Tensors and Relativity: Chapter 3
General properties of tensors
Tensors have the following general properties:
- If and are M/N tensors, so are , and for any and . Thus tensors of a given type form a vector space .
- If is a M/N tensor and is of type M'/N', then the outer product is a tensor of type M+M'/N+N'.
- For any tensor of type M/N, one can construct a tensor of type M-1/N-1 by contracting an upper index with a lower index for example is constructed from [ or ]. Note however that there may be several ways of contracting and they give different tensors.
- Tensors can be symmetric on pairs of upper or lower indices for example and , or anti- symmetric . However symmetric or anti- symmetric pairs of indices with one index up and the other down cannot be defined since the relationship need not hold in all frames.