T1: 2. Manifolds and Tensors Flashcards
What does it mean for spacetime to be ‘locally flat’?
Around every point, there is a coordinate system which maps a local neighbourhood to a point in the real plane.
Define a coordinate system (and hence, coordinates)
A map from a part of spacetime to an open region in the n-dimensional real plane. Coordinates are the collection of corresponding functions.
Define a coordinate transformation
The map between two overlapping coordinate systems across the n-dimensional real plane.
Define a chart (coordinate system) in the context of manifolds.
Given a set M, a chart is a function phi which forms a bijection between U (the subset of M) and the open subset of the n-dimensional real plane defined by the action of phi on an element of U.
Define an atlas
A collection of charts {(U_a, phi_a)}, such that:
- The union of all subsets U = M (i.e. the subsets cover the entire original set).
- If the intersection of a pair of subsets U_a and U_b is not the empty set, the successive map of the inverse phi_b and phi_a is differentiable where it is defined.
Define a maximal altas
The collection of all charts which satisfy the atlas conditions.
Define the n-dimensional manifold
The pair of a set M and a maximal atlas.
Define the three primary kinds of vectors and which is most suitable on a manifold
a. Displacement vector
b. Tangent vector
c. Normal vector
b - Manifolds consider locally flat regions and the tangent vector in infinitesimally small.
Why are displacement vectors not appropriate for use on a manifold.
They don’t form a vector space on a manifold: ‘tip to tail’ vector addition is not commutative.
How do we define d/dλ in flat space?
V^μ ∂_μ = (dx^μ/dλ) ∂/∂X^μ
Define a tangent vector field
A choice of vector V in the tangent vector space at each point p in M; a vector-valued function on M
Define tangent vector space T_p
The space of vectors V=d/dλ tangent to M at point p.
Define cotangent space T_p^*
The space of linear maps ω from the tangent space to the reals.
I.e. The space of vectors cotangent to M at point p.
How do we define the cotangent vector?
A differential function df maps from tangent space to real numbers. We can equate this to the action of the tangent vector on some function f:
df(V) = V(f)
Define cotangent vector field
A choice of cotangent vector ω∈ T_p^* at each point p in M.
What is the basis of tangent vectors?
∂/∂X^μ
What is the basis for covectors?
dx^μ
How do we coordinate transform components of tangent vectors?
By chain rule on the coordinates in the component.
V ̃^μ = V^ν (tilde on top)
Refer to notes
How do we coordinate transform components of covectors?
By chain rule on the coordinates in the basis vector.
ω ̃_μ = ω_ν (tilde on bottom)
Refer to notes
Define a tensor of type [r,s] at a point p.
An object which linearly maps r covectors and s vectors to the reals.
Define a type [r,s] tensor field
An object which gives an [r,s] tensor at each point p on M.
What type of tensor is a function?
Type [0,0]: scalar
What type of tensor is a vector (covector)?
Type [1,0] (type [0,1])
What type of tensor is Kroneker delta? What is special about it?
Type [1,1] with same components in any coordiante system: diag(1, … ,1).
What are the three tensor equation rules?
- Free/dummy indices
- Renaming indices
- Rearrange kernel symbols as desired
Why is a [1,0] tensor a vector?
r denotes the number of covectors the tensor takes as an argument and maps to the reals.
This tensor takes one covector and maps it to the reals; the action of a vector.
