The question
This question follows on from the use of projective coords for spacetime in Notation for Translation Group Generators . Under Felix Klein's Erlangen Program, Minkowski spacetime starts as a 4-d projective space. It is then made into an affine space (which has a notion of parallelism) by picking an invariant hyperplane at infinity; lines in the projective space which meet at infinity are parallel. An affine space admits dilatations which are expansions or contractions of space about the eigenpoint of the dilatation. When introduced projectively[1], dilatations are seen to be generalized translations because a translation is a dilatation in which the eigenpoint is on the hyperplane at infinity. Since one can smoothly change a dilatation into a translation by moving the eigenpoint onto the hyperplane at infinity, it is hard to avoid the conclusion that a dilatation is an active transformation on the same footing as translations, rotations and boosts. Yet we don't experience dilatations in everyday life. Why is this?
Some background observations
The general feeling in physics seems to be that dilatations are changes in the units of measurement, but this view is only possible because dilatations are introduced in an ad hoc way as scale transformations and the link with translations is missed.
In order to see that dilations are active transformations like translations, the translation generators of Notation for Translation Group Generators become, $$ [P_{\Gamma}]^{\Phi}_{\ \Psi}=\delta^{\Phi}_{\Gamma}\delta^{4}_{\Psi} $$ where indices $\Phi,\Psi,\Gamma=0,1,2,3,4$. The translations are $P_{0},P_{1},P_{2},P_{3}$ and the dilatation generator is $P_{4}$.
The attempt to construct classical relativistic Hamiltonian particle mechanics in Minkowski spacetime with dilatations fails unless the particles are massless. Massive particles only exist by removing the dilatations by hand in order to restrict the group to the ten parameter Poincare group.
Edit: An attempt to derive dilatations without having to read Coxeter [1].
Start with Minkowski spacetime as a 4-d projective space; points are rays in a 5-d vector space $x^{\Phi}\in V_{5}$ with upper case Greek indices like $\Phi=0,1,2,3,4$. The spacetime group is GL(5,R). Introduce parallelism by picking an invariant hyperplane $\delta^{4}_{\Phi}\in \tilde{V}_{5}$. Invariance means, $$ [D(g^{-T})]_{\Phi}^{\ \Psi}\delta^{4}_{\Psi}=\xi \delta^{4}_{\Phi} $$ where $\xi$ is a scale factor which is unimportant because the physical entities are rays. This implies that the group matrices are now have the block form, $$ \begin{equation} D(g)= \left[ \begin{array}{cc} \Lambda & a \\ 0 & 1+a^{4} \end{array} \right] \end{equation} $$ where $\Lambda^{\mu}_{\ \lambda}$ is a 4x4 matrix and $a^{\mu}$ is a 4x1 vector and lower case Greek indices run $\mu=0,1,2,3$. These affine group matrices factor as, $$ \begin{equation} D(g)= \left[ \begin{array}{cc} 1 & a \\ 0 & 1+a^{4} \end{array} \right] \left[ \begin{array}{cc} \Lambda & 0 \\ 0 & 1 \end{array} \right] \ . \end{equation} $$ The first matrix is a dilation. In tensor notation, the pure dilatations are, $$ [D(g)]^{\Phi}_{\ \Psi}=\delta^{\Phi}_{\ \Psi}+a^{\Phi}\delta^{4}_{\Psi} $$ and a point transforms under a dilatation as, $$ x'^{\Phi}=[D(g)]^{\Phi}_{\ \Psi}x^{\Psi}=x^{\Phi}+a^{\Phi}x^{4} \ . $$ This shows that a dilatation acts like the identity on points in the hyperplane at infinity (the points which we can write as $x^{\mu}$ because $x^{4}=0$) and the eigenpoint of the dilatation is $a^{\Phi}$ with eigenvalue $1+a^{4}$. The last equation implies that a dilatation is an expansion or contraction of spacetime about the eigenpoint because a dilatation is a linear combination of the point and the eigenpoint.
Consider two spacetime points $x^{\Phi}$ and $y^{\Phi}$. Transform them to $x'^{\Phi}$ and $y'^{\Phi}$ under a dilatation with eigenpoint $a^{\Phi}$. The dilatation moves the line joining $x$ and $y$ parallel to itself so that the line $xy$ (in an informal notation) is parallel to line $x'y'$. To see this, the point where the line $xy$ intersects the hyperplane at infinity is, $$ b^{\Phi}=\delta^{\Phi \Psi}_{[\Gamma\Delta]}x^{\Gamma}y^{\Delta}\delta^{4}_{\Psi} =x^{\Phi}y^{4}-x^{4}y^{\Phi} $$ and the dilatation acts like the identity on $b$ which is in the hyperplane at infinity, so line $x'y'$ also intersects the hyperplane at infinity at $b$ and so $xy$ and $x'y'$ are parallel lines. Now consider the quadrilateral $x'y'yx$. We have just seen that the sides $xy$ and $x'y'$ are parallel. The quadrilateral can be made into a parallelogram if the sides $xx'$ and $yy'$ are also parallel. These lines are parallel if they intersect at a point in the hyperplane at infinity. But, these lines both go through the eigenpoint $a$ of the dilatation. Therefore, $xx'$ and $yy'$ are parallel if the eigenpoint $a$ is moved onto the hyperplane at infinity. With the eigenpoint $a$ on the hyperplane at infinity, we've made a parallelogram by starting with a line $xy$ and moving it parallel to itself to $x'y'$ with $xx'$ and $yy'$ also parallel. This is how to construct a parallelogram with a translation. In other words, a translation is a dilatation with its eigenpoint on the hyperplane at infinity. This construction is why I cannot see a reason to exclude dilatations from the spacetime group, it seems dilatations must be just like rotations and boosts.
[1] Coxeter, "Introduction to Geometry", chapter 5.
