  The "amplituhedron" for tree-level scattering amplitudes in the bi-adjoint
$\phi^3$ theory is given by the ABHY associahedron in kinematic space, which
has been generalized to give a realization for all finite-type cluster algebra
polytopes, labelled by Dynkin diagrams. In this letter we identify a simple
physical origin for these polytopes, associated with an interesting
(1+1)-dimensional causal structure in kinematic space, along with solutions to
the wave equation in this kinematic "spacetime" with a natural positivity
property. The notion of time evolution in this kinematic spacetime can be
abstracted away to a certain "walk", associated with any acyclic quiver,
remarkably yielding a finite cluster polytope for the case of Dynkin quivers.
The ${\cal A}_{n{-}3},{\cal B}_{n{-}1}/{\cal C}_{n{-}1}$ and ${\cal D}_n$
polytopes are the amplituhedra for $n$-point tree amplitudes, one-loop tadpole
diagrams, and full integrand of one-loop amplitudes. We also introduce a
polytope $\bar{\cal D}_n$, which chops the ${\cal D}_n$ polytope in half along
a symmetry plane, capturing one-loop amplitudes in a more efficient way.
