Relativity without constancy of light speed

$\kappa > 0$ represents Euclidean geometry, in which the time axis is equivalent to (and freely interchangeable with) the spatial ones. In other words it acts like a fourth spatial dimension.

So you can for example take a left turn into the time axis and go forward, then turn around and go back in time. Many consider this non-physical, leaving the choice beweeen $\kappa = 0$ (Galilean Transformation) and $\kappa < 0$ (Lorentz Transformation).

As said in the answer of @m4r35n357 it is the case of Euclidean geometry. To see this, look at the transformations, that preserve the distance : $$ds^2 = dx^2 + dt^2$$

Among with the translations there are also rotations: $$ \begin{pmatrix} t^{'} \\ x^{'} \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} t \\ x \end{pmatrix} $$ Look, for example, at the first row. After defining $v = \tan \theta$, it is actually the $t$ transformation, with $\kappa = 1$: $$ t^{'} = \frac{t + v x}{\sqrt{1 + v^2}} $$ This geometry is obtained from the Minkowski space by Wick rotation $t \rightarrow i t$. It is mathematically consistent and alright, however, our world is described by the metric (in the flat case), with the minus sign between $dx^2$ and $dt^2$.

As stated in almost all the answers, $\kappa>0$ does indeed define a set of transformation that preserves the Euclidean metric, with the Lorentz group $SO(1,3)$ being replaced by $SO(4)$. Although this does make mathematical sense, there's an internal physical inconsistency, which shows that an universe with $\kappa>0$ is not possible.

To show the inconsistency, first note that homogeneity and isotropy of space along with the principle of relativity not only gives us the spacetime transformations, but also gives us the velocity addition rule$^*$, which looks like $$w=\frac{u+v}{1-\kappa uv} .$$ Since $\kappa>0$, assume $\kappa=1/c^2,\ c\in\mathbb R,\ c<\infty$. Then $$w=\frac{u+v}{1- uv/c^2}.$$ We first show that there exists a velocity greater than $c$. To show this, consider $u=c/2=v$. Then $w=c/(3/4)=4c/3>c$.

Now, let $\gamma_u=1/\sqrt{1+\kappa v^2}=1/\sqrt{1+v^2/c^2}$. The spacetime transformation tells us that $\gamma_0=1$ and hence the square root in $\gamma_u$ is the positive square root and thus $\gamma_u>0\ \forall u\in \mathbb R$.

The assumed postulates also lets us derive$^*$ $$\gamma_w=\gamma_u\gamma_v(1-\kappa uv)=\gamma_u\gamma_v(1-uv/c^2).$$

Consider $u,v$ such that $uv>c^2$. This is possible because there exist velocities greater than $c$ as we have shown. Then $1-uv/c^2<0\Rightarrow \gamma_w=\gamma_u\gamma_v(1-uv/c^2)<0\Rightarrow \gamma_w<0$ which is a contradiction.

This is the internal inconsistency. The postulates do not allow for an universe with $\kappa>0$.


$^*$ The derivation of these facts is very beautifully shown in this paper. It's simple and concise, and the proof of inconsistency is also in the paper.