> This is not correct. First, there is no concept of "energy conservation" that applies to the universe as a whole the way you describe.
If the concept of energy conservation does not apply to the entire Universe, then we must ask where the additional energy created comes from or is removed. I'm not aware of evidence to suggest that it is not conserved. If there was something like this, it seems like something that could be exploited.
> Second, our best current model of the universe, based on the equations of General Relativity, says it will keep expanding forever.
I'm aware that's what is currently supported, but the collapsing Universe followed by the expanding Universe loop ties everything up nicely (cyclic model [1]).
I'm also aware that this is essentially based on the second law of thermodynamics [2], but I think as long as all processes a system undergoes are fully reversible, it should in theory be able to reverse. It would be like observing a binary counter: 0000 is the start, then we observe numbers like 0110 and 1011 and say "see, entropy is increasing indefinitely", but eventually we tick over to 1111, then 0000.
Of course if any system variable changes irreversibly (i.e. space expands infinitely), then it would not work. But given there is ongoing debate about the Hubble constant [3] and I think there is still room to believe a cyclic model could be plausible.
> If the concept of energy conservation does not apply to the entire Universe, then we must ask where the additional energy created comes from or is removed.
No, we musn't, because the reason why the concept of energy conservation does not apply to the entire universe is that there is no well-defined concept of "energy of the universe as a whole" to begin with. So you can't even frame a well-defined question about whether this thing that doesn't exist is conserved or not.
> the collapsing Universe followed by the expanding Universe loop ties everything up nicely (cyclic model [1]).
This is a speculative model which at present has no evidence that favors it over our best current model. Of course we might discover more evidence in the future that would change that. If that did happen in future, then yes, we would have to reassess the "energy of the universe as a whole" question as well. I am not familiar enough with the details of cyclic models to know whether there is a well-defined "energy of the universe as a whole" in them or not.
According to the heat death of the universe theory which holds true if Einstein's cosmological constant is positive, every atomic / sub-atomic particle will eventually be in its own “observable universe”, spaced out enough that even moving at the speed of light away from any particle you would never reach another particle.
> if Einstein's cosmological constant is positive, every atomic / sub-atomic particle will eventually be in its own “observable universe”
This is not correct. Gravitationally bound systems will remain bound in a universe with a positive cosmological constant. A "Big Rip" scenario, where gravitationally bound systems get ripped apart and eventually every elementary particle is beyond the cosmological horizon of every other, requires "phantom energy", i.e., a density of dark energy that increases with time.
Light from sources at cosmological distances is redshifted, and we have several excellent lines of evidence for that, notably https://en.wikipedia.org/wiki/Lyman-alpha_forest which clearly shows that the emitted light was redshifted progressively as it reached intervening clouds of hydrogen on its way to our spectrographs.
Energy is directly proportional to wavelength. For light, E = hf, where h is Planck's constant. It's inversely proportional to wavelength, E = (hc)/λ, where c is the speed of light and λ is the wavelength. Redshifting means longer wavelengths, so less energy at the point of detection compared to the point of emission.
Energy conservation is local. At large scales, photons lose energy. Where does it go?
The connection to what I wrote above is that General Relativity guarantees that at every single point in a general curved spacetime there is a small patch of (quasi-)static flat spacetime. That patch can be ultramicroscopic deep within a black hole, or very large (in human terms) in interplanetary space in our solar system. In that flat patch energy conservation holds, as Carroll described. Outside that patch, we can see light redshift or blueshift through spacetime; the same energy shifts happen to massive particles like cosmic ray electrons, protons and neutrinos, too. It also happens to gravitational waves.
Event horizons break time-reversal symmetry. Nothing comes back from the other side. Generically, in the presence of a relativistic quantum field (like the 17 in the standard model of particle physics), an event horizon radiates greybody radiation comparable to a blackbody with a temperature inversely proportional to the horizon area. To save words, we just use that temperature (and go back to a calculation of the exact spectrum when greater accuracy is required). Black hole event horizons are really cold. Cosmological event horizons are ridiculously cold. And really it's the apparent horizon that has a measurable temperature. There may not be an event horizon in finite time, but in many circumstances the apparent horizon is indistinguishable by experiment. We have evidence of matter crossing to the other side of these horizons (e.g. from tidal disruption events and black hole/neutron star collisions); we have decent upper limits on horizon temperatures for a small handful of black hole horizons. (And someday we will have precision laboratory evidence of Unruh event horizon temperatures).
If the universe undergoes recollapse, black holes will tend to merge into ever bigger black holes as galaxy clusters freely-fall closer together (and then interact with each other, merging into gargantuan elliptical things with silly large velocity dispersions generating all sorts of collision & merger opportunities). So you need some unknown extra step to undo the time-reversal invariance violation of black hole horizons. Have any ideas?
The bright side is that black holes hold so much entropy that you aren't in any danger of violating the second law of thermodynamics via recollapse, unless your idea does that accidentally. The entropy is easy to see in a Boltzmann sense. Given the "no hair" conjecture, a black hole is fully described by a small number of parameters. Those parameters are a macrostate. A microstate is a configuration of things that fell into the black hole imparting its mass and spin and charge(s)). Entropy is proportional to the log of all the possible microstates for that macrostate. As more matter goes into a black hole, its entropy skyrockets. (The increase in BH entropy absolutely dwarfs the decrease in vacuum entropy; since hard vacuum is a macrostate, and every reasonable volume of hard vacuum is totally substitutable for every other similar volume of hard vacuum, there's a lot of entropy there too. As we fill the vacuum with ejected stars, radiation, high-metallicity dust, and so on as we squash galaxy clusters together during recollapse, that's a loss of entropy. But still nowhere near as low an entropy as the original cold gas clouds that were around after the cosmic microwave background formed. So your idea that recollapse means low entropy is on shaky ground, especially if you are taking your "binary counter ticks back to zero" idea seriously. You can take some comfort that lots of very smart people who can do the maths and have decent physical intuitions (Penrose comes to mind) have struggled with entropy in a cyclic cosmology.
Finally, the Hubble tension at its very strongest changes the age of the the oldest galaxies from almost 14 billion years to almost double that, not whether it the universe is expanding faster now than it was between one and four billion years ago. It's not something on which you want to rest a theory of cosmic recollapse.
While this is a good blog post on this topic, I would point out that Carroll does gloss over one crucial point. He says:
"[T]he best you can rigorously do is define the energy of the whole universe all at once, rather than talking about the energy of each separate piece."
What he is glossing over here is that you can't "rigorously" do this unless your standard of "rigor" is looser than it is supposed to be in GR. Any way of defining "the energy of the whole universe" will involve what are called "pseudo-tensors", which depend on particular choices of coordinates. The standard of rigor in GR is supposed to be that everything that has physical meaning is expressed in terms of tensors, i.e., objects that do not depend on particular choices of coordinates. By that standard, it is not possible to "rigorously" define "the energy of the whole universe", as I pointed out in another post upthread.
This comment started as a short scribble ("I mostly agree", in essence) but I kept getting distracted, going away, coming back and adding more. It seems to have grown in length but not in focus.
I completely agree for cosmological spacetimes. I largely agree more generally, other than thinking one shouldn't be afraid of Landau-Lifshitz and company for fear of lack of "rigour", or even resorting to vectors[1] in appropriate circumstances. Everyone should already know doing so can be a footgun. If not, someone will call them out (reviewer 2 if unlucky).
That said, it's just a blog posting aimed at non-scientists. As I know you know, Carroll has well-regarded lecture notes and now a textbook [§3.5 in the lead up to eqn 3.92 is roughly the meat of the blog post, and for the sentence you picked out cf §3.8's discussion of the use of Killing's equation, around eqns 3.178 - 3.181, defining a total energy after slicing certain spacetimes (but not the cosmological one, of course)] for anyone who wants more "rigour". :-)
Another nice -- in fact, imho better, because of how it gets to eqn (3) [and the side comment in square brackets immediately after it -- it's not a gravitational statement!] -- blog entry covering similar territory is by Michael Weiss (the diagonalargument.com one) & John C Baez: https://math.ucr.edu/home/baez/physics/Relativity/GR/energy_... Cherry picking their support for the point you make: 'These pseudo-tensors have some rather strange properties. If you choose the "wrong" coordinates, they are non-zero even in flat empty spacetime. By another choice of coordinates, they can be made zero at any chosen point, even in a spacetime full of gravitational radiation. For these reasons, most physicists who work in general relativity do not believe the pseudo-tensors give a good local definition of energy density, although their integrals are sometimes useful as a measure of total energy.'
I will admit that your notion of "rigor" is unusual to see in strongly relativistic astrophysics (e.g. NS-NS GW 170817 literature), and practically unheard of in the low speed weak field limit. So what? Is using GEM's vectors there somehow wrong just because it's a pain to deal with relatively boosted frames?
I think you're wrong that 'Any way of defining "the energy of the whole universe" will involve what are called "pseudotensors"'. There are other approaches. More germane to astrophysics -- although this is perhaps arguing more with Weiss&Baez than you? -- there is lots of work on trying to define quasilocal energy without pseudotensors. See especially e.g. the first paragraph and endnote [1] of Brown & York 1993 PRD vol 47 iss 4 <https://harvest.aps.org/v2/journals/articles/10.1103/PhysRev...> (direct PDF). And now I should really make myself time to trawl through the recent relevant work of self-styled "mathematician who pretends to be a physicist", Steve McCormick <https://www.quasilocal.com/#section2>.
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[1] For fun: "Linear algebra is easy. Just remember [that] real numbers are vectors, polynomials are vectors, integrable functions are vectors, matrices are vectors, tensors are vectors, and so on. As I said, easy." @j_bertolotti lighting a fire https://twitter.com/j_bertolotti/status/1786407421755134325 And a golden reply, "All tensors are vectors. They're not in the same vector space as the vectors they're maps for, but a tensor product of vector spaces is another vector space, satisfying all the axioms." https://twitter.com/roystgnr/status/1786449834498511130
> s using GEM's vectors there somehow wrong just because it's a pain to deal with relatively boosted frames?
No, that's not the issue. "GEM's vectors" still correspond to invariants--you can re-express them, if you like, as appropriate contractions of the stress-energy tensor or the Einstein tensor. Expressing them as vectors in a carefully chosen coordinate chart is just a convenience, to make the math easier to understand.
The same is not true of the "pseudo-tensors" that are used in attempts to define a "total energy of the universe as a whole". Those do not correspond to invariants. Expressing them in a carefully chosen coordinate system is not just a convenience; it's a necessity, because in other coordinate charts they either vanish or you can't even write them down at all. That's the issue.
> Brown & York 1993
The term "quasilocal" in the title should be an indicator that this method does not solve the issue I described above for the universe as a whole. It does give an alternate way of making sense of the concept of "gravitational binding energy" for a finite, bound system surrounded by vacuum, or of the idea of treating the mass of a black hole as a form of energy. But the universe as a whole is neither of these things.
If the concept of energy conservation does not apply to the entire Universe, then we must ask where the additional energy created comes from or is removed. I'm not aware of evidence to suggest that it is not conserved. If there was something like this, it seems like something that could be exploited.
> Second, our best current model of the universe, based on the equations of General Relativity, says it will keep expanding forever.
I'm aware that's what is currently supported, but the collapsing Universe followed by the expanding Universe loop ties everything up nicely (cyclic model [1]).
I'm also aware that this is essentially based on the second law of thermodynamics [2], but I think as long as all processes a system undergoes are fully reversible, it should in theory be able to reverse. It would be like observing a binary counter: 0000 is the start, then we observe numbers like 0110 and 1011 and say "see, entropy is increasing indefinitely", but eventually we tick over to 1111, then 0000.
Of course if any system variable changes irreversibly (i.e. space expands infinitely), then it would not work. But given there is ongoing debate about the Hubble constant [3] and I think there is still room to believe a cyclic model could be plausible.
[1] https://en.wikipedia.org/wiki/Cyclic_model
[2] https://en.wikipedia.org/wiki/Second_law_of_thermodynamics
[3] https://en.wikipedia.org/wiki/Hubble's_law#Hubble_tension