General Relativity 02 - Manifold and tensor fields
Last updated: 2017-04-19 21:55:03 PDT.
(This is from Charpter 2.2 of Wald)
In GR, we discuss about spaces that are not exactly flat. If we can split the space into parts and each part is continously corresponsive to a region of , then it is called a manifold. is called toe dimension of the manifold.
Normally, a finite dimensional manifold can be embeded in a higher dimensional Euclidean space. Whereas such embedding might not be natural. In GR, the spacetime does not naturally live in a higher dimensional space, so an abstract definition of a manifold is necessary.
Definition (Manifold): An n-dimensional, , real manifold is a set together with a collection of subsets satisfying the following properties:
- is a open cover of .
- For each , there is a one-to-one, onto, map where is an open subset of .
- For each and that , the map maps to . Then both sets must be open and the map must be .
Each map is called a "chart" or a "coordinate system". The definition of manifold and complex manifold simply is the same with some natural changes.
Through out this book, the manifold involed are all assumed to be Hausdorff and paracomact.
NOTE By this definition, the charts, by no means, are required to provide a "straight coordinate" to , by which I mean the tangent vector can change through out each .
Example Euclidean space with one chart and map to be identity function.
Example 2-sphere (a 2-dimensional spherical surface embedded in ).
In SR, each coordinate system is applicable to the whole universe. but in GR, a coordinate system is only usable inside its corresponding open set.
With the mapping to , we can easily define differentiability and smoothness of maps between manifolds.
Defintion (Diffeomorphism): Let and to be manifolds and and to be chart maps. A map is said to be if for each is in Euclidean spaces. If is , one-to-one and onto, then it is called a diffeomorphism, and and are said to be diffeomorphic.
NOTE diffeomorphism requires the manifolds to have the same dimension.
Tangent vector in a manifold (without embedding in ):
We can define tangent vectors as directional derivatives. In , the mapping between vectors and directional derivatives is one-to-one. defines derivative operator . Directional derivatives are characterized by Leibnitz rules.
Let denote the collections of all functions from manifold into . We define a tangent vector at point to be a map which is (1) linear and (2) obeys Leibnitz rules:
NOTE: Be very careful that the second rule also applies the function and at .
Proof.If , since linearlity, otherwise divide the equation by , and .
The maps of a tangent vectors of a point forms a vector space by adding this addition law: .
Theorem 2.2.1: Let be an -dimensional manifold. Let and let denote the tangent space at , then .
Proof. Given a chart of open set where , If , then is . For define functional by
This means, the th component of is a functional that takes the th partial derivative of function in , and apply the point to it. It is clear that is a derivative from the chain rule. Now we need to prove that .
For any function , if is ,especially, we have
Let , and we havewhere denotes , that is, picking the th element of the result.
For any , apply the functional,This means is a linear combination of .
The basis is called a coordinate basis, frequently denoted as simply . For each different chart chosen, there is a different coordinate basis , andWe can also get the vector transformation law from it:
A smooth curve, on a manifold is a map of into , . At each point , there is a tangent vector associated with as follows.Therefore the components of is given by
If and are on the manifold, there is no way to correlate them in a general manifold. Another construct ("connection", or "parallel transportation") must be introduced to do so. However, if the curvation is nonzero, the identification of with obtained in this manner will depend on the choice of curve.
A tangent field, , on a manifold is an assignment of a tangent vector, for each point . Despite the fact that the tangent spaces and at different points are different vector spaces, there is a natural notion of what it means for to vary smoothy from point to point.
A one-parameter group of diffeomorphisms is a map from . In particular:
- is a diffeomorphism, and
At each point , is a curve, called the orbit of which passes through at . Define to be the tangent vector at . Thus we can consider the vector field to be the generator of such a group.
Conversely, we can ask a question, that if given a vector field , can we find a family of curves s.t. for each point , there is one and only one curve that passes through the point with the tangent vector equals to . The answer is yes.
Therefore we have a one-to-one mapping between a tangent field and a diffeomorphism. We can thus consider a tangent field to be a mapping of type :
Given two smooth vector fields and , we can define the commutator field as follows: