"Energy is not conserved" vs. energy is conserved: Friedmann Equations

[timmdeeg]

TL;DR Summary

It seems there are two ways to think about the energy conservation in the universe.

First, "Energy is not conserved" as e.g. explained by Sean Carroll in https://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/ .

Second, the Friedmann Equations are expressed in energy conservation, e.g. https://core.ac.uk/download/pdf/25318877.pdf equation (16).

Do we talk about two different "kinds" of energy, one which is conserved and another one which isn't?

In fact, thinking of the universe as a closed adiabatic system then according to the first law of thermodynamics the energy is conserved:
dU + PdV = 0, whereby U is the internal energy of the universe. But this assumes that pressure is doing work which rises the question how this is possible as the pressure is the same everywhere.

Final question. It it correct that in an expanding universe time-translation invariance does not hold? Which would mean that energy is not conserved.

 

[Old Person]

Hi.

TL;DR Summary: It seems there are two ways to think about the energy conservation in the universe.

Second, the Friedmann Equations are expressed in energy conservation, e.g. https://core.ac.uk/download/pdf/25318877.pdf equation (16).

I have only glanced at that article. It seems that Eqn 16 is based on a conservation of mass and not a conservation of energy. It seems to be the analogue of what was developed as Eqn 5 - the conservation of mass - which was based on purely Newtonian assumptions and appeared earlier in the article.

Eqn 14 and eqn 15 in that article are the Friedmann equations. Overall, it doesn't look like they were trying to describe the Friedmann equations as something resembling a conservation of energy. However, I must make it clear again that I have only made a cursory inspection of the article.

TL;DR Summary: It seems there are two ways to think about the energy conservation in the universe.

Final question. It it correct that in an expanding universe time-translation invariance does not hold? Which would mean that energy is not conserved.

Yes, that is the usual final conclusion if you want it in just one short sentence. Our understanding of what "Energy" is and how it behaves should be based on Noether's theorem. So if the system is described with a Lagrangian that does not have time translation invariance then there is no conserved quantity like "Total Energy" that can be identified.

All the usual caveats apply: Most of physics involves small sub-systems of the entire universe and only consider their behaviour over small time scales (compared to cosmological time scales). As such those can be well approximated as systems with time translation invariance and then the law of conservation of energy applies.

Best Wishes.

 

[PeterDonis]

"Energy is not conserved" as e.g. explained by Sean Carroll

What Carroll means is that there is no global "energy" that is conserved in the spacetime that describes our universe (or indeed in any non-stationary spacetime).

the Friedmann Equations are expressed in energy conservation, e.g. https://core.ac.uk/download/pdf/25318877.pdf equation (16).

That equation is not an equation of global energy conservation. It is an equation of local stress-energy conservation. Local conservation of stress-energy is guaranteed by the Einstein Field Equation and applies to any spacetime.

Do we talk about two different "kinds" of energy, one which is conserved and another one which isn't?

Sort of. See above.

thinking of the universe as a closed adiabatic system

It isn't, at least not in the sense you are used to. You can do thermodynamics in curved spacetime, but it's more complicated than ordinary thermodynamics. And it will still end up telling you that there is no global conserved energy in a non-stationary curved spacetime.

 

[PeterDonis]

It seems that Eqn 16 is based on a conservation of mass and not a conservation of energy.

It's local conservation of stress-energy, which is not the same as either "mass" or "energy".

 

[timmdeeg]

Yes, that is the usual final conclusion if you want it in just one short sentence. Our understanding of what "Energy" is and how it behaves should be based on Noether's theorem. So if the system is described with a Lagrangian that does not have time translation invariance then there is no conserved quantity like "Total Energy" that can be identified.

All the usual caveats apply: Most of physics involves small sub-systems of the entire universe and only consider their behaviour over small time scales (compared to cosmological time scales). As such those can be well approximated as systems with time translation invariance and then the law of conservation of energy applies.

Thanks for your comments, very appreciated!

 

[timmdeeg]

That equation is not an equation of global energy conservation. It is an equation of local stress-energy conservation. Local conservation of stress-energy is guaranteed by the Einstein Field Equation and applies to any spacetime.

Ah, I see. Thanks, this clarifies my question.

 

[ohwilleke]

What Carroll means is that there is no global "energy" that is conserved in the spacetime that describes our universe (or indeed in any non-stationary spacetime). . . .

You can do thermodynamics in curved spacetime, but it's more complicated than ordinary thermodynamics. And it will still end up telling you that there is no global conserved energy in a non-stationary curved spacetime.

Honestly, I am surprised that the lack of global conservation of energy doesn't receive more attention as an "unsolved problem of physics" in the same sense that "problems" like the hierarchy problem, the strong CP problem, and the matter-antimatter asymmetry of the universe issue do.

You can say that it is full explained by the equations of GR with a cosmological constant, so it is "solved" but that's pretty dismissive when conservation of mass-energy both locally and globally is a bedrock principle in so many other areas of physics.

 

[PeterDonis]

I am surprised that the lack of global conservation of energy doesn't receive more attention as an "unsolved problem of physics"

I think the lack of attention is because most physicists do not consider it a problem, just a fact about curved spacetimes that don't have a well-defined ADM or Komar energy.

conservation of mass-energy both locally and globally is a bedrock principle in so many other areas of physics

Locally, yes, but GR has local conservation of stress-energy.

In what other areas of physics is global energy conservation, where "global" means "covering the entire universe", a bedrock principle?

 

[Vanadium 50]

Why would it be a problem?

You get energy conservation in systems that are time-translation invariant. The universe isn't.

 

[timmdeeg]

It's local conservation of stress-energy, which is not the same as either "mass" or "energy".

Why isn't "local conservation of stress-energy" pars pro toto if we remember the cosmological principle?

In other words, why can't I say if the adiabatic Friedmann Equation - which expresses conservation of energy - holds locally then this implies that it holds for the entire universe assuming the cosmological principle?

The Friedmann Equations "don't care" about about time-translation invariance, at least as I understand it. And Noethers theorem refers to the symmetry of the whole universe. This seems to mean that if energy is conserved locally but not globally doesn't exclude each other. Would you agree to that?

 

[PeterDonis]

why can't I say if the adiabatic Friedmann Equation - which expresses conservation of energy - holds locally then this implies that it holds for the entire universe assuming the cosmological principle?

Because it would be wrong. Global conservation is not the same thing as local conservation. Local conservation is a differential law, obtained by taking the covariant divergence of the Einstein Field Equation. It holds at every event--which is consistent with the cosmological principle--but it's still local.

Global conservation is an integral law: you have to have some invariant way of picking out an integral that describes the "total energy". The only spacetimes in which such an invariant exists are asymptotically flat spacetimes (but even then there are two such integrals, not one--the ADM energy and the Bondi energy) and stationary spacetimes (for which the Komar energy is the invariant). For other spacetimes, no such invariant integral exists.

The cosmological principle says that the universe should be the same locally everywhere. It makes no sense to say the universe is globally "the same everywhere", since a global property is not a property that could vary or not vary "from place to place"; it's an integral property.

The Friedmann Equations "don't care" about about time-translation invariance

If "don't care about" means "don't have", yes, this is correct. FRW spacetimes are not stationary, and they are not asymptotically flat. So they don't fall into either of the two classes of spacetimes above that have invariant global "total energy" integrals. So they don't have any.

Noethers theorem refers to the symmetry of the whole universe

Noether's theorem for energy requires a timelike Killing vector field, i.e., it requires the spacetime to be stationary. FRW spacetimes are not stationary, so Noether's theorem for energy does not apply to them.

This seems to mean that if energy is conserved locally but not globally doesn't exclude each other.

If you mean that energy in a spacetime that isn't stationary or asymptotically flat can be conserved locally but not globally, yes, this is correct.

 

[timmdeeg]

Thanks for these great answers!

Global conservation is an integral law: you have to have some invariant way of picking out an integral that describes the "total energy". The only spacetimes in which such an invariant exists are asymptotically flat spacetimes (but even then there are two such integrals, not one--the ADM energy and the Bondi energy) and stationary spacetimes (for which the Komar energy is the invariant). For other spacetimes, no such invariant integral exists.

So global energy conservation depends on the "kind" of spacetime, good to know how. So an example where global energy conservation holds would be Schwarzschild spacetime.

 

[PAllen]

Global conservation is an integral law: you have to have some invariant way of picking out an integral that describes the "total energy". The only spacetimes in which such an invariant exists are asymptotically flat spacetimes (but even then there are two such integrals, not one--the ADM energy and the Bondi energy) and stationary spacetimes (for which the Komar energy is the invariant). For other spacetimes, no such invariant integral exists.

But note, the Bondi energy is, by construction, not necessarily conserved even when it is well defined. The difference between the conserved ADM energy and Bondi energy is the energy of radiation (including gravitational) reaching infinity. Thus, the Bondi energy decreases for an isolated pair of co-orbiting bodies embedded in asymptotically flat spacetime, for example.

 

[PeterDonis]

So an example where global energy conservation holds would be Schwarzschild spacetime.

Yes. The "mass" $M$ in the metric is the ADM energy of the spacetime.

 

[cianfa72]

Local conservation is a differential law, obtained by taking the covariant divergence of the Einstein Field Equation. It holds at every event--which is consistent with the cosmological principle--but it's still local.

EFE is a tensorial equation. The covariant divergence of the tensor $T^{\mu \nu}$ should mean take the covariant derivative of the tensor and contract a pair of upper/lower indices i.e. contract respectively the lower index $\gamma$ with the upper index $\mu$ of $$\nabla_{\gamma} T^{\mu \nu} = (\nabla T)_{\gamma}{}^{\mu \nu}$$ to get $$(\nabla T)_{\mu}{}^{\mu \nu} = \nabla_{\mu} T^{\mu \nu}$$ Does it make sense?

 

[PeterDonis]

The covariant divergence of the tensor $T^{\mu \nu}$ should mean take the covariant derivative of the tensor and contract a pair of upper/lower indices i.e. contract respectively the lower index $\gamma$ with the upper index $\mu$ of $$\nabla_{\gamma} T^{\mu \nu} = (\nabla T)_{\gamma}{}^{\mu \nu}$$ to get $$(\nabla T)_{\mu}{}^{\mu \nu} = \nabla_{\mu} T^{\mu \nu}$$ Does it make sense?

Yes, that's correct.

 

[cianfa72]

Global conservation is not the same thing as local conservation. Local conservation is a differential law, obtained by taking the covariant divergence of the Einstein Field Equation. It holds at every event--which is consistent with the cosmological principle

So, local conservation of energy in GR tells us that at any event the flux of stress-energy tensor over the surface enclosing an infinitesimal volume of spacetime around it is null.

Global conservation is an integral law: you have to have some invariant way of picking out an integral that describes the "total energy". The only spacetimes in which such an invariant exists are asymptotically flat spacetimes (but even then there are two such integrals, not one--the ADM energy and the Bondi energy) and stationary spacetimes (for which the Komar energy is the invariant). For other spacetimes, no such invariant integral exists.

Noether's theorem for energy requires a timelike Killing vector field, i.e., it requires the spacetime to be stationary. FRW spacetimes are not stationary, so Noether's theorem for energy does not apply to them.

So, for a stationary spacetime, the Komar energy is actually the "quantity" that is conserved according the Noether's theorem.

 

[PeterDonis]

local conservation of energy in GR tells us that at any event the flux of stress-energy tensor over the surface enclosing an infinitesimal volume of spacetime around it is null.

Yes.

for a stationary spacetime, the Komar energy is actually the "quantity" that is conserved according the Noether's theorem.

Yes.