in this post is the following.

• Once a manifold exists, the collection $\Cf$ of smooth functions and the collection $A(M)$ of curves are automatically there, and the two fix the directional derivative $V_{\alpha,p}(f) = d(f \circ \alpha)/d\tau |_{0}$ at a point $p$.
• The collection of directional derivatives has scalar multiples by changing the curve’s parameter $\tau \to a\tau$ and addition by adding coordinates and subtracting $x(p)$, so it becomes a vector space over $\R$, the tangent space.
• Choosing a chart gives $V_{\alpha,p} = v^i \partial_i |_{x(p)}$ by the multivariable chain rule, and $\partial_i$ is the coordinate basis. The $v^i$ are the coordinate derivatives of the curve.
• The evaluation coordinates tagged on the basis distinguish the tangent spaces point by point, and the dual basis $dx^i$ is fixed by $dx^i(\partial_j) = \delta^i_j$.

This construction is the foundation that leads into the Manifold series.