Oh no, I’ve completely forgot about bivectors from the last time I read one of these articles. iirc, the main takeaway was that they think bivectors are easier than curl (because they don’t involve a right-hand rule), whereas I think they’re about the same.
using oriented tiles to illustrate torque-exerting fields, rather than a vector along the axis of rotation, is immediately more obvious, ill give them that. oh! i have notes from the last one i read!
ah okay yeah i remember now. bivectors: - wedge products don’t have a cyclic symmetry to keep track of, so they reflect like normal - are readily generalisable to higher-dimensions, unlike the cross-product - are difficult to add together (place two edges of the bivectors along the same edge, then complete the shape, and the new face is the new bivector – or just add up the associated cross products) - relate to pairs of vectors by that pair being such that the cross product is at a normal to the bivector and has the same magnitude
there are probably benefits to a clear geometric difference between electric and magnetic fields, since they are so different. why not go the rest of the way and treat the electric field as the 1-form it mathematically-formally is and the magnetic field as the 2-form? well, that’s much harder to explain! (ironic heh)
Bivectors and Current Loops
Okay, let’s start with the physics.
A bivector whose attitude is in the plane of a current loop has the same orientation as the current loop and magnitude equal to the field at that point. This is true also for the bivectors associated with each of the 4 straight segments.
At points outside the plane, this latter rule guides the way. At that point, attitude the bivector so that one edge is parallel to the current segment, and the other side is along the ray to it. Then we can add up all the bivectors (oh no) to get the total field!
ok yeah, the coordinates of a bivector are also oriented planes. bxy is the projection onto the xy plane. the sign of the projection depends on if the orientation of the bivector matches the rotation from x to y. hence the compont labels are antisymmetric: bxy = − byx and so all the bxx ones are zero. huh! even though these are not lengths, they do combine pythagoreanly to give the bivector’s magnitude! |b| = bxy2 + byz2 + bzx2 and these components can be written in an antisymetric matrix (a second rank tensor?)
the traditional magnetic field vector is normal to the bivector. its components are given by the somewhat unobvious relation (until you look at the diagram): Bx = byz, By = bzx and Bz = bxy. Snazzy! You know, I do like this formalism.
since the magnetic field is not a true (polar) vector but a pseudo (axial) vector, if you reflect it in a plane, you get weird minus signs in the terms not involving that plane (for simple planes). for bivectors, because everything is plane based, the minus signs are where you’d expect.
Bivectors and moving charges
The bivector Biot-Savart law is: $b = \frac{\mu_0}{4\pi} \frac{q}{r^2} (\vec{v} \wedge \hat{r})$ so that’s the wedge product. ie, the associated bivector lies in the plane spanned by the velocity and displacement vectors placed tip-to-tail and the arrow directions define the bivectors orientation. this matches the intuition established previously the bivector magnetic field lies in the plane formed by v and r, and the orientation of the near edge matching the flow of positive charge. the magnitude is simply the area of the parallelogram, like cross products.
then they do a bunch of calculus and algebra to derive the current loop result. it’s a sunday, im not doing that.
Magnetic forces and the matrix product
ie, how do bivectors cause charges to move? presumably normally through their plane, right?
wha- the bivector dot product used to calculate forces is order dependent. well then this whole thing just boils down to do you care about the Lorentz force or about fields. apparently this dot product formally corresponds to “tensor index contraction” which sounds made up, but then again, so are all words.
hmm it seems at this point we pick up a bunch more arbitrary formulas and lose a lot of the elegance of the approach.
relativistic magnetism
in relativistic electromagnetism, electric and magnetic fields are treated as complicated tensors. luckily, bivectors already have a matrix representation, and it’s quite straightforward to extend it into a 4x4 matrix with a time-row and time-column that matches. then, comparing this to the typical 4x4 EM field tensor of relativity, we can see that the additional time rows and columns correspond neatly to the parts of the electric field!
these get mixed up in Lorentz boosts, but not ordinary rotations. and this is ensured by the symmetries of the matrix and so on. nifty!!
other stuff generalises, but now we introduce metrics and other bits and bobs. we use a flat metric to define the inner product – that’s interesting.
ok that’s it! not reading the appendices soz.
what do i think?
- this is pretty cool! glad i know about it
- don’t think its particularly useful or advantageous…