Bivector

In Algebra, the term of bivector indicates a antisymmetric Tenseur of order 2, i.e. a quantity X which can be written

{\ mathbf {X}} = X_ {ab} {\ mathbf {\ Omega}} ^a \ wedge {\ mathbf {\ Omega}} ^b ,
where the quantities ω has is linear forms and \ wedge signs it indicates the external Produit.

A bivector can be seen like a Linear application acting on the Vecteur S and transforming them into linear forms. The coefficients X can be seen like formant a antisymmetric matrix.

The bivecteurs are abundantly used in General relativity, where several tensors can be connected to bivecteurs. In particular, the electromagnetic Tenseur is a bivector, and the Tenseur of Weyl can be seen like an application acting on the bivecteurs. This fact is besides with the orginie of a classification of various spaces according to the characteristics which present their tensor of Weyl in this context: it is about the Classification of Petrov.

Varied definitions

Simple Bivector

A bivector X is known as simple if it can be expressed in the shape of the product external of two linear forms U and v , i.e. if there is

{\ mathbf {X}} = {\ mathbf {U}} \ wedge {\ mathbf {v}} ,
or, in term of components,
X_ {ab} = \ frac {1} {2} \ left (u_a v_b - v_a u_b \ right).
In the case of a simple form, the quantity X_ {ab} X^ {ab} is known as of time kind, of kind spaces or of light kind according to its value (respectively positive, negative and null if the Convention of sign of metric the is (- +++) and respectively negative, positive and null in the case of convention reverses (+---)).

Dual Bivector

In a space with four dimensions on which a Métrique riemannienne is defined, one can use the Tenseur of Levi-Civita to associate a bivector {\ mathbf {X}} with his dual bivector, noted \ tilde {\ mathbf {X}} , according to the formula

\ tilde X_ {ab} = \ frac {1} {2} \ epsilon_ {abcd} X^ {Cd} .
The dual one of a dual bivector corresponds to the sign close with the vector of origin:
\ left (\ tilde X_ {ab} \ right) {} \ tilde {} = - X_ {ab} .

Two bivecteurs X and Y satisfy using their duaux bivecteurs some cleanlinesses like

X_ {ab} \ tilde Y^ {ab} = \ tilde X_ {ab} Y^ {ab} ,
X_ {ac} Y_b {} ^c - \ tilde X_ {bc} \ Y_a^c tilde = \ frac {1} {2} g_ {ab} X_ {Cd} Y^ {Cd}

Bivector autodual

A complex bivector is known as autodual if it satisfies

\ tilde {\ mathbf {X}} = - I {\ mathbf {X}} .
Any bivector X can see himself associating a bivector autodual X
  • by combining it with its dual, according to the formula
    {\ mathbf {X}} ^* = {\ mathbf {X}} + I \ tilde {\ mathbf {X}} .

    Three-dimensional vector complexes associated with a bivector

    The physical significance of a bivector autodual appears by noticing that the six components independent of a real bivector can be transformed into a complex three-dimensional vector. It is enough for that to choose a vector of time kind, U and to define the quantity X by

    X_a = X^*_ {ab} u^b.
    A simple calculation immediately makes it possible to reconstitute the original bivector, by
    X_ {ab} ^* = 2 u_ {X_ {B} + I \ epsilon_ {abcd} u^c X^d = 2 \ left (u_ {X_ {B} \ right) ^*.

    An example: the electromagnetic tensor

    The electromagnetic Tenseur is an antisymmetric tensor of order 2. It is thus a bivector. The vector X calculated by the method above gives

    X^j = E^j - I C B^j.

    Reference

    • , pages 47 to 49.

    Note

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