Tensors and Relativity: Chapter 3

General properties of tensors

General properties of tensors

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Tensors have the following general properties:

  • If tex2html_wrap_inline1915 and tex2html_wrap_inline1917 are M/N tensors, so are tex2html_wrap_inline1921 , tex2html_wrap_inline1923 and tex2html_wrap_inline1925 for any tex2html_wrap_inline1927 and tex2html_wrap_inline1929 . Thus tensors of a given type form a vector space .
  • If tex2html_wrap_inline1915 is a M/N tensor and tex2html_wrap_inline1935 is of type M'/N', then the outer product tex2html_wrap_inline1939 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 tex2html_wrap_inline1947 is constructed from tex2html_wrap_inline1949 [ tex2html_wrap_inline1951 or tex2html_wrap_inline1953 ]. 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 tex2html_wrap_inline1955 and tex2html_wrap_inline1957 , 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.