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

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