In mathematics, a bivector is the vector part of a biquaternion. For biquaternion, w is called the biscalar and is its bivector part. The coordinates w, x, y, z are complex numbers with imaginary unit h:
x=x1+hx2, y=y1+hy2, z=z1+hz2, h2=-1=i2=j2=k2.
(x1i+y1j+z1k)+h(x2i+y2j+z2k)
r1=x1i+y1j+z1k
r2=x2i+y2j+z2k
q=xi+yj+zk=r1+hr2.
The Lie algebra of the Lorentz group is expressed by bivectors. In particular, if r1 and r2 are right versors so that
| 2 | |
| r | |
| 1 |
=-1=
| 2 | |
| r | |
| 2 |
Now, and the biquaternion curve is a unit hyperbola in the plane The spacetime transformations in the Lorentz group that lead to FitzGerald contractions and time dilation depend on a hyperbolic angle parameter. In the words of Ronald Shaw, "Bivectors are logarithms of Lorentz transformations."[2]
The commutator product of this Lie algebra is just twice the cross product on R3, for instance,, which is twice .As Shaw wrote in 1970:
Now it is well known that the Lie algebra of the homogeneous Lorentz group can be considered to be that of bivectors under commutation. [...] The Lie algebra of bivectors is essentially that of complex 3-vectors, with the Lie product being defined to be the familiar cross product in (complex) 3-dimensional space.[3]
William Rowan Hamilton coined both the terms vector and bivector. The first term was named with quaternions, and the second about a decade later, as in Lectures on Quaternions (1853).[1] The popular text Vector Analysis (1901) used the term.[4]
Given a bivector, the ellipse for which r1 and r2 are a pair of conjugate semi-diameters is called the directional ellipse of the bivector r.[4]
\begin{pmatrix}hv&w+hx\\-w+hx&-hv\end{pmatrix}
Ludwik Silberstein studied a complexified electromagnetic field, where there are three components, each a complex number, known as the Riemann–Silberstein vector.[5] [6]
"Bivectors [...] help describe elliptically polarized homogeneous and inhomogeneous plane waves – one vector for direction of propagation, one for amplitude."[7]