Relativity, complex numbers, and gyrovectors

Posted on by John

The previous post discussed an unusual algebraic structure on the real interval (-1, 1) inspired by (and applied to) special relativity. We defined an addition operator ⊕ by

x \oplus y = \frac{x + y}{1 + xy}

How might we extend this from the interval (-1, 1) to the unit disk in the complex plane? The definition won’t transfer over unmodified because it does not map the unit disk to the unit disk. For example,

\frac{i/2 + i/2}{1 + (i/2)^2} = 4i/3

which is outside the unit disk. But if we conjugate the x in the denominator then we get an operation that does map the unit disk to itself.

x \oplus y = \frac{x + y}{1 + \bar{x}y}

Since the complex conjugate of a real number is itself, this definition coincides with the original definition on the interval (-1, 1).

Complex relativistic addition as defined above has the right range, but it’s an odd form of addition. It’s asymmetric in x and y which rightly suggests that ⊕ is not commutative. In fact, ⊕ is not associative either. However, there is a way to compensate for this lack of symmetry.

Define the gyration of x and y by

\mbox{gyr}[x, y] = \frac{1 + x\bar{y}}{1 + \bar{x}y}

Then we have replacements for commutativity

x \oplus y = \mbox{gyr}[x, y] y \oplus x

and for associativity

\begin{align*} x \oplus (y \oplus z) &= (x \oplus y) \oplus \mbox{gyr}[x, y]\, z \\ (x \oplus y) \oplus z &= x \oplus (y \oplus \mbox{gyr}[x, y]\, z) \end{align*}

It’s also possible to extend scalar multiplication by real numbers.

r \odot x = \frac{(1 + |x|)^r - (1 - |x|)^r}{(1 + |x|)^r + (1 - |x|)^r} \, \frac{x}{|x|}

When we define vectors as complex numbers in the unit disk, define vector addition by ⊕ and scalar multiplication by ⊙, we do not get a vector space, but something analogous called a gyrovector space. The axioms of a gyrovector space are similar to those of a vector space, but with gyrations sprinkled in.

In some sense gyrovector spaces are to hyperbolic geometry what vector spaces are to Euclidean geometry.

References

Michael K. Kinyon and Abraham A. Ungar. The Gyro-Structure of the Complex Unit Disk. Mathematics Magazine , Oct., 2000, Vol. 73, No. 4 (Oct., 2000), pp. 273–284

Abraham A. Ungar. The holomorphic automorphism group of the complex disk. Aequationes Mathematicae 47 (1994) pp. 240–254

2 thoughts on “Relativity, complex numbers, and gyrovectors”

  1. Fred Bacon

    Why not $x \oplus y = \frac{x + y}{1 + \frac{\bar{x}y + x\bar{y}}{2}}$ ? It obviously commutes and reduces to the original equation when x and y are real.

  2. John

    The definition proposed in your comment sometimes maps values from the unit disk to values outside the unit disk.

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