Friday, 22 January 2021

Tensor Tricks

 Cat gets Carroll
The file Commentary Tensor Tricks.pdf contains some useful equations for tensor manipulation which I have collected.

The very first on tensor rank comes from early in the book (page 21) where we are told that: a tensor $T$ of type (or rank) $\left(k,l\right)$ is a multilinear map from a collection of dual vectors and vectors to $\mathbf{R}$:
$$T:\left(T_p^\ast\times\cdots\times T_p^\ast\right)_{k\ times}\times\left(T_p\times\cdots\times T_p\right)_{l\ times}\rightarrow\mathbf{R}$$
That is Carroll's 1.56 and I am pretty sure he has that the wrong way round. It should say: a tensor $T$ of type (or rank) $\left(k,l\right)$ is a multilinear map from a collection of $k$ vectors and $l$ dual vectors to  $\mathbf{R}$:$$T:\left(T_p\times\cdots\times T_p\right)_{k\ times}\times\left(T_p^\ast\times\cdots\times T_p^\ast\right)_{l\ times}\rightarrow\mathbf{R}$$But it turns out that you don't really need to know what $k$ and $l$ are separately in General Relativity (because we always have a metric). You only need to know the total rank $k+l$!

Contents

• What tensor rank?
• Multi-dimensional Chain Rule
• Partial derivative gives Kronecker delta: Coordinates, Vectors, Tensors
• Partial derivatives commute
• Metric is always symmetric
• Contracting with metric lowers / raises index
• You can lower or raise indices on a tensor equation
• Swap indices with metric or any similar tensor
• Inverse of a matrix
• The determinant of the inverse is reciprocal of the determinant
• Determinant of a tensor in terms of Levi-Civita symbol
• Inverse tensor
• A relationship for the derivative of the determinant
• Fully contracted symmetric × antisymmetric tensor vanishes
• Symmetrising a tensor equation
• Two formulas involving four-velocity
• Second formula
• The projection tensor on four-velocity
• Contra / co-variant tensor transformation matrices
• Tensor contractions using matrices