Tensor taper

The tensor taper or contraction is a mathematical term from linear algebra . It is a generalization of the trace of a linear mapping to tensors that are at least singly covariant and singly contravariant.

definition

Let be a finite-dimensional vector space and be ${\displaystyle V}$

${\displaystyle T_{s}^{r}(V):=\underbrace {V\otimes \cdots \otimes V} _{r{\text{-mal}}}\otimes \underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{s{\text{-mal}}}}$

the tensor space of -fold contravariant and -fold covariant tensors (short: -Tensors) over . ${\displaystyle r}$ ${\displaystyle s}$ ${\displaystyle (r,s)}$ ${\displaystyle V}$

The linear mapping is called the tapering or contraction of a tensor (more precisely: contraction) ${\displaystyle (k,l)}$

${\displaystyle C_{l}^{k}:T_{s}^{r}(V)\rightarrow T_{s-1}^{r-1}(V)}$

with and , which by ${\displaystyle 1\leq k\leq r}$ ${\displaystyle 1\leq l\leq s}$

${\displaystyle v_{1}\otimes \cdots \otimes v_{r}\otimes \xi _{1}\otimes \cdots \otimes \xi _{s}\mapsto }$

${\displaystyle \xi _{l}(v_{k})(v_{1}\otimes \cdots \otimes v_{k-1}\otimes v_{k+1}\otimes \cdots \otimes v_{r}\otimes \xi _{1}\otimes \cdots \otimes \xi _{l-1}\otimes \xi _{l+1}\otimes \cdots \otimes \xi _{s})}$

can be defined. There is an element of . Not every element of is of this form, but the elements of this form create the tensor space and the mapping is well defined. If one sets , then a tensor -th level becomes a tensor of level . ${\displaystyle v_{1}\otimes \cdots \otimes v_{r}\otimes \xi _{1}\otimes \cdots \otimes \xi _{s}}$ ${\displaystyle T_{s}^{r}(V)}$ ${\displaystyle T_{s}^{r}(V)}$ ${\displaystyle n:=r+s}$ ${\displaystyle n}$ ${\displaystyle n-2}$

Examples

• If one interprets a matrix as a simply co- and contravariant tensor, the taper of a matrix is ​​its trace. This can be seen very quickly if the matrix is ​​represented as a linear combination . Here one forms the basis of and that forms the dual basis of . If one now applies the function , one obtains. This shows that the tensor taper is a generalization of the trace operator known from linear algebra . For this reason, the mapping is also called track formation. ${\displaystyle A\in \operatorname {End} (V)\cong V\otimes V^{*}}$
  ${\displaystyle A=\sum _{i,j}\lambda _{i}^{j}\,v_{i}\otimes \xi _{j}}$
  ${\displaystyle v_{i}}$ ${\displaystyle V}$ ${\displaystyle \xi _{j}}$ ${\displaystyle V^{*}}$ ${\displaystyle C_{1}^{1}}$
  ${\displaystyle C_{1}^{1}(A)=C_{1}^{1}(\sum _{i,j}\lambda _{i}^{j}\,v_{i}\otimes \xi _{j})=\sum _{i,j}\lambda _{i}^{j}\delta _{ij}=\sum _{i}\lambda _{i}^{i}=\operatorname {Spur} (A).}$
  
• The Ricci tensor is obtained from the Riemann curvature tensor by tapering . ${\displaystyle R_{ijk}^{l}}$ ${\displaystyle R_{ik}=R_{ijk}^{j}}$

literature

• R. Abraham, JE Marsden, T. Ratiu: Manifolds, Tensor Analysis, and Applications (= Applied Mathematical Sciences 75). 2nd edition. Springer-Verlag, New York NY et al. 1988, ISBN 0-387-96790-7 .

Individual evidence

1. ↑ Ulrich E. Schröder: Special Theory of Relativity . German, Frankfurt am Main 2005, ISBN 3-8171-1724-8 , pp. 51 .