Quadruple tensor

Four-tensor is a term from the theory of relativity . A four-tensor is a tensor over the 4-dimensional vector space of Minkowski space-time and its dual space , or in general relativity over the tangential space to spacetime , a four-dimensional Riemannian manifold . ${\displaystyle M}$ ${\displaystyle M^{*}}$

A quadruple tensor of level is an element of the tensor product ${\displaystyle (k,l)}$

${\displaystyle \underbrace {M\otimes M\otimes \dotsb \otimes M} _{k{\text{ mal}}}\otimes \underbrace {M^{*}\otimes M^{*}\otimes \dotsb \otimes M^{*}} _{l{\text{ mal}}}}$

Such a tensor of level is called -fold contravariant and -fold covariant. Four-tensors of level or are also called contravariant or covariant four-vectors . ${\displaystyle (k,l)}$ ${\displaystyle k}$ ${\displaystyle l}$ ${\displaystyle (1,0)}$ ${\displaystyle (0,1)}$

First-order quad tensors can be represented by a vector with four entries. Examples:

• Speed ​​of four
• Quadruple pulse

Fourth-order tensors of the second order can be represented by a matrix . Examples: ${\displaystyle 4\times 4}$

• metric tensor
• electromagnetic field strength tensor
• Energy-momentum tensor

A fourth-order quadruple tensor can be represented by entries. Example: ${\displaystyle 4^{4}=256}$

• Riemann curvature tensor

Web links

• Quadruple tensor in the lexicon of physics