Newtonian Mechanics in 4+1 Dimensions

In the higher-dimensional universe of the world of Ord, most of Newtonian mechanics generalizes to 4 spatial dimensions (and 5 total dimensions when you include time--hence the 4+1 in the title) just fine. 

 is still true when F and a are 4-component vectors instead of 3-component vectors, and so is 

 for linear momentum. Squaring vectors still produces scalar quantities, so

is still true, and 

still works just fine. Rotation occurs in a plane with some fixed center for all numbers of dimensions, so the formula for moment of inertia in a given plane, 

is also still valid.

But when it comes to angular momentum and torque, we've got a problem. 

and 

contain cross products, which only exist in exactly 3 dimensions. Usually, these are explained as creating a unique vector that is perpendicular to both of the inputs; but in less than 3D, there is no such vector, and in 4 dimensions or more, there is a whole plane (or more) of possible vectors. In reality, angular momentum and torque are not vectors--they are bivectors, oriented areas rather than oriented lines, which exist in any space of more than 1 dimension. It just happens that planes and lines are dual in 3D--for every plane, there is a normal vector, and for every vector there is a perpendicular plane, so we can explain the cross product as producing the normal vector to the plane of the bivector.

In 4D, you can't implicitly convert a bivector into its dual vector and back, so we have to deal with the bivectors directly. Bivectors are formed from the outer product or wedge product (denoted ∧) of two vectors, or the sum of two other bivectors. Thus, we can write the angular formulas for a point particle in any number of dimensions as 

and 

And those a good for orbital momentum and torque about an external point on an arbitrary body as well. To get spin, we need a sum, or an integral, all of the components of an extended body. That means we need to be able to sum bivectors! That's easy to do in 2D and 3D; in 2D, bivectors can be represented by a single number (their magnitude and sign), and we know how to add numbers; in 3D, as we saw, bivectors can be uniquely identified with their normal vectors, and we can add normal vectors. In either case, you always get a simple bivector (also called a blade) as a result; i.e., for any bivector in 2D and 3D space, you can find a pair of vectors whose wedge product is that bivector. But in 4 dimensions and above, that is no longer true. This is because, once you identify a plane in 4+ dimensions, there are still 2 or more dimensions left over in which you can specify a second completely perpendicular plane which intersects the first at exactly one point (or zero or one points in 5+ dimensions), and there is no set of two vectors that can span multiple planes. This also means that there can be two simultaneous independent rotations, with unrelated angular velocities, and the formulas for angular momentum and torque must be able to account for arbitrary complex bivector values. You could, of course, just represent sums of bivectors as... sums of bivectors, with plus signs in between them. But that's really inconvenient, and if you can't simplify sums of bivectors, then those formulas aren't very useful for predicting how an object will spin after a torgue is applied to it!

Fortunately, even though the contributions of multiple not-necessarily-perpendicular and not-necessarily-parallel simple bivectors will not always simplify down to a single outer product, it turns out that in 4 dimensions, any bivector can be decomposed into the sum of two orthogonal simple bivectors--and most of the time, the result is unique. Unlike vector / bivector addition in 3D, this is not a simple process of just adding together the corresponding components, but there are fixed formulas for computing the two orthogonal components of any sum of two bivectors. They are complicated and gross, but at least they exist! So, we can, in fact, do physics!

The result of bivector addition does not have a unique decomposition exactly when the two perpendicular rotations have exactly the same magnitude. This is known as isoclinic rotation. With isoclinic rotations, you can choose any pair of orthogonal planes you like as a decomposition. Once you pick a coordinate system to use, there are exactly 4 isoclinic rotations, depending on the signs of each of the two component bivectors. In isoclinic rotation, every point on the surface of a hypersphere follows an identical path, and there is no equivalent of an equator or pole. Meanwhile, simple rotation results in a circular equator, but also a circular pole--i.e., a circle of points that remain stationary as the body spins. That circle is also the equator for the second plane of rotation, so the ideas of "equator" and "pole" become effectively interchangeable for any object in non-isoclinic complex rotation. One plane's equator is the other plane's pole, and vice-versa.

Looking ahead a little bit to quantum mechanics, particle spin in 4D is still quantized, still inherent, still divides particles into fermions and bosons--but has two components, just like the angular momentum of a macroscopic 4D object. Whether or not a particle is a boson or a fermion depends on the sum of the magnitudes of the two components. If the sum is half-integer, the particle is a fermion. If the sum is integer, then its a boson. Thus, bosons can (but need not necessarily) have isoclinic spins, and the weird feature of quantum mnechanics that the spin is always aligned with the axis you measure it in would not be so weird, because that's the case for isoclinic rotation of macroscopic objects, too! Fermions, on the other hand, can never have isoclinic spins! Because if one component has a half-integer magnitude, the other must not. In both cases, however, there end up being four possible spin states for all particles with complex spins, allowing fermions to pack more tightly than they do in our universe; 2 spin states (as in our universe) for particles with simple spins; and of course only a single spin state for particles with zero spin.