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The group of Mobius transformations, denoted by ${\rm Mob}(2,\mathbb{C})$, is isomorphic to ${\rm SL}(2,\mathbb{C}))/\mathbb{Z}_2$ which in turn is isomorphic to the Lorentz group ${\rm SO}^+(3,1)$.

This connection, to me, seems very intriguing. After all, Mobius transformation is the most general, one-to-one, conformal map of the Riemann sphere to itself, given by $$w=f(z)=\frac{az+b}{cz+d}\tag{1}$$ where $a,b,c,d$ are arbitrary complex constants satisfying $(ad-bc)=1$. Apparently, (1) has nothing to do with spacetime transformations.

But the aforementioned isomorphism makes me curious whether there is any deep physical consequence(s) related to this isomorphism.

Qmechanic
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2 Answers2

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  1. A future-directed light-ray $$n^{\mu}~=~(1,{\bf p}/|{\bf p}|), \qquad {\bf p}~\in~\mathbb{R}^3\backslash\{\bf 0\}, \tag{1}$$ can be identified with a non-zero future-directed light-like 4-vector $$p^{\mu}~=~(E,{\bf p})\tag{2}$$ if we mod out with the energy $E\equiv |{\bf p}|>0$.

  2. Therefore the set of future-directed light-rays (through a fiducial point) can be identified with the Riemann sphere $$\mathbb{C}P^1~\cong~S^2~\cong~(\mathbb{R}^3\backslash\{\bf 0\})/\mathbb{R}_+.\tag{3}$$

  3. The restricted Lorentz group $SO^+(3,1)$ acts transitively on the set of future-directed light-rays, cf. e.g. my Phys.SE answer here. Hence they can be identified with Moebius transformations.

  4. For a proof of $SO^+(3,1)\cong SL(2,\mathbb{C})/\mathbb{Z}_2$, see e.g. this Phys.SE post.

Qmechanic
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You'll find an intuitive way into your question via the Wiki article on Möbius.

"In physics, the identity component of the Lorentz group acts on the celestial sphere in the same way that the Möbius group acts on the Riemann sphere. In fact, these two groups are isomorphic. An observer who accelerates to relativistic velocities will see the pattern of constellations as seen near the Earth continuously transform according to infinitesimal Möbius transformations. This observation is often taken as the starting point of twistor theory."

See also the late section on Applications, which discusses the isomorphism of the Möbius group with the Lorentz group, SO+(1,3) and SL(2,C). Also the classification table near the end, which I find suggestive.

Hope this helps.

iSeeker
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