B Redshift & Energy Conservation: Questions Answered

[Jimmy87]

Hi,

I have read over several threads already on this and have a few questions if someone could please answer that would be great:

1) The threads seem to suggest that energy is not conserved (or at least it isn't a requirement) on the scale of the universe. Why does it not have to be conserved under GR? I have studied GR so if possible a reasonably qualitative explanation would be good if one exists!

2) I haven't seen any threads specifically link this problem to the CMB. From a bit of surfing around the CMB apparently has 0.1% of its original emission energy (no surprise when it is only 3 Kelvin!). Surely this energy has gone somewhere? Surely GR can just wish all this energy away. These photons were originally gamma and are now 3K microwaves!

3) I have found some threads (some on StackExchange and some on PF) that say you can loosely liken this to a classical analogy with gas pressure. The energy lost by the photons does work on expanding the universe and if the universe were to start contracting the photons would all blue shift? I know it is much more involved than this but is this a reasonable 'classical' explanation that is somewhat true or is it completely incorrect and not worth ever using?

Thanks!

 

[PeroK]

Energy is not a "thing" that has to be somewhere. It's a dynamic property of things. And it's frame dependent. A ball thrown out of the back of a car has a lot of KE from the reference frame of the car, but may have very little KE in the reference frame of the road.

Likewise, if a source of light is moving away from you, then the energy of the photons in the reference frame of the source is greater than in your reference frame as the receiver. That energy doesn't go anywhere. The red-shift is a function of the relationship between the source and receiver.

And, in an expanding universe the relationship between the source and receiver depends on the expansion of space between them. This also leads to a red-shift between the source and the receiver.

You could try here:

https://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

 

[Dale]

Why does it not have to be conserved under GR?

Conservation of energy in GR is a little tricky. Locally it is always conserved based on the EFE, but globally it is only conserved in a special class of spacetimes where there is a symmetry in time. By Noether’s theorem that symmetry in time is associated with conservation of energy. In particular, the FLRW spacetime used in cosmology does not possesses that symmetry, so the most important example of energy non-conservation is cosmology.

Surely this energy has gone somewhere? Surely GR can just wish all this energy away.

Non-conservation means precisely that the energy doesn’t have to go anywhere. It simply disappears. If it went somewhere then it would be conserved.

These photons were originally gamma and are now 3K microwaves!

Note that because of the local conservation of energy if you follow an individual bit of CMB light (a null geodesic) you can attribute the change to ordinary Doppler shift between the surface of last scattering and the receiver. But when you add that up globally you get non-conservation.

The energy lost by the photons does work on expanding the universe and if the universe were to start contracting the photons would all blue shift?

Well, that is half true. It doesn’t do work expanding the universe (if it did then energy would be conserved). But yes, if the universe were contracting then we would have non-conservation in the opposite direction.

 

[Orodruin]

Surely GR can just wish all this energy away. These photons were originally gamma and are now 3K microwaves!

A photon cannot intrinsically be a gamma or a microwave photon. It depends on the reference frame.

 

[Jimmy87]

Conservation of energy in GR is a little tricky. Locally it is always conserved based on the EFE, but globally it is only conserved in a special class of spacetimes where there is a symmetry in time. By Noether’s theorem that symmetry in time is associated with conservation of energy. In particular, the FLRW spacetime used in cosmology does not possesses that symmetry, so the most important example of energy non-conservation is cosmology.

Non-conservation means precisely that the energy doesn’t have to go anywhere. It simply disappears. If it went somewhere then it would be conserved.

Note that because of the local conservation of energy if you follow an individual bit of CMB light (a null geodesic) you can attribute the change to ordinary Doppler shift between the surface of last scattering and the receiver. But when you add that up globally you get non-conservation.

Well, that is half true. It doesn’t do work expanding the universe (if it did then energy would be conserved). But yes, if the universe were contracting then we would have non-conservation in the opposite direction.

Thanks for such detailed answers people. So when you say I am half right about the classical analogy bit do you mean that some of the photon energy goes into driving the expansion or that side of it is completely wrong. I know that some of the expansion is caused by dark energy.

 

[PAllen]

Another way to look at it, my preference, is that before you can talk of energy conservation, you have to be able to talk about total energy. However, total energy in GR (including gravitational) cannot be defined in covariant form except for asymmptotically flat spacetime or stationary spacetime (which has a timelike kvf, or symmetry). If total energy cannot be defined globally or over a large region, then the question of energy conservation cannot even be asked . Thus rather than say "energy conservation is violated" or "energy is not conserved" in GR, in general, I prefer "total energy cannot be defined" in GR, in general, so asking about conservation is meaningless.

 

[PAllen]

A point related to redshift can be made, though. Consider, rather than a realistic cosmology, an isolated explosion (say, a supernova) in an otherwise empty universe. Here, there would be gravitational redshift of emitted photons that would be compensated unambiguously with gravitational energy using the ADM formulation of total energy for asymptotically flat spacetime. However, for an isotropicaly expanding universe, there is no way unambiguous way to total gravitational energy to even attempt to balance it against redshift.

 

[Dale]

So when you say I am half right about the classical analogy bit do you mean that some of the photon energy goes into driving the expansion or that side of it is completely wrong.

You were right that in a contracting universe we would get blue-shift rather than red-shift. There is no work involved, that was the wrong part.

The photons don't drive the expansion in terms of pushing and doing work. However, they do have stress-energy so they contribute to the overall evolution of the spacetime as any source of stress-energy does.

 

[Jimmy87]

You were right that in a contracting universe we would get blue-shift rather than red-shift. There is no work involved, that was the wrong part.

The photons don't drive the expansion in terms of pushing and doing work. However, they do have stress-energy so they contribute to the overall evolution of the spacetime as any source of stress-energy does.

Thanks. I don’t know much of the specifics of GR - could you explain what you mean by photons having stress-energy? Also, is it true that the expansion of the universe is driven by dark energy - so can you say that the dark energy is doing work to expand the universe?

 

[phinds]

Also, is it true that the expansion of the universe is driven by dark energy

yes (sort of ... see the next answer)

so can you say that the dark energy is doing work to expand the universe?

No. "Dark Energy" is really just a holding phrase, short for "we don't know WHAT the hell is causing it but SOMETHING is and we need a name for it just so we can talk about it". If we knew for sure that it was a form of energy then the answer would, of course, be yes. It could just be something about the geometry of space-time that we don't yet understand.

 

[PeterDonis]

could you explain what you mean by photons having stress-energy?

"Stress-energy" is a general term in GR for any kind of "stuff" that affects the spacetime geometry. The mathematical object involved is the stress-energy tensor:

https://en.wikipedia.org/wiki/Stress–energy_tensor

The "electromagnetic stress-energy tensor" described in that article is the one that would be applicable to photons.

 

[PAllen]

Thanks. I don’t know much of the specifics of GR - could you explain what you mean by photons having stress-energy? Also, is it true that the expansion of the universe is driven by dark energy - so can you say that the dark energy is doing work to expand the universe?

Please, forget completely about work in relation to cosmological expansion. Dark energy changes nothing about this. The galaxies receding from each other are in free fall, with no forces, so there is no work. Period. It also takes no work to “expand space” because vacuum is not something you can do work on. One analogy I can offer is to think of an inverted cone. Each higher horizontal slice through it is bigger, but no work is needed to make this happen. It is just a feature of the geometry of the cone. Same with spacetime - the existence of accelerated expansion rates is a feature of the spacetime geometry. Dark energy is (possibly) a field that determines this geometry, not a force that does work.

 

[Ibix]

Thanks. I don’t know much of the specifics of GR - could you explain what you mean by photons having stress-energy? Also, is it true that the expansion of the universe is driven by dark energy - so can you say that the dark energy is doing work to expand the universe?

The stress-energy tensor is the thing in GR that does the same job (conceptually) that mass does in Newtonian gravity - it's the source. It's a moderately complicated object, including elements that correspond to stresses and energies. Radiation obviously carries energy and momentum, so it is a source term for gravity (although often in a rather different way than matter).

Dark energy does not drive expansion any more than light does. Its presence just means that the expansion rate as a function of time is different to that of a universe without dark energy. Or, to put it another way, that the observed expansion rate over time cannot be accounted for by any combination of matter and radiation, so we propose an as-yet-unknown type of stuff labelled "dark energy" that makes the maths work out. That makes it sound a bit like a fudge factor - but it ends up explaining at least one other observation (the "Integrated Sachs-Wolfe effect", something to do with the way light behaves as it crosses over and under-dense regions - @kimbyd knows more than me [not difficult!]). So it's speculative, but pretty solid speculation.

 

[pervect]

There are circumstances under which energy is conserved in GR. The most important (IMO) is the case of an asymptotically flat space-time. The OP says he is "familiar with GR", so perhaps this means something to him as-is.

In this case, where you have asymptotically flat space-time, we have a good definition of conserved energy, the so-called ADM energy.

In case "asymptotically flat" doesn't mean anything to the OP, I'll add some comments, that are not however rigorous.

Suppose one has a vacuum, and suppose this vacuum is flat, as in special relativity. If you put a finite lump of "stuff" in this empty space-time, far enough away from the lump of stuff, space-time will go back to being nearly flat, and you'll have an asymptotically flat space time.

In this case, we have a well-defined notion of energy, which is called the ADM energy.

An infinite universe, filled with mass-energy, however, doesn't have this property of asymptotic flatness. So, we can't use this notion of energy to define a conserved energy.

The problem with energy in GR was (I believe) first noticed by Hilbert. He asked his friend, Emily Noether, to investigate the issue, resulting in what physicists call "Noether's theorem", which relates energy conservation to time-translation symmetries.

 

[Jimmy87]

Brilliant - thanks to everyone for taking their time for writing such detailed and understandable answers!