If we consider a medium where perturbations always take the same time to reach every point in space when measured from an arbitrary observer's frame of reference then how would length have to contract to make up for the fact that when given two observers the time it takes for a wave that propagates radially outwards to reach them (even when they're standing at different distances away from the source) is the same, call it $\Delta t_c$?
In lorenzian manifolds we state that the fastest velocity possible would be the same for every reference frame, yet things get weird if we said that the maximum time it takes for a perturbation to arrive to every point in space is the same. If we consider two points in a manifold that behaves like this, call them $x$ and $y$ at distances $l_{x}$ and $l_{y}$ away from the source then that would mean that the speed at which they measure the propagation of the waves would be: $$v_{x}= \frac{l_{x}}{\Delta t_c} \\ v_{y}= \frac{l_{y}}{\Delta t_c}$$ The source would see that its perturbations radiate outwards with the same velocity in all directions, so in order for the time it takes for them to reach $x$ qand $y$ to be the same, ditances would have to contract or expand until they both are the same distance away from the source. If we pick $l_{x}$ to stay fixed then that must mean that: $$\frac{l_x}{v_x}=\frac{l_y}{v_y}$$ $\implies$ $$l_{x}=l_{y} \frac{v_y}{v_x}$$ The new length for $y$ after its contraction or expansion would have to be the same as $l_x$ so $l_{y}'=l_x$ is the new contracted or expanded length of between $y$ and the source. If we assume that the change in length is linear, as is the case with lorenz manifolds then $l_{y}'=\lambda l_{y}$, or rather $l_x=\lambda l_{y}$. By substituting this relationship into our previous equation we have: $$\lambda = \frac{v_y}{v_x}$$
This gets considerably more complicated if we consider a second source. If the time it takes to reach $x$ and $y$ is also $\Delta t_c$ and the speed at which the perturbation propagates through the medium is the same as the one our first source measured then the observers at $x$ would measure different speeds for the perturbations. In fact if we carefully select the position of a number of sources it would seem that in order for $\Delta t_c$ to be the same for all points (other than the sources) all would have to converge to $x$.
Is there any manifold which behaves in this manner?