Tensor #15. Coordinate Transformations텐서 #15. 좌표 변환

These are my notes on XylyXylyX’s What is Tensor lectures. This post covers lecture 15 of the playlist, Coordinate Transformations. In the previous post we fixed the bases of the vector spaces at all points at once from the coordinates, so this time I sort out the two ways to change that basis.

Only the names change

Points of spacetime are named by coordinates $x^\mu$. The vector space at a point P is now called the tangent space, and the dual space is called the cotangent space. The structure stays the same and only the names change. An element of the cotangent space is a map that takes a vector and returns a real number, and it is called a one-form. Two-forms and higher can be built from one-forms too, but the method comes much later. It also stays true that every point has its own tangent space, cotangent space, and tensor product spaces.

Why call it tangent

A vector in the tangent space is a differential operator $L = a^\mu\partial_\mu$. Apply this operator to a function $f(x^0,x^1,x^2,x^3)$ on spacetime and we get the rate at which f changes at the point P in the direction $a^\mu$, which is the directional derivative. A directional derivative does what a tangent line does, so the name tangent space fits.

It may look like a mere renaming, but there is content in it. If $f = (x^0)^2 + \sin x^3$ and $L = \partial_0 + \partial_3$, we get the following.

Even with the same formula, at a point P we get the number with P’s coordinates put in, and at a point Q we get a different number with Q’s coordinates put in. So it is not a swap of symbols. It is an object that gives a different result at each point. All of this holds only because a coordinate system exists first. Cartesian or cylindrical does not matter, but we have to lay down coordinates before we can think about functions and fix bases.

The first way: frame fields

The tangent space is also a vector space, so we may choose any four independent operators as a basis.

$\Lambda$ is the basis change matrix we saw earlier. What is new is that the tangent spaces are separate at each point, so a transformation made in one tangent space has no effect on any other. To give a transformation at every point, the components of $\Lambda$ must be functions of spacetime. Choosing a basis point by point like this is called a frame field, and it is a topic of intermediate general relativity. The basis of the cotangent space changes by the inverse matrix, as always.

The second way: coordinate transformations

The other way a basis changes is to change the coordinate system of spacetime. A coordinate transformation is a bundle of functions.

There are four functions that give the new coordinates from the old ones, and four inverse functions. Going from Cartesian to spherical coordinates is an example. All that changes is the name by which we call the point P, and the point and its tangent space stay as they are. So a function on spacetime must also give the same number at the same point.

The function f becomes a different function $f'$ of the new coordinates, but its value at the point P must be the same. P does not care whether it calls itself x or x’. This requirement is one of the deepest foundations of general relativity.

Q. Does the tangent space change when the coordinates change?
A. The tangent space stays as it is. Only the name of the point has changed. Since we fixed the basis by leaning on the coordinates, the operators used as the basis change from $\partial_\mu$ to $\partial'_\mu$. It is the same space read with a different basis.

The Jacobian

Apply the new basis $\partial'_\mu$ to an arbitrary function and use the chain rule. We get $\partial'_\mu f = \dfrac{\partial x^\nu}{\partial x'^\mu}\,\partial_\nu f$, and the function f can be dropped, because a differential operator can equal another differential operator.

The $\Lambda$ of the first way is here not a matrix we choose freely but is fixed by the partial derivatives of the coordinate transformation functions. Since its components are functions of the coordinates, it is also a different matrix at each point. This matrix is called the Jacobian. (The name Jacobian is added to help understanding, and it was not covered in the lecture.)

In the toy below, pick a transformation and drag the point P. The new coordinate curves are drawn over the old grid, and you can see how the new basis $\partial'_0, \partial'_1$ sits at P. The vector A stays the same and only the basis changes, and the components change the opposite way so that the same arrow is built again.

The dual basis is the inverse matrix

The one-form basis changes by the inverse matrix, as always.

By the chain rule the two matrices are inverses of each other.

So $\map{dx'^\mu}{\partial'_\nu} = \delta^\mu{}_\nu$ keeps holding in the new basis. For the promise of the dual basis not to break, the two have to match up like this.

Whatever changes the way the basis vectors change is defined as covariant, and whatever changes by the inverse matrix is defined as contravariant. The basis $\partial_\mu$ of the tangent space is covariant by definition, and the one-form basis $dx^\mu$ is contravariant. The vector $A = A^\mu\partial_\mu$ is an object independent of coordinates, so when the basis changes one way, the components must change the other way.

Q. Is a basis chosen as a frame field the same as one obtained from a coordinate transformation?
A. A coordinate transformation fixes $\Lambda$ by partial derivatives, so it is a special case of a frame field. Conversely, not every frame field comes from some coordinates. For example, the basis $\partial_r,\ \tfrac{1}{r}\partial_\theta$ with lengths set to 1 in polar coordinates is not the partial-derivative basis of any coordinates. (This explanation is added to help understanding and was not covered in the lecture.)

Q. A vector should be covariant, so why are the components of a vector called contravariant?
A. Covariant and contravariant are names attached to how the basis and the components change. The basis vectors are covariant and their components are contravariant. The vector as a whole is the object that does not change when the two are put together. For a one-form it is the other way round, with a contravariant basis and covariant components. The names leaning toward the components is the cause of the confusion. (This explanation tidies the muddled remarks near the end of the lecture into the standard definition and was not covered in the lecture.)

Where the formalism is skipped

Strictly speaking, spacetime is a manifold, and the collection of all the tangent spaces at all points is the tangent bundle. A vector field picks one vector from each tangent space, and this is called a section of the tangent bundle. This series does not go deep into this formalism and starts from laying down coordinates. When this post says tangent space or vector field, it often means the tangent spaces at all points at once.

Where textbooks begin

Many general relativity textbooks write the component transformation formulas above right in chapter one, and define that whatever changes this way is a contravariant vector and whatever changes that way is a covariant vector. Without knowing the reason, it looks like a rule set by force. Now the reason is visible. That formula is the rule that the components of a vector obey, the basis is hidden, and index gymnastics manages that hidden basis for us.

Changing coordinates is like changing clothes in general relativity. We can change them freely, and the laws of physics must be the same in any coordinates. This constraint is general covariance, and it tightly binds the form of the laws of nature. The remaining problem is that the tangent spaces are separate spaces apart from each other, so we cannot compare vectors at different points. This is solved in a later post. In the next post we put the metric tensor on top of this structure.