Questions about paper "How closed is cosmology"

[KleinMoretti]

TL;DR Summary

A new paper discusses how a closed system like the universe can exhibit features of open systems.

In the paper, the authors argue that a closed system can exhibit features of an open system one of the being the non-conservation of energy, my questions how can a closed system have non-conservation of energy. (I know in GR, conservation of energy is subtle issue but it doest seem like the authors are referring to the type of non-conservation of energy present In GR)

 

[PeterDonis]

in GR, conservation of energy is subtle issue but it doest seem like the authors are referring to the type of non-conservation of energy present In GR

Yes, they are. They are not talking about "closed systems" in general. They are talking about a closed cosmology, i.e., a closed universe--where "closed" here means "spatially finite but unbounded". For example, an FRW universe containing only matter with density greater than the critical density. They are then simply choosing a definition of the global "energy" of such a universe that results in it not being conserved. Which is just an example of the subtle issue in GR with global conservation of energy.

 

[KleinMoretti]

the reason I said that is because they say that non conservation of energy is a feature of their model as oppose to a regular closed system where energy is conserved. "Throughout this work, we have investigated different inequivalent ways in which dynamical systems can be considered closed. These different notions include closure in the sense of conserved energy, conserved measure density and dynamical autonomy. While we found merits for each, no single notion of closure was completely adequate for all purposes. In particular, we found conven- tional notions of conserved energy and measure to be wanting when applied to the cosmological setting. In that context, we compared a closed (in the sense of preserved measure density and Hamiltonian) symplectic description of cosmology to an open contact formulation. We found that the latter should be taken to give a better complete description of the system.

Central to our argument was a requirement to treat as physical only those structures that are strictly necessary for maintaining empirical adequacy. The symplectic ‘closed’ descriptions violate this principle because they treat as distinct states that differ only by the overall scale of the universe (as determined by v), which is not empirically relevant. In contrast, the contact description does satisfy this principle as it excludes v. While this description does not conserve energy or measure density, it is nevertheless autonomous in the empirically sufficient variables, and is thus ‘closed’ in the third sense discussed above."

 

[PeterDonis]

they say that non conservation of energy is a feature of their model as oppose to a regular closed system where energy is conserved

That's because they are choosing a very unusual definition of a "closed" system.

For example, they are defining a damped harmonic oscillator as "closed" on the grounds that they can define its dynamics to be "autonomous", by which they mean a self-contained set of equations. But that set of equations includes a damping constant, which they are treating as just a magical property of the system. In any actual oscillator, however, damping arises because of interactions with other systems (for example, friction), and those interactions exchange energy. The paper is simply ignoring this and pretending that they can still treat such as system as "closed". In ordinary physics terminology, such a damped oscillator would not be closed, and non-conservation of its energy would simply be a result of energy being dissipated to other systems through the damping interactions.

 

[KleinMoretti]

yes I saw the their damping oscillator example and that I can understand, what im confused about is how they use that same logic in cosmology, in the oscillator example like you said there is a dissipation of energy due to damping interactions, but when talking about the universe where there is no interactions with an outside environment, what is the physical consequence of the non-conservation of energy present in their model.

 

[PeterDonis]

what is the physical consequence of the non-conservation of energy present in their model.

I'm not sure, and I'm not sure the authors of the paper have considered that question. The usual answer in GR is none--that if you find a definition of "total energy of the universe" that isn't conserved, that's just a consequence of the issues in GR that you referred to before, and it doesn't have any physical meaning.

 

[KleinMoretti]

I'm not sure, and I'm not sure the authors of the paper have considered that question. The usual answer in GR is none--that if you find a definition of "total energy of the universe" that isn't conserved, that's just a consequence of the issues in GR that you referred to before, and it doesn't have any physical meaning.

another confusing thing is the supposed reason why their model is better, "Note that the Hamiltonian is constraint and must be zero — a fact that follows from the time reparametrization invariance of
general relativity. From the existence of these structures it is easy to verify that the ‘energy’ (the Hamiltonian) and the measure Ω = ω1+n where n is the number of scalar fields, are conserved over time. Furthermore, from Hamilton’s equations we can find an equations of motion of each of the phase-space variables written in terms of itself and the other phase-space variables. Our system is therefore autonomous. It would hence seem apparent that cosmology is a closed system following all three notions that we have discussed.
There is, however, a catch: The algebra generated by the set of variables just described is strictly larger than what is needed for empirical adequacy. In particular, the volume v can be set to any value by re-scaling its initial condition without affecting any empirical predictions of the theory. This is because such predictions only ever rely on the value of v relative to some reference value (usually taken to be an initial or final condition).
It is clear from these that the dynamical algebra closes in terms of the φ and H without ever needing to refer to v. We should therefore seek a description of our system that never makes reference to v in the first place."

it seems that to me that the presence of non-conservation in their model is intentional.
"In contrast, the contact description does satisfy this principle as it excludes v. While this description does not conserve energy or measure density, it is nevertheless autonomous in the empirically sufficient variables, and is thus ‘closed’"

 

[billtodd]

Yes, they are. They are not talking about "closed systems" in general. They are talking about a closed cosmology, i.e., a closed universe--where "closed" here means "spatially finite but unbounded". For example, an FRW universe containing only matter with density greater than the critical density. They are then simply choosing a definition of the global "energy" of such a universe that results in it not being conserved. Which is just an example of the subtle issue in GR with global conservation of energy.

I don't understand this claim. if it's spatially finite then it should be bounded, if it's unbounded then it cannot be spatially finite.
Do you physicists also think logically?
Sorry if I offended anyone by this question.

 

[PeterDonis]

if it's spatially finite then it should be bounded, if it's unbounded then it cannot be spatially finite.

No. A 3-sphere is spatially finite but unbounded.

Do you physicists also think logically?
Sorry if I offended anyone by this question.

The question is not offensive, but it does show that you should be really, really hesitant about making such remarks, since any lack of logic in this case is on your side.

 

[Bandersnatch]

No. A 3-sphere is spatially finite but unbounded.

Finite but >without a boundary<. The term 'bounded' when referring to metric spaces is tantamount to finite. A 3-sphere is boundaryless and bounded. E.g.:
https://mathworld.wolfram.com/BoundedSet.html
https://en.wikipedia.org/wiki/Bounded_set

 

[PeterDonis]

Finite but >without a boundary<.

Yes, that would be the precise term.

 

[billtodd]

No. A 3-sphere is spatially finite but unbounded.


The question is not offensive, but it does show that you should be really, really hesitant about making such remarks, since any lack of logic in this case is on your side.

Well the sphere is bounded by the fact that it's the boundary between the interior of a 4-d ball and the outside of the ball.

But the universe if it's a closed system then it's all what there is if it's finite in space or infinite we cannot really know.

 

[billtodd]

Finite but >without a boundary<. The term 'bounded' when referring to metric spaces is tantamount to finite. A 3-sphere is boundaryless and bounded. E.g.:
https://mathworld.wolfram.com/BoundedSet.html
https://en.wikipedia.org/wiki/Bounded_set

one should be really careful with words and their context.
As I am quite acquainted with both maths and physics.

 

[Ibix]

Well the sphere is bounded by the fact that it's the boundary between the interior of a 4-d ball and the outside of the ball.

...but by making this argument you are presupposing a 5-or-more dimensional background in which the universe is embedded, just as the 3d surface of the Earth is embedded in a 3d space in which we are free to move. That is not the model under discussion, which is one where the 4d universe (closed or otherwise) is all there is. There is no "inwards" or "outwards" direction, so no boundary.

 

[PeterDonis]

Well the sphere is bounded by the fact that it's the boundary between the interior of a 4-d ball and the outside of the ball.

First, that's not the sense in which I was using the term "bounded". I was using to mean "without boundary", which @Bandersnatch correctly pointed out is the more precise term.

Second, as @Ibix pointed out, our best current model of our universe does not include an embedding in any higher dimensional space, and we have no evidence for any such embedding, so to our best current knowledge there is no "4-d ball" or "outside".

But the universe if it's a closed system then it's all what there is if it's finite in space or infinite we cannot really know.

Our best current model says that our universe is spatially infinite. However, it is still possible (though considered unlikely) that it is spatially finite but very much larger than our observable universe.

 

[Ken G]

First, that's not the sense in which I was using the term "bounded". I was using to mean "without boundary", which @Bandersnatch correctly pointed out is the more precise term.

Yes, I thought it was completely clear that in the process of addressing this extremely subtle article that brings up a lot of new ways to think about the meaning of the terms "open" and "closed" in gravitational physics, you were saying that they mean "closed" as in unable to exchange anything across a boundary because there is no boundary, even though it is finite in size. Being finite is important because that's the only way to have a global energylike quantity that is not infinite, thus avoiding troublesome issues. So @PeterDonis 's herculean effort to bring cogence to this obviously very advanced and mathematical article should not get hung up on such a simple issue as the semantic difference between "bounded" and "having a boundary," though of course the distinction is important to make

Our best current model says that our universe is spatially infinite. However, it is still possible (though considered unlikely) that it is spatially finite but very much larger than our observable universe.

I think it would actually be rather ironic to even attempt to make that distinction, given how much importance the article attributes to what it calls "empirical adequacy." They seem to be saying, somewhat akin to Einstein saying that physics should be built entirely from invariants, that it should also be built entirely from things that we can actually observe. I think I hear a minimal principle being invoked that any physical characteristic (like a total energy) should only be regarded as relevant to identify if it plays a role in what can be observed, whereas if "empirical adequacy" can be achieved without that quantity, then that quantity should not exist in the theory either. Hence, the distinction between a universe that is truly infinite, versus one that is finite but "without boundary", cannot be made if the distinction occurs beyond any ability to observe it. By that logic, any two theories distinguished only by that difference should never be regarded as distinct theories, and no parameters should be included in those theories that make such a distinction. It then follows that if there is an energylike quantity that is valuable even if not conserved, then it should be restricted to applying to what we can observe, i.e., the energy of the observable universe only. In that case, there very clearly is a boundary, so the only way to rule out transfer across that boundary is to invoke homogeneity, which most cosmologies do anyway, making the whole "boundary" issue rather moot.

 

[pbuk]

one should be really careful with words and their context.
As I am quite acquainted with both maths and physics.

Then you should be familiar with the meaning of unbounded in this context. I believe this was first used by Einstein in the title of Chapter 31 of Relativity: The Special and General Theory (1920): 'The Possibility of a “Finite” and Yet “Unbounded” Universe'.

Edit: the original German title of this chapter is "Die Möglichkeit einer endlichen und doch nicht begrenzten Welt", where "nicht begrenzten" can be translated literally as "unlimited". I am not sure whether the words limit, bound or indeed begrenzt, had the Mathematical meaning in 1920 that they do now.

 

[jtbell]

the original German title of this chapter is "Die Möglichkeit einer endlichen und doch nicht begrenzten Welt", where "nicht begrenzten" can be translated literally as "unlimited".

In German, the word Grenze means "border" or "boundary". To me, nicht begrenzt carries the literal meaning "not bordered" or "not bounded". I don't know whether it necessarily also carries the connotation of "unlimited" or "infinite".

 

[Ken G]

I think what's interesting about attributing such crucial importance to "empirical adequacy" (meaning that no physics theory should ever try to predict more than we could ever possibly observe, and any that do are by definition being equipped with extraneous features that are of no value and should therefore be removed) is that, although GR seems to allow a distinction between "infinite" and "having no boundary", no such distinction should ever exist in any theory of cosmology. This follows from the fact that we already know that any distinction that is made between our universe being infinite and having no boundary would have to be a distinction that exists entirely beyond the observable horizon, hence does not respect this principle of empirical adequacy that the above article attributes such crucial importance to. Put differently, the principle says that the "cosmological principle" that the universe is homogenous on the largest scales can only ever be applied to the observable universe, and there would never be any valid purpose in extending it beyond the observable horizon.

This makes perfect sense, who cares about the difference between a universe that is homogeneous out to the observable horizon, and then does something completely different beyond that horizon, versus one that continues to be homogeneous everywhere. That would be a classic example of angels on a pin, something physics cannot falsify. Therefore, it follows that the distinction between "infinite" and "having no boundary" in our universe can forever be put to bed as an entirely moot, and therefore physically meaningless, distinction, and that particular chapter can be set aside as having no further physical importance.

But one potential weakness in this principle of "empirical adequacy" is it sounds a lot like a principle of "denial of the importance of hypotheticals." That makes sense if one thinks physics should work like saying "given that the universe is the way it is and obeys the laws it does, what is the probability distribution for some given observation." But others think that physics can also include questions like "what is the probability distribution for the universe to be the way it is." In that case, one needs hypotheticals that go beyond empirical adequacy, i.e., one includes in one's models hypotheticals that could never actually be observed (a la multiverse thinking). Whether or not multiverse thinking counts as valid physics is an important question, and I think it challenges the concept of empirical adequacy.