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I'm watching this minutephysics video on Lorentz transformations (part starting from 2:13 and ending at 4:10). In my spacetime diagram, my worldline will be along the $ct$ axis and the worldline of an observer moving relative to me will be at some angle w.r.t. the $y$ axis.

When we switch to the other observer's spacetime diagram, the observer's worldline will be along the $y$ axis and my worldline would flip over to the other side, but the angle between the worldlines would be preserved. Then the video goes on to mention three possibilities:

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The even at $(2,4)$ ends up getting mapped to $(0,T)$, where $T<4$, $T=4$ or $T>4$. And it seems like the video is suggesting only one possible transformation for each case, giving a total of only three possible transformations. For an angle preserving transformation $A$, given any two unit vectors $u_1,u_2$,

$$[u_1]^TM[u_2]=[u_1]^TA^TMA[u_2]\implies M=A^TMA$$

where $M$ is the metric we're assuming for the space. Seems like (though I'm not a 100% sure) the $T>4$ possibility corresponds to rotation (Euclidean, sure about this one), $T=4$ to Galilean boost (Galilean) and $T<4$ to Lorentz boost (Minkowski metric).

But why is it being suggested that only these three transformations (satisfying the angle preservation and linearity properties) are possible? Is it possible to find any other transformations than rotation for the case of $T>4$, or other than Lorentz for $T<4$, etc.? If not, can anyone direct me to a proof or explanation of why only three transformations are possible?

Edit: Specifically a mathematical argument/proof of why Euclidean, Galilean and Lorentz must be the only linear angle-preserving transformations in flat geometry.

Shirish Kulhari
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  • To understand why only these transformations are conformal (i.e. angle preserving) and no others, read about Minkowski diagrams in : Andrew Steane's Relativity Made Relatively Easy (He is also on PhyStackExchange). It will help you understand the reason behind it. It is linked to the factor $\beta$ called rapidity which defines a hyperbolic angle between the world lines of two observers in different relative frames. – Ishika_96_sparkle Jun 20 '20 at 14:05
  • @Ishika_96_sparkle: Unfortunately I don't have access to that book. If possible/convenient, could you write the proof/justification on why only these 3 transformations are conformal, in the form of an answer? – Shirish Kulhari Jun 20 '20 at 14:16
  • @Ishika_96_sparkle: Thanks a lot for the references! I had a look at the google books version of Steane's book. While it certainly covers why Lorentz transformation must be the linear conformal transformation we're looking for, it doesn't seem to address why Lorentz, Galilean and rotation are the only linear conformal transformations for flat space. For example, see: https://physics.stackexchange.com/questions/113656/why-does-minkowski-space-provide-an-accurate-description-of-flat-spacetime. One of the answers says that such an argument can be made with differential geometry...(cont'd) – Shirish Kulhari Jun 20 '20 at 14:30
  • @Ishika_96_sparkle: (cont'd)...and that is the kind of argument I'm looking for. Without even invoking any physics, it seems we can mathematically prove that these 3 are the only linear conformal transformations. I'll edit the question to make it less ambiguous. Thanks again! – Shirish Kulhari Jun 20 '20 at 14:32
  • Indeed Shirish, the proof would take into account the Poincare group of Hyperbolic space rotations parameterized by rapidity and boosts. Mathematically, The structure of 1D Lorentz transformation the matrix with hyperbolic elements $\cosh \beta$ and $\sinh \beta$ defines a rotation in hyperbolic space. ,i would try to write up some basic explanation in a few days. – Ishika_96_sparkle Jun 20 '20 at 14:44
  • Please have a look here https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations

    The part Using the geometry of spacetime

     – Ishika_96_sparkle Jun 20 '20 at 14:51
  • @Ishika_96_sparkle: Thanks again! But to reiterate, I'm already convinced that Lorentz transformations are conformal. I've studied the basics of SR - including derivation of LT, its connection with hyperbolic geometry, etc. and there's no doubt that it's conformal - angles being hyperbolic can be considered a consequence of the Minkowski metric. What I'm actually looking for is any justification/proof (or reference to one) on why, apart from the 3 I listed, there can be no other conformal transformation. – Shirish Kulhari Jun 20 '20 at 15:05
  • Hi, were you able to construct something from the possibility in our chat? Ref: The relation $\beta=Tanh^{-1}(v/c)$ for v<<c (Galilean) and v=0.9c(Lorentz). The domain of $Tanh^{-1}$ is the unit interval (−1, 1) and the range is $\mathbf{R}$. The conformal mapping is $-1<v/c<1$ onto the interval $-\infty < \beta <\infty$. Any transformation that obeys this mapping would indeed be angle preserving. – Ishika_96_sparkle Jun 22 '20 at 05:12

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