This is interesting, but I'm going to be one of those people for a second...
I hate headlines that assume what I do and don't know. Just call 'em 10 physics facts that most people aren't aware of and are pretty interesting. Because I actually did know a bunch of those facts from school and I was a liberal arts major...
Yes, it's annoying and, as other users pointed out, broke the site guidelines. There's the magic number thing but also the linkbait 'you'. We've reworded it to follow them instead.
> If the original title begins with a number or number + gratuitous adjective, we'd appreciate it if you'd crop it. E.g. translate "10 Ways To Do X" to "How To Do X," and "14 Amazing Ys" to "Ys." Exception: when the number is meaningful, e.g. "The 5 Platonic Solids."
It is offered but the student must elect to take the course, it is possible to pass through an American public school system without taking any advanced topics. Someone who does not enroll in courses of at least college prep level will arrive at university and be required to take some remedial credits to catch up with their peers.
In a closed physical system energy is conserved.
For example, if you have container that has Helium on one side and Oxygen on the other. Both gases will mix, i.e. maximize entropy.
Minimizing energy would mean, the gas would reach absolute zero.
I don’t know if it’s always true in the context he mentions - particle physics - but I’m pretty sure it is not always true in e.g. chemical physics. Both entropy and enthalpy matter. The reaction of potassium vapor with bromine vapor is spontaneous even though the solid crystalline potassium bromide product has lower entropy than the gas phase reactants. In the comments on the article Milkshake also gives an example of an entropically driven spontaneous reaction: ammonium nitrate dissolving in water. You need enthalpic and entropic terms to predict if a reaction is thermodynamically spontaneous.
And if you want to calculate if a thermodynamically spontaneous reaction will appreciably take place, you also need to understand the actual intermediates between A + B -> C, and find reaction barrier heights for all of them, and weight them appropriately. This is still very hard to do so information of this sort is mostly determined by experiment rather than calculation.
While it is true that a system doesn't minimize the energy of the entire system, as energy IS conserved, except apparently for a miniscule amount due to the universe expanding, it is also true that locally it often looks like minimizing energy in some sense.
If you have a vibrating spring on a table, after a while it will have stopped because it has created motions in the air and in the table, spreading out at the speed of sound in whatever material it encounters, and some of it also at the speed of light in the form of thermal radiation.
So, looking only at the spring, it is minimizing it's energy by returning to a stable quiscent state. In the same way, if we look at a sphere with radius as "the speed of light times the time from the start of the experiment", all energy can still be accounted for within that volume.
One system, several perspectives.
I think the minimization model is often used because the statistical model of entropy can be hard to explain, as it simultaneously allows complex relatively ordered systems like us humans to exist, while at the same time explains the almost complete disorder of thermal energy and how it disperses.
Thus, since the energy minimization of the local state can be used as a relatively truthful proxy, we learn that model first. With the possible bonus that it also allows one to temporarily avoiding some rather (afaict) gnarly math until you actually have the tools you need to understand it.
There are variations on this that are more problematic (which I've never seen solutions to).
1) It looks like it takes forever, to an outside observer, for anything to reach the event horizon. So if no matter appears to reach event horizons, how can they (event horizons) form in the first place (i.e. exist in our universe)?
2) While it takes forever for matter to fall in (again to outside observers), it takes finite time for black holes to evaporate via Hawking radiation. So not only do black holes would take forever to form, they seem to have to extinguish before being able to exist (again in the sense of event horizons).
I'm not a physicist, but my speculation is that Black Holes don't really exist, they're just a limit of a process that approaches but doesn't converge to a singularity, which I think is unphysical.
I'm also not a physicist but I think I can guess a plausible answer to these?
1. Because if you don't have a black hole yet then there is no event horizon to prevent you from falling in and creating one. I think the event horizon wouldn't form at exactly the same point whose crossing would finally increase density enough to create it, so this shouldn't be a problem. (Although, again, since the event horizon doesn't exist yet, I think it might still not be a problem even in that case?) I would also expect that quantum fluctuations can also inject matter into a black hole, just as they can remove matter from it.
2. Again, they wouldn't take forever to form -- see above.
I'm pretty sure black holes have been observed indirectly, so the idea that they actually don't exist would require a pretty rock solid alternative explanation!
> I'm pretty sure black holes have been observed indirectly
What we have observed indirectly are compact objects that emit no light, but contain a large enough mass in a small enough volume that they can't be anything else but a black hole, if we only take classical GR into account.
As I responded in another post just now, it is possible that black holes, in the sense of objects with actual event horizons (boundaries of regions from which light will never escape, even in the infinite future), cannot exist when quantum gravity effects are taken into account. If that is the case (and it is not clear whether it is--as I said, this is an open area of research), then the objects we call black holes based on our current observations won't have actual event horizons--eventually, in the very far future, light will escape from those compact regions. But they will still have apparent horizons, i.e., surfaces from which outgoing light is not escaping now (or for a very long time in the future). And that, in itself, is sufficient to show that the issues raised by darkmighty are not valid.
> There are variations on this that are more problematic (which I've never seen solutions to).
Then you haven't spent much time looking at actual textbooks on GR, since all of these issues are addressed there. (Not to mention in many peer-reviewed papers in the field.)
> if no matter appears to reach event horizons, how can they (event horizons) form in the first place (i.e. exist in our universe)?
This is the same fallacy as the fallacy that nothing can actually fall into a black hole because it looks like it takes forever from the outside. Oppenheimer and Snyder published a mathematical model way back in 1939 that shows how a black hole can form in a finite time from the gravitational collapse of a massive object, as seen by an observer falling inward on the surface of the object. The collapse appears to take forever as seen by a distant observer, but this is an optical illusion caused by the effect of spacetime curvature on the paths of light rays. This has been studied for decades and is thoroughly understood.
> While it takes forever for matter to fall in (again to outside observers), it takes finite time for black holes to evaporate via Hawking radiation.
Wrong. First, if matter is falling in, the hole is gaining mass, not losing it, so it will never evaporate even though, in principle, it is emitting Hawking radiation (in practice this radiation is many, many orders of magnitude too faint to detect for any black hole we can observe).
Second, if you include quantum effects, and therefore Hawking radiation in your model, you've changed the model, and it is no longer true that a distant observer will never see anyone falling into the hole. Instead, the distant observer will see light signals emitted by objects falling through the hole's horizon at the same time the distant observer sees the hole evaporate. (At least, that is the case in the simplest model, the one Hawking used when he first published his prediction of Hawking radiation. More complicated models have been developed since, and we won't know which, if any, of them is really correct until we have an experimentally confirmed theory of quantum gravity.)
> I'm not a physicist, but my speculation is that Black Holes don't really exist, they're just a limit of a process that approaches but doesn't converge to a singularity, which I think is unphysical.
Your speculation is uninformed and wrong. It is possible that an actual black hole, with an actual event horizon, is impossible when all of the laws of physics, including the laws of quantum gravity, are taken into account. This is an open area of research. But if it turns out that actual black holes cannot exist, it won't be for any of the reasons you give.
> This is the same fallacy as the fallacy that nothing can actually fall into a black hole because it looks like it takes forever from the outside. Oppenheimer and Snyder published a mathematical model way back in 1939 that shows how a black hole can form in a finite time from the gravitational collapse of a massive object, as seen by an observer falling inward on the surface of the object. The collapse appears to take forever as seen by a distant observer, but this is an optical illusion caused by the effect of spacetime curvature on the paths of light rays. This has been studied for decades and is thoroughly understood.
But isn't equating the two observers incorrect? That is, when speaking of 'formation time', we are speaking of the formation time by observers at infinity, free from the effects of local curvature. What I understand is that all observers must agree on reality, thus on whether the BH forms at all: if the infalling observer makes it through an EH and into a singularity, then those exist (by definition) as unique reality. But evaporation seems to complicate this classical GR view understood for a while: since now outside observers see finite-time quasi-evaporation, infalling observers must observe this phenomenom while infalling, and actually before reaching the EH.
There's no way I can see that:
1) EHs and singularities (i.e. Black Holes as a steady-state phenomenon) exist;
2) All observers agree on reality of their formation and evaporation.
I defiantly conclude they don't exist (or I'm missing something major) :)
That is, black holes in nature would be something akin to a extremely time-dilated "Object trap" without singularties (although perhaps it must gain an apparent horizon?).
> when speaking of 'formation time', we are speaking of the formation time by observers at infinity
No, we aren't. The hole isn't formed at infinity. The time that is being talked about for observers at infinity (or more correctly, very far away) is the time it takes light rays, coming outward from the object that is collapsing into a black hole, to reach those observers. The curvature of spacetime near the horizon is such that it takes longer and longer for those light rays to get out to the observer far away, the closer to the horizon they are emitted. At the horizon, outgoing light rays do not move outward at all, due to the curvature of spacetime there, so they never reach the observer far away--hence that observer never sees the horizon form. But, as I said, that is an optical illusion caused by the curvature of spacetime.
> What I understand is that all observers must agree on reality, thus on whether the BH forms at all
All observers agree on observable events and measurement results that they can all observe, yes. But if an observer cannot observe a particular event (such as the horizon forming) because spacetime curvature prevents light rays from that event from ever reaching him, then he can say nothing at all about whether that event happens or not.
> evaporation seems to complicate this classical GR view understood for a while
It complicates it in the sense I said before: that observers far away, instead of never seeing infalling objects cross the horizon, now see those objects cross the horizon (more precisely: see light rays emitted outward by those objects when they crossed the horizon) at the same time they see the hole evaporate (more precisely: see light rays from the hole's final evaporation). It does not mean what you claim, that infalling observers will see the hole evaporate before they fall in. The spacetime geometry of an evaporating black hole (i.e., including quantum effects) is different from the spacetime geometry of a classical black hole that never evaporates. The statement that a distant observer never sees an object cross the horizon is only true of the latter, not the former.
> I defiantly conclude they don't exist (or I'm missing something major) :)
The second is the correct choice. You need to stop being defiant and start learning what the models of black holes in physics actually say.
> black holes in nature would be something akin to a extremely time-dilated "Object trap" without singularties (although perhaps it must gain an apparent horizon?)
To us far away, an apparent horizon looks the same as an event horizon, and works the same--all the things I said above would still be true (at least for the time we have been observing--see below for how long it would take for differences to become observable) if the objects we observe and call black holes actually have only apparent horizons (because of hypothetical quantum gravity effects that, as I've already said, are an open topic of research), not event horizons. And all of your criticisms would still be wrong. (In at least some of the hypothesized quantum gravity models, there is no singularity and no event horizon, only an apparent horizon; but to us far away, the object looks and works the same into the very far future; it takes something like 10^70 years for the difference to be observable.)
> No, we aren't. The hole isn't formed at infinity. The time that is being talked about for observers at infinity (or more correctly, very far away) is the time it takes light rays, coming outward from the object that is collapsing into a black hole, to reach those observers. The curvature of spacetime near the horizon is such that it takes longer and longer for those light rays to get out to the observer far away, the closer to the horizon they are emitted. At the horizon, outgoing light rays do not move outward at all, due to the curvature of spacetime there, so they never reach the observer far away--hence that observer never sees the horizon form. But, as I said, that is an optical illusion caused by the curvature of spacetime.
I think you have a misconception about GR here. In GR, spacetime isn't just an "optical device" that delays rays cast in an absolute reference. In relativity, the behavior of light rays is actually closely related to interpretation of time and space. You can't separate the two. In fact, the definition of (relative) local time is given exactly by the rate a "light clock" ticks when communicating periodically to a "light clock" located elsewhere. Approaching the event horizon, the rate of clock ticking goes to 0 relative to an outside observer (to the the limited extent that coordinate frames can be defined in GR). So classically you can't say that to the outside observer the BH forms in finite time, I don't think. What matters in the end is what the observer sees, which, according to you, would be (for Bob, far away):
Alice falls into BH -> Alice approaches EH -> Alice's BH evaporates -> Alice finishes falling into the BH
That's not possible! The light rays from evaporation, which happens, in Alice's own frame, after she falls into the BH, cannot precede the light rays from her fall. So she cannot truly go past the EH.
No, you do. Everything I am saying is taken straight from GR textbooks--mainly Misner, Thorne, & Wheeler (1973) and Wald (1984)--and peer-reviewed papers, such as Hawking's original paper on black hole evaporation. You need to go read them.
> In GR, spacetime isn't just an "optical device" that delays rays cast in an absolute reference.
I never said it was. The paths of light rays in GR are determined by the geometry of spacetime. "Optical illusion" is one way of trying to describe the effect of that in the case under discussion. But the effect is there regardless of what you call it.
> In relativity, the behavior of light rays is actually closely related to interpretation of time and space. You can't separate the two.
This is just another way of saying that the paths of light rays in GR are determined by the geometry of spacetime. Which is true, but to figure out what that means in a specific scenario, you need to properly understand the spacetime geometry in that scenario. You evidently do not.
> In fact, the definition of (relative) local time is given exactly by the rate a "light clock" ticks when communicating periodically to a "light clock" located elsewhere.
There is a limited set of scenarios in which this works, but that limited set does not include the case under discussion (Alice falls into a black hole while Bob remains far away). In general in GR, there is no well-defined way of comparing "times" for spatially separated observers. The only well-defined time is local--proper time along a particular observer's worldline.
> Alice falls into BH -> Alice approaches EH -> Alice's BH evaporates -> Alice finishes falling into the BH
I never said that was the sequence of events. It isn't. The correct sequence of events is (note: this is what happens to Alice, Bob does not see all of these events--see my other post in response)
Alice falls into BH -> Alice approaches EH -> Alice finishes falling into BH -> Alice's BH evaporates
If you don't understand how that happens, you need to go study GR textbooks. The model that gives the sequence of events I just described has been well understood for several decades now--Hawking published the first paper using it in the 1970s.
Well you've convinced me that, if Hawking's model is as you say, it's obviously incorrect :P
I don't know if this is just a mathematical simplification that doesn't impact conclusions about existence of EHs and singularties, but it all looks fishy.
> you've convinced me that, if Hawking's model is as you say, it's obviously incorrect
Why? Note that I'm not claiming (and Hawking never claimed) that his model is experimentally verified--that's obviously impossible since Hawking radiation from any black hole we can observe is many orders of magnitude too faint to observe, if it exists and has the intensity predicted by the model. But the model is perfectly self-consistent.
I don't have time to explore this topic in depth right now (way too much on my plate already, and I'm not that fast or brilliant), but I suspect there's literature on this topic. Something I could find is this:
Basically, my intuition is more or less in line with this firewall phenomenon and that no particle crosses a true event horizon, even in their own frame, and no certainly no singularity.
The firewall hypothesis is one of the quantum gravity hypotheses that is an open area of research, yes. But even if the firewall hypothesis ends up being confirmed (which won't happen any time soon), that does not mean it's the only consistent hypothesis. Your claim was that the classical GR model, and Hawking's original model, were inconsistent. They're not. They might end up not being correct when quantum gravity effects are taken into account, but that doesn't make them inconsistent.
The Firewall as spelled out in the original paper is widely considered to be unlikely, not in the least because it violates the equivalence principle. There are weaker versions of the firewall based in M-Theory which exist because there is no singularity, and the event horizon is actually the surface of the fuzzball which takes its place, but that’s a different sack of weasels.
> What matters in the end is what the observer sees, which, according to you, would be (for Bob, far away):
Not what you said. What Bob sees is:
Alice falls into BH -> Alice approaches EH -> Alice reaches EH/Alice's BH evaporates
In other words, as I said, Bob sees Alice crossing the EH at the same time he sees the BH evaporate. Bob still never sees any light signals from Alice that were emitted below the EH; those signals are trapped inside the hole and never escape. In Hawking's original model, those signals, along with Alice herself, hit the singularity and are destroyed; they cease to exist, so they never come out again even when the hole evaporates away. An open question is what the correct model will end up being when we have a full understanding of quantum gravity; but whatever that model ends up being, it won't validate the particular issues you are raising.
What do you mean by at the same time? Those two photons reach Bob at exactly the same instant?
We had already agreed that General Relativity should imply Bob never sees Alice reach EH, regardless of quantum effects (at least not in finite time). We also agreed that Bob does see the BH evaporate (in finite time to any desired fraction), as seemingly dictated by QM. Thus it cannot be that the two are observed simultaneously!
Moreover, there cannot be a chain of events whereby Alice evaporates into photons (emanating from EH), and those photons (originating from Alice's mass-energy) hit Bob before the photons of Alice when she was near, but did not reach the EH yet (again because Alice cannot be observed to reach EH). It would violate causality.
It can't be that both Alice falls into the EH and we receive Alice's energy from evaporation: either the BH does not evaporate or Alice doesn't cross the EH (even in her own observation). Assuming Hawking radiation really does exist, then some kind of evaporation would need to occur as objects approach the EH, it cannot come directly from the EH boundary.
Could it be that Alice herself is seen to evaporate -- the evaporation might stem directly from Alice, before reaching the EH? My claim that the EH/singularity never forms would be valid.
I could be terribly mistaken of course, but your arguments did not convince me unfortunately (quite the contrary).
I wonder if what I'm missing could be related to the expansion of the EH as a body approximates.
> What do you mean by at the same time? Those two photons reach Bob at exactly the same instant?
In the original idealized model of Hawking, yes. That is what the spacetime geometry says. One way of thinking about it is that, with this spacetime geometry, the path of any light ray emitted outward at the event horizon travels along that horizon into the future, but the area of the horizon decreases as the hole evaporates. At the instant of the hole's final evaporation, all of the light rays emitted anywhere along the horizon are at the same point as the final evaporation of the hole, and they all go outward from there together. This is a highly idealized model; but that's what the model says.
> We had already agreed that General Relativity should imply Bob never sees Alice reach EH, regardless of quantum effects
No, we haven't. You claimed that, and I pointed out that you were wrong. Bob never sees Alice reach the EH in the classical GR model, with no quantum effects. Quantum effects change the model. In the model including quantum effects, Bob does see Alice reach the EH. See above and my previous posts.
> there cannot be a chain of events whereby Alice evaporates into photons (emanating from EH), and those photons (originating from Alice's mass-energy) hit Bob before the photons of Alice when she was near
This is correct. And it is also what the model of a hole evaporating by Hawking radiation says. All of the light signals emitted by Alice before she reaches the EH will reach Bob before he sees the hole's final evaporation (and the light signal emitted by Alice when she crossed the EH).
> I could be terribly mistaken of course, but your arguments did not convince me
You are terribly mistaken, and you didn't read my arguments carefully enough. See above.
> the expansion of the EH as a body approximates
I don't know what you're talking about here. As a black hole evaporates by Hawking radiation, the area of its horizon decreases. It doesn't increase.
1. Because an 'event horizon' is not a physical boundary, it's more like a boundary that separates events that we can observe and events that we will never see in this universe, ever - that is, in this universe we can't know anything about events that happen beyond this horizon.
I mean, if you're a single point this is true. The more realistic situation is you're torn apart first.
The point (and this is the main point behind SR, see my comment elsewhere in this thread) is that in your rest-frame, you are, well, at rest. In some ways, it's sort of like an extension of the classical mechanical view that the easiest frame to do physics in is an inertial frame, while in moving frames, you have to add a velocity or a non-inertial frame, you have pseudo-forces. Newton still contented that there was some universal rest frame, somewhere, out there. Einstein essentially made the "inertial frame" a general thing: the best place to do physics is the rest frame for a system, but there is no "universal rest frame" and every observer lives in their own rest frame.
Technically, in GR, space time is also flat where you are, you only notice curvature when you look across distances, so, if you're dilly-dallying in your own rest frame as you fall into a black hole, well, you notice nothing because you're in your rest frame. You already experience this in free fall (which was Einstein's insight), you're just at rest. It's just other observers accelerating with respect to you (say, people on the ground, or the ground itself) measure you as accelerating with respect to them. When you hit the ground is when you are essentially violently accelerated into the ground's frame.
Of course, it's complicated in real life, because people aren't points, and if the scale of curvature is smaller than you're body's height, well, different parts will accelerate differently from each other (say, your eyes observing your feet), so you'll be torn apart.
To evaluate the validity of those facts, whom can I ask for advice? Because, I cannot afford years of study in physics to understand and assess the relevance of those facts to graduate school.
Besides that, when we talk about facts, why is that I cannot find any solid scientific sources to those claims in that article?
I looked for footnotes, haven't found any, besides some other blogspot links and an aeon article.
The title is misleading, there are little to none sources, the whole thing smells like an opinion and pop-science. I might not be a physicist, but I'm not lacking competency in reading.
It's a blog post, not a scientific paper, and the statements it makes are very general, of the sort you will find somewhere in many textbooks, but not in one specific place that can be referenced. What you're asking for is kind of like asking for a specific footnote for the definition of a prime number in a blog post about number theory.
If what you are looking for is a quick way to know whether what the article says is true without having to take any time to actually study science, there isn't one. There is no royal road to learning.
It would be more accurate to say the wormholes are hypothetical than to say that they are fictional. They are an idea someone came up with of something that might really exist, we just don't have any evidence of them.
Wormholes could potentially exist; there is even a well-known non-crackpot theory called ER=EPR that roughly suggests entangled particles are connected by wormholes.
What is fiction is wormholes you can send information or people though.
I think it’s misleading to characterise Sabine Hossenfelder’s blog as “some blogspot blog”. She is a serious and respected theoretical physicist, not some random blogger.
What do you mean? Even if the information happens to be correct in this case, it's still not a good idea to believe everything you read. There's plenty of bad info about physics out there, and it's not that easily to tell the difference.
>> dismissing everything with [citation needed] isn't a great idea
> it's still not a good idea to believe everything you read
This is not a binary option. In particular my point is you should click a couple of links and do some holistic assessment of how/why something might be wrong before knee-jerking with "citation needed".
I see "citation needed" as a terse and (outside Wikipedia) rude way of asking for sources. In the article, some list items have links and others don't. It's reasonable to ask politely for sources.
The phrasing of "citation needed" was very much not the point here. It wasn't even an actual quote for us to debate its rudeness or politeness. If you read my original response, I merely used it as a terse placeholder representing the reaction in the top comment, which is the topic I was actually addressing. The point was that a lack of sources does not grant the reader an automatic license to dismiss and/or disparage the text without further investigation.
She's a pop-science writer. She is a random blogger. She has a job as a physicist. It doesn't make her opinion a fact, just because she's a theoretical physicist with a job and a blog.
These are commonly accepted theories. It would be like citing Darwin every time you talked about evolution–nobody does this because you can easily do this yourself if you were in the unlikely event of being unaware of the science behind it.
If you are saying that the issue apperently most contentious to you is a hypothesis, why did you call the entire article opinion to begin with? Seems contradictory to me.
As far as I can tell, she's really only saying is that we would have easily seen QG effects of some sort if it wasn't because it's only relevant at high curvatures, for some measure of high. Hence all of the relevant theories in the QG fields can with high probability only be relevant at high curvatures, because we should have seen them otherwise. Which part of that do you consider to be incorrect?
I think the coolest fact that I didn't grok until undergrad is that gravity relates to the stress-energy tensor and is not just due to rest mass alone. Which means that anything that has energy associated with it (i.e., everything we've ever observed) affects the gravitational field.
Examples: A compressed spring weighs more than the same spring in an uncompressed state. A box of light (e.g., a box with perfectly reflective mirrored walls) weighs more than the same box without light, despite the fact that photons have zero rest mass. The bulk of a proton's "effective mass" is due to the kinetic energy of the quarks that comprise it. The joint earth + moon system weighs less than if you added up each component weighed in isolation.
The other interesting fact is that the Heisenberg Uncertainty Principle is commonly misunderstood to mean that you can't simultaneously measure the position and momentum of a particle. In fact, you can indeed obtain partial information on both properties at the same time, and there's quite a few papers out there describing how to perform joint measurements of incompatible observables.
What HUP more accurately entails is that, for a quantum state corresponding to a specific system, you cannot obtain complete information about the two properties — no matter what you do. This is because performing a position measurement destroys your ability to perform a subsequent momentum measurement (on the same system) and vice versa. One of the postulates of quantum mechanics is that measuring a system collapses it into an eigenstate of the observable corresponding to the type of measurement that was performed. Since the position and momentum operators don't commute, it's not possible to put the two into a joint eigenbasis (they can't be simultaneously diagonalized).
Since you obviously like this sort of thing, let me point out that quantum uncertainty isn't simply a factor of measurement/observer effects[1][2]. (The only part of what you said that I am addressing by this is the "this is because" section.)
I'll temper your statement about uncertainty. You can measure whatever you want, but if your observables are incompatible, you can't say it's still in the eigenstate you measured it in afterward each measurement. Then again, people have to explain what it means when they say "measure at the same time" means. If it means "system A is in eigenstate |1> and eigenstate |a>" where |1> and |a> are eigenstates of different (non compatible) operators the answer is no.
> I think the coolest fact that I didn't grok until undergrad is that gravity relates to the stress-energy tensor and is not just due to rest mass alone. Which means that anything that has energy associated with it (i.e., everything we've ever observed) affects the gravitational field.
And not just energy, but tension. And tension is a signed quantity!
Imagine that you go out into deep (flat) space and build a large disk out of unobtanium, an extremely strong and rigid material. The ratio of the circumference to the disk to the radius is exactly pi. Now imagine trying to bring the disk to Earth's surface. Here space is curved by gravity, so the radius of a disk is just a bit longer relative to its circumference. So the disk has to flex! If it is rigid enough, it will want to spring back into flat space to minimize energy. You could imagine adding some gearing that would let you control this (anti-)gravity effect at will.
Calculating the material strength needed to make this effect usefully strong is left as an exercise to the reader.
(Disclaimer: this makes sense to me, but I am not a licensed antigravity engineer)
I like the first part, entropy is essentially a measure of degeneracy for a state (the elementary definition is the log of the number of states multiplied by a unitful constant). Technically, if you're a supposed super-intelligence, that is good at remembering detail, every state of a system can be distinguishable and thus have small entropy. For example, consider a finite number of legos in a room. A computer could potentially remember where every lego is placed, while person can't do as well but could distinguish between a state where the legos are strewn about or built into a castle. So, the person would lump all the disordered states into just one state (the "mess" state) and give it a high entropy compared to the number of organized states (castles or ships made out of the legos).
I guess I always knew this, and this is sort of what we mean when we say "high entropy" but it's kind of fun to say it out explicitly like this. Most of the others seem like conflating the strict applicability of a theory vs. it's practical limits, like that QM doesn't really mean "small" or (equivalently) high energy, it just means when you're near the Heisenberg uncertainty limit for the observations you're making, which in most cases means small.
Allow me to add another one: "Special Relativity means when you go faster, time slows down for you!"
And another (controversial may be): "In Schrodinger's cat, the cat is both dead and alive at the same time!"
> every state of a system can be distinguishable and thus have small entropy
Which corresponds quite well to the fact that the entropy of a pure state is zero (since S=−tr[ρlnρ]). Entropy is better described as a property that arises between systems.
> For example, consider a finite number of legos in a room.
This works with legos, but not subatomic particles. No matter how smart you are, you can't distinguish between two states which differ only by having two electrons swapped.
Fair. This is mainly for macroscopic systems, not fundamental particles or other species that cannot be distinguished at all, then all you can do is lump the states together and give it a degeneracy.
Right, according to this blogspot post, one could figure that what we've learned about entropy is completely wrong! But no! The net amount of energy inside an isolated system does not change. It dissipates evenly through the system, and the system as a whole reaches it's maximum entropy/equilibrium. That means, each part of the system which can exchange energy, has the same amount of energy.
Conservation of energy is not a physical law. It's a consequence of mathematical theorem called Noether's theorem. All systems with certain differentiable symmetries have a corresponding conservation laws.
In a system that is not time translation invariant, energy don't have to be conserved. Expansion of universe breaks the time symmetry.
I'll say that most systems are time invariant, so conserve energy. One should always know the system one works with. For example, first year undergrad problems near the surface of the earth for example since they ignore the earth do not conserve momentum.
Note that Noether's second theorem still applies, so any timelike vector field will yield a conservation law due to diffeomorphism invariance. In that sense, energy conservation still holds even in the absence of Killing fields, though some people argue this doesn't count as a 'proper' law of energy conservation as it includes contributions by the gravitational field, for which an energy-momentum tensor cannot be defined...
Thanks, you're answering my question. It's not a random distribution - it's separate substances (butter separate from flour separate from egg), so their order is actually higher. Then I agree - it's strange the author goes for the dough.
I don’t understand her point either. And for number 3, entropy for a single particle (as far as such a thing can be defined) is quite different from entropy for a macroscopic system. I’m not sure it makes sense to say that particles decay “to reach a state of highest entropy”.
I think that's meant to be an illustration of the naïve tendency to think of something purposeful - an intended state - as being more ordered than something broken or scattered. A dough is a thing you made for a reason, a mess on the floor is just a mess on the floor. Don't take it as a physicist thinking in physical terms, but as the immediate impression of an ordinary person in ordinary circumstances - you need to reflect and understand to see the truth.
Not only that: Think about a canvas with some random splatters of paint, compared to a canvas uniformly coloured with a homogeneous mixture of that paint. What's more 'orderly'?
I, for one, thought that conservation of energy was iron clad. It turns out that "except for the expansion of the universe" proviso, it is. As a religious adherent of the conservation laws this makes me unhappy. And brings up new questions. I'll have to sleep on this one.
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[ 3.1 ms ] story [ 98.4 ms ] threadI hate headlines that assume what I do and don't know. Just call 'em 10 physics facts that most people aren't aware of and are pretty interesting. Because I actually did know a bunch of those facts from school and I was a liberal arts major...
https://news.ycombinator.com/newsguidelines.html
Quantum mechanics and Feynman diagrams in first year of highschool?
Like "School of Engineering and Mathematical Sciences" etc.
I did all biology and chemistry in highschool, I didn't have my first physics class until college.
Very interesting, I feel somewhat misled if this is true.
And if you want to calculate if a thermodynamically spontaneous reaction will appreciably take place, you also need to understand the actual intermediates between A + B -> C, and find reaction barrier heights for all of them, and weight them appropriately. This is still very hard to do so information of this sort is mostly determined by experiment rather than calculation.
While it is true that a system doesn't minimize the energy of the entire system, as energy IS conserved, except apparently for a miniscule amount due to the universe expanding, it is also true that locally it often looks like minimizing energy in some sense.
If you have a vibrating spring on a table, after a while it will have stopped because it has created motions in the air and in the table, spreading out at the speed of sound in whatever material it encounters, and some of it also at the speed of light in the form of thermal radiation.
So, looking only at the spring, it is minimizing it's energy by returning to a stable quiscent state. In the same way, if we look at a sphere with radius as "the speed of light times the time from the start of the experiment", all energy can still be accounted for within that volume.
One system, several perspectives.
I think the minimization model is often used because the statistical model of entropy can be hard to explain, as it simultaneously allows complex relatively ordered systems like us humans to exist, while at the same time explains the almost complete disorder of thermal energy and how it disperses.
Thus, since the energy minimization of the local state can be used as a relatively truthful proxy, we learn that model first. With the possible bonus that it also allows one to temporarily avoiding some rather (afaict) gnarly math until you actually have the tools you need to understand it.
https://en.wikipedia.org/wiki/Minimum_total_potential_energy...
https://en.wikipedia.org/wiki/Principle_of_minimum_energy
This one is really interesting! I've definitely never seen anyone emphasize this before.
1) It looks like it takes forever, to an outside observer, for anything to reach the event horizon. So if no matter appears to reach event horizons, how can they (event horizons) form in the first place (i.e. exist in our universe)?
2) While it takes forever for matter to fall in (again to outside observers), it takes finite time for black holes to evaporate via Hawking radiation. So not only do black holes would take forever to form, they seem to have to extinguish before being able to exist (again in the sense of event horizons).
I'm not a physicist, but my speculation is that Black Holes don't really exist, they're just a limit of a process that approaches but doesn't converge to a singularity, which I think is unphysical.
1. Because if you don't have a black hole yet then there is no event horizon to prevent you from falling in and creating one. I think the event horizon wouldn't form at exactly the same point whose crossing would finally increase density enough to create it, so this shouldn't be a problem. (Although, again, since the event horizon doesn't exist yet, I think it might still not be a problem even in that case?) I would also expect that quantum fluctuations can also inject matter into a black hole, just as they can remove matter from it.
2. Again, they wouldn't take forever to form -- see above.
I'm pretty sure black holes have been observed indirectly, so the idea that they actually don't exist would require a pretty rock solid alternative explanation!
What we have observed indirectly are compact objects that emit no light, but contain a large enough mass in a small enough volume that they can't be anything else but a black hole, if we only take classical GR into account.
As I responded in another post just now, it is possible that black holes, in the sense of objects with actual event horizons (boundaries of regions from which light will never escape, even in the infinite future), cannot exist when quantum gravity effects are taken into account. If that is the case (and it is not clear whether it is--as I said, this is an open area of research), then the objects we call black holes based on our current observations won't have actual event horizons--eventually, in the very far future, light will escape from those compact regions. But they will still have apparent horizons, i.e., surfaces from which outgoing light is not escaping now (or for a very long time in the future). And that, in itself, is sufficient to show that the issues raised by darkmighty are not valid.
Then you haven't spent much time looking at actual textbooks on GR, since all of these issues are addressed there. (Not to mention in many peer-reviewed papers in the field.)
> if no matter appears to reach event horizons, how can they (event horizons) form in the first place (i.e. exist in our universe)?
This is the same fallacy as the fallacy that nothing can actually fall into a black hole because it looks like it takes forever from the outside. Oppenheimer and Snyder published a mathematical model way back in 1939 that shows how a black hole can form in a finite time from the gravitational collapse of a massive object, as seen by an observer falling inward on the surface of the object. The collapse appears to take forever as seen by a distant observer, but this is an optical illusion caused by the effect of spacetime curvature on the paths of light rays. This has been studied for decades and is thoroughly understood.
> While it takes forever for matter to fall in (again to outside observers), it takes finite time for black holes to evaporate via Hawking radiation.
Wrong. First, if matter is falling in, the hole is gaining mass, not losing it, so it will never evaporate even though, in principle, it is emitting Hawking radiation (in practice this radiation is many, many orders of magnitude too faint to detect for any black hole we can observe).
Second, if you include quantum effects, and therefore Hawking radiation in your model, you've changed the model, and it is no longer true that a distant observer will never see anyone falling into the hole. Instead, the distant observer will see light signals emitted by objects falling through the hole's horizon at the same time the distant observer sees the hole evaporate. (At least, that is the case in the simplest model, the one Hawking used when he first published his prediction of Hawking radiation. More complicated models have been developed since, and we won't know which, if any, of them is really correct until we have an experimentally confirmed theory of quantum gravity.)
> I'm not a physicist, but my speculation is that Black Holes don't really exist, they're just a limit of a process that approaches but doesn't converge to a singularity, which I think is unphysical.
Your speculation is uninformed and wrong. It is possible that an actual black hole, with an actual event horizon, is impossible when all of the laws of physics, including the laws of quantum gravity, are taken into account. This is an open area of research. But if it turns out that actual black holes cannot exist, it won't be for any of the reasons you give.
But isn't equating the two observers incorrect? That is, when speaking of 'formation time', we are speaking of the formation time by observers at infinity, free from the effects of local curvature. What I understand is that all observers must agree on reality, thus on whether the BH forms at all: if the infalling observer makes it through an EH and into a singularity, then those exist (by definition) as unique reality. But evaporation seems to complicate this classical GR view understood for a while: since now outside observers see finite-time quasi-evaporation, infalling observers must observe this phenomenom while infalling, and actually before reaching the EH.
There's no way I can see that:
1) EHs and singularities (i.e. Black Holes as a steady-state phenomenon) exist;
2) All observers agree on reality of their formation and evaporation.
I defiantly conclude they don't exist (or I'm missing something major) :)
That is, black holes in nature would be something akin to a extremely time-dilated "Object trap" without singularties (although perhaps it must gain an apparent horizon?).
There is no "equating the two observers".
> when speaking of 'formation time', we are speaking of the formation time by observers at infinity
No, we aren't. The hole isn't formed at infinity. The time that is being talked about for observers at infinity (or more correctly, very far away) is the time it takes light rays, coming outward from the object that is collapsing into a black hole, to reach those observers. The curvature of spacetime near the horizon is such that it takes longer and longer for those light rays to get out to the observer far away, the closer to the horizon they are emitted. At the horizon, outgoing light rays do not move outward at all, due to the curvature of spacetime there, so they never reach the observer far away--hence that observer never sees the horizon form. But, as I said, that is an optical illusion caused by the curvature of spacetime.
> What I understand is that all observers must agree on reality, thus on whether the BH forms at all
All observers agree on observable events and measurement results that they can all observe, yes. But if an observer cannot observe a particular event (such as the horizon forming) because spacetime curvature prevents light rays from that event from ever reaching him, then he can say nothing at all about whether that event happens or not.
> evaporation seems to complicate this classical GR view understood for a while
It complicates it in the sense I said before: that observers far away, instead of never seeing infalling objects cross the horizon, now see those objects cross the horizon (more precisely: see light rays emitted outward by those objects when they crossed the horizon) at the same time they see the hole evaporate (more precisely: see light rays from the hole's final evaporation). It does not mean what you claim, that infalling observers will see the hole evaporate before they fall in. The spacetime geometry of an evaporating black hole (i.e., including quantum effects) is different from the spacetime geometry of a classical black hole that never evaporates. The statement that a distant observer never sees an object cross the horizon is only true of the latter, not the former.
> I defiantly conclude they don't exist (or I'm missing something major) :)
The second is the correct choice. You need to stop being defiant and start learning what the models of black holes in physics actually say.
> black holes in nature would be something akin to a extremely time-dilated "Object trap" without singularties (although perhaps it must gain an apparent horizon?)
To us far away, an apparent horizon looks the same as an event horizon, and works the same--all the things I said above would still be true (at least for the time we have been observing--see below for how long it would take for differences to become observable) if the objects we observe and call black holes actually have only apparent horizons (because of hypothetical quantum gravity effects that, as I've already said, are an open topic of research), not event horizons. And all of your criticisms would still be wrong. (In at least some of the hypothesized quantum gravity models, there is no singularity and no event horizon, only an apparent horizon; but to us far away, the object looks and works the same into the very far future; it takes something like 10^70 years for the difference to be observable.)
I think you have a misconception about GR here. In GR, spacetime isn't just an "optical device" that delays rays cast in an absolute reference. In relativity, the behavior of light rays is actually closely related to interpretation of time and space. You can't separate the two. In fact, the definition of (relative) local time is given exactly by the rate a "light clock" ticks when communicating periodically to a "light clock" located elsewhere. Approaching the event horizon, the rate of clock ticking goes to 0 relative to an outside observer (to the the limited extent that coordinate frames can be defined in GR). So classically you can't say that to the outside observer the BH forms in finite time, I don't think. What matters in the end is what the observer sees, which, according to you, would be (for Bob, far away):
Alice falls into BH -> Alice approaches EH -> Alice's BH evaporates -> Alice finishes falling into the BH
That's not possible! The light rays from evaporation, which happens, in Alice's own frame, after she falls into the BH, cannot precede the light rays from her fall. So she cannot truly go past the EH.
No, you do. Everything I am saying is taken straight from GR textbooks--mainly Misner, Thorne, & Wheeler (1973) and Wald (1984)--and peer-reviewed papers, such as Hawking's original paper on black hole evaporation. You need to go read them.
> In GR, spacetime isn't just an "optical device" that delays rays cast in an absolute reference.
I never said it was. The paths of light rays in GR are determined by the geometry of spacetime. "Optical illusion" is one way of trying to describe the effect of that in the case under discussion. But the effect is there regardless of what you call it.
> In relativity, the behavior of light rays is actually closely related to interpretation of time and space. You can't separate the two.
This is just another way of saying that the paths of light rays in GR are determined by the geometry of spacetime. Which is true, but to figure out what that means in a specific scenario, you need to properly understand the spacetime geometry in that scenario. You evidently do not.
> In fact, the definition of (relative) local time is given exactly by the rate a "light clock" ticks when communicating periodically to a "light clock" located elsewhere.
There is a limited set of scenarios in which this works, but that limited set does not include the case under discussion (Alice falls into a black hole while Bob remains far away). In general in GR, there is no well-defined way of comparing "times" for spatially separated observers. The only well-defined time is local--proper time along a particular observer's worldline.
> Alice falls into BH -> Alice approaches EH -> Alice's BH evaporates -> Alice finishes falling into the BH
I never said that was the sequence of events. It isn't. The correct sequence of events is (note: this is what happens to Alice, Bob does not see all of these events--see my other post in response)
Alice falls into BH -> Alice approaches EH -> Alice finishes falling into BH -> Alice's BH evaporates
If you don't understand how that happens, you need to go study GR textbooks. The model that gives the sequence of events I just described has been well understood for several decades now--Hawking published the first paper using it in the 1970s.
I don't know if this is just a mathematical simplification that doesn't impact conclusions about existence of EHs and singularties, but it all looks fishy.
Why? Note that I'm not claiming (and Hawking never claimed) that his model is experimentally verified--that's obviously impossible since Hawking radiation from any black hole we can observe is many orders of magnitude too faint to observe, if it exists and has the intensity predicted by the model. But the model is perfectly self-consistent.
> it all looks fishy
Why?
https://en.wikipedia.org/wiki/Firewall_(physics)
Basically, my intuition is more or less in line with this firewall phenomenon and that no particle crosses a true event horizon, even in their own frame, and no certainly no singularity.
Not what you said. What Bob sees is:
Alice falls into BH -> Alice approaches EH -> Alice reaches EH/Alice's BH evaporates
In other words, as I said, Bob sees Alice crossing the EH at the same time he sees the BH evaporate. Bob still never sees any light signals from Alice that were emitted below the EH; those signals are trapped inside the hole and never escape. In Hawking's original model, those signals, along with Alice herself, hit the singularity and are destroyed; they cease to exist, so they never come out again even when the hole evaporates away. An open question is what the correct model will end up being when we have a full understanding of quantum gravity; but whatever that model ends up being, it won't validate the particular issues you are raising.
We had already agreed that General Relativity should imply Bob never sees Alice reach EH, regardless of quantum effects (at least not in finite time). We also agreed that Bob does see the BH evaporate (in finite time to any desired fraction), as seemingly dictated by QM. Thus it cannot be that the two are observed simultaneously!
Moreover, there cannot be a chain of events whereby Alice evaporates into photons (emanating from EH), and those photons (originating from Alice's mass-energy) hit Bob before the photons of Alice when she was near, but did not reach the EH yet (again because Alice cannot be observed to reach EH). It would violate causality.
It can't be that both Alice falls into the EH and we receive Alice's energy from evaporation: either the BH does not evaporate or Alice doesn't cross the EH (even in her own observation). Assuming Hawking radiation really does exist, then some kind of evaporation would need to occur as objects approach the EH, it cannot come directly from the EH boundary.
Could it be that Alice herself is seen to evaporate -- the evaporation might stem directly from Alice, before reaching the EH? My claim that the EH/singularity never forms would be valid.
I could be terribly mistaken of course, but your arguments did not convince me unfortunately (quite the contrary).
I wonder if what I'm missing could be related to the expansion of the EH as a body approximates.
In the original idealized model of Hawking, yes. That is what the spacetime geometry says. One way of thinking about it is that, with this spacetime geometry, the path of any light ray emitted outward at the event horizon travels along that horizon into the future, but the area of the horizon decreases as the hole evaporates. At the instant of the hole's final evaporation, all of the light rays emitted anywhere along the horizon are at the same point as the final evaporation of the hole, and they all go outward from there together. This is a highly idealized model; but that's what the model says.
> We had already agreed that General Relativity should imply Bob never sees Alice reach EH, regardless of quantum effects
No, we haven't. You claimed that, and I pointed out that you were wrong. Bob never sees Alice reach the EH in the classical GR model, with no quantum effects. Quantum effects change the model. In the model including quantum effects, Bob does see Alice reach the EH. See above and my previous posts.
> there cannot be a chain of events whereby Alice evaporates into photons (emanating from EH), and those photons (originating from Alice's mass-energy) hit Bob before the photons of Alice when she was near
This is correct. And it is also what the model of a hole evaporating by Hawking radiation says. All of the light signals emitted by Alice before she reaches the EH will reach Bob before he sees the hole's final evaporation (and the light signal emitted by Alice when she crossed the EH).
> I could be terribly mistaken of course, but your arguments did not convince me
You are terribly mistaken, and you didn't read my arguments carefully enough. See above.
> the expansion of the EH as a body approximates
I don't know what you're talking about here. As a black hole evaporates by Hawking radiation, the area of its horizon decreases. It doesn't increase.
The point (and this is the main point behind SR, see my comment elsewhere in this thread) is that in your rest-frame, you are, well, at rest. In some ways, it's sort of like an extension of the classical mechanical view that the easiest frame to do physics in is an inertial frame, while in moving frames, you have to add a velocity or a non-inertial frame, you have pseudo-forces. Newton still contented that there was some universal rest frame, somewhere, out there. Einstein essentially made the "inertial frame" a general thing: the best place to do physics is the rest frame for a system, but there is no "universal rest frame" and every observer lives in their own rest frame.
Technically, in GR, space time is also flat where you are, you only notice curvature when you look across distances, so, if you're dilly-dallying in your own rest frame as you fall into a black hole, well, you notice nothing because you're in your rest frame. You already experience this in free fall (which was Einstein's insight), you're just at rest. It's just other observers accelerating with respect to you (say, people on the ground, or the ground itself) measure you as accelerating with respect to them. When you hit the ground is when you are essentially violently accelerated into the ground's frame.
Of course, it's complicated in real life, because people aren't points, and if the scale of curvature is smaller than you're body's height, well, different parts will accelerate differently from each other (say, your eyes observing your feet), so you'll be torn apart.
Besides that, when we talk about facts, why is that I cannot find any solid scientific sources to those claims in that article?
You could ask other physicists. But you would not know whether to believe them any more than you know whether to believe the author of this article.
> when we talk about facts, why is that I cannot find any solid scientific sources to those claims in that article?
Where have you looked?
The title is misleading, there are little to none sources, the whole thing smells like an opinion and pop-science. I might not be a physicist, but I'm not lacking competency in reading.
It's a blog post, not a scientific paper, and the statements it makes are very general, of the sort you will find somewhere in many textbooks, but not in one specific place that can be referenced. What you're asking for is kind of like asking for a specific footnote for the definition of a prime number in a blog post about number theory.
If what you are looking for is a quick way to know whether what the article says is true without having to take any time to actually study science, there isn't one. There is no royal road to learning.
It would be more accurate to say the wormholes are hypothetical than to say that they are fictional. They are an idea someone came up with of something that might really exist, we just don't have any evidence of them.
What is fiction is wormholes you can send information or people though.
"Aspects of theoretical physics not always taught in school" would be more appropriate.
The Feynman Lectures on Physics is not well sourced, but people eat it up year after year without any skepticism :)
> it's still not a good idea to believe everything you read
This is not a binary option. In particular my point is you should click a couple of links and do some holistic assessment of how/why something might be wrong before knee-jerking with "citation needed".
He just blogged educational insights.
As far as I can tell, she's really only saying is that we would have easily seen QG effects of some sort if it wasn't because it's only relevant at high curvatures, for some measure of high. Hence all of the relevant theories in the QG fields can with high probability only be relevant at high curvatures, because we should have seen them otherwise. Which part of that do you consider to be incorrect?
Examples: A compressed spring weighs more than the same spring in an uncompressed state. A box of light (e.g., a box with perfectly reflective mirrored walls) weighs more than the same box without light, despite the fact that photons have zero rest mass. The bulk of a proton's "effective mass" is due to the kinetic energy of the quarks that comprise it. The joint earth + moon system weighs less than if you added up each component weighed in isolation.
The other interesting fact is that the Heisenberg Uncertainty Principle is commonly misunderstood to mean that you can't simultaneously measure the position and momentum of a particle. In fact, you can indeed obtain partial information on both properties at the same time, and there's quite a few papers out there describing how to perform joint measurements of incompatible observables.
What HUP more accurately entails is that, for a quantum state corresponding to a specific system, you cannot obtain complete information about the two properties — no matter what you do. This is because performing a position measurement destroys your ability to perform a subsequent momentum measurement (on the same system) and vice versa. One of the postulates of quantum mechanics is that measuring a system collapses it into an eigenstate of the observable corresponding to the type of measurement that was performed. Since the position and momentum operators don't commute, it's not possible to put the two into a joint eigenbasis (they can't be simultaneously diagonalized).
[1] https://www.nature.com/news/quantum-uncertainty-not-all-in-t... [2] https://en.wikipedia.org/wiki/Uncertainty_principle
And not just energy, but tension. And tension is a signed quantity!
Imagine that you go out into deep (flat) space and build a large disk out of unobtanium, an extremely strong and rigid material. The ratio of the circumference to the disk to the radius is exactly pi. Now imagine trying to bring the disk to Earth's surface. Here space is curved by gravity, so the radius of a disk is just a bit longer relative to its circumference. So the disk has to flex! If it is rigid enough, it will want to spring back into flat space to minimize energy. You could imagine adding some gearing that would let you control this (anti-)gravity effect at will.
Calculating the material strength needed to make this effect usefully strong is left as an exercise to the reader.
(Disclaimer: this makes sense to me, but I am not a licensed antigravity engineer)
I guess I always knew this, and this is sort of what we mean when we say "high entropy" but it's kind of fun to say it out explicitly like this. Most of the others seem like conflating the strict applicability of a theory vs. it's practical limits, like that QM doesn't really mean "small" or (equivalently) high energy, it just means when you're near the Heisenberg uncertainty limit for the observations you're making, which in most cases means small.
Allow me to add another one: "Special Relativity means when you go faster, time slows down for you!"
And another (controversial may be): "In Schrodinger's cat, the cat is both dead and alive at the same time!"
Which corresponds quite well to the fact that the entropy of a pure state is zero (since S=−tr[ρlnρ]). Entropy is better described as a property that arises between systems.
A macroscopic state is not a physical object. The same microscopic state may correspond to different macroscopic states for different observers.
This works with legos, but not subatomic particles. No matter how smart you are, you can't distinguish between two states which differ only by having two electrons swapped.
Conservation of energy is not a physical law. It's a consequence of mathematical theorem called Noether's theorem. All systems with certain differentiable symmetries have a corresponding conservation laws.
In a system that is not time translation invariant, energy don't have to be conserved. Expansion of universe breaks the time symmetry.
0. https://en.wikipedia.org/wiki/Noether%27s_theorem
Why the dough is of higher entropy than the (random) distribution of dough components?
> I’d go for the dough.
Strange.