Using homogeneity of space, isotropy of space and the principle of relativity (without the constancy of light speed), one can derive:
$$x' = \frac{x-vt}{\sqrt{1+\kappa v^2}}$$
$$t' = \frac{t+\kappa vx}{\sqrt{1+\kappa v^2}}$$
$\kappa = 0$ denotes Galilean and $\kappa < 0$ denotes Lorentz Transformation.
What does $\kappa > 0$ denote? Is it physically possible? I was told that it is self-inconsistent. Can somebody help me with a proof of this?
$\kappa > 0$ 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 $\kappa>0$.
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$^*$ 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.
There has been much discussion on one-way speeds of light and simultaneity etc. in the philosophy of physics literature. Most famously, Reichenbach introduced a parameter $\epsilon$, which gives the (one-way) light speed in opposite directions as $c/2\epsilon$ and $c/2(1-\epsilon)$. Here $c$ is the "two-way" speed of light, which can actually be experimentally measured.
One way of interpreting this discussion is as ordinary relativity but described on a "tilted" hyperplane: one which is not orthogonal to the observer's 4-velocity. This is the approach of papers like Ungar 1991, see equation 9 for the one-way Lorentz transformations. I haven't analysed your $\kappa$ parameter specifically. But it is certainly consistent to describe relativity using coordinates that are tilted relative to given observer(s).
