How can metric expansion of space redshift light? Doesn't that violate conservation of momentum, since the redder light has lower momentum?
And adding on to that: Will light inside a box redshift? If I weigh the box (i.e. weigh the light inside the box), then wait a bit for the light to redshift, then weigh the box again?
You are correct! Cosmological redshift does violate conservation of momentum (and energy as well [1]). But conservation of energy and momentum does not actually apply if spacetime itself is globally changing.
The underlying reason for this is that Noether's theorem tells us that every physical symmetry implies a conservation law for some physical quantity. Conservation of energy and momentum comes from the fact that the physical laws are the same throughout time and space. However, cosmological expansion violates that assumption, so there is no reason that energy and momentum should still be conserved. [2]
[1]: One side note here is that relativistically, energy and momentum are not really separate physical quantities, but instead two components of the same underlying physical quantity. Unfortuantely, this quantity does not really have a good name (despite Taylor & Wheeler's attempt to call it "momenergy"). It ends up being called the momentum 4-vector, but the temporal component of this 4-vector is energy.
[2]: This is only true globally. Locally, the laws are approximately the same from one moment to the next, so conservation of energy and momentum hold for small distances and short times.
And adding on to that: Will light inside a box redshift? If I weigh the box (i.e. weigh the light inside the box), then wait a bit for the light to redshift, then weigh the box again?