Two-vector

A two-vector or bivector^[1] is a tensor of type $\scriptstyle {\binom {2}{0}}$ and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars).

The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of vectors, especially a linear combination of tensor products of pairs of basis vectors. If f is a two-vector, then^[2]

\mathbf {f} =f^{\alpha \beta }\,{\vec {e}}_{\alpha }\otimes {\vec {e}}_{\beta }

where the f ^{α β} are the components of the two-vector. Notice that both indices of the components are contravariant. This is always the case for two-vectors, by definition.

An example of a bivector is the stress–energy tensor. Another one is the orthogonal complement^[3] of the metric tensor.

References

^ Penrose, Roger (2004). The road to reality : a complete guide to the laws of the universe. New York: Random House, Inc. pp. 443–444. ISBN 978-0-679-77631-4. Note: This book mentions “bivectors” (but not “two-vectors”) in the sense of $\scriptstyle {\binom {2}{0}}$ tensors.
^ Schutz, Bernard (1985). A first course in general relativity. Cambridge, UK: Cambridge University Press. p. 77. ISBN 0-521-27703-5. Note: This book does not appear to mention “two-vectors” or “bivectors”, only $\scriptstyle {\binom {2}{0}}$ tensors.
^ Penrose, op. cit., §18.3

[1] Penrose, Roger (2004). The road to reality : a complete guide to the laws of the universe. New York: Random House, Inc. pp. 443–444. ISBN 978-0-679-77631-4. Note: This book mentions “bivectors” (but not “two-vectors”) in the sense of $\scriptstyle {\binom {2}{0}}$ tensors.

[2] Schutz, Bernard (1985). A first course in general relativity. Cambridge, UK: Cambridge University Press. p. 77. ISBN 0-521-27703-5. Note: This book does not appear to mention “two-vectors” or “bivectors”, only $\scriptstyle {\binom {2}{0}}$ tensors.

[3] Penrose, op. cit., §18.3

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See also

References