"What I think that means is there is no holy era of time. It emerged. If, in the distant future, we find ourselves in a universe where all the stars have burned out, and all the black holes have evaporated, and all the radiation has been diluted by the dark energy that expanded our universe, and all we have is some very cold bath of photons here and there—basically thermal equilibrium; de Sitter space, as we call it—there will be no sense of time anymore. There will nothing you can do to determine whether time is going one way or the other. Time will then have un-emerged again. It will be like the poem, This is the way the world ends / Not with a bang but a whimper."
I am not a cosmologist, but if all of space reached thermal equilibrium, if space does have any mass (perhaps a tiny amount from the dark energy it contains?), and the entire universe is essentially one uniform mass, wouldn't that mass then collapse upon itself, in a kind of big bounce?
If that is true, then time never really ends, it just "slows down" a lot (not much can happen, and not terribly quickly) until the next sparkup, whereupon time speeds up dramatically (inflationary theory).
> if ... the entire universe is essentially one uniform mass, wouldn't that mass then collapse upon itself, in a kind of big bounce?
de Sitter space is a vacuum solution for Einstein's field equations.
What this means is that the stress-energy tensor is basically "canceled out" everywhere by the uniform energy distribution. That means no point in spacetime experiences any field influence. That means no gravity, that means no collapse, just stasis.
I remember some science show where they said a cup of hot water technically has slightly more mass than a cup of cold water, due to the mass contribution from the (higher/hotter) energy. Are you saying this is not the case? Would a near-absolute-zero cup of water (ice) have the same mass as a cup of boiling water?
Since we're talking about General Relativity (the top of the thread introduces the stress-energy tensor, and a particular solution to the Einstein Field Equations) let's talk in those terms. We can also talk in those terms because it is the more-fundamental theory from which Special Relativity (which gives the famous E=mc^2) can be derived.
The "stress-energy tensor" determines the curvature of spacetime. It goes by other names: the matter tensor, the energy-momentum tensor, and so forth, largely depending on how one wants to interpret the tensor components in a particular problem.
Three quick paragraphs with some reference to the mathematics:
You can write the energy-momentum tensor's components as a 4x4 matrix where each cell represents a flux of momentum from one direction to another. Momentum which "hangs around" flowing only from the past to the future corresponds to the m in "E=mc^2" either very locally or more globally in a spacetime which has no gravitation whatsoever. Since a nonzero "m" represents a nonzero value of the energy-momentum tensor, and since the energy-momentum determines the curvature of spacetime, "E=mc^2" is really just an excellent approximation that is better when m is small.
The fuller Special Relativity relation is E^2= (mc^2)^2 + (pc)^2 where we square to get rid of sign problems and where p represents linear momentum; if there is no momentum and the mass stays at the spatial origin (moving only in time) in our reference coordinates, then all but one component of the energy-momentum tensor is zero, and the one remaining one is totally determined by "m".
If we (non-gravitationally) impart momentum onto our m then p becomes nonzero and so does at least one other component of the energy-momentum tensor.
Summing up: mass sources energy-momentum, which determines curvature. Moving mass sources even more energy-momentum, and so greater curvature.
In your question about the cup, let's apply a restriction: we use the same number of water molecules at all times.
As we heat the ice or water, the motion of the water molecules relative to the centre of the coffee cup (or the overall centre-of-mass or the overall centre-of-momentum) increases. Increased motion means increased momentum. This in turn means a greater curvature is sourced by the heated molecules, or if you like, that there is a greater "active" gravitational interaction for the hot water than for the cold water or ice. "Active" in the sense of small things falling towards it. It virtually certianly also has an identically larger "passive" gravitational interaction, where "passive" is in the sense of falling towards an object that makes it seem like a very small thing. (We have excellent experimental evidence that passive and active interaction strengths are effectively identical, and the underlying theory demands it, ignoring some details about gravitational back-reaction which even experts won't want to think too hard about.)
So if we use a very sensitive scale we can measure that the hot water with exactly N molecules of hot water weighs more than the exactly N molecules of cold water. Ignoring gravitation again, we see that E is greater in the hot water case, but because there is a larger average value for p (momentum) for each of the molecules. The "m" remains effectively the same in the hot and cold water. Rest mass (m_0) is properly determined by the count of water molecules when they are completely free of momentum (including internal momentum, right down to the momentum of photons, electrons, quarks and gluons), which gives you the answer to when E is lowest for the cup of exactly N molecules of water. (It's at absolute zero!)
Back to gravitation: the minimum influence on the curvature of spacetime by the N molecules of water is when they are at absolute zero.
The difference is very tiny at the sorts of temperatures you're asking about. However, if we could somehow confine the N molecules of water into a magical box of the volume of the coffee cup, and heat the water molecules to the point where the molecular bonds break, the atoms all ionize, the oxygen nuclei disintegrate into protons, the protons disintegrate into a quark-gluon plasma, and keep going through several orders of magnitude (wherein we may discover new subatomic physics!) then as we go the "active" and "passive" gravitational interactions of the confined matter will grow substantially. It will become very heavy (passive interaction: hard to hold above the lab's floor, assuming the ultra high energy stuff didn't vapourize everything around it) and noticeably start affecting the trajectories of ever larger things (dust, pencils, lab assistants...). In the extreme, we can add so much momentum to to the stress-energy-momentum tensor the magical container encloses that it collapses into a black hole.
In essence this is what happens in the cores of neutron stars when they collapse into black holes: the internal pressure and temperature gets so high that an event horizon forms. The "container" is a few solar masses worth of incredibly dense "nuclear pasta" and other exotic stuff we don't know much about yet, and is more the size of the city the magical-container lab is in than the lab itself.
One last thing: the total value of the stress-energy tensor at a given point is observer-independent -- an ultrarelativistic observer passing by the lab and a scientist standing still in the lab will both agree on the total stress-energy-momentum of each container of water. However, different observers may prefer to split the total tensor value into its sixteen components in different ways; indeed, any single observer is free to do so because the splitting is coordinate-dependent and one is free to select from an infinite set of systems of coordinates in describing spacetime or a small patch thereof.
Because of this sort of coordinate- and observer- independence ("general covariance", technically) modern physics uses tensor fields as fundamental objects and either copes with the heavyweight mathematics that requires or reduces tensors by imposing special conditions on coordinates and on how coordinate-dependent vector-values are extracted from the tensors.
Finally, the stress-energy-momentum tensor is contributed to by all the fields of matter, so this would (where not excluded for convenience) include the classical electromagnetic tensor field, or the fields of quantum electrodynamics, or the fields of the Standard Model of particle physics. The contribution to stress-energy for known matter is always non-negative. So when considering classical or quantum matter fields, where there are nonzero field-values there is nonzero stress-energy. In heating up the water molecules we are also creating [a] more photons and [b] higher-momentum photons. Photons carry nonzero momentum and so contribute to the "p" term in E^2=(mc^2)^2+(pc)^2 or more fundamentally they add to the total tensor-value of the energy-momentum tensor.
I hope this is a helpful answer.
PS: I should have said that absolute zero is probably not physically achievable in our universe, but to the extent things can get very close to absolute zero (attokelvins or colder) we can always describe a relatively-moving observer who will think the object is warmer than someone at rest with respect to it (and some ultrarelativistic observers might think it's rather hot, spraying out a thermal bath of kilokelvin photons!). Nobody knows what the quantum chromodynamics equivalent of absolute zero would be, so that's always happening, and the momenta of the quarks and gluons thus don't completely vanish.
As someone who isn't that familiar with the physics behind it, this comment was just as an intriguing read, if not even more than the original article. Thanks for taking the time to write an in-depth response.
> de Sitter space is a vacuum solution for Einstein's field equations.
Yes, an exact solution.
> What this means is that the stress-energy tensor is basically canceled out" everywhere by the uniform energy distribution
No, it means that the stress-energy tensor is zero at every point in the whole spacetime. There is no matter to source any curvature; the curvature is specified by the theoretician writing down the vacuum solution.
In Lorentzian 4-dimensional vacuum de Sitter spacetime ("dS" below) there is no useful concept of energy anywhere. The parameter \Lambda is incorporated directly into the Einstein curvature tensor and so is interpreted geometrically. One would have to perturb the dS vacuum in order to have a useful interpretation of \Lambda as a source of gravitational energy.
Vacuum solutions can be useful in understanding physical systems in which gravitation is important. For example, one can add non-zero stress-energy by hand and use perturbation theory to study the consequences rigorously. Alternatively, one can "cut out" parts of a vacuum solution and work with the rest which closely resembles a physical system (such as deep space far from the matter in and around galaxies). As the expansion of the universe dilutes away the matter contributing to a non-zero stress-energy tensor, our universe will resemble the sparser parts of a de Sitter spacetime. However, from our perspective (i.e. with data in our sky) there are much better approximations at various scales than de Sitter spacetime perturbed by matter fields: there is a lot of obviously non-zero stress-energy under our feet, lighting up our sky day and night, and causing heat to flow from one place to another.
> That means no gravity
There can be extremely strong spacetime curvature in de Sitter space!
In general we can time-orient dS: in one direction objects which only interact gravitationally and do not generate significant stress-energy will always converge, in the other they will always diverge, depending on the value of the \Lambda parameter, which is always positive for dS. If \Lambda is large, all such objects leave the others' causal cones much more quickly than they could, even in principle, in flat spacetime.
If we perturb dS with a dense ball of nonrelativistic (i.e., slowly-moving) dust that feels only the gravitational interaction, the dust will tend to collapse if \Lambda is sufficiently small, and will locally approximate a Lemaître-Tolman-Bondi (LTB) collapsing dust metric, which in turn resembles a perturbation of vacuum Schwarzschild or Kerr (which are eternal black hole models; there is no matter to collapse).
A scattering of such collapsing dust-balls with significant gaps between them has been studied by a number of relativists as a family of "swiss-cheese" cosmologies very similar to the one developed by Einstein & Strauss in 1945, who represented the balls of dust as already-collapsed vacuum Schwarzschild solutions, calling them "holes" in the "cheese" of the surrounding expanding spacetime. The Einstein-Strauss-de Sitter model approximated some features of our universe at intermediate ranges, but has since been superseded by the the Friedmann-Lemaître-Robertson-Walker (FLRW) model of the modern standard model of cosmology that among other things has representations of nonzero stress-energy that the vacuum Einstein & Strauss model did not, and which better matches observables in our sky than models that added more realistic matter fields to Einstein & Strauss vacuum.
One feature of Einstein-Strauss-de Sitter style expanding swiss-cheese models is that a small "test" amount of sparse low-mass nonrelativistic dust scattered in the LTB "holes" will stay confined within the hole while the hole separates from everything else. However, an identical test dust scattered in the "cheese" part well outside of holes will never find its way into a hole. Ultimately the former test dust will converge while the latter will diverge.
We can get highly similar results by perturbing the FLRW model. (The observable universe is well approximated by an almost FLRW model in which there are primordial density fluctuations: over-dense areas collapse gravitationally, comparably to the matter in the "holes" of a non-vacuum Einstein-Strauss-de Sitter universe.)
Locally in all these cases, what we have are the tensor fields contributing to the Einstein Field Equations, and general covariance. We can talk about the values of these tensors at any given point (and its neighbourhood), but General Relativity does not lend itself to universal definitions of energy or even energy-density. We can reduce the tensors by using gauge fixing and other techniques, and "demote" the cosmological constant in FLRW or in dS into an energy. (That's really all Dark Energy is in the standard cosmology). In such a gauge we can talk about the work matter interactions (e.g. the electromagnetic interaction) does "against" dark energy. However, the real physics are in the tensors, and applying coordinates and selecting particular observers, while often able to make things much easier to calculate, can be highly misleading too.
The idea that the cosmological constant is "cancelled" out by matter-matter gravitational interactions (and electromagnetic and nuclear interactions) is a statement that the geometrical nature of the cosmological constant is "demoted" into an extremely weak force field, and leads to all sorts of statements that work in that "demotion" context but not generally. Typical statements are that the solar system is expanding, or the galaxy or cluster is expanding, because of the tiny local force on matter by dark energy.
It is better to retreat to "swiss-cheese" and say that the geometry of our solar system is not well-approximated by expanding metrics like dS or expanding Robertson-Walker, and that consequently it is a better fit to observations that in our "hole" (which the matter of our solar system, galaxy and cluster sources) the cosmological constant simply vanishes.
Better still would be to advance to inhomogeneous metrics, but those are a subject of research beyond the scope of this comment.
Finally, your last paragraph can also be read as a request to be directed to the concept of the Jeans Instability in the Expanding Universe. If in the very early universe there was truly uniform and dense distribution of stress-energy, the slightest perturbations would magnify, leading to structure formation. As I wrote above primordial fluctuations of an otherwise FLRW model approximates modern cosmological observations very well. (One could alternatively introduce dissipation, quantum uncertainties, and so forth, as a source of gravitational Jeans Instability causing the collapse or fragmentation of a uniform distribution of stress-energy).
I'm not an expert, but I don't think that quote is quite right. There is still a notion of time, there is just no notion of a direction of time. For example, if you go deep into empty space, then one patch of space looks just like the next one. That doesn't mean there is no longer a physical concept of space. Similarly, just because one point in time doesn't look very different from the next, doesn't mean that there isn't a notion of time.
The main point of thermodynamic time is to describe the idea that time appears to be moving forward. According to fundamental physics, as far as we can tell, time is symmetric (with a few caveats). So going forward in time should look similar to going backwards in time, just like going to the left in deep space is the same as going to the right. But the fact that we are in a pocket of high entropy means that time has an apparent direction.
Speaking of this, does anyone know of an explanation for why the direction in spacetime in which entropy decreases is parallel to the time dimension (the one with a different sign in the relativistic metric) of spacetime?
Maybe I'm missing something, but I've never seen a link made between the two.
As opposed to entropy decreasing as you travel, say, toward the galactic north? It's a good question. One answer could be the mostly spatially uniform initial conditions of the big bang.
Also, a spatial entropy gradient would gradually diffuse over time to become uniform again.
But in general, the arrow of time is a consequence of systems that store a memory of their pasts, which is possible because the past is lower entropy than the future. (The past can be more accurately predicted by querying the state of the data in memory than the future can). You could probably set up a carefully designed system with a spatial entropy gradient, and somehow get a memory where the arrangement of atoms on the left always "remembers" the arrangement of atoms on the right, but not the other way around. Maybe there are even systems like this found in nature? Interesting to think about.
Thanks for the answer. I mean exactly that! Thoughts on your thoughts:
> One answer could be the mostly spatially uniform initial conditions of the big bang.
That's fair, but in that case, to continue the galactic north analogy, is there a reason why the Big Bang couldn't have happened uniformly in the dimension-3 hyperplane of (time, plane of the milky way) instead of the space hyperplane?
> a spatial entropy gradient would gradually diffuse over time to become uniform again.
But wouldn't that be "diffuse over direction-of-decreasing entropy time", not "diffuse over relativistic-time-dimension time"? In that case, I don't think it would help.
Apologies if I'm missing your points or not making sense, I can't claim I understand any of this well. I just feel like I see a lot of circular arguments around this :).
> is there a reason why the Big Bang couldn't have happened uniformly in the dimension-3 hyperplane
It's a really good question. I think it has to do with how space expanded after the big bang (which is to me still a mysterious subject).
To avoid confusion, let me call "time" the arrow of time as perceived from within a system, and "TIME" the coordinate axis of spacetime that has the opposite sign from the other three.
The early days of the universe were full of heat and radiation everywhere throughout space, which seems like a high entropy scenario for sure. But entropy was able to increase as the universe cooled because space itself expanded adiabatically (and so the configuration space became larger).
In other words, we end up with an entropy gradient that points along the TIME axis because the volume of space expands along that axis. Like space-time is shaped like a pyramid rather than a cube, with the pointy axis in the TIME direction.
I don't know enough general relativity to say for sure, but I expect that that pyramid shape (expanding towards later times) is probably a consequence of TIME behaving differently than other spatial dimensions. If spacetime shrank volumetrically toward the galactic north, in a fairly uniform way throughout all times and throughout the universe, then we'd have a different conclusion. (This might seem like the case near a black hole).
I guess another thing to remember is that TIME and space really are different things. For example, conservation of angular momentum means the universe has constant angular momentum at all times, but not necessarily in all directions of space.
> In other words, we end up with an entropy gradient that points along the TIME axis because the volume of space expands along that axis.
That's a good insight, I hadn't thought of it that way.
> I guess another thing to remember is that TIME and space really are different things. For example, conservation of angular momentum means the universe has constant angular momentum at all times, but not necessarily in all directions of space.
Yes, I realize they are different, but I feel like the issue is more that don't understand the relationship between the special properties of TIME and those of time.
That's a pretty difficult case to reason about. But setting aside the time warping effects, I think a black hole's entropy would be too high to support any kind of memory-bearing structure in its vicinity.
I'm not sure, and this is a lot of speculation, but I feel like it has to do with the fact that mass/energy is conserved over time (but not in space) [1]. The mass now corresponds to the mass in the past/future, which, for example, allows memories to form. In any case, time is still a qualitatively different dimension than space.
Low entropy systems are likely to transition into high entropy systems, and the reverse is unlikely to occur. It is largely a statistical argument; low entropy states are by definition less likely to occur by chance than high entropy states.
(You might then ask why we care what happens 'by chance'; the answer is either that physics is fundamentally unpredictable, or that the scale of the universe so large that the influence of the whole on any particular subsystem must necessarily be modeled with noise.)
Here is the crux. Time IS motion. We measure this motion of objects relative to one another by comparing their motion to a standard set of things in motion called a clock. There is nothing else to time except that; but it is real, as real as anything, it is not an illusion.
This is strangely approaching Nietzsche's notion of Eternal Return of the Same with the caveat that he additionally speculated that if particles and their configurations are finite, then moments in lived experience necessarily repeat themselves "at some point". This to him was the most gargantuan challenge to the Christian moral tradition since this eliminates any notion of Providence as well as the more general moral notion of "You could've done otherwise" since there's no value to considering timelines, not even from a "Best Possible World of All Worlds" sense.
>that if particles and their configurations are finite, then moments in lived experience necessarily repeat themselves
I don't know much about Nietzsche, but this statement does not seem correct. One can have infinite configurations of a finite number of particles. The only way a finite number of particles would imply finite configurations would be (a) space is finite, and (b) space is not continuous. That is, if you imagine the universe as composed of discrete slots in which particles must exist, and the number of slots is finite, then his idea makes sense. But, is there a reason to think this?
That quote also shows how difficult it is to even discuss the idea without invoking the notion of a flow from a past to a future.
Science provides both a way of calculating what is going on, and also, apparently, explanations (up to the limits of our knowledge) for what happens. To eliminate the notion of time's flow from discussions of physics, would we have to abandon the concept of causality from our explanations?
That same quote confuses me. If there are no events to measure time, why does that mean that time stopped? Can't it keep flowing just as space keeps expanding regardless of measurement? More so, photon decay is not completely ruled out yet, so events could keep happening in that cold photon bath.
It has to do with what time means. A coloiquial understanding of time has almost everything to with the way our mind and our memories work.
Our mind and memories won't be of much use at the end of the universe, so we have to depend on a more scientific definition. The best scientific way of describing and measuring time outside of our minds basically comes down to measuring systems as they increase entropy. As the universe ages, and these systems approach maximum entropy, the ability to measure time, and the ability to describe in any concrete way what time means, becomes increasingly difficult.
> I am not a cosmologist, but if all of space reached thermal equilibrium, if space does have any mass (perhaps a tiny amount from the dark energy it contains?), and the entire universe is essentially one uniform mass, wouldn't that mass then collapse upon itself, in a kind of big bounce?
Nor am I, but as long as the universe keeps expanding and everything keeps flying apart the average density will probably continue to decrease and gravity will be too weak to cause a collapse.
Or are you saying dark energy (intrinsic to space itself) would eventually dominate and concentrate?
Dark energy would simply accelerate the asymptotic approach to equilibrium by spreading everything out. When everything is evenly spread out and in thermal equilibrium, the aggregate effect of gravity could take over, if the mass/energy of the universe was truly uniformly distributed. Or this is my armchair/idiot idea anyway.
> there will be no sense of time anymore [...] This is the way the world ends
If there is no sense of time why does the world "end"?
> If that is true, then time never really ends, it just "slows down" a lot (not much can happen, and not terribly quickly) until the next sparkup, whereupon time speeds up dramatically (inflationary theory).
Time slowing down or speeding up is still a "sense of time". Moreover time slowing down or speeding up is a human perception, and not a fact of the universe.
No, the rest of the universe could still effect that coffee externally. A universe at thermal equilibrium would encompass everything (which naturally also means there could not be anyone or anything around to measure anything either)
I love that story, alas it doesn't give me any additional insight on these questions.
Plus, the state of the art has progressed much since then. I'm curious what the latest science says about when the universe is only empty space in thermal equilibrium, would the space itself have any mass, and would that mass be self-attractive? Would there be an even distribution of particles that don't decay (proton?), or if all particles decay, then an even distribution of energy, which also has mass?
Imagine a ball rolling down a huge basin. Very deep, ball gets a terminal velocity, rolls down at constant speed. Then the basin shallows, and finally the ball reaches the bottom. From the ball's point of view, "downnwardness", an ineluctable constant, has ceased to exist.
"What I think that means is there is no holy era of time. It emerged. If, in the distant future, we find ourselves in a universe where all the stars have burned out, and all the black holes have evaporated, and all the radiation has been diluted by the dark energy that expanded our universe, and all we have is some very cold bath of photons here and there—basically thermal equilibrium; de Sitter space, as we call it—there will be no sense of time anymore. There will nothing you can do to determine whether time is going one way or the other. Time will then have un-emerged again. It will be like the poem, This is the way the world ends / Not with a bang but a whimper."
I am not a cosmologist, but if all of space reached thermal equilibrium, if space does have any mass (perhaps a tiny amount from the dark energy it contains?), and the entire universe is essentially one uniform mass, wouldn't that mass then collapse upon itself, in a kind of big bounce?
If that is true, then time never really ends, it just "slows down" a lot (not much can happen, and not terribly quickly) until the next sparkup, whereupon time speeds up dramatically (inflationary theory).