Can a finite universe end in heat death?

[durant35]

Hello guys,

I was reading some models about the topology and size of the universe (always a controversial topic), then a question came to my mind.

It is predicted that our universe will expand until it reaches heat death. Can a closed, finite universe also reach heat death and be described by the standard de Sitter cosmology?

Or in other words, can a closed, finite universe reproduce the standard flat lambda model which prioritizes dark energy as the source of accelerated expansion?

Thanks in advance

 

[phinds]

Speaking from a base of almost pure ignorance, I'd say that it MUST be possible because "finite" does not constrain the size, really, so things could get far enough apart that it would amount to heat death.

 

[Jorrie]

Speaking from a base of almost pure ignorance, I'd say that it MUST be possible because "finite" does not constrain the size, really, so things could get far enough apart that it would amount to heat death.

Yes, even if our universe was mildly closed today (e.g. Omega = 1.5), but with the presently observed cosmological constant, it would eventually have evolved towards a de Sitter universe with Omega =1 and heat death would still follow. This is because the Omega_Lambda would eventually dominate anyway.

 

[Chronos]

A finite universe would imply it is a closed system. Thermodynamics push closed systems towards thermal equilibrium [i.e., heat death]. Thus, heat death is inevitable in a finite universe. Of course, as usual, there is a caveat. Under the rules of thermodynamics a closed system must have a boundary and we already know that boundary conditions and infinity do not play well together. What kind of boundary conditions logically apply to all that exists [the universe]? What is outside/separate from everything that exists, nothing? Nothing cannot possesses any definable properties without forfeiting its claim to be nothing, and would make a terrible excuse for a boundary between all that exists and all that does not exist. Boundary condition get very messy where the universe is involved.

 

[Chalnoth]

Under the rules of thermodynamics a closed system must have a boundary and we already know that boundary conditions and infinity do not play well together.

I don't think this is true. A toroidal-shaped universe would be finite but without any boundary in space. I don't think thermodynamics has anything to say about the potential existence of that possibility.

 

[Chronos]

You can make that argument so long as you are willing to relax the notion of boundaries on a closed system. A system without boundaries is like a definite integral without limits: Difficult to solve and not necessarily continuous across all values. I would agree thermodynamics has little to say about such a possibility because it is no more meaningful than any other singularity that arises in a mathematical function.

 

[durant35]

You can make that argument so long as you are willing to relax the notion of boundaries on a closed system. A system without boundaries is like a definite integral without limits: Difficult to solve and not necessarily continuous across all values. I would agree thermodynamics has little to say about such a possibility because it is no more meaningful than any other singularity that arises in a mathematical function.

How about a closed, spherical universe?

Yes, even if our universe was mildly closed today (e.g. Omega = 1.5), but with the presently observed cosmological constant, it would eventually have evolved towards a de Sitter universe with Omega =1 and heat death would still follow. This is because the Omega_Lambda would eventually dominate anyway.

How can a closed universe evolve to de Sitter space? Can de Sitter have curvature?

 

[Chronos]

A closed spherical universe has a boundary, hence is spared thermodynamic paradoxes. Being finite, however, imposes a heavy enough burden on any closed universe model [what lays beyond everything in existence?]. A de Sitter space, by definition has positive curvature, and positive curvature strongly implies a closed universe- although there are ways closure can be avoided, as Chalnoth noted.

 

[durant35]

A closed spherical universe has a boundary, hence is spared thermodynamic paradoxes. Being finite, however, imposes a heavy enough burden on any closed universe model [what lays beyond everything in existence?]. A de Sitter space, by definition has positive curvature, and positive curvature strongly implies a closed universe- although there are ways closure can be avoided, as Chalnoth noted.

Just for curiosity, what paradoxes did you have in mind while typing the first sentence?

I thought that de Sitter space by definition has zero curvature, hence it is what will happen in an expanding flat universe. Can you expand why de Sitter has positive curvature and does that in someway imply that the whole universe should be finite (and closed)? And what happens to the cosmological horizon in the case of de Sitter space which has curvature? Objects go away from us on curved trajectories?

Sorry for the accumulation of questions, but I think they are all pretty much connected. Thanks for the answers

 

[Bandersnatch]

A closed spherical universe has a boundary, hence is spared thermodynamic paradoxes. Being finite, however, imposes a heavy enough burden on any closed universe model [what lays beyond everything in existence?]. A de Sitter space, by definition has positive curvature, and positive curvature strongly implies a closed universe- although there are ways closure can be avoided, as Chalnoth noted.

This post doesn't sound right. Admittedly, I'm operating beyond the edge of what I understand, so feel free to educate me.
How does a sphere (or hypersphere) have a boundary if it's not embedded in a higher-dimensional space? Your response to Chalnoth's counterexample of toroidal space didn't clarify what you meant.
Why would there be a 'heavy burden' of the type you describe? Finite and closed means that it makes no sense to ask about anything beyond.
Why does positive space-time curvature (i.e. de Sitter space) imply spatially closed universe? It can be sliced into spatial slices of positive, negative or flat curvatures. Using FLRW metric the slicing of de Sitter space along constant t gives flat 3D space, or so I'm told.
I thought Chalnoth's toroidal space (which is flat) example shows that a closed space doesn't need a boundary - you seem to be saying that it shows that a positively curved space doesn't need to be closed. Can you explain what you mean here?

 

[PeterDonis]

A closed spherical universe has a boundary

It does? A spatial slice of such a universe has topology ##S^3##, which is a compact manifold without boundary.

 

[PeterDonis]

I thought that de Sitter space by definition has zero curvature

That is not correct. It has positive curvature, as Chronos said. More precisely, it has positive spacetime curvature (the Ricci scalar is positive everywhere--it's just a multiple of the positive cosmological constant). There are slicings of de Sitter spacetime with positive, zero, and negative spatial curvature.

 

[Jorrie]

A de Sitter space, by definition has positive curvature, and positive curvature strongly implies a closed universe- although there are ways closure can be avoided, as Chalnoth noted.

There may be some confusion between the meanings of "de Sitter space" and a "de Sitter universe". AFAIK, a de Sitter universe is spatially flat, but is expanding exponentially, while de Sitter space is spatially static (and closed). Or do I misunderstand?

 

[PeterDonis]

AFAIK, a de Sitter universe is spatially flat, but is expanding exponentially, while de Sitter space is spatially static (and closed). Or do I misunderstand?

You misunderstand. A de Sitter universe is just de Sitter space by a different name. One possible coordinate chart on de Sitter space corresponds to a spatially flat, exponentially expanding universe. But there is another chart on the same space (same manifold, same geometry) which is spatially static and closed. (These two charts do not cover the same portion of the entire manifold.)

 

[durant35]

You misunderstand. A de Sitter universe is just de Sitter space by a different name. One possible coordinate chart on de Sitter space corresponds to a spatially flat, exponentially expanding universe. But there is another chart on the same space (same manifold, same geometry) which is spatially static and closed. (These two charts do not cover the same portion of the entire manifold.)

Can it be closed and exponentially expanding? Which in some sense highlights the main question in my thread.

 

[PeterDonis]

Can it be closed and exponentially expanding?

These are properties of slicings, not of the spacetime. The spacetime admits a closed slicing (which is static) and an exponentially expanding slicing (which is spatially flat). But it's the same spacetime in both cases.

 

[durant35]

These are properties of slicings, not of the spacetime. The spacetime admits a closed slicing (which is static) and an exponentially expanding slicing (which is spatially flat). But it's the same spacetime in both cases.

I am afraid that I don't understand the thesis presented in the current way.

If the whole universe is not infinite in size (space that you mentioned), but finite - it curves back on itself, therefore we have a closed and finite universe. Right?Now in de Sitter case, space is exponentially expanding. You said that it implies that the topology is flat. Can the space be curved (so that it closes on itself) and still exponentially expanding?

 

[Jorrie]

These are properties of slicings, not of the spacetime. The spacetime admits a closed slicing (which is static) and an exponentially expanding slicing (which is spatially flat). But it's the same spacetime in both cases.

Thanks, I think I understand. Is it then correct to say that the slicing used for the LCDM cosmological model will evolve towards the latter case, as matter density is vanishing? Or must one be more exact than this?

 

[PeterDonis]

Is it then correct to say that the slicing used for the LCDM cosmological model will evolve towards the latter case, as matter density is vanishing?

In the LCDM model, the flat slicing is the only one that is homogeneous and isotropic; the presence of matter density breaks the symmetry of exact de Sitter spacetime. As the matter density gets smaller and smaller compared to the cosmological constant, other possible slicings on de Sitter spacetime will get closer and closer to being homogeneous and isotropic slicings on the actual spacetime of the universe (as they are on exact de Sitter spacetime).

 

[Chalnoth]

You can make that argument so long as you are willing to relax the notion of boundaries on a closed system. A system without boundaries is like a definite integral without limits: Difficult to solve and not necessarily continuous across all values. I would agree thermodynamics has little to say about such a possibility because it is no more meaningful than any other singularity that arises in a mathematical function.

Typically it's actually simpler. You just use periodic boundary conditions, with the periodicity dependent upon precisely how the universe is connected.

 

[durant35]

Since I haven't been given an appropriate answer to my previous question, I will ask another.

How can de Sitter be static and expanding at the same time? Each pach is static and it would seem to the observer that nothing is expanding but the space as a whole would be increasing in size.

 

[PeterDonis]

How can de Sitter be static and expanding at the same time?

Because, as I said, "static" and "expanding" are properties of coordinate charts, not the spacetime itself.

Or, if that's too abstract, think of "static" and "expanding" as describing two different families of observers in de Sitter spacetime. "Static" and "expanding" are properties of the families of observers, not the spacetime itself. The two families of observers split spacetime into "space" and "time" in different ways, so they see a different behavior of "space" with "time".

Each pach is static and it would seem to the observer that nothing is expanding but the space as a whole would be increasing in size.

No, that's not the way it works. See above.

 

[durant35]

Because, as I said, "static" and "expanding" are properties of coordinate charts, not the spacetime itself.

Or, if that's too abstract, think of "static" and "expanding" as describing two different families of observers in de Sitter spacetime. "Static" and "expanding" are properties of the families of observers, not the spacetime itself. The two families of observers split spacetime into "space" and "time" in different ways, so they see a different behavior of "space" with "time".

Okay, now I understand better what this means. But does it make sense to say that the universe is expanding at all if we use this criteria of observer-dependence like you did?

 

[PeterDonis]

does it make sense to say that the universe is expanding at all if we use this criteria of observer-dependence like you did?

Yes, if a particular family of observers is picked out by the spacetime geometry. In the case of our actual universe, which is not exact de Sitter spacetime because it has ordinary matter/energy density present, there is such a family of observers: the "comoving" observers, who see the universe, including its ordinary matter/energy density, as homogeneous and isotropic. No other family of observers does. Those observers see the universe as expanding, and when we say "the universe is expanding", those observers are what we are referring to.

In an exact de Sitter spacetime, with only a positive cosmological constant present (no ordinary matter/energy density), there is no single family of observers that is picked out the way "comoving" observers are in our actual universe: there are multiple families of observers who see the spacetime as homogeneous and isotropic, and they do not all see the universe as "expanding". So it doesn't make sense to say de Sitter spacetime is "expanding" the way we normally use that term in referring to our actual universe.

 

[Chronos]

It is not easy to remove observer dependence from the data. If all observers tend to agree the data suggests something akin to expansion, it is difficult to imagine how the question can be settled in any absolute sense.

 

[Chalnoth]

Since I haven't been given an appropriate answer to my previous question, I will ask another.

How can de Sitter be static and expanding at the same time? Each pach is static and it would seem to the observer that nothing is expanding but the space as a whole would be increasing in size.

de Sitter spacetime is static because it is exactly the same at every point in time. It's also exactly the same at every point in space.

There is no matter at all in de Sitter space-time, but if you place two test particles in the space-time that are far enough away, they will tend to recede from one another.

 

[grauitate]

Hello
as I was invited to see me here,
"Hi grauitate, We look forward to seeing you at Physics Forums"

It is predicted that our universe will expand until it reaches heat death. Can a closed, finite universe also reach heat death and be described by the standard de Sitter cosmology?

I'll try to respond to this interesting question from my point of view, i.e. from a consequent application of the EP (equivalence principle) and Einstein's universe:
It needs some time for me to get aware that a finite and closed universe is adiabatic, i.e. it is thermally a perfectly isolated closed system, i.e. the heat content of mass and energy is constant (also if mass converts into energy and vice versa). At the event horizon of the closed universe any observer at any place at any time sees the CMB (cosmic microwave background), where the photons of the distant objects arriving at the observer are shortly before gravitationally eliminated to zero (z≅1.000...1.500) and vanishing in the far IR.
The expansion of such a closed system is not reasonable and should not take place as it would need an extra generation of matter and energy coming out of nothing.
This is a direct outcome of the definition of a gravitationally closed system (comparable to a huge black hole with the radius R of the event horizon of 13.7 billion lightyears (≅ 1.3e26 m), a mass of M = Rc²/2G ≅ 8,7e52 kg and a density of ρ = M/V = M/(2π²R³) ≅ 2.0e-27 kg/m³. Keeping this a closed system every year of expansion would need an extra ΔM ≅ 6.4e42 kg, (i.e approx. 3.2e12 x sun mass) or an equivalent energy of ΔE = ΔMc² ≅ 5.8e59 J. This painfully hurts the 1. law of energy conservation of gravitationally closed systems. So, expansion is not a reasonable property of such an universe. So never fear, the answer is that a closed, finite universe cannot reach heat death. And an idealistically supposed massless de Sitter universe here is of no relevance.
Please don't ask for any references as this is a straight, logic and axiomatic consequence of Einstein's universe, first developed in 1917 and unfortunately abandoned in the 1930s, when redshift was decided to be identified by expansion (although there were a lot of doubts on this until today) instead by gravitation.
I hope I could answer your question appropriately.

 

[PeterDonis]

At the event horizon of the closed universe

A closed universe has no event horizon.

This is a direct outcome of the definition of a gravitationally closed system (comparable to a huge black hole with the radius R of the event horizon of 13.7 billion lightyears

This is not a closed universe. A black hole is an asymptotically flat spacetime--it is of infinite extent. The concept of an event horizon only makes sense in such a spacetime. And such a spacetime is neither spatially closed nor isolated in the sense you are using the term.

The rest of your post doesn't make sense because of the above errors.

Please don't ask for any references

Sorry, that's not a valid position to take here. Please read the PF rules.

 

[grauitate]

A closed universe has no event horizon.

Any observer looking from insight of the closed universe to the CMB is getting the photons from near the event horizon, where z → ∞ and where the time run (from the observers view and time scale) is stopping and any periodic processes will cease (this is, I think, the general and simply definition of an event horizon).
This conception is reasonable and is observationally manifoldly verified. It is equivalent to the time retardation effects observed e.g. at high z SNe, still today explained by the expansion of the standard model..

This is not a closed universe. A black hole is an asymptotically flat spacetime--it is of infinite extent.

The closed universe has a finite radius, density and mass (energy) content - but its interior, as you say, is infinite or unlimited, you can surve around like on the Earth's surface and you will never find any border, and from the observers point of view, it looks like you are always in the center. And may be, as you say, you are surfing in a flat spacetime.

Sorry, that's not a valid position to take here. Please read the PF rules.

Sorry, things which are immediately logic and axiomatic (no proof needed) should be allowed to be mentioned here - otherwise one will never get any concise and definite base for discussion and everything ends in a terrible mismatch and confusion.

 

[PeterDonis]

where the time run (from the observers view and time scale) is stopping and any periodic processes will cease (this is, I think, the general and simply definition of an event horizon

No, it isn't. The definition of an event horizon is: the boundary of a region of spacetime that cannot send outgoing light signals to future null infinity. It is still perfectly possible for timelike worldlines to cross the horizon in the ingoing direction, and time does not "stop" there and periodic processes do not cease. Also, in order for there to be an event horizon at all, as you can see from the definition, there has to be a future null infinity, and a closed universe does not have one.

And may be, as you say, you are surfing in a flat spacetime.

An asymptotically flat spacetime, which is not the same as a closed universe: the two are mutually inconsistent.

things which are immediately logic and axiomatic (no proof needed) should be allowed to be mentioned here

Nothing you have said is "immediately logic and axiomatic". In fact much of what you have said is simply wrong. See above.

Also, asking for references is not asking for a logical proof. It's asking for evidence that your claims are based on mainstream science. Which yours are not, as far as I can see.

 

[grauitate]

It is still perfectly possible for timelike worldlines to cross the horizon in the ingoing direction, and time does not "stop" there and periodic processes do not cease. Also, in order for there to be an event horizon at all, as you can see from the definition, there has to be a future null infinity, and a closed universe does not have one.

I'll try to answer as carefully as possible, because I assume there is a fundamental misunderstanding:
Any inside observer in an finite and closed universe can observe the wall of the CMB at z ≅ 1,000...1,500, where all periods of processes are observed slower in the same range as well. Only from the observers view shortly behind this CMB (approx. 10...100 kpc) wall the redshift gallops against z →∞ (but not detectable any more) - of course, that space area is the same like everywhere insinde the universe and does not differ from anywhere inside - this wall, as the word "horizon" suggests, is only an appearance of the observer and has nothing to do with BHs phenomena when radiation or matter is coming from outside in.
Of course, this wall or horizon does not exist in reality in the apparent local area.
The finite and closed universe has no outside, there is only inside. Its dynamic is not the expansion, but it is the splendid and naturally balancing flow processes of structure creation and dissolution taking place visibly and everlasting in the range of our eyes and telescopes from radio to gamma ray frequencies.

Regarding mainstream science, this is not a science covered with concrete. My understanding is that the "salt in the soup" of science should be discussion of proposals at the edge of mainstream or outside to create theories which can avoid the severe problems still existing e.g. in the standard model.

Although invited, I admit that this forum may be is not the appropriate for me. Nevertheless, thank you for your responses.

 

[phinds]

Regarding mainstream science, this is not a science covered with concrete. My understanding is that the "salt in the soup" of science should be discussion of proposals at the edge of mainstream or outside to create theories which can avoid the severe problems still existing e.g. in the standard model..

Yes, that's true, it's just NOT the purview of this forum. Here the focus is ONLY on mainstream science, not beyond the boundaries of it.

 

[PeterDonis]

I assume there is a fundamental misunderstanding

I agree, but I think you're mistaken about whose misunderstanding it is. See below.

Any inside observer in an finite and closed universe can observe the wall of the CMB at z ≅ 1,000...1,500, where all periods of processes are observed slower in the same range as well.

Yes.

Only from the observers view shortly behind this CMB (approx. 10...100 kpc) wall the redshift gallops against z →∞ (but not detectable any more)

No. There are at least two errors here:

(1) The 10...100 kpc you are quoting is not an actual distance; it's a notional number that doesn't correspond to anything anyone actually measures. It certainly does not correspond to a "distance behind the CMB" that some event horizon is at (since there is no event horizon in a closed universe).

(2) In an idealized closed universe model, ##z \rightarrow \infty## corresponds to the initial singularity. In other words, as seen by any observer, any event in the universe (i.e., after the initial singularity) has a finite ##z##. This is just another way of saying that there is no event horizon in a closed universe.

this wall

There is no "wall" in a closed universe. See above. You are fundamentally mistaken about what a closed universe model says.

The finite and closed universe has no outside, there is only inside.

This is true of any FRW spacetime, not just a closed one.

My understanding is that the "salt in the soup" of science should be discussion of proposals at the edge of mainstream or outside to create theories which can avoid the severe problems still existing e.g. in the standard model.

PF has a "Beyond the Standard Model" forum where hypotheses about how to extend existing physics can be discussed. But even there you have to have some basis for discussion: some concrete proposal (usually published in a peer-reviewed paper) that makes testable predictions.

However, that is irrelevant to this discussion in any case, because the closed universe model is not an extension of existing physics; it's built using standard General Relativity and the Standard Model and requires no assumptions beyond what is currently known. But in order to discuss it, you have to understand what it actually says, and you don't appear to, as I've explained above.