  There is a possibility that spacetime itself is ultimately an emergent
phenomenon, a near-universal "low-energy long-distance approximation", similar
to the way in which fluid mechanics is the near-universal low-energy
long-distance approximation to quantum molecular dynamics. If so, then direct
attempts to quantize spacetime are misguided - at least as far as fundamental
physics is concerned. Based on this and other considerations, there has
recently been a surge of interest in the notion of energy-dependent and
momentum-dependent "rainbow'' geometries. In the present article I will not
discuss these exotic ideas in any detail, instead I will present two specific
and concrete examples of situations where an energy-dependent "rainbow''
geometry makes perfectly good mathematical and physical sense. These simple
examples will then serve as templates suggesting ways of proceeding in
situations where the underlying physics may be more complex. The specific
models I will deal with are (1) acoustic spacetimes in the presence of
nontrivial dispersion, and (2) a mathematical reinterpretation of Newton's
second law for a non-relativistic conservative force, which is well-known to be
equivalent to the differential geometry of an energy-dependent conformally flat
three-manifold. These two models make it clear that there is nothing wrong with
the concept of an energy-dependent "rainbow'' geometry per se. Whatever
problems may arise in the implementation of any specific
quantum-gravity-inspired proposal for an energy-dependent spacetime are related
to deeper questions regarding the compatibility of that specific proposal with
experimental reality.
