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3.4 Bivectors 353.4 BivectorsBivectors are antisymmetric tensors of second order, or 2-forms,X = X ab ω a ∧ ω b . (3.27)A simple bivector, X ab = u [a v b] , represents a 2-surface element spannedby the two tangent vectors u = u a e a and v = v a e a . This surface elementis spacelike, timelike or null according to whether X ab X ab is positive,negative or zero, respectively.Taking a particular orientation of (a neighbourhood in) M, we definethe Levi-Civita 4-form ε to be −4! √ −g ω 1 ∧ ω 2 ∧ ω 3 ∧ ω 4 , where g is thedeterminant of the matrix g ab of metric tensor components with respectto a positively-oriented basis {e a }. Its components are written ε abcd , and,if the positively-oriented basis is an orthonormal tetrad {E a } as in (3.7),are defined byε 1234 = −1. (3.28)This amounts to a choice of orientation of the four-dimensional manifoldin which the basis {E a } represents a Lorentz frame with E 4 pointingtoward the future and with a right-handed spatial triad as {E α }. If thatbasis is related to the complex null tetrad (3.8) by the formula (3.12),then (3.28) can be written asε abcd m a m b l c k d =i. (3.29)The corresponding three-dimensional tensor obtained by contraction witha timelike unit vector u = E 4 , giving components ε abcd u d , will be denotedε αβγ , where as usual α, β, γ =1, 2, 3.With the aid of the Levi-Civita 4-form we define the dual bivector X,∼in index notation, by˜X ab ≡ 1 2 ε abcdX cd . (3.30)To avoid confusion we emphasize that the concepts of dual basis anddual bivector have entirely distinct meanings. Repeated application ofthe duality operation (3.30) givesA bivector is called null (or singular) if( ˜X ab ) ˜ = −X ab . (3.31)X ab X ab =0=X ab ˜X ab (3.32)holds. Two bivectors X and Y satisfy the identitiesX ac Y b c − ˜X bcỸa c = 1 2 g abX cd Y cd , ˜X ab Y ab = X abỸ ab , (3.33)