Tensors & Index Notation Flashcards
(40 cards)
What is a tensor?
A multilinear object that transforms according to a rule under coordinate transformations.
What is a (1,0) tensor?
A contravariant vector — transforms like dx^μ.
What is a (0,1) tensor?
A covariant vector — transforms like ∂_μ.
What is a (2,0) tensor?
A rank-2 contravariant tensor: T^{μν}.
What is a (0,2) tensor?
A rank-2 covariant tensor: T_{μν}.
What is the transformation law for a (1,1) tensor?
T’^μ_ν = (∂x’^μ/∂x^α)(∂x^β/∂x’^ν) T^α_β
What does index contraction mean?
Summing over one upper and one lower index, producing a scalar or lower-rank tensor.
What is the Kronecker delta δ^μ_ν?
Identity operator on indices: 1 if μ = ν, 0 otherwise.
What does it mean to raise an index?
Use the inverse metric: V^μ = g^{μν}V_ν.
What does it mean to lower an index?
Use the metric: V_μ = g_{μν}V^ν.
How does the metric tensor transform?
Like a (0,2) tensor: g’{μν} = (∂x^α/∂x’^μ)(∂x^β/∂x’^ν)g{αβ}
What is the meaning of tensor rank?
The number of indices a tensor has — total of covariant and contravariant indices.
What is a scalar field?
A (0,0) tensor — invariant under coordinate transformations.
How is a vector defined in GR?
As a quantity transforming under coordinate transformations like dx^μ or ∂/∂x^μ.
Why is the metric a (0,2) tensor?
Because it has two covariant indices: g_{μν}.
What is the significance of symmetry in tensors?
It can reduce the number of independent components and encode physical properties.
What does antisymmetric mean for a tensor?
T_{μν} = -T_{νμ}.
How many independent components does a symmetric (0,2) tensor have in 4D?
10 components.
How many independent components does an antisymmetric (0,2) tensor have in 4D?
6 components.
What is a mixed tensor?
A tensor with both covariant and contravariant indices, e.g., T^μ_ν.
What is a tensor product?
An operation combining two tensors into a higher-rank tensor.
Is the product of two tensors always a tensor?
Yes, if both transform properly under coordinate transformations.
What operation reduces tensor rank?
Index contraction.
What is the dual of a vector?
A covector (1-form); maps vectors to scalars.
