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- 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

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

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:

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

When the Andromeda Galaxy heads towards us, the light from it appears blue-shifted. By

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*R*as you'd expect. However, the density of radiation drops by^{3}*R*(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.^{4}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

*mc*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!^{2}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?