# Does the redshifting of photons from the Universe's expansion violate conservation of momentum?

The energy-momentum relation,

$$E^2 = m^2c^4 +p^2c^2,$$

lets us derive the momentum of a massless particle:

$$p = \frac{E}{c} = \frac{h\nu}{c}$$

However, the expansion of the Universe redshifts light. This should decrease the momentum of photons. Where would the momentum go, in order for conservation of momentum to hold?

• Is this a different question to astronomy.stackexchange.com/questions/18613/… ? – ProfRob Nov 24 '16 at 16:33
• @RobJeffries Yes, because as far as I know, conservation of energy does not hold in GR. I'm asking about momentum. – Sir Cumference Nov 24 '16 at 16:35
• Light blue-shifts as it falls into super-clusters, then red-shifts a little less as it climbs out of the ever-expanding cluster's gravity well. Basically expansion causes universe to not act like a closed system energy or momentum-wise; but this is only on extremely large scales. – Wayfaring Stranger Nov 26 '16 at 16:00

## 1 Answer

In relativity you can think of a single conservation law that unites conservation of energy and momentum -- conservation of four-momentum. Energy and momentum are the zeroth and the first to third components of the four-momentum respectively. Such conservation laws arise from invariance of the Lagrangian with respect to a translation in space-time coordinates.

In General Relativity these conservation laws are local concepts that (most people think) can only be applied in local, inertial (flat) frames of reference. In particular, they cannot be applied in changing space-times and so cannot be applied to situations involving the expansion of the universe.

• The second paragraph is a little muddled and makes it sound as though there is a controversy among experts, when there isn't. The issue isn't whether the spacetime is "changing" or has a timelike Killing vector. That's just the condition for test particles to have a conserved energy. The issue is whether the spacetime is curved. (As a side issue, not relevant here, there is a way to make a globally conserved mass-energy in the case of asymptotically flat spacetimes.) – Ben Crowell Dec 17 '20 at 16:13