Hyperbolic Geometry, continued from part 1, in which we learned about critical distances and critical angles.
Quickie review:
There. You are caught up, now.
Let's start with a useful fun fact that follows from our observations so far:( Read more... )
So this comes up in Relativity, it having been noted that, in a space-time diagram, velocities are essentially slopes and, as such, don't add/subtract the way you'd think, that the Right Thing with velocities in one dimension is to project them onto the 1-Year Later hyperbola and measure the distances along that — what we've been calling the "velocity angles" — i.e., adding/subtracting those quantities instead.
One is then inspired to ask what happens in two (and three) spatial dimensions where you can now have velocity vectors with some (ordinary) angle between them that's not 0° or 180°; how do you combine those? Normally, we'd be doing trigonometry at this point (two magnitudes, angle between them, Law of Cosines, turn the crank, done), but it's going to need to be a different kind of trigonometry, because the hyperboloid thing that the 2D velocities project onto isn't flat,…
… just like combining displacements along a sphere needs spherical trigonometry.
Except this isn't a sphere, but maybe there's some other kind of geometry out there.
Anyway, that would be a use-case. If you badly want to see a worked example up front, I did one a while back.
(Or you could just be curious about what's going on in all of those M. C. Escher drawings. Whatever works.)
But, in fact, we won't be depending on that. This will be more of an axiom wanking approach parallel to what we did here for Spherical Geometry,
… and, to be completely lazy about it, we'll just copy-paste in the initial bits of text we need from there. Like this:
Start with a line ℓ and a point A not on it. How do you put a line through A that doesn't intersect ℓ?
(blah blah blah … feel free to go back and read everything here about the parallel postulate up to "Taking Red Pill #1" since it's probably been, like, 7 years since you read it.)
… fast forward through 2000 years of Greeks, Romans, Arabs, and basically everybody, beating their heads against the wall trying to prove [the proposition that there will always be exactly one line through A that doesn't intersect ℓ] … and failing … perhaps because (drumroll…) there exist ways to have this Not Be True.
Once you start thinking along these lines, it's not actually that hard to come up with alternatives — everything is, of course, easy once somebody has thought to do it and figured out how. So let's try out an alternative and see where it gets us. (… cue Twilight Zone music …)
[There is no such line]. All pairs of lines intersect. No exceptions. [All straight lines are actually circles with radius 90°, etc… blah blah blah… Spherical Geometry. And there was much rejoicing.]
(… insert record-scratch noise.
Uh oh. McCoy found the Guardian of Forever,
went back in time,
Edith Keeler is no longer road-kill,
and now the Universe is completely different.
Shit …)
We'll give you multiple lines, i.e., we'll not only give you a second line, we'll let you tilt it through a whole range of angles, none of which will make it intersect the first line.
Oddly enough, this will also have consequences.
(…and now the pink elephants can really come screaming out of the walls.)
( Read more... )Special Relativity continued from part 5, in which we learned about translating coordinates, the unit-later hyperbola, and velocity angles
Or you can start from Part 1
Today's topic is constant acceleration, which, in Special Relativity, is much weirder than you might expect. ( Read more... )
My Special Relativity series continued from part 4.
Today's topic is how to translate Moving People coordinates to Stationary People coordinates or to other Moving People and how it's really like a rotation except where it isn't, and how there's this thing that's like a rotation angle but different.
So I needed a survey of my backyard to find out where everything is. The ISO (Stationary) People who I was originally going to hire were busy that day and also expensive, so I had the Rotated (Stationary) People do it, instead. ( Read more... )
My explanation of Special Relativity, trying to keep it geometric, continued from part 3 wherein I describe the Interval between two events, a kind of "spacetime measurement" that everyone agrees on and that, depending on the trajectory between the events, can represent either a (proper) distance or a (proper) time lapse.
It is time to expand our universe a bit.
Let's try adding a dimension. We move from the John Hancock Tower to the Flatiron Building (which, if you haven't seen it, is this mostly flat thing standing on end, hence the name). Each floor is a single south-to-north corridor of offices.
( Read more... )My explanation of Special Relativity, trying to keep it almost completely geometric, continued from part 2 wherein I describe
We now take a moment to introduce, out of left field, a new definition that will turn out to be very useful:
Give a pair of events, separated by a distance and elapsed time , the interval between them is defined as the quantity
which looks like a squared distance except for an annoying minus sign,
which seems to depend on whose coordinates we're using (Stationary People vs. Moving People vs. somebody else), but we will now show that:
My revamped explanation of Special Relativity, trying to keep it completely geometric and not invoking Walls of Math, continued from here where I have now completely beaten to death the concept of "Stationary" and it's time to
Actually, we're going to be particular about who we associate with. We want Moving People who will likewise be able to say that they don't think of themselves as moving, meaning they can't be accelerating or spinning, either. Which leaves having them coast at some constant velocity that is non-zero (because otherwise they'd be stationary and One of Us) and upwards (just to pick a direction).
We also want them moving slower than lightspeed (STL), for Reasons.
( Read more... )(yeah, ok, I don't actually know Latin)
So, here's another go at explaining Special Relativity. I remain annoyed at how few people really get it, even amongst avid SF consumers, entirely too accustomed to generations of SF writers papering over FTL issues with technobabble, 'cause we need that galactic empire, don'tcha know.
Also, it's long past time to dump poorly motivated 1930s pedagogy that real physicists abandoned long ago (e.g.,. "Your mass increases as you go faster", "What? why?", "Fuck you, it just does" [spoiler alert: just No; forget you ever heard that. And if, in 2026, anyone is still trying to teach it that way, someone needs to sit them down for A Talk]).
All we need is basic geometry you knew or could have learned about in 6th grade plus a bit of algebra (up to Pythagorean Theorem). I think I can get by without using a single square-root sign.
But there will be fewer handwaves this time. Because we have to be clear why things have to Not Be The Way You Expected and not leave wiggle room. Here goes:
Shine a flashlight off of a moving boat. How fast does the light go?
( Read more... )(...so, if I post this now
so that it is nearer the top of the page, will hachyderm.io light up the dreamwidth entry on my profile the way it's supposed to?)
Update: Survey sez, "Yes!" If my theory is correct, then the cutoff is somewhere between 30656 and 192123 characters, except the byte numbers are probably bigger since we're in UTF-8 Land. A 65536-byte limit (on how much Mastodon will read before giving up) would not surprise me.
(this is Part 5; Part 1 through Part 4 contain possibly-necessary background, but maybe not)
It is time to start putting the pieces together.
I'll start with my first cardinal sin, which is that I've only seen the TV series, not read the books, so if I missed huge reams of infodump where the authors spelled out their actual numbers, so be it. Then again, authors who are not Andy Weir tend not to do this anyway, the cardinal rule of storytelling being that no matter how much work you put into world-building your iceberg, you need to leave most of it underwater, in order to not inspire your readers to beat you to death. At least not until after the series is hugely successful, to the point where publishers can be assured of there being enough rabid nutjobs out there that copies of a Chris-Tolkien-style dump of all research, drafts, cocktail napkins with half-begun stories on them, etc… accumulated during the life of the author will actually sell.
Perhaps what I'll be complaining about is where the TV writers indeed screwed things up. This seems unlikely to me, but if so, this wouldn't be the first time this sort of thing has happened (and, while you might feel sorry for the book authors, nobody was putting a gun to their heads and saying they had to take the TV deal).
Onward…
This is mainly because I wanted to have all of the formulas in one place. Also curious to see how much I can compress the derivation and still have it vaguely make sense. Also I wanted to learn more MathML.
We have two bodies with mass. Gravity attracts them to each other. How do they move?
( Read more... )(This is Part 4; continued from Part 3, or you can start at Part 1
Recall how rockets work: Every seconds we toss out some bit of mass , with velocity in the exact wrong direction, giving the rest of the ship, whose total mass is , a small kick in the direction we want to go. The momentum accountants then tell us:
( Read more... )(this is Part 3, more background for critiquing The Expanse; here are Part 2 and Part 1)
Let's suppose we have Best Possible Rocket, as much free fuel/reaction-mass/whatever as we want, … and we just want to get there as fast as possible? Interestingly, there will still be limits.
( Read more... )(This is "Deconstructing The Expanse, Part 2", continued from Part 1, here.)
As luck would have it, I decided to go back and watch part of an episode to check something, mainly to try to pinpoint where exactly the show goes off the rails, and aside from saving myself from a really embarrasing blunder, it also reminded me of a topic I meant to cover but forgot to.
It's Season 3, Episode 7, the one where Maneo, Speed Demon Belter Guy in his tiny racing pinnace decides he's going to be the first one through The Ring and arranges this whole elaborate sequence of "slingshots" to get there first (what happens when he gets there is a fun scene that I liked because I'm actually 12, but I won't spoil it).
And then I realized…
People have multiple misconceptions about gravitational maneuvers. The "slingshot" is badly named. I blame Star Trek.
( Read more... )Just to get something out of the way up front:
I am now mostly convinced that, barring unexpectedly early, lucky results to our terraforming experiments, colonizing the solar system will for a long time remain an absurd enterprise that will only make sense if we manage to screw up the Earth really, really, really bad — at which point we've probably already gone extinct — and that SF premised on this should be regarded as a variety of steampunk (i.e. seeing what stories we can make out of improbable combinations of technology).
But never mind that. I do eventually expect an assortment of random human-crewed research/mining-supervision/etc stations scattered about needing to engage in commerce of a sort. E.g., just as today we maintain South Pole Station at fantastic expense and there's no way in hell anybody expects that to turn into a colony any time soon (or ever, or, at least, not until after the Antarctic ice-sheet has completely melted), but there will always be a few nutjobs wanting to live there. They may never comprise more than an infinitesimal percentage of Earth's population, but it will be sufficient to make the rest of this post/sub-series not entirely useless.
Having finished a re-watch of (the good parts of) The Expanse, I have to say, I really appreciate how they evidently did give thought to the solar system being way, way bigger than people give it credit for and are actually rather successful in getting this across in ways previous shows completely punted on — the Star Trek: TNG opening that zips by Mars, Jupiter, and Saturn in a matter of seconds being one of those "No, Just No" moments.
( Read more... )(…a curious observation about relativistic light-sails, … followed by a teentsy bit of obligatory, if undeserved, "Stargate: Universe" bashing.)
A couple of questions that come up in trying to imagine how the interstellar transit economy/infrastructure is all going to work:
Reviewing what I said earlier on the subject of mirrors/sails:
[We'll need them to be] extremely lightweight, totally reflective, micrometeoroid-tolerant, blah-blah-blah. There will be some engineering tradeoff to determine the size (i.e., the smaller we make it, the more accurate the rocket/laser thing has to be, and the more energy density it has to withstand, but the less vulnerable it'll be), which I'll leave the engineers to figure out.
My original thought was that our two applications for mirror-building don't really have much to do with each other beyond their reflecting (pardon the pun) the supply and demand sides of our energy economy, that once we got into the details, the differing goals (energy collection vs. propulsion) would assert themselves and ultimately there'd be very different requirements/designs.
For one thing, for solar satellites, why care about making them lightweight? We're just putting them in orbit, and, beyond occasional questions of servicing and refurbishment, they're not going anywhere. And, if anything, we want them sturdy enough to stand up to the stress of being in however close to the sun we need them to be.
Lightweight only matters in that the less material we use, the more we can make, and we will ultimately have to be making millions of them.
It's not that any of this is wrong; we're just missing an important detail is all.
( Read more... )It seems we can do shit like and maybe also . Wheee…
(And if this isn't looking right, you may need to switch/update your browser: Google just released Chrome 109 which includes MathML support for the first time in a long while. Firefox has evidently had this since forever. Probably pointless waiting for Microsoft Edge to catch up (Update (2023-01-14): Edge caught up). And Opera is owned by the Chinese now, so you probably shouldn't be using it anymore. Bleah.)
Looks like I get to go update my math-heavy posts.
I keep thinking I should put up a pinned post with pointers to my various "Space" posts and other things which people keep trying to find (or, rather, which I keep pointing people to when I get tired of explaining stuff in other forums)
And then I discover these Dreamwidth options I hadn't seen before (and maybe you hadn't either). And now, as the guy said in the Dr.Who "Blink" episde, "Look to your right -->" (*)
I am, of course, open to better ways to do this; this is still going to be cumbersome to maintain. (sure would be nice to be able to turn off auto-formatting for the Custom Text text; oh well...)
(*)Edit: well okay, not necessarily. You need to be viewing the full post on its own rather than seeing it as part of a feed (in which case it'll be whatever modules you have selected rather than me) — and I suppose if you're using your own format/stylesheet to view my posts, you may also lose... bleah.
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timeasmymeasure
