In Tristan Needham's wonderful book Visual Complex Analysis, the chapter on Möbius transformations begins by asserting that they correspond one-to-one with the Lorentz transformations from special relativity. I tried to work out the details of this correspondence and I would like my results to be confirmed or corrected. I am pretty sure of my work but wanted to ask because it differs slightly from what Needham asserts.

I am working with the following definitions:

(1) The Möbius transformations are transformations of the complex plane of the form $z \mapsto \frac{az+b}{cz+d}, a,b,c,d \in \mathbb{C}, ad-bc \neq 0$. I'll refer to the group of these as $P(1,\mathbb{C})$, the notation in Ahlfors.

(2) The Lorentz group is the group of linear transformations of $\mathbb{R}^4$ that preserve the Lorentz form $t^2-x^2-y^2-z^2$. Thus it is represented by the set of real 4x4 matrices satisfying $M^tI_{1,3}M=I_{1,3}$ where $I_{1,3}$ is the matrix of the Lorentz form, i.e. $1, -1, -1, -1$ on the main diagonal and zero elsewhere. For the group I'll use the notation $O_{1,3}$ from Artin.

(3) A "space-like interval" is a vector in $\mathbb{R}^4$ on which the Lorentz form is negative; a "time-like interval" is a vector in $\mathbb{R}^4$ on which the Lorentz form is positive. I am adopting the convention that time is the first coordinate.

After many hours of work I now believe the following:

(a) The isomorphism with $P(1,\mathbb{C})$ really only belongs to a subgroup of $O_{1,3}$ of index 4, call it $M$. There is a subgroup of index 2 of $O_{1,3}$ that has a surjective homomorphism to $P(1,\mathbb{C})$, but it is not bijective because the kernel is $\{I,-I\}$. The Lorentz transformations in the coset of the index 2 subgroup actually map to orientation-reversing transformations of $\mathbb{C}$ which are thus not Möbius transformations.

(b) $M$ is the path component containing $I$.

(c) Also, $M$ is the set of Lorentz transformations that preserve the direction of time on time-like intervals and also preserve the orientation of 3-space given by the space-components of sets of 3 linearly independent space-like intervals.

(d) However, even the transformations in this subgroup $M$ are capable of reversing time on space-like intervals or the orientation of space given by the space components of sets of 3 time-like intervals.

Is all this correct? If/where incorrect, what is true instead?


Qiaochu Yuan pointed me to Wikipedia where all this is discussed in the Lorentz group article.


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