After a month of silence, I’ve returned to sharing my thoughts with two new posts: “June Circles” and “From Spheres and Circles to Spacetime — Evolving Coordinates.”
But to be honest, it didn’t go well. That month away seems to have cost
me my coherence. A few loyal readers were glad to see me back—but less
enthusiastic about the content. What I wrote came out muddled and
confusing. So, it’s time to begin again—from scratch.
Interestingly, in theoretical chemistry, the phrase ab initio
("from the beginning") often appears in paper titles. I rarely see it
in physics or mathematics, but here, it feels just right. This will be
my third attempt, and I’m starting ab initio, from chaos toward order. As the saying goes, “On the third knock, the door opens.” Let’s see if it does.
My plan is to discuss the infinity.
The infinity point of space and time. It all started with projective
geometry of an ordinary two-dimensional plane. Parallel lines should
meet at some point "at infinity". Different bundles of parallel lines
meet at infinity at different points. Thus projective geometry added
"the line at infinity". If we replace 2D plane by 3D space, we need to
add a point at infinity for each bundle of parallel lines in ℝ3. This is widely used in computer aided graphics. Here is an excerpt from the paper "Beyond the Celestial Sphere: Oriented Projective Geometry and Computer Graphic" by Kevin G. Kirby, Mathematics Magazine, Vol. 75, No. 5 (Dec., 2002), pp. 351-366.
To infinity ...
You are driving a virtual car in a computer game. Look out through the windshield. The trees that line the highway are rushing past you. They are nicely displayed in perspective: they begin
far away as dots, but they grow taller as you race toward them. Next,
look ahead at the horizon. You see the sun. Unlike the trees, the sun
never gets closer to you. Still, it is certainly subject to some
transformations: you turn your car left, and the sun veers right.
The trees and the sun need to be represented inside the program somehow.
Ultimately, this depends on attaching parts of them to points in a
virtual space. One might think at first that a useful way to represent
any point would be to use Cartesian coordinates, a triple (x, y, z) of
real numbers. Taking your car to be located at the origin, a tree might
be centered at, say, (-20.42, 10.63, -94.37). Where is the sun? It is
very far away, perhaps at (9.34⨉109, 2.71⨉109, -1.23⨉1011).
But there is something strange about these large numbers. It seems
pointless to waste time (and numerical precision) decrementing such big
numbers by a few tens as our car drives on toward the sunset. We would
like to simplify things by somehow locating the sun at infinity.
One
uniform way to represent both ordinary points and points at infinity is
to use four numbers instead of three. Here's how it works. Take the
point (2, 3, 4). Instead of representing it as a column vector in ℝ3, we tack a 1 on the end and represent it as a column vector in ℝ4 : x = [2 3 4 1]T.
This representation is not meant to be unique: we can multiply this
column vector by any positive number and we will say it represents the
same point.(...)
 |
| You are driving a virtual car in a computer game |
That may be good for traveling in
space. But we want to travel in space and in time. For lines in space-time
we sometimes use the term world line. There are three kinds of
world lines: we can travel with a speed that is slower than the speed of
light, faster than light, or with exactly the speed of light. There
will be, perhaps, three kinds of infinity points. There are two known
mathematical ways to achieve that, using a trick similar to that used in
projective geometry. The first way, the standard one, is to start with
space-time and add two extra dimensions, that is to add a hyperbolic
plane with signature (+1,-1), and then to study the projective null cone there.
This is called Möbius geometry. The second way, the way I am following,
is the Lie sphere geometry. We start with 3D space alone, and time
emerges automatically, related to the radius of a sphere. We add three
extra dimensions to the three dimensions of space, we add extra
dimensions with signature (+1,-1,-1), and take the projective null cone
there. The end result is mostly the same - we end up with six
dimensions and signature (+1,+1,+1,-1,+1,-1). But, in the Lie sphere
geometry approach, we obtain a reacher structure. I like this second approach
better, as it fits the possibility of taking into account the possible
'ether', or 'quantum vacuum', with a preferred reference frame.
In the previous two posts I was
oscillating between circles and spheres. Finally I decided to take
spheres, and restrict to circles only when it will be more convenient
for graphical illustrations. So, let us start, ab initio.
There are two ways of playing with
geometrical constructs. The first way goes back to Descartes - we use
coordinates. The second way goes back to Euclid - it is coordinate-free.
Nowadays, in practice, we often first use coordinates, and only after
the result is obtained, we work on expressing our result in a
coordinate-free way. I will follow this path. We will be using
coordinates at first, and only then search for a way to understand what
have been done in a more elegant and, perhaps, deeper way.
Notation.
Let V be a 6-dimensional real vector space with the quadratic form Q of signature (4,2).
Note: In previous
post I have used Q to denote the Lie quadric. But since now we start ab
initio, therefore the symbol Q is being cleared from its previous
meaning. We will use X,Y,... to denote vectors of V. The scalar product in V will be written as X·Y. Thus
Q(X)=X·X. (1)
We denote by N the null cone in V:
N = {X∈V: Q(X) = 0}. (2)
In V⟍{0} (V with the removed origin) we introduce the equivalence relation
X∼Y if and only if Y = λX, λ>0. (3)
We denote by PV and PN the sets of equivalence classes of vectors in V⟍{0} and in N⟍{0}:
PV = V⟍{0} / ℝ×>0, (4)
PN = N⟍{0} / ℝ×>0, (5)
where ℝ×>0 is the multiplicative group of all strictly positive real numbers.
We denote by π the natural projection from V to PV. It maps every non-zero X in V to its equivalence class [X].
V is 6-dimensional, N⟍{0} is
5-dimensional, and so PN is 4-dimensional. That is the central object of
our studies. As it will be seen in the future, it is the (doubly)
compactified Minkowski space, diffeomorphic to the product S3⨉S1. It was denoted Q+ in previous posts.
We can use π to equip PN with a natural topology and with a differentiable structure. We will do it later.
The flat Minkowski space-time M will be
identified as a particular open, dense set in PN. First we will do it
using coordinates. For this we will use orthonormal bases for M and for
V. Using an orthonormal basis in M we identify M with ℝ3,1. Thus the scalar product in M can be written as
x·y = xTηy, (6)
where η is the diagonal matrix η=diag(1,1,1,-1). The quadratic form q of M is then
q(x) = x·x = (x1)2 + (x2)2 + (x3)2 - (x4)2. (7)
Note: sometimes we may use the letters x1=x, x2=y, x3=z, x4=t. We will also use the notation x = (x,t), where x is a vector in ℝ3, and t is a real number (time). In this case x·x = x2 - t2.
A basis EA in V, (A=1,2,...,6), is called orthonormal if EA·EB =GAB, where G is the diagonal matrix
G = diag(1, 1, 1, -1, 1, -1). (8)
We denote by XA the coordinate of X with respect to such a basis: X = XA EA.
The embedding
We will now define the embedding of M into PN. It is defined by the following map from M to V
X(x) = ( x, ½(1 - q(x, t)), -½(1 + q(x, t)) ). (9)
Then the embedding, which will be denoted by τ is defined by
τ(x) = [X(x)]. (10)
Exercise 1. Prove that if τ(x) = τ(x'), then x=x'.
We notice that if X = X(x), q = q(x,t), then X5 - X6 = 1, X5 + X6 = -q. We define
X0 ≐ X5 - X6. (11)
X∞ ≐ X5 + X6, (2)
Thus the embedding formula looks simpler if we use, instead of the two basis vectors E5 and E6, vectors E0 and E∞ defined as
E0 = ½(E5 - E6), (13)
E∞ = ½(E5 + E6). (14)
Using these coordinates the embedding formula takes the form:
X(x) = xμEμ - q(x)E∞ + 1 E0. (15)
The vectors E0 and E∞ are now in N (Why?), therefore they define two points p0 = [E0], and p∞ = [E∞] in PN. We also notice that
E0·E∞ = 1/2. (16)
The point p0 is the image τ(0) of the origin x = 0 of M. The point p∞ does not correspond to any point in M, it is not in the set τ(M). It is one of the points of the infinity set, which we define as
M∞ = {[X]: X0 = 0} = {[X]: X5 = X6}. (17)
In the next post we see that PN is a disjoint union of three sets
PN = M+ ∪ M- ∪ M∞, (18)
where
M+ = {[τ(x)]: x ∈ M}, (19)
M- = {[-τ(x)]: x ∈ M}. (20)
Thus PN, the doubly compactified Minkowski space, consists of two copies of M, and of the infinity set M∞. We will discuss in details the structure of M∞, and also the intuitive meaning of these 'points at infinity'.
In all this I am borrowing ideas from
what is called 'oriented projective geometry', as described in the
computer graphics review paper by Kirby, mentioned at the beginning. In
oriented projective geometry one cares about the direction of the line.
For space-time that means that we want to distinguish, in particular,
between the future and the past. This is not usually done in the
standard conformal compactification of the Minkowski space. I am not so
sure about the necessity of distinguishing between left and right, but
if it gives better algorithms for the computer graphics, perhaps, for
efficiency reason, it is also exploited in the Nature.
