Some prerequisites: Differential geometry. E.g. a course “897602 Geometrical Methods of Physics” at IF.
Drawings as you read would help.
2. Set up and coordinate maps
Let be an dimensional manifold, and let be a coordinate chart. This means that is an open region in and is a bijection
It is convenient to use the maps defined via
That is, for
In most physics (and perhaps also mathematics) textbooks, is normally regarded as a place holder for a number. For example, we would encounter them in an equation
The solution for this would be For us however, writing or would refer to maps. So in this post, writing something like would be meaningless (unless agreed in advance).
We call as a coordinate map. For simplicity of the discussions, in this post we restrict ourselves to only one coordinate chart, and hence we use only one coordinate map. This means that although we are discussing transformations in this post, we do not make use of any coordinate transformations(!)
Let us consider a map
- is bijective.
- and can be differentiated infinitely many times.
In this case, the map is called a “diffeomorphism”.
In principle, we need to be more precise what it means by “differentiation” or “derivative” for the map After all, we only know how to take a derivative on a function from to But we do not know (suppose we have not yet studied differentiable manifold) how to take a derivative on a function from one manifold to another.
The idea is that, roughly speaking, we need to represent part of by coordinate charts, and represent by maps between two images of coordinate maps. If all the representatives of are differentiable, then is said to be differentiable. The same goes for differentiable infinitely many times. More details might appear in later versions……
For physicists, I would say that the concept of “differentiable infinitely many times” is related to the concept of “smoothness” (well, I think this is just the definition by mathematicians). Although I do not say much about what it means by “differentiable infinitely many times”, I think physicists have the ability to accept the concept of “smoothness” anyway even if we do not know the exact definitions. So let us simply move on.
If we collect all the diffeomorphisms on and use the map composition operation then we have a group called “diffeomorphism group” The proof that it is a group is left as an exercise for keen readers. Let me give some hints that the identity element of is simply the identity map, and that the inverse of as a group element is simply given by the inverse map
So each element of the group maps a point to another point To avoid later clutter of notation, when the context is clear, we will write instead of
For a point we call
the th-coordinate (in the chart ) of the point Let us consider for Its th-coordinate is given by
Note that is the th component of coordinate representation of So we write
Roughly speaking, whereas is a map (more precisely, it is a map ).
Note that the expression resemblance the one ( or more often is written as ) we often seen in physics textbooks (especially general relativity). Such expression is almost always interpreted as coordinate transformation from to So it looks to me that this point of view is the passive transformation. However, in this entire post, I am trying to discuss active transformation. The end result should be no different from the discussion involving passive transformation.
4. Transformation on scalar field
Consider a scalar function
We have learned in the previous section that each element of the group is a map between points in By an appropriate definition, each element induce a transformation on That is, for each element define a map (note: means a set of smooth functions on )
By the above definition, also forms a group. Consider
So we have
That is we have the closure relation. Furthermore, the multiplication rule above suggests that preserves the group operation and gives a one-to-one correspondence between and . The set of all such under the map composition forms a group. Technically, we say that this group is (group) isomorphic to
where is coordinate representation of and are such that
By writing which is the coordinate representation of the rule for transformed scalar field (14) can be rewritten as
This rule resemblance (by resemblance, I mean, we simply make some renamings of the notations and abuse some others) which is often interpreted as the result of coordinate transformation
5. Transformation on vector field