# Why does universe expansion introduce redshift - doppler effect should not apply

If we think of a projectile being launched from a distant object to us, and the object is not moving away from us, then the projectile will arrive with the same velocity local to us - whether the universe is expanding or not. The expansion of the universe does not introduce a Doppler effect. So why is this explanation used to explain redshift of light from distant galaxies?

in a bit more detail:

SCENARIO 1

distant object O is stationary with regard to us at X. Universe is not expanding. projectile P launched from O at 300mph relative to O towards X.

start: speed of P relative to X is 300mph
half way: speed of P relative to X is 300mph
end: speed of P relative to X is 300mph

SCENARIO 2

distant object O is stationary with regard to us at X. Universe is expanding at rate that moves O away from X at 200mph. projectile P is launched from O at 300mph relative to O towards X.

start: speed of P relative to X is 100mph
half way: speed of P relative to X is 200mph
end: speed of P relative to X is 300mph

in both cases the projectile arrives with the same local velocity, even if it would have taken longer to get here in the expanding universe.

## 1 Answer

If we think of a projectile being launched from a distant object to us, and the object is not moving away from us, then the projectile will arrive with the same velocity local to us - whether the universe is expanding or not.

This premise is incorrect. Assuming both source and observer are at rest relative to the Hubble flow, the measured momentum of massive particles will also 'redshift' according to $$\frac{p_\text{observer}}{p_\text{source}} = \frac{a(t_\text{emission})}{a(t_\text{absorption})}$$ The handwaving explanation is that same as electromagnetic waves, a particle's de Broglie wavelength will be subject to spatial expansion.

For a more explicit calculation, you need to parallel transport the particle's momentum vector at time of emission along the particle geodesic and expand it in the observation frame. Explicit calculations can be found on this page, specifically problems 15 and 16.