The reasonable things that continue happening each day in our universe would be extremely unlikely if we are just Boltzman brains. Every bit of sensible reality would be coincidental. The very continuance of that reality is an experiment constant proving the falsehood of Boltzman brains, at a rate of oh maybe millions of sigmas of confidence per second.

Now, if you believe the universe came to an initial state due to pure thermodynamic coincidence, millions of sigmas per second is laughably small compared to the chance that a whole universe outside your brain popped into existence, so Boltzman brains are the most believable thing and you should believe in them.

This completes a pretty direct argument: Believing the initial state of the universe was a thermodynamic coincidence forces you to believe in Boltzman brains, Boltzman brains force you to believe reality should collapse immediately, and reality does not collapse immediately. Therefore you simply can't believe the first assumption, that initial state of the universe was a thermodynamic coincidence.

Accepting this is often called the "Past Hypothesis". It's spoken of in deferential terms and said that it can't ever be proven... But to me this is rock-solid proof, with more sigmas of evidence than any other scientific discovery and increasing by the second! Can't we just call it the Past Theorem already?

> and reality does not collapse immediately

How do you know that reality does not collapse immediately? At any given instant you could be a fresh brain that just came into existence, all your previous memories which imply a life lived up to this point also formed in that same instant.

i.e. a time stepped simulation, absent external reference, doesn't know how long it's been between the actual steps - could be seconds, could be hours, could be years.

EDIT: Like the real issue with "death" is that it's not "eons on darkness" - which is why I think people get afraid of it (or one of the reasons) - but that actual, literal non-existence is inconceivable even though we all did it - 13 billion years of not-existing in the universe, then suddenly you.

So after you die the same problem re-emerges: the conscious experience of "you" ends...but then from the subjective blink of an eye if something happens to restart that information process just right, suddenly again, you - and it has to be you, and no one else, because if it wasn't then well, it would be someone else - i.e. why am I me, and not my wife or son for example?

For a Boltzmann brain there is no real past or future - your reality does indeed collapse immediately and you'll never know it; the idea that 'reality doesn't collapse immediately' is not a verifiable fact, because the only evidence you have to the contrary is encoded in and perceived by your brain...

You could even live kind of a "life" by randomly popping into existence once every million years in some differrent galaxy, experiencing one planck time and collapsing (and the only thing that connects the instances to each other is that the next instance by random chance has memories consistent with the previous instance).

They could even appear in non-chronological order.

I don't think it's likely, but it's more likely than having the one randomly generated brain experience stuff "in real time".

With mesh faces now supported in BREP, I'm more optimistic about a mixed modeling approach, where you can do the braindead find-the-cylinders conversion but keep the rest mesh, not needing to force it into some eldritch contortion of BSurfs.

The main advantages of BREP are:

(1) You capture design intent because of the explicit dimensioning and constraint model, which of course is still not used enough and 2D drawings are useful.

(1a) This intent is often needed (even if just implicitly) during the manufacturing process as machinists, toolmakers, etc. + their software (CAM, G-Code, etc.) convert them into physical parts.

(2) They are human understandable and editable.

(3) The legacy ecosystem uses old CAD, and it's very hard to switch - ie Boeing isn't putting 777 in a new CAD unless it has some massive advantage.

So having BREP, or perhaps a mixed approach like you suggest with the feel of BREP (feature tree, direct modeling, etc.) approach would ease the transition.

This multiplies, so after a million days a clock on the moon will read 58 seconds behind a clock on the earth.

In an Earth-centered inertial frame, the Moon is moving faster than the Earth, but it is also at a very high altitude, and the altitude effect, which speeds up the Moon clock relative to Earth, is much larger than the speed effect, which slows it down.

(Note that the above only takes into account the effects of the Earth's gravity. The paper also takes into account the effects of the Moon's gravity, which don't change the above answer qualitatively but do add small corrections numerically, so the 58.721 microseconds per day is not the actual value the paper ends up with.)

I'm assuming this affects clocks (and things) but not time, right? Time itself is no different on the moon (it's not the future there).

I know this must be true otherwise we would be surrounded by time travellers by now. So, where does this cancels out?

My intuition says that if we have two clocks, each clock with a display and a laser pointing to each other, and we put one of them on the moon and the other on earth, someone observing it from a third equidistant point would see both lasers blinking at the same rate.

If that's true (I don't know if it is) both of your answers are kinda right, aren't they? From earth, the moon clock ticks slower compared to earth clocks, therefore it lags behind. From the moon, the moon clock ticks faster compared to earth clocks, therefore it skips ahead, and vice-versa. I am not sure though, I feel like I'm missing something.

This is false. Time itself is in fact progressing at a different rate in both frames, and a second in one is not the same as a second in the other.

But, time still "works the same way" for both reference frames in the sense that its not like the clocks in one appear to move in "slow motion". An observer on Earth or the Moon still sees clocks ticking at one second per second, and objects appear to move and react in the ordinary physical ways at the same rate you expect them to. But this is because the observer is also an object in that reference frame, so their perception of everything is in the same "slow motion" as the objects they observe.

Now, if an observer on the Moon was to watch an observer on Earth with a telescope or vice versa, then they would indeed see "the video playing at a slightly wrong speed". The effect is relative between observers, not local to an observer. That's why its called relativity.

The 3rd party seeing blinks at the same rate was just a guess, I'm happy to learn that this guess is false.

Otherwise, as you have been told, your intuition was trully false, and spacetime behaves in some unintuitive ways. Whats weird about it is that it seem to behave differently depending on where you are observing something from, thats cruical. Its all about point of view, literally. Wherever you are (earth, moon, space, near of blackhole), time and space will work normaly for you because you are there. But if you are looking how things are happening somewhere else, it suddenly starts looking very weird.

You said that you would be surrounded by time travellers, and you indeed literally are. The boring answer is we are all tkme travelers, but the interesting part of it that if we accelerate differently (one going up the tower while other waits down), their times literally start moving different amounts, and when they meet, one is younger than another. The reason we dont notice is because we keep moving at very, very, very low fractions of speed of light where time dilatation is easy to ignore (must be mathematically compensated for GPS sattelites to even work, tho), but that doesnt mean its not ever-presently there.

If I modify that scene a little bit, it serves to showcase my doubts.

4 astronauts are on orbit around the water planet, but Brand is sick. Cooper, Brand and Doyle go down so Romily can spend 20 years researching a cure.

After an hour, Cooper, the sick Brand and Doyle go back to orbit and reunite with Romily. For Romily, 20 years passed and he completed the sythesis for the cure. Romily has "stolen" some information about the future by "walking it" in a relatively different pace, and it was able to share that information with someone that "walked it" in a slower pace.

Now, this is for gravity. This was possible because of Gargantua. But gravity is not the only component of the "microseconds per day" formula, there's seems to be more.

So I'm interested perhaps in an illustrative scenario where the speed is highlighted (whatever the outcome is, doesn't matter), not the gravity, and there is an attempt at information exchange (don't care if can be successful or not, as long as the example actually explains it).

That would be something like the standard twin paradox. You could set that up so that, for example, Cooper, Brand, and Doyle go out in their spaceship at very high speed and don't come back until 20 years have passed on Earth, where Romily has spent that time finding a cure. With the right numbers for the speed of the spaceship and the distance it travels out and back, it could be arranged that only an hour would pass for those aboard the ship. The downside of this is that the rocket power you would need to boost the ship to the required speed, and then turn it around and come back and decelerate to land on Earth again, would be huge, way, way beyond what would normally be considered reasonable.

The reason a supermassive black hole was used instead in Interstellar was to allow Cooper, Brand, and Doyle to be able to travel at much more reasonable speeds using much more reasonable rockets, and take advantage of the spacetime geometry of the hole to get the extreme time dilation effect. The reason a rapidly rotating hole was used (aside from the fact that we believe most supermassive holes in our universe are rapidly rotating in this way) was that it allows free-fall orbits to exist very close to the hole's horizon that have very large time dilation factors like 1 hour to 20 years, so that a planet in a free-fall orbit could plausibly exist in that region. For a non-rotating hole that is not possible; to get that close to the horizon and get the time dilation factor you would need, you would have to use rockets to "hover" and the rocket power involved would be of the same order of unreasonableness, if not more so, as what would be needed for the "twin paradox" trip I described above.

What's the difference?

> otherwise we would be surrounded by time travellers by now

I'm not sure what your reasoning is here.

> where does this cancels out?

I don't understand the question. What is supposed to cancel out?

> someone observing it from a third equidistant point would see both lasers blinking at the same rate.

No, they wouldn't.

> From earth, the moon clock ticks slower compared to earth clocks

No, it ticks faster. That was my point.

  - I get into a spacecraft, go into orbit around earth.
  - Once in orbit, I start accelerating until my clock ticked faster for long enough to be 1s ahead of earth.
  - Once I'm 1s ahead of earth, I capture transmissions my buddy sent to me of the stock exchange rates.
  - My buddy, on the ground, has a telescope and is ready to invest or sell stocks depending on whether I stop or keep accelerating.
  - My buddy now has knowledge about the future, I land and share the money we stole from the future with him.

Let me re-describe your scenario properly:

Your buddy sends you information about stock exchange rates when his clock reads 12 noon exactly. You are one light-second away from him, so his message is received by you at 12 noon + 1 second, his time. That is also 12 noon + 2 seconds, your time, but that doesn't matter; it's still information about rates when his clock said 12 noon, not rates when his clock said 12 noon + 1 second, 1 second after he sent the message.

In other words, the fact that your clock ticks faster and gets "ahead" of your buddy's does not mean you receive information from your buddy's future.

Going further than this is way above my skillset :D Thanks for the time to indulge my curiosity, I'll study more and try to develop a better intuition.

If you mean the 1 second light travel time is the same as the 1 second clock difference, you can increase the clock difference by just spending longer in orbit. For example, you could wait until the clock difference was 1 hour. Then, if your buddy sent you a light signal at 12 noon by his clock, it would arrive at 1 pm + 1 second (1 second light travel time) by your clock. But it would still be telling you stock exchange rates at 12 noon by your buddy's clock, not 1 pm by your buddy's clock.

Note that this is wrong: you don't have to accelerate to make your clock tick faster. You just have to be in orbit at a high enough altitude for the speedup due to altitude to outweigh the slowdown due to your free-fall orbital speed.

From the paper:

> Φ0 is the effective gravitational potential in the rotating frame, which is the sum of the static gravitational potential of the Earth, and a centripetal contribution

This is just so I can catch up with a specific desired dilation relative to earth.

I don't want to make my orbit higher, on the contrary, the less distance the better so communication is faster.

It does.

> therefore accelerating matters.

No, it doesn't. In a free-fall orbit, proper acceleration is zero; you are weightless. "Accelerate" would mean firing your rockets to change your orbit. You don't want or need to do that.

It is true that there is a coordinate acceleration for a body in a circular orbit, in coordinates centered on the Earth, but coordinate acceleration is irrelevant to what we are discussing.

> This is just so I can catch up with a specific desired dilation relative to earth.

As I said, you do that by staying in an appropriate free-fall orbit for a long enough time. You don't need or want to fire your rockets.

> I don't want to make my orbit higher, on the contrary, the less distance the better so communication is faster.

But the lower your orbit, the less your clock speeds up relative to Earth clocks. And if your orbit is low enough, your clock will actually run slow compared to Earth clocks (because the altitude effect no longer outweighs the effect of your orbital speed). For example, clocks on the ISS run slow compared to Earth clocks.

That's almost what I meant! Spacecraft, acceleration to pick up speed (not to go higher), stock-exchange cheaters.

I don't get the "don't need" or "don't want". It is part of my scenario. I also don't want a twin falling into a black hole, but it is a thought experiment that helps put things in perspective, specially for layman like me.

I always heard of the scenario of the twin at the speed of light that remains younger. I am introducing an element of communication (originally, clocks with laser beams then stock exchange rates) into that scenario and trying to understand what the offsets mean.

This is all above my paygrade, I know, so don't worry! I know I'm far from getting it and I don't want to bother :)

That would mean you would no longer be in a free-fall orbit, you would be moving faster than free-fall orbit speed, and your clock would run slower. Depending on how much you sped up, you might even end up having your clock run slower than Earth clocks.

> stock-exchange cheaters

Nothing you can possibly do with your rocket will enable you to cheat on the stock exchange. No matter what you do, you can't have your buddy's light signals contain information from his future.

> I don't get the "don't need" or "don't want".

You don't need or want to fire rockets to speed up if your objective is to have your clock run as fast as possible relative to Earth's at a given altitude. Indeed, if you really want your clock to run as fast as possible relative to Earth's clock at a given altitude, you should use your rocket to "hover" motionless (meaning zero speed relative to Earth's center of mass) at that altitude, not to speed up relative to free-fall orbital speed.

I'm not trying to come up with an experiment that enables stock-exchange cheating. I'm trying to come up with a thought experiment that highlights the effect of speed on time dilation, with the purpose of understanding what the accumulation of "microseconds per day" means, and in the spirit of the paper posted I want to put an element of information/communication there (clocks with laser beams, stock exchange, doesn't matter what it is, for this purpose they're equivalent).

> Depending on how much you sped up, you might even end up having your clock run slower than Earth clocks.

So there is at least a component of the "microseconds per day" offset formula that contributes to a slowing down compared to what is being orbited, is that correct?

If I got this right, speed matters but it is negligible compared to other factors for objects like the moon or a human made satellite. That's OK. Like I said, I don't want to actually cheat in the stock exchange, I want to understand that effect.

I'm not sure what you mean by "compared to what is being orbited". We are comparing a clock in a free-fall orbit to a clock on Earth.

> speed matters but it is negligible compared to other factors for objects like the moon or a human made satellite

For the Moon, yes, the speed effect is small compared to the altitude effect.

For human made satellites, it isn't. For example, as I have already said, clocks on the ISS run slow compared to Earth clocks--because in low Earth orbit, the slowdown due to speed is greater than the speedup due to altitude.

Your answer seems to indicate that yes. So, in principle, I should be able to come up with a thought experiment that highlights what that speed means.

Would this hypothetical faster moon, billions of years orbiting earth at a different speed, look geologically younger than our regular speed moon? (genuine don't know, but that's the kind of coloquial illustrative example I'm looking for).

Meaning, if you attached rockets to the Moon to speed it up while not changing its orbital radius? In that case the speedup relative to Earth clocks would be less, yes.

> in principle, I should be able to come up with a thought experiment that highlights what that speed means

I gave you one upthread, in the subthread where the movie Interstellar was being discussed.

> Would this hypothetical faster moon, billions of years orbiting earth at a different speed, look geologically younger than our regular speed moon?

Meaning, if you compared the two after billions of years of Earth clock time had elapsed? Yes.

Let's try this scenario:

You are in your spaceship in free-fall orbit around the Earth at the altitude of the Moon. Your buddy remains on Earth, and every day, when you in your ship pass directly overhead, he sends you a light signal with a time stamp. More precisely, he will do this every 24 hours and 50 minutes by his clock, since that's how long it takes from one overhead passage of the Moon to the next. (We'll ignore all the variations in the Moon's orbit and assume your ship is in a circular orbit at the appropriate altitude and in the right plane to pass directly over your buddy each day.)

Let's say that on day 0, you pass over your buddy at 12 noon precisely by his clock. He sends you his signal at that instant, and you receive it 1.25 seconds later, the light travel time to the Moon. And let's say you've adjusted your clock so it reads exactly 12 noon + 1.25 seconds when you receive that signal.

Then, 24 hours and 50 minutes later, you pass over your buddy again and he sends you his next signal at 12:50 pm by his clock. You receive it at 12:50:01.250058 pm by your clock, i.e., 1.25 seconds of light travel time + 58 microseconds that your clock gained. But, as I've said, the information in the light signal is still the information from 12:50 pm by your buddy's clock, not 12:50:000058.

24 hours and 50 minutes after that, you pass over your buddy again and he sends you another signal, this time at 1:40 pm by his clock. You receive it at 1:40:01.250116 by your clock, since your clock has now gained 116 microseconds. But, again, the information in the signal is still from 1:40 pm by your buddy's clock, not from 1:40:000116.

And so on, for as long as you want to run the experiment.

In other words, the light signals your buddy sends you are "markers" in spacetime: they govern what information your buddy can send you. The difference in your clock rates is not in "how fast you go into the future"--your clock moving faster doesn't mean you get your buddy's light signals "in advance". All it means is that there are more ticks of your clock--58 microseconds more--in between each "marker" light signal. Or, to put it in the geometric language that is common in General Relativity, the length of your worldline between two successive "markers" is 58 microseconds longer than the length of your buddy's worldline between those same two "markers". But the causal structure of spacetime--what information you and your buddy can exchange--doesn't depend on your clock rates, it only depends on the "markers", the light signals, between you.

Thanks, this helps a lot! I think I have just one more question :D

I can send him a signal too, right?: I can stop accelerating, or beam him back at the speed of light.

When I receive the signal at 12:50:01.250058 I stop accelerating. 1.25 seconds later at 12:50:02.500000 (his clock) my buddy sees that I sent him a response by stopping my acceleration. Or does he see me stopping when his clock is at 12:50:02.500058?

If he sees it at 12:50:02.500000, no one gets information in advance, but I gained 58 microseconds that I could use to run a computer program for a little longer, as an example, and make a better statistical prediction than he could do on the ground. Or do I lose the ability to use those 58 microseconds in some way?

Yes. Rather than stop accelerating (you weren't in the first place, you were in a free fall orbit in my scenario), you can just send a light signal back, which is a lot simpler--and uses a lot less rocket fuel. :-)

Suppose you receive your buddy's signal at 12:50:01.250058 and immediately send one back. Your buddy will receive that signal at 12:50:02.5000000 by his clock. No 58 microsecond gain, since his clock didn't gain that time, only yours did.

> gained 58 microseconds that I could use to run a computer program for a little longer, as an example

Yes. The Interstellar scenario discussed upthread gives a much more extreme example (and in my response I gave a similar scenario but involving speed instead of using the gravity of a supermassive rotating black hole to get the time dilation effect).

> the twin at the speed of light that remains younger

Neither of these are actual scenarios in relativity. I'm not sure where you are getting them from but your information appears to be garbled.

There is a so-called "twin paradox" in relativity (not actually a paradox so the name is a misnomer), where two twins who take different trips (in the original scenario, one stays at home and one travels out to a distant star and back again at high speed) can end up with one younger than the other when they meet up again. But neither twin can travel at the speed of light; that's impossible for an ordinary object like a person. And neither twin can fall into a black hole, because if they did they could never come back out to meet up with the other twin.

[0]https://chromewebstore.google.com/detail/unhook-remove-youtu...

1) In the current version, you can actually set very flexible notification schedules and use a medium of your choice (email, webpush) to receive the notifications about new videos.

2) You can create lists by cloning YouTube’s playlists and then keep track of your progress by marking them Watched/Unwatched. This is good if you’re taking a course on YouTube.

3) I am working on adding the “tags” functionality which will suggest similar channels to users based on the kind of videos & channels they’ve added.

4) I will also be adding a video summary feature which will send a short summary of the video to you.

I have many more features in mind, just need to get some initial validation of the idea.

A high school sophomore! I'm thoroughly impressed, not just by the clarity of this explanation, but by the tone, showing appreciation for where and how the elegance pops up, and empathy for a reader who's never heard of these concepts. Throw in the minimal web design I was sure I was reading some graybeard.

Excellent writing, you should be proud of this.

For the rest of us (and because I've previously heard that in films etc. about university students and never really known exactly what it meant except that it's one of the years) that's the second year of high school, which in a middle-school-having system (as is the norm in the US) is year 10, which in the US is two years (not three as it would be in the UK for example) of high school away from university.

Just a bit of a tangent to understand. Impressive indeed.

   Type mismatch error: operation ‘+’ can not be applied to values ‘5’ and ‘“sophomore”’

0. https://en.wikipedia.org/wiki/Level_of_measurement

1. https://en.wikipedia.org/wiki/Private_(rank)

School districts in the US almost always start without established middle schools. Instead, there's so much elementary school capacity and so much secondary school capacity in a district. And then, whenever the population of students in the district changes, the choices of the district school board to respond to that are either to:

1. build one additional elementary school for every N additional grade 1-7 students, and/or one additional secondary school for every M additional grade 8-12 students (and continue to pay these CapEx costs even if student population goes down; and agree to deal with the fact that growth in elementary students will inevitably propagate to growth in secondary students 7 years later);

2. add temporary capacity to existing elementary + secondary schools with portable classrooms (which can easily be reduced back down if populations decrease — but which don't come with commensurate increases in central infrastructure, eventually straining the capacity of school facilities);

3. or, at one point, you look at the numbers of students in each year in your district, treat the years as vnodes, and then rebalance a contiguous set of those vnodes to a new "shard" called "middle schools". I.e. you pick a set of school-years, where extracting students from that set of years from one or two elementary and secondary schools in the district into a newly-built middle school, will result in a "good" amount of capacity being freed from those schools, and the resulting middle school instantly having a "good" utilization.

Once you've chosen option 3 one time, it sets a precedent in that district for what years middle school should cover in that district. Most districts only do the study for what years would be best to extract out the first time they build a middle school. After that, they just go with what they did before, even if it's no longer optimal (because skipping the study saves time and money.)

In theory, school districts could actually build successive middle schools that "pull" different years of utilization out of the elementary+secondary schools around them than previously-built middle schools have. The only issue would be in administrative assignment of students to schools — this would require you to place middle schools with different year coverage so they never have overlapping catchment area.

People have been specifying locations in terms of NS/EW coordinates since the greeks. Celestial navigation ensures we always have a clear idea where north is, and when two locations are at the same latitude. It's the most natural way we've understood and discussed far-away places.

I don't think it's fair to say mercator invented this projection so much as he famously published maps which used it.

(btw, I agree this line is a complete retrospective coincidence, just not with this particular argument)

Mercator is special projection that was not used before him. Different projections give different distortions.

I think the biggest issue with group assignment is the fact that you just can't type while the wrong group is selected. I don't care what tools you have for narrowing down letters, if I see a solution in my mind, I should be able to type it out like I could on a piece of paper. Assignment feels more like adding flags in minesweeper (optional, helpful) than part of the solution.

It's not a trial of Sam, it's an autopsy of the board.

Although an individual reader's gut and other information might conceivably lead one to take all those claims at face value, I don't think it meets a journalism standard of evidence for "reveals" in a headline.

IIUC, headlines and soundbites have a lot of influence on our understanding of the world.