Energy conservation in Doppler (NOT cosmological) redshifts?

[Amaterasu21]

TL;DR Summary

There are lots of popular explanations for whether or not energy is conserved in an expanding universe. But what about Doppler shifts in a stationary space? Where do photons lose their energy to (redshifts) or gain energy from (blue-shifts)?

Hi all,

My question is about Doppler redshifts, but I'm going to mention cosmological redshifts first because I'm a lay person as far as cosmology's concerned (I'm an amateur astronomer and did a few introductory astrophysics/cosmology courses at university, but my degree focus was planetary science, and I haven't formally studied GR at all) and I want to make sure I've got my ideas about how it relates to conservation of energy right. If I've got anything wrong, please let me know! If you just want to see my question on Doppler shifts, skip to the last two paragraphs. I'll put all the cosmology stuff in green so you know what to ignore.

As I understand it, in an expanding Universe where the scale factor R increases, the density of matter drops off by R³ as you'd expect. However, the density of radiation drops by R⁴ (due to the frequency of photons getting lower through redshift) and the density of dark energy remains constant. A naive look at that suggests energy is not constant - radiant energy is being destroyed while dark energy is being created.

Based on the popular articles and videos I've seen, to take just three examples:
this one from Ethan Siegel:
this one from Sean Carroll: https://www.discovermagazine.com/the-sciences/energy-is-not-conserved
this video from Nick Lucid:

cosmologists' responses seem to fit into two categories:

1) Noether's Theorem tells us conservation laws come from symmetries of nature, and energy conservation comes from time symmetry. An expanding universe is not time-symmetric, so conservation of energy does not apply. However, this doesn't mean it's a free-for-all where free energy and perpetual motion machines are just around the corner! Energy and momentum change in precisely defined ways as space expands, and energy conservation is the special case of that when space is stationary, just as special relativity is a special case of general relativity when there's no acceleration or gravity to consider.

2) It depends on how you define energy - because it changes in a predictable way you can still define a quantity called "energy" which is conserved in an expanding universe, as long as you add a few extra places for that energy to come from and go to. When photons get cosmologically red-shifted, their energy gets dumped into the gravitational field, similar to lifting a stone out of a well and increasing gravitational potential energy there. With dark energy, the negative work done by the existing dark energy as space expands offsets the positive energy produced in the new space.

Different physicists and cosmologists agree on the physics/mathematics, but argue about how best to translate this for the general public. Those in camp 1) say energy is not well defined in GR, and the energy of the gravitational field can only be defined for the universe as a whole, not at each point, so it doesn't have a density and doesn't show up in GR calculations like the energy-momentum tensor. So it's best to leave it out and say energy isn't conserved - if space isn't time-symmetric, Noether tells us energy shouldn't be conserved anyway!
Those in camp 2) say it's better to talk about the energy of the gravitational field or work done by space so energy can still be conserved. We've historically added new forms of energy to the list to save energy conservation before (heat was originally defined as energy by Rumford if I recall correctly to save the failure of the law of conservation of mechanical energy, and rest energy was added so we could keep a single law of conservation of energy rather than a messy "law of conservation of energy plus another quantity equal to mc² for all masses"), and it preserves the idea of energy changing in a consistent, defined way that can be defined as conserved. If the public hear "energy conservation can be violated," their first thought will probably be free energy and perpetual motion machines, and that's definitely not what GR says!

It's a bit like the old debate over the definition of mass - older physics texts talk about relativistic mass, define mass the same way Newton would, and say mass increases as you approach the speed of light. Newer ones say mass is a constant and only refers to rest energy, instead talking about how momentum, force and kinetic energy no longer obey Newtonian definitions at high speeds. The physics itself hasn't changed, it's just physicists have decided the new definition of mass is more useful. The question "is energy conserved in an expanding universe?" is similar - physicists agree on the physics, but they disagree on the semantics.

Am I right about all this or have I picked up a misunderstanding somewhere?

I'm also under the impression gravitational redshifts involve photons losing their energy to the gravitational field, again like how a stone loses kinetic energy and stores it as gravitational potential energy as it rises against gravity.

So, with that said, on with my main question:
What about Doppler redshifts? How is energy conserved there?
At least cosmological and gravitational redshifts involve GR, but Doppler redshifts can involve sources of light moving through a flat, non-expanding space that knows nothing of GR or cosmology. That's motion in the traditional sense, through a time-symmetric space. In that scenario energy IS conserved and defined just the way we define it in high school physics.

When the Andromeda Galaxy heads towards us, the light from it appears blue-shifted. By E = hf, the energy of the photons it emits increase on their way to us. Where does this energy come from? When a star in the Milky Way moves away from us, its light appears red-shifted. Where does the energy of their photons go as their frequency drops?

 

[Vanadium 50]

Geez that's a long message. Fortunately, the answer is short: energy is conserved but it is not invariant.

 

[Ibix]

How much kinetic energy does a car of mass ##m## have, (a) as measured by a pedestrian who says its velocity is ##v##, and (b) by its driver who says its velocity is zero?

Energy is frame dependent even in Newtonian physics. So the energy of light being measured by someone at rest with respect to Andromeda doesn’t "come from" or "go to" anywhere when measured by someone at rest with respect to the Milky Way. It's just different.

Relativity does provide an interesting explanation of this fact: the energy is one component of the energy momentum four vector. If you measure in a different frame then that component is different, for the same reason the z component of a three vector changes if you change your axes.

 

[PeroK]

When the Andromeda Galaxy heads towards us, the light from it appears blue-shifted. By E = hf, the energy of the photons it emits increase on their way to us. Where does this energy come from? When a star in the Milky Way moves away from us, its light appears red-shifted. Where does the energy of their photons go as their frequency drops?

In addition to the above answers: a star must recoil when it emits a photon, so the total energy is conserved. Imagine the non-relativistic case where a large mass ##M## emits a small mass ##m##. In the original rest frame of the large mass the small mass has kinetic energy ##\frac 1 2 mv^2##, where ##v## is the speed of the small mass in the original rest frame. In the new rest frame of the large mass, the small mass has energy ##\frac 1 2 m(v + v_M)^2##, where ##v_M## is the (small) recoil speed of the large mass.

In any case, the kinetic energy of the small mass is greater in one frame than the other. This is because, as mentioned above, energy is not frame invariant.

 

[Orodruin]

Geez that's a long message. Fortunately, the answer is short: energy is conserved but it is not invariant.

To add to this, this statement is true in Newtonian physics as well as in relativity.

 

[Amaterasu21]

Ah, all of what you're saying about this being due to kinetic energy not being frame invariant makes a lot of sense. Thanks everyone! That's answered to my satisfaction. I hope I got my message about the cosmological redshift correct as well!

 

[Ibix]

I hope I got my message about the cosmological redshift correct as well!

The point there is that there's no clear consensus on how to define "energy of the gravitational field" in the general case in general relativity, or even if it can be done. Specific cases where it can be done are known, but FLRW spacetimes are not one of them.

It's not at all clear that any of that matters to cosmological redshift. As in the kinematic version, the point is that the relationship between your local reference frame and the light's four-momentum is different from the relationship between the emitter's local reference frame and the light's four-momentum, so you are measuring different things. The difference from kinematic redshift is that the reason for the different relationships is tangled up with the curvature of spacetime, whereas pure kinematic redshift is just about relative velocity. I think you can try to attribute the changing spatial curvature to some sort of exchange with the energy of the gravitational field, but I don't think it's ever been done in a way that convinces even a large fraction of cosmologists.

 

[Orodruin]

The difference from kinematic redshift is in the reason for the different relationships is tangled up with the curvature of spacetime, whereas pure kinematic redshift is just about relative velocity.

I’d argue that the ”tangling up” is just a result of a quite ad hoc split of the only relevant effect, the inner product between the light’s parallel transported 4-momentum and the 4-velocities of the observers.

 

[Ibix]

I’d argue that the ”tangling up” is just a result of a quite ad hoc split of the only relevant effect, the inner product between the light’s parallel transported 4-momentum and the 4-velocities of the observers.

I see your point here - in fact all Doppler shift is just variation in that inner product. But I think there is a qualitative distinction to be made between the case of relative motion in flat spacetime and that of curved spacetime where relative motion isn't uniquely defined. In the former case there's an obvious family of observers who see the same redshift, but in FLRW spacetime an inertial observer won't even see constant redshift from an inertial source. Or, to put it another way, the inner product comes to be what it is for quite different reasons in the two cases.

 

[Orodruin]

I see your point here - in fact all Doppler shift is just variation in that inner product. But I think there is a qualitative distinction to be made between the case of relative motion in flat spacetime and that of curved spacetime where relative motion isn't uniquely defined. In the former case there's an obvious family of observers who see the same redshift, but in FLRW spacetime an inertial observer won't even see constant redshift from an inertial source. Or, to put it another way, the inner product comes to be what it is for quite different reasons in the two cases.

The only differences I see all follow from that the Minkowski spacetime is a flat affine space.

It is also true in Minkowski space that you generally will not see constant redshift from an inertial source.

 

[Ibix]

The only differences I see all follow from that the Minkowski spacetime is a flat affine space.

Yes.

It is also true in Minkowski space that you generally will not see constant redshift from an inertial source.

Fair enough. But is there any inertial observer in an FLRW spacetime (not Milne!) who sees constant redshift from an inertial source that isn't co-located?l

 

[Orodruin]

not Milne!

Damn it! I have spread the word on my favorite coordinates on Minkowski space too well ...

Fair enough. But is there any inertial observer in an FLRW spacetime (not Milne!) who sees constant redshift from an inertial source that isn't co-located?l

In FLRW, probably not (unless Milne). However, any flat spacetime should do (ie, does not need to be an affine space).

 

[Ibix]

Damn it! I have spread the word on my favorite coordinates on Minkowski space too well ...

I probably do fall for the same thing twice, but I try not to...

However, any flat spacetime should do (ie, does not need to be an affine space).

I don't think I understand the distinction you're making. Isn't flat spacetime a synonym for Minkowski, and hence an affine space?

 

[Orodruin]

I don't think I understand the distinction you're making. Isn't flat spacetime a synonym for Minkowski, and hence an affine space?

No. You can have different global structures like a cylinder etc.