Four dimensions

In four dimensions the basis elements for the space Λ²ℝ⁴of bivectors are (e12, e13, e14, e23, e24, e34), so a general bivector is of the form

A = a12e12+ a₁₃e13+ a₁₄e14+ a₂₃e23+ a₂₄e24+ a₃₄e34.

31.6.1 Orthogonality

In four dimensions bivectors are orthogonal to bivectors. That is, the Hodge dual of a bivector is a bivector, and the space Λ²ℝ⁴is dual to itself in Cℓ4(ℝ). Normal vectors are not unique, instead every plane is orthogonal to all the vectors in its Hodge dual space. This can be used to partition the bivectors into two 'halves’, for example into two sets of three unit bivectors each. There are only four distinct ways to do this, and whenever it’s done one vector is in only one of the two halves, for example (e12, e13, e14) and (e23, e24, e34).

31.6.2 Simple bivectors in 4D

In four dimensions bivectors are generated by the exterior product of vectors in ℝ⁴, but with one important difference from ℝ³and ℝ². In four dimensions not all bivectors are simple. There are bivectors such as e12+ e34that cannot be generated by the external product of two vectors. This also means they do not have a real, that is scalar, square.

In this case

(e12+e34)²=e12e12+e12e34+e34e12+e34e34=−2 + 2e1234.

The element e1234is the pseudoscalar in Cℓ4, distinct from the scalar, so the square is non-scalar.

All bivectors in four dimensions can be generated using at most two exterior products and four vectors. The above bivector can be written as

e12+e34=e1∧ e2+e3∧ e4.

Similarly, every bivector can be written as the sum of two simple bivectors. It is useful to choose two orthogonal bivectors for this, and this is always possible to do. Moreover, for a generic bivector the choice of simple bivectors is unique, that is, there is only one way to decompose into orthogonal bivectors; the only exception is when the two orthogonal bivectors have equal magnitudes (as in the above example): in this case the decomposition is not unique.^[1]

The decomposition is always unique in the case of simple bivectors, with the added bonus that one of the orthogonal parts is zero.

31.6.3 Rotations in R

As in three dimensions bivectors in four dimension generate rotations through the exponential map, and all rotations can be generated this way. As in three dimensions if B is a bivector then the rotor R is e^B/2and rotations are generated in the same way:

v^′= RvR⁻¹.

The rotations generated are more complex though. They can be categorised as follows:

simplerotations are those that fix a plane in 4D, and rotate by an angle “about” this plane.

doublerotations have only one fixed point, the origin, and rotate through two angles about two orthogonal planes. In general the angles are different and the planes are uniquely specified

isoclinic rotations are double rotations where the angles of rotation are equal. In this case the planes about which the rotation is taking place are not unique.

These are generated by bivectors in a straightforward way. Simple rotations are generated by simple bivectors, with the fixed plane the dual or orthogonal to the plane of the bivector. The rotation can be said to take place about that plane, in the plane of the bivector. All other bivectors generate double rotations, with the two angles of the rotation equalling the magnitudes of the two simple bivectors the non-simple bivector is composed of. Isoclinic rotations arise when these magnitudes are equal, in which case the decomposition into two simple bivectors is not unique.^[21]

A 3D projection of antesseractperforming anisoclinic rotation.

Bivectors in general do not commute, but one exception is orthogonal bivectors and exponents of them. So if the bivector B = B1+ B2, where B1and B2are orthogonal simple bivectors, is used to generate a rotation it decomposes into two simple rotations that commute as follows:

R = e^B1+B22 = e^B12 e^B22 = e^B22 e^B12

It is always possible to do this as all bivectors can be expressed as sums of orthogonal bivectors.

31.6.4 Spacetime rotations

Spacetimeis a mathematical model for our universe used in special relativity. It consists of threespacedimensions and onetimedimension combined into a single four-dimensional space. It is naturally described using geometric algebra and bivectors, with theEuclidean metricreplaced by aMinkowski metric. That is the algebra is identical to that of Euclidean space, except thesignatureis changed, so

ei2= {

1, i = 1, 2, 3

−1, i = 4

(Note the order and indices above are not universal – here e4is the time-like dimension). The geometric algebra is Cℓ₃,₁(ℝ), and the subspace of bivectors is Λ²ℝ^3,1.

The simple bivectors are of two types. The simple bivectors e23, e31 and e12 have negative squares and span the bivectors of the three-dimensional subspace corresponding to Euclidean space, ℝ³. These bivectors generate ordinary rotations in ℝ³.

The simple bivectors e14, e24and e34have positive squares and as planes span a space dimension and the time dimen-sion. These also generate rotations through the exponential map, but instead of trigonometric functions, hyperbolic functions are needed, which generates a rotor as follows:

where Ω is the bivector (e14, etc), identified via the metric with an antisymmetric linear transformation of ℝ^3,1. These areLorentz boosts, expressed in a particularly compact way, using the same kind of algebra as in ℝ³and ℝ⁴. In general all spacetime rotations are generated from bivectors through the exponential map, that is, a general rotor generated by bivector A is of the form

R = e^A2.

The set of all rotations in spacetime form theLorentz group, and from them most of the consequences of special relativity can be deduced. More generally this show how transformations in Euclidean space and spacetime can all be described using the same kind of algebra.

31.6.5 Maxwell’s equations

(Note: in this section traditional 3-vectors are indicated by lines over the symbols and spacetime vector and bivectors by bold symbols, with the vectors J and A exceptionally in uppercase)

Maxwell’s equationsare used in physics to describe the relationship betweenelectricandmagneticfields. Normally given as four differential equations they have a particularly compact form when the fields are expressed as a spacetime bivector from Λ²ℝ^3,1. If the electric and magnetic fields in ℝ³are E and B then the electromagnetic bivector is

F = 1

cEe4+ Be123,

where e4is again the basis vector for the time-like dimension and c is thespeed of light. The product Be123yields the bivector that is Hodge dual to B in three dimensions, asdiscussed above, while Ee4as a product of orthogonal vectors is also bivector valued. As a whole it is theelectromagnetic tensorexpressed more compactly as a bivector, and is used as follows. First it is related to the4-currentJ, a vector quantity given by

J = j + cρe4,

where j iscurrent densityand ρ ischarge density. They are related by a differential operator ∂, which is

∂ =∇ − e4

1 c

∂

∂t.

The operator ∇ is adifferential operatorin geometric algebra, acting on the space dimensions and given by ∇M =

∇·M + ∇∧M. When applied to vectors ∇·M is thedivergenceand ∇∧M is thecurlbut with a bivector rather than vector result, that is dual in three dimensions to the curl. For general quantity M they act as grade lowering and raising differential operators. In particular if M is a scalar then this operator is just thegradient, and it can be thought of as a geometric algebraicdeloperator.

Together these can be used to give a particularly compact form for Maxwell’s equations in a vacuum:

∂F = J.

This when decomposed according to geometric algebra, using geometric products which have both grade raising and grade lowering effects, is equivalent to Maxwell’s four equations. This is the form in a vacuum, but the general form is only a little more complex. It is also related to theelectromagnetic four-potential, a vector A given by

A = A +1 cVe4,

where A is the vector magnetic potential and V is the electric potential. It is related to the electromagnetic bivector as follows

∂A =−F,

using the same differential operator ∂.^[22]