Why is it interesting?

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\begin{eqnarray} {\rm Translations}\quad (4\ {\rm param.})\quad & x^ i & \rightarrow\> \tilde{x}^ i =\, x^ i +a^ i \,,\tag{transl} \\ {\rm Lorentz\ transf.}\quad (6\ {\rm param.})\quad & x^ i & \rightarrow\>\tilde{x}^ i =\,\Lambda ^ i {}_ j \, x^ j \,,\tag{lor}\\ {\rm dila(ta)tion}\quad (1\ {\rm param.})\quad & x^ i & \rightarrow\>\tilde{x}^ i =\,\rho\, x^ i \,,\tag{dil}\\ {\rm prop.\ conf.\ transf.}\quad (4\ {\rm param.})\quad & x^ i & \rightarrow\> \tilde{x}^i =\,{\frac{x^i + \kappa ^i\,x^2}{1+ 2\kappa_j\,x^j +\kappa_j\kappa^j\,x^2}}.\tag{proper} \end{eqnarray} Here $a^i, \Lambda^i{}_j, \rho, \kappa^i$ are 15 constant parameters, and $x^2 := g_{ij}x^ix^j$.

• The Poincare subgroup (transl), (lor) leaves the spacetime interval $ds^2 = g_{ij}dx^idx^j$ invariant
• Dilatations (dil) and proper conformal transformations (proper) change the spacetime interval by a scaling factor $ds^2 \rightarrow \rho^2 ds^2$ and $ds^2 \rightarrow \sigma^2 ds^2$, respectively (with $\sigma^{-1} := 1+ 2\kappa_j\,x^j +\kappa_j\kappa^j\,x^2$).
• The Weyl subgroup, is generated by transformations (transl)-(dil).

The light cone $ds^2 =0$ is left invariant by all these transformations.

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Examples

Example1

Example2:

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