Projective interpretation of some doubly special relativity theories

A class of projective actions of the orthogonal group on the projective space is being studied. It is shown that the Fock-Lorentz and Magueijo-Smolin transformations known as doubly special relativity are such transformations. The formalism easily leads to new types of transformations.

Figure 1

The orbits, under the action of $SO(2)$, when $SO(2)$ is acting on $\mathbb{R}{P}^2$ ordinarily, that is, when it is $L_{\infty }$ which is mapped into itself. The line $\Delta$, with equation $X=1$, is being mapped into $L_{\infty }$ by the projective map $(X,Y)\mapsto (\frac{X}{1-X},\frac{Y}{1-X})$. The touching circle is the circle with $\mu =1$. Note that circles with $\mu >1$ cross $\Delta$ at two points.

Figure 2

The orbits, under the action of $SO(2)$, when $SO(2)$ is acting on $\mathbb{R}{P}^2$ such that the line $\delta$, with equation $x=-1$, is mapped into itself. The image of the touching circle is now the parabola $\mu =1$. The line $\delta$ is invariant under this action. As is seen from the two-sheet hyperbola, for some values of the rotation parameter $\theta$, points on one side of $\delta$ will move to the other side. Also note that when $\mu \to \infty$, the two sheets of the hyperbola will approach $\delta$, and will cover it twice.

Figure 3

Orbits of $SO(1,1)$ in $X^{\mu }$ space, for $a^{\mu }=(-1,0)$. The dashed line with equation $T=1$ is the line $\Delta$. The touching hyperbola is the one tangent to the dashed line. Note that all the orbits are asymptotic to the null directions.

Figure 4

Orbits of $SO(1,1)$ on $\mathbb{R}{P}^2$, for $a^{\mu }=(-1,0)$. The invariant plane is $\delta$ (the set $t=-1$). For $\mu <-1$ the orbit is an ellipse (labeled $a$). For $\mu =-1$ the orbit is a parabola ($b$). For $-1<\mu <0$ the orbit is a spacelike hyperbola of two sheets ($c$ and $c^{\prime }$). For $\mu =0$ the orbit is a cone ($d$). For $\mu >0$ the orbit is a timelike hyperbola of two sheets ($e$). Note that all these orbits intersect at the points $P$ and $P^{\prime }$, which are the images (under $f^{-1}$) of the null directions in the $X^{\mu }$ space, and remember that all the orbits of $SO(1,1)$ in $X^{\mu }$ space are asymptotic to the null directions.

Figure 5

The case $a^{\mu }=(-1,1)$.

Figure 6

The orbits of $SO(1,1)$ under the projective action for $a^{\mu }=(-1,1)$.