Inertial Transformations and the Nonexistence of Tachyons for Spacetime Dimension Greater than Two

We consider real linear transformations between two inertial frames with constant relative speed $v$ in a $d$-dimensional spacetime where light moves with constant speed $c=1$ (for some chosen units) in all frames. For $d=2$ we show that the standard relative velocity formula holds and that any associated anisotropic conformal factor is multiplicative under composition of inertial transformations for $|v|\neq 1$. Assuming that the inertial transformation matrix is continuous in a neighbourhood of $v=0$ and differentiable at $v=0$, we determine the conformal factor for all $|v|\neq 1$. For an isotropic spacetime, the general solution reduces to the standard $d=2$ Lorentz transformation for $|v|<1$ or to a Tachyonic transformation for $|v|>1$, first described by Parker in 1969. For $d>2$ we show that no Tachyonic-like inertial transformations exist which are compatible with constant light speed.