From Spheres and Circles to Spacetime — Evolving Coordinates

Here we continue from the June circles.  There will be changes in the notation. Changes always create temporary chaos. We have to learn how to survive when our environment undergoes change. Survival of the fittest — that rule seems to govern all evolution in the universe. Survival of the fittest and adaptation. A cactus that retains water well survives in arid deserts. A person who learns from failure may “survive” mentally and socially better than better than someone who rigidly resists change. Will my readers survive after this post?

But why the changes? Here is the reason. The Lie sphere geometry has been discovered while studying the geometrical properties of spheres. It was pure mathematics, but, surprisingly, it has found application in relativistic physics. The idea goes as follows. Consider the simple equation of a circle on the plane:

x² + y² = r².

Now rewrite it as

x² + y² - r²=0.

Now substitute r=ct:

x² + y² - c²t²=0.

This is the equation of the light cone of special relativity. The radius of the circle expands with the velocity of light. The value of this radius can serve as a time coordinate of an event. If a circular wave was created at a point x and time t, when it reaches the origin x=0, its radius has the value r=ct. Therefore it should not be surprising that studying circles we study spacetime of special relativity. This is will how it will go.

We will denote the coordinates of the 2+1 dimensional Minkowski space by x,t, and coordinates in the 5-dimensional space R^3,2 of Lie spheres by ξ = (ξ⁰,ξ¹,ξ²,ξ⁴,ξ⁵) with the quadratic form

ξ·ξ = (ξ⁰)² + (ξ¹)²  + (ξ²)²  - (ξ⁴)² - (ξ⁵)².                 (0)

With

q(x,t) = x²-t²,             (1)

the formula (12) of the previous post embeds the Minkowski space in the projective quadric Q⁺ would read:

ξ(x,t) =  ½(1 − q(x, t)) e₀ + x  + t e₄ + ½(1 + q(x, t)) e₅,                 (2)

Note: In what follows I  will use the term 'projective' to mean 'semi-projective', where instead of lines we consider half-lines in R^3,2 (λ>0, instead of λ≠0 in the equivalence relation).

Environmental change — the fittest survive.

To obtain (2) we used a particular convention for representing oriented spheres (circles in our case) as points of a space with 1+2 more dimensions. Let us compare our formula with the formula from the paper "On organizing principles of discrete
differential geometry. Geometry of spheres" by A.I. Bobenko and Yu. B. Souris, Russian Math. Surveys 62:1 1–43. Here is the corresponding extract from this paper:

To relate it to our Eq. (2) we replace, in the formula (4) of the paper, c by x, r by t, and substitute (3) to obtain

ξ= - ½(1 − q(x, t))e_N+1 + x + te_N+3 + ½(1 + q(x, t))e_N+2.

We see e_N+3 corresponds to our e₄, e_N+2 corresponds to our e₅, and e_N+2 corresponds to our -e₀.  It will be more convenient for us in the future to follow the convention used in this paper, particularly regarding the sign of e₀. On the other hand, it will also be convenient to number the indices as 1,2,3,4,5. Thus, in what follows, we will use the basis E₁=e₁, E₂=e₂, E₃=e₄, E₄=e₅, E₅=-e₀. We will write X=(X^A), A=1,...,5):

X = X^AE_A, 

with

X·X = (X¹)² + (X²)² -(X³)²+(X⁴)²- (X⁵)².

For this reason in the future we will replace our embedding formula (2) with the following one:

X(x,t) =  x  + t E₃ + ½(1 + q(x, t)) E₄ - ½(1 − q(x, t)) E₅ ,                 (2a)

In our coordinates it reads:
X¹(x, t)  = x¹, 
X²(x, t)  = x²,                 (3)
X³(x, t) = t,
X⁴(x, t) = ½(1 + q(x, t)),
X⁵(x, t) = -½(1 − q(x, t)) ,         

We notice that

X⁴(x, t) - X⁵(x, t) = 1.                (4)

However, this last formula is not invariant under selecting an element from the equivalence class defining the projective space. The invariant formula is, instead:

X⁴(x, t) - X⁵(x, t) > 0.                (5)

Notice that we have

X(x,t)·X(x,t) = 0,                (6)

and this equation defines the projective null cone in the projective space — our Q⁺ universe. It then follows from the definitions that Q⁺ splits into a disjoint union of three sets:

Q⁺ = M₊ ∪ M_- ∪ M_∞,                (7)

where

M₊ = {[X]: X⁰+X⁵ >0},            (8)

M_- = {[X]: X⁰+X⁵ <0},            (9)

M_∞  = {[X]: X⁰+X⁵ = 0}.            (10)

The sets M₊ and M_- are  open in Q⁺, the set M_∞  is closed (Why?). The map (x,t) ⟼ [X(x,t)] is bijective from R⁴ to M₊. Similarly the map  (x,t) ⟼ [-X(x,t)] is bijective from R⁴ to M_-. The map [X] ⟼ [-X] is a bijection from M₊ onto M_-.

Exercise 1. Prove the above statements.

Exercise 2. Prove that M₊ ∩ M_- = ∅.

June circles

After a break - return. Back to April 4, to the post "Lie Sphere Geometry Part 3: oriented circles". We will take off from there. We like circles. As a young boy I was crazy with hoop rolling.

Much like on the picture above I was running with the bicycle rim hoop, as fast as I could,  along the streets of my town, attempting, not always successfully, not to run into pedestrians. So, I do like circles. To have more fun we consider circles on the sphere S²:

S² = {x∈R³:  x² = 1}                (1)

For every m∈S², and for every r∈[0,π] the unoriented circle S_r(m)is defined as

S_r(m) = {x∈S²:  x·m = cos r}.                (2)

We orient the circle by defining its normal unit vector:

n(x) = (m - cos(r) x)/sin(r).                (3)

As in Lie Sphere Geometry Part 3: oriented circles and Seneca, we find that (m,r) and (-m, r + π mod 2π) define the same oriented circle - which bothers me a little. We will get rid of this identification soon. While it is the fact of life that we can read only one side of a page at a time, the book pages are usually printed on both sides. The same, that is my crazy idea, is with our universe. It has two sides, and with sufficient knowledge and technology, we should be able to read what is on the other side. If you think this is inadmissible fantasy, you can always consider it as a pure mathematics, much like imaginary numbers. Do they "exist"? Does not matter. They are, at least sometimes,  useful.

The whole point of Lie Sphere Geometry is to represent every oriented circle as a point of some "space" of more dimensions. But, as physicists, we think of a circle dynamically. We think of it as an expanding or contracting "wave", like those created by a piece of rock when it falls on the surface of a lake. So, each nonpoint circle either was or will be just a point. It expands or contracts. How fast? Say, with the velocity of light. So the parameter r of the circle is related to "time". That we should keep in our minds. At least this is what I am keeping in my mind. So it should not be a great surprise if following this way we will rediscover some aspects of special relativity, with Lorentz transformations acting in the space of circles. Except that, since I have decided to play with circles rather than spheres, our "space-time" will have only  2+1 dimensions, instead of the usual 3+1. It will be easier to imagine graphically. That is why this summer we will play with circles rather than spheres. But let us proceed step by step.

We have introduced the Lie Sphere Geometry in Lie Sphere Geometry Part 5: Lie Quadric with the sentence:

"According to Sophus Lie the Universe is 6-dimensional 6 = 3 + 1 + 2."

But now, for the present purposes,  we reduce our ambitions to 5 = 2+1+2, and we adapt accordingly:

We take five-dimensional real vector space R⁵ with coordinates x⁰, x¹, x², x⁴, x⁵. There we introduce the indefinite scalar product

(x,y) = x⁰y⁰ + x¹y¹ + x²y² - x⁴y⁴ - x⁴y⁵.                (4)

We denote by R^3,2 the resulting inner-product space. We endow R^3,2 with orientation and denote by e₀,e₁,e₂,e₄,e₅ the corresponding orthonormal basis in R^3,2. Thus 

(e₀,e₀)=(e₁,e₁,)=(e₂,e₂)=1, (e₄,e₄)=(e₅,e₅)= -1.                (5)

Now we go to projective space by introducing in R^3,2 the equivalence relation

x∼y if and only if there exists  a real λ>0 such that y=λx.

The equivalence classes [x] form the projective space P(R^3,2). It is a compact 4-dimensional manifold.

Definition. The Lie quadric Q⁺⊂P(R^3,2) is the smooth quadric hypersurface

Q⁺ = {[u]∈P(R^3,2): (u,u) = 0}.                (6)

We notice that Q is well defined: if λ>0 then (u,u)=0 if and only if (λu,λu) = 0. The condition defining Q⁺ takes away one dimension from the four dimensions of P(R^3,2). Thus Q⁺ is a three dimensional and compact.

Comparing to Part 5, you can think that we have just put there x³=0. We are simply freezing one space dimensions. We have also introduced Q⁺ instead of the standard Q right away.  i.e. we used λ>0  instead of λ≠0 in the definition of the equivalence relation .

The following Proposition from Part 6 remains essentially unchanged except that now spheres have been replaced by circles, and m is now a unit vector in R³:

Proposition 1. The following formula defines an explicit isomorphism between the space of oriented circles  and Q⁺:

(m,r) ⟼ [m+cos(r)e₄+sin(r)e₅],               (7)

0 ≤ r < 2π.                    (8)

So far so good, but if we want to relate our geometrical model to physics, and to special relativity in particular, we need to descend from thr sphere S² to the plane R². Special relativity assumes that space is flat. Which, on one hand, is good, because it simplifies the mathematics, but on the other hand it may be bad, because this assumption seems to be questionable on a very small and very large scales. Anyway we have all needed tools ready - we use the stereographic projection. Which takes us first to Part 9, where we have defined oriented spheres in R³. Now we descend to R²:

Definition 1. The oriented circle with center p in R² and signed radius ρ, 0≠ρ∈R, is

S_ρ(p) = {y∈R²: (y-p)²=ρ²}                (9)

with unit normal vector field

n'(y) = (p-y)/ρ.                (10)

We now use the inverse stereographic projection to map a circle in R² to a circle in S², and then use (7) to represent it as a point in Q⁺. We have already did it for spheres in Part 13. We adapt the corresponding Proposition from there:

Proposition 2. With p = p¹e₁ + p²e₂ ∈ R² , ρ ∈ R, and

q(p, ρ) = p² − ρ²,                             (11)

the correspondence, denoted by τ_s between oriented circles S_ρ(p) in R² and points of Q⁺
is given by:

τ_s(p, ρ) = [ ½(1 − q(p, ρ)) e₀ + p  + ρ e₄ + ½(1 + q(p, ρ)) e₅ ].                 (12)

We can now use the formula above for embedding the 2+1 dimensional Minkowski spacetime in Q⁺ by setting ρ=ct, where t is time, and c is the speed of light. Choosing units in which c=1, we simply set ρ=t, then q(p, t) = p² − t². Here p represents space coordinates of an event.

We will continue in the next post where we will decompose Q+ into a set theoretic sum of two copies of the Minkowski spacetime (spacetime and anti-spacetime)and "the conformal infinity". While the geometry of the Minkowski spacetime is well known, the geometry of the conformal infinity is largely either unknown or just neglected.