Hyperbolic angles are measured in areas, though. Which is also why the inverse hyperbolic functions are denoted with ar- instead of arc-. Or they should be, if people could just keep their mathematical etymologies straight.
7:14 I am extremely upset that I've never before seen all the trig functions laid out so logically at the same time! This would have really helped high school me!
The quality of teachers makes a massive difference. Sadly there is no real qualification for math teachers to understand math well. When there is 2 or more ways to demonstrate any process a good teacher will show as many examples as are needed for every student in their class to understand. Thankyou Mrs Vandenberg 1987 Ringwood High School.
I have been looking for a decent explanation for this for a long time. This finally did it. There is barely anything on the internet discussing rapidity and how hyperbolic trig functions can be used to express and manipulate velocities. Thanks for doing this!
I've found 4 or 5 explanations of the "speed of light" that describe it more completely. As a speed of causality, not a particular speed. Technically, that speed is just the fastest ANYTHING can go. Because we use "speed" is the problem like he says.
I heard it described well in this way: "The speed of light is not about light. Or the speed of it. It's the speed limit of the universe, of causality. Technically, you and I, and everything in the universe are "traveling at the speed of light." We use this "speed of light to make sense of the relationship between space and time."
One thing I absolutely LOVE about Nick is how he oftentimes fact-checks *himself* in his own videos, pointing out things he did wrong while correcting himself. It’s amazing and shows just how good of a teacher he is.
Speeds that fast do exist, but not as physical things moving through space. Two objects in space may be far enough apart, and due to all of the space in between expanding, those two points are moving apart, relative to each other, faster than the speed of light.
But could they exist in eventual other universes? Would the difference only make sense when you compare absolute speeds in one universe compared to another one? I didn't get everything. It's 7AM, I just got up and The sound was too low... I'll just re-watch it later.
Man I love Nick's videos. They're just a tad bit more mathy than most of the edutaiment content on youtube, but this allows him to dive into topics others can't afford to go.
While watching, I noticed the velocity addition formula looked strangely like the tangent angle sum formula, so I paused and looked it up. I was disappointed to see it didn’t match because of an incorrect sign. I continued to watch and then it was revealed it was actually hyperbolic tangent! Genius! That makes great sense given the connection between hyperbolic geometry and Minkowski spacetime. Great video!
@@ScienceAsylum I have been somethat confused recently regarding speed of light invariance in all reference frames. As far as I know we do not have any light travelling at c due to some matter interfering in the vacuum, so even neutrino can move faster than light in some cases. So, let's imagine we have a spaceship travelling from observer A trying to catch up to the light from laser beam observer A emitted and the speed of spaceship is slightly higher than of light in interstellar medium (like speed of neutrino), so from observer A perspective the spaceship will catch up to light at some point, therefore from spaceship crew's perspective they must catch up to light as well, right? So, the speed of light in this medium would not be the same c from the spaceship crew's perspective. So, why is the speed of light in our experiments the invariant despite being less than c due to a medium and not invariant in this hypothetical situation? Am I missing something?
New viewer here. I love how your videos haven't changed much in over 5 years. If it ain't broke don't fix it as they say! This has become one of my favorite science channels. And even though I follow stuff like this casually, you've been really making me want to learn math like you know it. I'm nearing 40 but I wonder if I could do it! Also random, but I love the music you use for the sponsored segments. Anyway keep up the awesome work Nick!
I have to say, this is one of my absolute favorites of your videos. It's right up there with "How Special Relativity Fixed Electromagnetism". In fact, I think it's a tie between that one and this one which one is my favorite, and that's saying something, because I didn't think anything could top or even match the sheer _insight_ that video connecting SR and E&M gave me. But _this_ video not gives such good insight into WHY the speed of light is the maximum speed, but also connects it _so elegantly_ to hyperbolic trig. There are other videos that give the same insight about the speed of light being the maximum speed due to geometry, but they don't make the math _elegant_ by bring in the hyperbolic trig and showing how it simplifies the otherwise ugly looking algebra that's more typically used. That sheer mathematical _elegance_ isn't something I see a lot of in educational physics content, so it's a real breath of fresh air compared to all the textbook style stuff that's so focused on just algebraically deriving equations and then doing lots of plugging and chugging.
I'll leave my comment from another video here: There is a reason that the speed of light has to be finite. People make this mistake because they associate the speed of light with.................well, light. When in fact it's the speed of causality. The value of this speed is arbitrary, but it must be finite. If it were infinite, all of the interactions in the universe would happen simultaneously, and time wouldn't exist. Think of the entire history of the universe until its end like a movie. The length of this movie is determined by how fast you play it. This is analogous to the speed of causality. The faster the speed, the shorter the movie. If the speed were infinite, it would no longer be a movie. It'd just be a photo. Time exists because there is a delay between something happening here and its effect being felt in another part of the universe i.e. effects cannot propagate instantaneously. This inability is what is manifested as the speed limit of causality. Light just happens to be one of the things that can travel at this speed. It's also the reason why all observers must agree on the speed of light. Because it's a law of the universe and it must be the same for all observers. This is actually the essence of special relativity.
I think this adds another perspective to the notion that light is "slow" compared to the size of the universe. A lot of stuff has to happen for us to exist and we exist in a timescale much less than that of this stuff, isn't it kind of to be expected that the universe looks big to us compared to the speed of causality? (as in, for cause to become effect across the universe it would take a timescale much larger than our own)
It's pretty interesting to try and square it with hyperbollically infinite c, as a kind of metaphysical affirmation of linear velocity vector-space. I guess that has to mean something like hyperbollic time has to refer to something very different to regular time, else it must be some sort of singularity - at least there's no shot at perceiving hyperbolic time, since perception wouldn't make much sense.
Wow! Really nicely put. This comment is a great spark of insight. I always thought, the speed of light is just another reflection of some underlying symmetry of the universe. I feel like what that symmetry might be is hidden inside this comment of yours.
@@user-sl6gn1ss8p Wow! Great explanation. Great addition to this thread too.
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I always though of causality being the Universe's equivalent of the clock-speed. It is what dictates how fast all the other things are. The remaining question for me is: what is time them? Because if you think about it, in a computer, the faster the clock, more computing can be done (work). Could the same be said about our time? E.g. if we are moving super fast, like 1/2 causality speed, time slow down for us. That means that we could get more work done? Or is it the opposite? If we are moving super fast, means that we are consuming part of the causality quota, and because of that we have less time to do work?
Since rapidity looks like the natural way to measure movement, i made this table to get accustomed: average walking rapidity: 17 µrad average running: 40 µrad average bicycle: 83 µrad average car in town: 170 µrad max legal rapidity for a car in most countries: about 430 µrad average high rapidity train: 1000 µrad sound rapidity at ground: 4000 µrad minimal satellisation rapidity: 95000 µrad escape rapidity: 135000 µrad
Hyperbolic angles are not measured in arc length of the unit circle, like euclidean angles. They are measured in area on the unit hyperbola. Which is why the inverse hyperbolic trig functions are denoted as ar- rather than arc-. Or they should be, if people could just keep their mathematical etymologies straight. So I don't think radian is the right name for them.
I have been studying Special theory and general theory for quite some time now. Still never thought of "Velocities" this way... Its amazing.. And also pretty elegant.
Naah. It's actually even simpler. He's just using the wrong time. Light doesn't change speed from our perspective because no time is passing. It gets stuck at light speed. From the perspective of the person on the spacecraft that is moving at 90 percent light speed it all works fine.
"He's just using the wrong time." Yep, when I discovered the concept of "rapidity", they said it was observer-measured distance traveled per unit of traveler-measured time.
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The one and only physics channel to so effortlessly foster intuitive understanding of complex issues using math! Another great video, as always.
i love so much that someone finally tackled this. i had an intuition about this a couple years back but didn't really have the language to express it (although when i saw your video thumbnail, something snapped in place and i realised it had to do with hyperbolic geometry.. my brain be weird like that at times) it's similar to when you pointed out that particles are _always_ waves.
The speed of light is also infinite in another sense. Objects that travel at the speed of light do not experience the flow of time. They exist everywhere on their path through spacetime at the same time. In fact, for objects which are moving very close to the speed of light, their clocks run slower and the surrounding universe appears flattened, hence shortening travel time. So as it turns out, while light speed isn't infinite, it nonetheless makes travel time near-instantaneous (at the expense of also causing the rest of the universe age very rapidly relative to the traveler).
Objects that travel at the speed of light do not even have a valid reference frame. Photons for example. It doesn't make sense to talk about what they experience.
Yeah... no... Try coming back down to reality and then look at that statement. First off, if that was even remotely true then the whole concept of a Lightyear would be utter garbage. In order for something to exist at all points along its path through spacetime simultaneously it would actually have to move at infinite speed. Lightspeed isn't infinite. It can be conceptualized as infinite to make sense of it acting as a universal speedlimit, but the simple fact is it does actually have a measurable speed. If it can be measured, it is not infinite.
Best video on special relativity I have ever seen. So so so so much more enlightening to think of relativity meaning an addition of angles in a velocity time graph rather than an addition of slopes than to think of things like the traditional spaceship clock thing. So cool.
When I took Physics III at Community College, our professor (my favorite professor ever by the way) took us through the Minkowski hyperbolic graphs and even gave us Hyperbolic graph paper to work things out in. One big regret I have is, despite getting an A in the class, I never grasped that part. It was so mind bendingly confusing! This video is a great start for me to hopefully finally understand it after not trying for so many years since college. I think I'll try and find my old notes. This is fascinating stuff!
4:45 This reminds me of when I first learned about the geometric interpretation of complex number multiplication and how it just adds the angles. That blew my mind and seemed so _weird,_ but looking back, I'm not really sure why it blew my mind, as it's actually really simple.
Complex numbers are basically just "spherical numbers." To go along with them are the hyperbolic numbers, also called the split-complex numbers. The difference is that with spherical numbers i^2 = -1, while hyperbolic numbers have j^2 = 1. If you want to continue this, there also exist "euclidean numbers," better known as the dual numbers which are defined by 𝛆^2 = 0. Where e^iθ = cos(θ) + isin(θ), e^jφ = cosh(φ) + jsinh(φ). If you try using dual numbers, then you actually get e^𝛆φ = 1 + 𝛆φ, meaning you can technically define a "euclidean sine" as the identity, and a "euclidean cosine" as the constant 1.
@@angeldude101 Cool! I'd heard of the split complex numbers before and even watched a video about them, but, unlike the complex numbers which have an insane amount of applications, the split version just seemed like a mathematical curiosity. But if they're actually a hyperbolic version of the complex numbers, that makes them much more interesting, IMO.
@@Lucky10279 Yup! Aside from there existing real numbers that already square to 1, the difference between complex numbers and split-complex numbers is the difference between spherical geometry and hyperbolic geometry. Something I remembered since posting my previous reply is that the part about multiplying complex numbers is the same as adding their angles is that it's basically the definition of the exponent rule that says that (e^x)(e^y) = e^(x+y). For complex numbers, this means adding spherical angles, while for split-complex numbers it means adding hyperbolic angles. If you do this with dual numbers, you come out with the multiplication being equivalent to adding euclidean "angles," which are really just distances.
@@angeldude101 holy crap, I've heard of the split complex numbers and the dual numbers, but I've never heard of the geometric interpretations they're tied to. I actually stumbled into the dual numbers while browsing nlab. All of this was initially set into motion when I began learning the Haskell programming language and began to learn abstract algebra in order to understand what the hell a monad is. Group theory connects everything, and it really boggles my mind that basically nobody outside of hardcore academia is taught it. Everywhere you look you can find beautiful connections between seemingly disparate parts of math.
This is very clever, getting rid of the units and turning the focus on asymptotes insteqd, which uis the truth and true nature of speed. It's like you're doing a reverse Zeno paradox, Einstein style, solving it rather than confusing us instead.. Love it!
A few years back on a physics video somewhere on TH-cam, I made a comment like "why don't we look at light as being infinite speed (since from its own perspective, travel takes zero time)" and got an insanely large number of replies telling me why I'm stupid and that makes no sense. I wish I could find that again so I could link them here.
Nick is the best science communicator on TH-cam. I understand the concept immediately, instead of having to watch over and over again. Bad for Nick, good for me!
Now it makes sense. Time slows down the faster you move. For anything moving at the speed of light, time stops. So time doesn't pass for a photon moving from any point A to B, meaning its speed it infinite, at least from the point of view of the photon
So if you were to move move or experience movement at the speed of light and you saw something moving 99.999999 etc % the speed of light would that thing seemingly not move at all?
@@husk2853 For you, moving at infinite speed without time passing, everywhere you go will be the same exact moment. You wouldn't experience anything, there's no reference of time to even see a thing stand still
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for a photon, time does not pass. Which makes its speed virtually infinite. But it still goes at speed C
That means, from the start of the universe to its future destruction, for example the big crunch.. all will happen in exactly 0 seconds for a photon. So for a photon, the bigbang never happened and the age of the entire universe is zero. That basically means that from the causality's point of view, the universe does not exist. The true point of view is the universe's point of view, so we can say nothing exists. The only truth is nothingness.
@@husk2853 if you would move at the speed of light everything would "look" speeded up by a factor of infinity not slowed down. You cant really look at that speed, but who cares :) There is also a problem with the 99.99999% of the speed of light. Even if we say we have a million 9s after the decimal point, the "rapidity" is still infinitely far away from the "rapidity" of light. So it's still infinitely slower than light. And if you reached the rapidity of light, you can divide it by a million and still be at the rapidity of light. This could give you an intuitive explanation, why massless particles cant be slower than the speed of light. (I considered light to be infinitely fast for a while now and it can explain a lot of things, but this video seems to be the first that uses this analogy...)
Thanks for this correct explanation! I read in the comments that there are more explanations of rapidity on YT, but this is the first one I encountered. To summarize: speed = c * tanh(rapidity) gives max speed = c * tanh(infinity) = c
7:08 casually as an aside, lays out the greatest explanation of trigonometry I have ever seen in my life. This is the power of intuitive instruction. Teachers like you are the future of education and I am truly glad to be living in the tomorrow Thanks again for making another muddled math/physics concept *Lucid* once more
9:52 _"... traditional units suck."_ Did you just endorse *Planck Units?* Ma man! I also see that you're using a lot more SI than Imperial / US Customary - and that too is a huge improvement IMO. Absolutely fabulous 👍
@@ScienceAsylum Your preferences exactly match mine! I've always dreamed of our civilization redefining all the physical quantities as multiples of these fundamental constants creating modern physical units. Like if intelligent aliens exist in our sphere of observation, exchanging information would be much easier since these constants are assumed to be the same everywhere in the observable universe. It'd be only a matter of modifying just the mathematical base and learning their numerical symbols then done.
Another great video! Where was this video 5 years ago when I was struggling to teach space-time graphs to young Physic students and trying to explain to them why the speed of light is a maximum!
technically, this video didn't explain why the speed of light is a maximum. Showing a graph and formula that are based on the speed of light to be the maximum to begin with, doesn't actually explain why it's the maximum.
It's the maximum because it's the speed at which time stops. It's actually very simple. It seems like teachers don't really understand this simple explanation. I had to figure that out myself.
@@Dragon-Believer why does time stop at the speed of light? All the explanations I've seen of it just doesn't seem logical, all the explanations resolve around "because the light of some other object can never reach you, then that means time stopped". Which is an illogical statement.
@@resresres1 well, we don't know why time stops at light speed. Which shouldn't be surprising because we have no idea what time actually is. So yes I agree those are dumb explanations. Time is a property of space? So moving through space affects the passage of time? I mean, we could be moving really fast or really slow through time. How would we know? Time affects all of our perceptions equally. We could be moving backwards. What I wonder is this... If you were moving at 90 percent light speed it change the speeds of light around you? Would light you were traveling towards slow down? My thinking is that it would not. It would still be moving at light speed in every direction relative to you. Because it's actually accelerating to infinity but it just appears to stop at light speed from our perspective. But the stars and planets would seem to move faster because they are at a relatively fixed speed. Light is accelerating at infinite rate to infinity but appears to stop at light speed. Because of time dilation. Light just interacts with time much differently than we do.
A short summary of me watching this video today: 1) A new video, nice!! 2) Oh, alright, but what does that mean? 3) Amazing, Nick always explains exactly what I have just been asking myself a second ago. 4) Alright, alright. Now I am lost. 5) Okay, back on track. I get it! 6) WHAT? That is amazing! *Mind has been blown* Again! Thank you for another great video. :)
Yeah, I can't do Physics and shit like that too well. It makes no sense. It's why I use magic in my cartoon, and break what science in the "Outside of consistent results" form
Once again a really nice video! You blew my mind when you said that small velocities' addition is just small angle approximation! I'm still a bit puzzled by what it means for spacetime to be hyperbolic, I think it would make a great video too.
Interesting. On earth and at low speeds we use the “small angle approximation” of trig functions. Love the relationship between math concepts and the real world
10:05 Wait, how does that even work mathematically? Aren't angles by definition bounded between 0 and 2 pi? Or what does the angle represent if it doesn't represent the angle our triangle makes with the horizontal axis?
What I particularly like is how you can easily see that Euclidean spacetime wouldn't make sense, since if you add angles in Euclidean space you can add two angles in the first quadrant and end up in second, for example, meaning you could run and throw object so fast that it ends up going in wrong direction or back in time.
I consider the speed of light to actually be slow in the grand scheme of things, while Daddy Nick Lucid is over here claiming it's infinite! :) Incidentally, this is the first explanation that truly clicked for me on why relativistic speeds don't "add up" as expected, and I've been consuming this information for 20+ years.
I think the same way. It's actually what frustrates me most about the way relativity describes frames of reference in the passage of time. objectively speaking if light is a historical record of information then we should only describe time as the defined points of objects minus the delay that is the time light takes to travel. If light was instantaneous this would be a different conversation.
Ok that was amazing. Thank you very much. I've encountered Lorenz transformations and all the stuff around adding velocities but the concept of rapidity works wonderfully to clarify it. Truly wonderful stuff. You're an excellent person!
I had a thought about doing this after i saw a video explaining that "Everything is moving at the speed of light, but it's a combination of doing it through space and time. light moves through space at C and not through time, and a theoretical stationary object would move through time at C but not through space." and then i thought, huh, would could measure speed as an angle. now i want to do it, but i'm certain it would confuse too many people.
Been imagining of “piggybacking” light, past, present, future would be meaningless… kinda the definition of “infinite” but Never thought of this way.. very logical & easy to understand.. thanks Nick!
This is exactly what I wanted to know, other videos treat light in a euclidean way, which doesnt fully explain why the speed of light is the upper limit of speed.
This doesn't really "explain" it either. It can't be explained. It can only be observed. And once it has been observed, it needs a mathematical framework to live in. If we want to cram it into geometry we intuitively understand, the mathematical constructs need to be contrived. If we want the mathematical constructs to be nice, we need unintuitive geometry. Neither is better or more explanatory. They are just different frameworks to use on the same physics.
@@MasterHigure I was about to say the same thing, that the video doesn't explain why there's an upper limit of speed. You mentioned "it can't be explained", but I don't believe this either. Everything can be explained, just that we may not understand it.
@@resresres1 Ultimately, there is no way to explain why our universe chose to be the exact way it is. We can only observe that it turned out to land on the particular rules it happens to have. This goes for the particular values of physical constants, like the speed of light, and in my opinion it goes for more big-picture stuff, like special relativity as a whole.
this is a really interesting way of putting it! never seen the speed of light represented geometrically, let alone as a hyperbola. When I saw the title I was expecting something about "relativistic momentum" and how that value approaches infinity as you approach the speed of light, or how at that speed any nonzero mass has infinite kinetic energy The reasoning that always helped me wrap my head around this was remembering that time slows down as you speed up. helps with both the "why you can't go faster" and the "why it's the same in all reference frames". and actually, I think this "time slowing down" is directly linked to the hyperbolic geometry you mentioned. would love to see a follow up relating this to time dilation and/or momentum. great video man!
I love your perspective on things.. It bends my mind but helps me to see things in a new way.. So lucky to have your perspective..!! Very grateful and thankfull
Ultimately, hyperbolic trig is very closely linked to spherical trig, and both are also very closely linked to the exponential function. Hyperbolic trig is like the mirror image of spherical trig, and both are different forms of the exponential function, just split into symmetric and antisymmetric parts. There is technically a version of trig that lies on the mirror line between hyperbolic and spherical trig, which I sometimes call "euclidean trig," but it's significantly more boring (or perhaps interesting specifically _because_ it's so boring).
@Mr Fl0v That's exactly why I call complex numbers "spherical numbers," split-complex numbers "hyperbolic numbers," and dual numbers "euclidean/flat numbers." In every case, the "angle" is twice the area of a sector.
Huh, I always thought of it like time dilation. The faster you go you slow down time and at the speed of light you stop time. So technically light speed is infinite because in the photons perspective, any two points that are any distance apart is the one and the same. Because it takes 0 time to get there and back.
If you replace "logically consistent" with "reasonable" you can then apply this phrase to any aspect of life. Logic is reason in a closed system where all variables are accounted for. Reason is an open system where evidence and axioms dictate one's synthesis. The only difference is the amount that perception must to change before we reevaluate our paradigm.
Yes, i want rapidity now instead of speed.....adding the rapidities would just help intuit the universe, and then measuring "speed" without a unit, just radians, how radical!
10:14 "...because a billion meters per second doesn't exist..." Those speeds... they simply don't exist in this reality... BOOOOM my mind was blown completely. I've been working on Relativity for years and somehow this just made this concept click for me.
3:47 Oh my gosh, the expression on the clone's face has me laughing so hard. Don't think I noticed it when I watched this video previously. It's especially hilarious that there two of you, side-by-side, one of you looking absolutely _flabergasted_ and _horrified,_ while the other is just friendly and enthusiasticly explaining. If I didn't know better, I'd say you must have a twin, because otherwise doing that shot must have been tricky. (Though I suppose you've had a lot of practice with the clone shots over the years -- it's still one of my favorite edutainment gags.)
In my admittedly limited understanding, the FSC is an observed value that our models do not yet constrain. The main difference is that the so-called 'speed of light', as this video states, is a result of hyperbolic geometry, and its 'value' is a result of the permittivity and permissivity of free space, which are 'more fundamental' in a sense, in that they act as the interpretation of that infinite rapidity into a measurable amount of space and time, given a reference frame. The speed of light can *only* be what those underlying values allow it to be. In contrast, the FSC is a dimensionless constant, a factor that 'scales' some part of the equations in the model, and can only be determined through matching observations to the model - there is nothing in the model that makes a specific value of the FSC 'fall out naturally'. There are a fair handful of these 'free parameters' in the current model, which is why the hunt is ongoing to find a model which is both effective at predicting observation, and which naturally produces values for some or all of these free parameters. When the 'fine tuning' problem is brought up in the contest of physics, this is what it refers to, the idea that these parameters appear to be free to vary, and yet have very specific values for as-yet-unknown reasons; this leads into things like the Anthropic Principle ("The values are what they are because we wouldn't exist to observe them in any universe in which they weren't those values"), which is unsatisfying to say the least.
There is no reason that we know for the speed of light or the FSC to have the exact values that they have. Same with Planck's constant, Newton's gravitational constant and a few others. Our universe just so happened to land where it landed.
@@HeavyMetalMouse nice comment.. or satisfying, perhaps? Given that we exist because we can, I don't see a problem with the Anthropic principle. I might be missing something?
Broke : v = s / t Woke : v = p / m But seriously, the more I study physics the more it feels like, momentum is a much more fundamental quantity than velocity. (Same goes for angular momentum)
And, as I understand it, photons do not experience subjective time. Like as soon as they are formed in, say, like the core of a sun-- bouncing around for however many millions of years to finally emerge triumphantly out of said Sun --and then come down to, say Earth, and help give a person some sunburn, or maybe knock off of a few free radicals, the whole journey would seem instantaneous to the photon if it was conscious... I think that's how it goes? 🤔 Physics is so weird! Cheers 🙂🙂🙂
I love your channel! It’s my favorite on here. I always walk away feeling like I get what was being talked about instead of feeling confused. You’re a great science and math communicator
I couldn’t get this intuitively until I thought about it like “things can’t go faster that the speed of light in the same way that things can’t be more stationary than stationary”
Wow! Another amazing one! Just like the earlier video about superposition, this one knocked it out of the park! So much clearer than the standard explanations! Thanks! (!!!)
Something I've been trying to relate for decades! What you named "rapidity," I've been calling, "the velocity parameter," and using the Greek letter alpha for it. β = v/c = tanh α γ = 1/√[1-(v/c)²] = cosh α βγ = sinh α I think I like your name better. But the key thing is, the rapidity is what adds when compounding velocities (such as your basketball thrown from a moving car). And so, it's what goes to ∞ with ever-increasing speeds; e.g., with acceleration that's constant in the co-moving frame. Thanks, Nick, for explaining & illustrating it superbly! I guess it took another physics geek to independently come up with all that. Fred
I have found to be extremely helpful this explanation highlighting the contrast between the High School Vs Hyperbolic geometry for understanding c, as well as another analogy of Poisson's equation for gravity & Einstein's equation, used in another video of yours, which talked about how elastic spacetime was.
Why didn't you spin up one your mind-blow clones for this video? You totally blew it out of the park with the woah factor in this one. Keep up the great work! this video is definitely in my top 3 favorite videos from your channel because it makes me completely rethink some of the ways I view the world. Thanks for making difficult subjects so much more... lucid. You probably hear that pun a lot lol. Your videos are enjoyable, high quality, and your teaching/editing style keeps me engaged and paying attention 100% of the time. Now if you'll excuse me, I now have to try to re-learn Trig again now that you've made the subject feel so fascinating and approachable.
Very nice overview of rapidity from simple concepts! Great work! I’m to far from intro level to know if it’s helpful. 😆 Nice that you mentioned hyperbolic geometry. I suspect viewers who are not familiar with that will remain very confused about what that is and how it works exactly. At least they now now a new word to google to find out more.
Yeah, this video wasn't meant to be a comprehensive explanation of hyperbolic geometry. It's just something I use in explanations occasionally. I leave the deep dive to the math channels.
Fantastic video. I'll admit, as a fellow physicist, I started crafting an emotional, strongly worded comment about the angle addition at the 4 minute mark with a long polemic about "you need to learn about rapidity and hyperbolic angles!" Lessons: a) watch the whole video b) Nick Lucid knows his stuff.
One of your best videos. Thank you! This filled in some gaps in my knowledge that I never even really knew were there. You have a special gift for presenting teaching in a way that cuts through all the extraneous noise and gets to the heart of the information.
Yo.. the way you desibed the angle of a slope, and how the higher the angle then the greater the output difference between two adjacent angles.. dude you just took the derivative of relativity and made me intuitively understand the 'resistance' (for lack of a better word) of spacetime with a perfect descption a dimension (or two) lower. Basically you just gave a very good model of 4d space in 2d. Wow thank you, that was just elegant in its simplicity and profoundness. 😲👏👏👏
Thanks for this explanation! This would have helped tremendously in High School Trig and Physics. This gets me closer to understanding the math BUT I’m still having trouble with the cosmological constant and relativity, and finding it hard to just accept that C is C, and C is the speed limit because physicists say it is. I sure wish I had a better physics teacher in 11th grade. I still love this stuff though. Thanks again!
Hi Nick! I can answer the apparent confusion as to why infinite angle looks like 45 degrees near the end. The most natural definition of angles in both Euclidean and hyperbolic geometry is made using the arc length parametrization of the unit circle/hyperbola. For the circle, that's why radians are the natural unit. On the unit hyperbola, If you use arc length from a fixed point (say (0,1)) on the unit hyperbola, then the angle of another point P on the unit hyperbola is defined to be the length of the path from the reference point to P. That way, rapidities corresponding the light-lines intersect the unit hyperbola at "the point at infinity" which is infinitely far away from the reference point!
Correction at 4:31 : The one in the brackets is km/h
Oops! I'm honestly surprised you're the first person to notice the typo.
"Do you know how fast you were going?"
"Uhh... about 0.4 radians?"
"This is a 0.25 radian zone. You're in trouble, buddy."
The speeding fine will be 2*pi in SI Dollars... LOL... But why so much? To account for universal inflation!
I laughed and accidently spilled out some of my lunch I was eating. Thank you lol
@@raymitchell9736 Can I pay it in tau?
0.4 radians relative to what tho
Hyperbolic angles are measured in areas, though. Which is also why the inverse hyperbolic functions are denoted with ar- instead of arc-. Or they should be, if people could just keep their mathematical etymologies straight.
7:14 I am extremely upset that I've never before seen all the trig functions laid out so logically at the same time! This would have really helped high school me!
Same
Yup
You see very well exactly how the co-variations really mirror their non-co counterparts.
The quality of teachers makes a massive difference. Sadly there is no real qualification for math teachers to understand math well. When there is 2 or more ways to demonstrate any process a good teacher will show as many examples as are needed for every student in their class to understand.
Thankyou Mrs Vandenberg 1987 Ringwood High School.
You’ll LOVE 3blue1brown then.
I have been looking for a decent explanation for this for a long time. This finally did it. There is barely anything on the internet discussing rapidity and how hyperbolic trig functions can be used to express and manipulate velocities. Thanks for doing this!
Glad it helped 🤓
I've found 4 or 5 explanations of the "speed of light" that describe it more completely. As a speed of causality, not a particular speed. Technically, that speed is just the fastest ANYTHING can go.
Because we use "speed" is the problem like he says.
I heard it described well in this way:
"The speed of light is not about light. Or the speed of it. It's the speed limit of the universe, of causality. Technically, you and I, and everything in the universe are "traveling at the speed of light."
We use this "speed of light to make sense of the relationship between space and time."
@@ScienceAsylum what. it is impossible to light can travel inifinite speed
@@ScienceAsylum dang how you are so good at physics.
One thing I absolutely LOVE about Nick is how he oftentimes fact-checks *himself* in his own videos, pointing out things he did wrong while correcting himself. It’s amazing and shows just how good of a teacher he is.
"Speeds that fast don't exist" is actually a really novel way of putting it, thank you for this insight, Nick!
You can still get faster speeds, but only for non-physical things (like the speed of the viewing area of a quasar beam as it sweeps past earth).
I've always liked the expression "There can be no instantaneous interactions in nature, so there must be a maximum rate of interaction."
Speeds that fast do exist, but not as physical things moving through space. Two objects in space may be far enough apart, and due to all of the space in between expanding, those two points are moving apart, relative to each other, faster than the speed of light.
But could they exist in eventual other universes? Would the difference only make sense when you compare absolute speeds in one universe compared to another one? I didn't get everything. It's 7AM, I just got up and The sound was too low... I'll just re-watch it later.
It really reminds me of when Michael from Vsauce said "The object would become hotter... Than temperature..."
Man I love Nick's videos. They're just a tad bit more mathy than most of the edutaiment content on youtube, but this allows him to dive into topics others can't afford to go.
This is without a doubt the most educational, entertaining and fun science channel on TH-cam. Thanks for all of your hard work.
Thanks! 🤓
While watching, I noticed the velocity addition formula looked strangely like the tangent angle sum formula, so I paused and looked it up. I was disappointed to see it didn’t match because of an incorrect sign. I continued to watch and then it was revealed it was actually hyperbolic tangent! Genius! That makes great sense given the connection between hyperbolic geometry and Minkowski spacetime. Great video!
It's fun having insights. Glad you had a chance to have it before I mentioned it 🤓
@@ScienceAsylum I have been somethat confused recently regarding speed of light invariance in all reference frames. As far as I know we do not have any light travelling at c due to some matter interfering in the vacuum, so even neutrino can move faster than light in some cases. So, let's imagine we have a spaceship travelling from observer A trying to catch up to the light from laser beam observer A emitted and the speed of spaceship is slightly higher than of light in interstellar medium (like speed of neutrino), so from observer A perspective the spaceship will catch up to light at some point, therefore from spaceship crew's perspective they must catch up to light as well, right? So, the speed of light in this medium would not be the same c from the spaceship crew's perspective. So, why is the speed of light in our experiments the invariant despite being less than c due to a medium and not invariant in this hypothetical situation? Am I missing something?
New viewer here. I love how your videos haven't changed much in over 5 years. If it ain't broke don't fix it as they say! This has become one of my favorite science channels. And even though I follow stuff like this casually, you've been really making me want to learn math like you know it. I'm nearing 40 but I wonder if I could do it! Also random, but I love the music you use for the sponsored segments. Anyway keep up the awesome work Nick!
Thanks! My pace has slowed down a bit in those 5 years and I've improved my animation skill, but that's about it as far as changes go.
I have to say, this is one of my absolute favorites of your videos. It's right up there with "How Special Relativity Fixed Electromagnetism". In fact, I think it's a tie between that one and this one which one is my favorite, and that's saying something, because I didn't think anything could top or even match the sheer _insight_ that video connecting SR and E&M gave me. But _this_ video not gives such good insight into WHY the speed of light is the maximum speed, but also connects it _so elegantly_ to hyperbolic trig. There are other videos that give the same insight about the speed of light being the maximum speed due to geometry, but they don't make the math _elegant_ by bring in the hyperbolic trig and showing how it simplifies the otherwise ugly looking algebra that's more typically used. That sheer mathematical _elegance_ isn't something I see a lot of in educational physics content, so it's a real breath of fresh air compared to all the textbook style stuff that's so focused on just algebraically deriving equations and then doing lots of plugging and chugging.
Thanks! This is one of my "passion" videos and I'm so happy that it performed so well.
I'll leave my comment from another video here:
There is a reason that the speed of light has to be finite. People make this mistake because they associate the speed of light with.................well, light. When in fact it's the speed of causality. The value of this speed is arbitrary, but it must be finite. If it were infinite, all of the interactions in the universe would happen simultaneously, and time wouldn't exist. Think of the entire history of the universe until its end like a movie. The length of this movie is determined by how fast you play it. This is analogous to the speed of causality. The faster the speed, the shorter the movie. If the speed were infinite, it would no longer be a movie. It'd just be a photo. Time exists because there is a delay between something happening here and its effect being felt in another part of the universe i.e. effects cannot propagate instantaneously. This inability is what is manifested as the speed limit of causality. Light just happens to be one of the things that can travel at this speed. It's also the reason why all observers must agree on the speed of light. Because it's a law of the universe and it must be the same for all observers. This is actually the essence of special relativity.
I think this adds another perspective to the notion that light is "slow" compared to the size of the universe. A lot of stuff has to happen for us to exist and we exist in a timescale much less than that of this stuff, isn't it kind of to be expected that the universe looks big to us compared to the speed of causality? (as in, for cause to become effect across the universe it would take a timescale much larger than our own)
It's pretty interesting to try and square it with hyperbollically infinite c, as a kind of metaphysical affirmation of linear velocity vector-space. I guess that has to mean something like hyperbollic time has to refer to something very different to regular time, else it must be some sort of singularity - at least there's no shot at perceiving hyperbolic time, since perception wouldn't make much sense.
Wow! Really nicely put. This comment is a great spark of insight. I always thought, the speed of light is just another reflection of some underlying symmetry of the universe. I feel like what that symmetry might be is hidden inside this comment of yours.
@@user-sl6gn1ss8p Wow! Great explanation. Great addition to this thread too.
I always though of causality being the Universe's equivalent of the clock-speed. It is what dictates how fast all the other things are.
The remaining question for me is: what is time them?
Because if you think about it, in a computer, the faster the clock, more computing can be done (work). Could the same be said about our time? E.g. if we are moving super fast, like 1/2 causality speed, time slow down for us. That means that we could get more work done? Or is it the opposite? If we are moving super fast, means that we are consuming part of the causality quota, and because of that we have less time to do work?
Since rapidity looks like the natural way to measure movement, i made this table to get accustomed:
average walking rapidity: 17 µrad
average running: 40 µrad
average bicycle: 83 µrad
average car in town: 170 µrad
max legal rapidity for a car in most countries: about 430 µrad
average high rapidity train: 1000 µrad
sound rapidity at ground: 4000 µrad
minimal satellisation rapidity: 95000 µrad
escape rapidity: 135000 µrad
µrad? I like it 👍
Hyperbolic angles are not measured in arc length of the unit circle, like euclidean angles. They are measured in area on the unit hyperbola. Which is why the inverse hyperbolic trig functions are denoted as ar- rather than arc-. Or they should be, if people could just keep their mathematical etymologies straight. So I don't think radian is the right name for them.
@@MasterHigure how about rad² ?
@@MasterHigure How are they even _angles_ at all if they're measured in units of area?
Out of curiosity, how did you compute those numbers?
I have been studying Special theory and general theory for quite some time now. Still never thought of "Velocities" this way... Its amazing.. And also pretty elegant.
Naah. It's actually even simpler. He's just using the wrong time. Light doesn't change speed from our perspective because no time is passing. It gets stuck at light speed.
From the perspective of the person on the spacecraft that is moving at 90 percent light speed it all works fine.
"He's just using the wrong time."
Yep, when I discovered the concept of "rapidity", they said it was observer-measured distance traveled per unit of traveler-measured time.
The one and only physics channel to so effortlessly foster intuitive understanding of complex issues using math! Another great video, as always.
My thoughts too. He gives most intuitive and correct explanations of relativity than all the other channels combined.
reed thar bibel and gose to cherch on sundey
@@friedgreenaliernwomerns2600 lmao
That's great for some. For someone who still struggles with basic Algebra, Trigonometry is way outta my league.😶🌫️
Nick Lucid G.O.A.T!
i love so much that someone finally tackled this. i had an intuition about this a couple years back but didn't really have the language to express it (although when i saw your video thumbnail, something snapped in place and i realised it had to do with hyperbolic geometry.. my brain be weird like that at times)
it's similar to when you pointed out that particles are _always_ waves.
Teacher: what is the speed of light?
Me: "fast fast"
Teacher: "Well, you're not wrong."
blind person: light doesn't exist!
@@Blox117 Nah, it's just too fast for your eyes to process
@@Me_549 no its too fast for your brain to process
@@Blox117 are you blind?
The exact speed of light is not only not a guess, we've defined metres (and in a way seconds) to have it be that exact number.
But the speed of light (as any universal constant) is not a number. In the context of specific units, its magnitude is a certain number.
@@eljuanman999 it's measured in units if time and space are measured in different units
Yes, but... how do we define meters? Or seconds?
@@joshuahillerup4290 yes, but I was talking about what the speed of light is, not how it's measured
It is not in any way a guess.
The speed of light is also infinite in another sense. Objects that travel at the speed of light do not experience the flow of time. They exist everywhere on their path through spacetime at the same time. In fact, for objects which are moving very close to the speed of light, their clocks run slower and the surrounding universe appears flattened, hence shortening travel time. So as it turns out, while light speed isn't infinite, it nonetheless makes travel time near-instantaneous (at the expense of also causing the rest of the universe age very rapidly relative to the traveler).
Só in other words it’s not time that creates speed it’s speed that creates time
@@luisandrade2254 Makes sense, snort some speed and you will have more than 60 minutes per hour :)
Objects that travel at the speed of light do not even have a valid reference frame. Photons for example. It doesn't make sense to talk about what they experience.
I believe it is the same sense.
Yeah... no... Try coming back down to reality and then look at that statement. First off, if that was even remotely true then the whole concept of a Lightyear would be utter garbage. In order for something to exist at all points along its path through spacetime simultaneously it would actually have to move at infinite speed. Lightspeed isn't infinite. It can be conceptualized as infinite to make sense of it acting as a universal speedlimit, but the simple fact is it does actually have a measurable speed. If it can be measured, it is not infinite.
Best video on special relativity I have ever seen. So so so so much more enlightening to think of relativity meaning an addition of angles in a velocity time graph rather than an addition of slopes than to think of things like the traditional spaceship clock thing. So cool.
When I took Physics III at Community College, our professor (my favorite professor ever by the way) took us through the Minkowski hyperbolic graphs and even gave us Hyperbolic graph paper to work things out in. One big regret I have is, despite getting an A in the class, I never grasped that part. It was so mind bendingly confusing! This video is a great start for me to hopefully finally understand it after not trying for so many years since college. I think I'll try and find my old notes. This is fascinating stuff!
Glad I could help! 🤓
Just found this channel and I need to say: This guy is utterly insane... in a very positive way. Very good stuff... both thumbs up!
4:45 This reminds me of when I first learned about the geometric interpretation of complex number multiplication and how it just adds the angles. That blew my mind and seemed so _weird,_ but looking back, I'm not really sure why it blew my mind, as it's actually really simple.
Complex numbers are basically just "spherical numbers." To go along with them are the hyperbolic numbers, also called the split-complex numbers. The difference is that with spherical numbers i^2 = -1, while hyperbolic numbers have j^2 = 1. If you want to continue this, there also exist "euclidean numbers," better known as the dual numbers which are defined by 𝛆^2 = 0.
Where e^iθ = cos(θ) + isin(θ), e^jφ = cosh(φ) + jsinh(φ). If you try using dual numbers, then you actually get e^𝛆φ = 1 + 𝛆φ, meaning you can technically define a "euclidean sine" as the identity, and a "euclidean cosine" as the constant 1.
@@angeldude101 Cool! I'd heard of the split complex numbers before and even watched a video about them, but, unlike the complex numbers which have an insane amount of applications, the split version just seemed like a mathematical curiosity. But if they're actually a hyperbolic version of the complex numbers, that makes them much more interesting, IMO.
@@Lucky10279 Yup! Aside from there existing real numbers that already square to 1, the difference between complex numbers and split-complex numbers is the difference between spherical geometry and hyperbolic geometry.
Something I remembered since posting my previous reply is that the part about multiplying complex numbers is the same as adding their angles is that it's basically the definition of the exponent rule that says that (e^x)(e^y) = e^(x+y). For complex numbers, this means adding spherical angles, while for split-complex numbers it means adding hyperbolic angles. If you do this with dual numbers, you come out with the multiplication being equivalent to adding euclidean "angles," which are really just distances.
@@angeldude101 holy crap, I've heard of the split complex numbers and the dual numbers, but I've never heard of the geometric interpretations they're tied to.
I actually stumbled into the dual numbers while browsing nlab. All of this was initially set into motion when I began learning the Haskell programming language and began to learn abstract algebra in order to understand what the hell a monad is.
Group theory connects everything, and it really boggles my mind that basically nobody outside of hardcore academia is taught it. Everywhere you look you can find beautiful connections between seemingly disparate parts of math.
This is very clever, getting rid of the units and turning the focus on asymptotes insteqd, which uis the truth and true nature of speed. It's like you're doing a reverse Zeno paradox, Einstein style, solving it rather than confusing us instead.. Love it!
The timing! Thank you so Much!!! I love hyperbolic geometry and I was just studying the special relativity concepts in this video.
A few years back on a physics video somewhere on TH-cam, I made a comment like "why don't we look at light as being infinite speed (since from its own perspective, travel takes zero time)" and got an insanely large number of replies telling me why I'm stupid and that makes no sense. I wish I could find that again so I could link them here.
Nick is the best science communicator on TH-cam. I understand the concept immediately, instead of having to watch over and over again. Bad for Nick, good for me!
Now it makes sense. Time slows down the faster you move. For anything moving at the speed of light, time stops. So time doesn't pass for a photon moving from any point A to B, meaning its speed it infinite, at least from the point of view of the photon
So if you were to move move or experience movement at the speed of light and you saw something moving 99.999999 etc % the speed of light would that thing seemingly not move at all?
@@husk2853 For you, moving at infinite speed without time passing, everywhere you go will be the same exact moment. You wouldn't experience anything, there's no reference of time to even see a thing stand still
for a photon, time does not pass. Which makes its speed virtually infinite. But it still goes at speed C
That means, from the start of the universe to its future destruction, for example the big crunch.. all will happen in exactly 0 seconds for a photon. So for a photon, the bigbang never happened and the age of the entire universe is zero. That basically means that from the causality's point of view, the universe does not exist. The true point of view is the universe's point of view, so we can say nothing exists. The only truth is nothingness.
@@husk2853 if you would move at the speed of light everything would "look" speeded up by a factor of infinity not slowed down. You cant really look at that speed, but who cares :) There is also a problem with the 99.99999% of the speed of light. Even if we say we have a million 9s after the decimal point, the "rapidity" is still infinitely far away from the "rapidity" of light. So it's still infinitely slower than light. And if you reached the rapidity of light, you can divide it by a million and still be at the rapidity of light. This could give you an intuitive explanation, why massless particles cant be slower than the speed of light. (I considered light to be infinitely fast for a while now and it can explain a lot of things, but this video seems to be the first that uses this analogy...)
Thanks for this correct explanation! I read in the comments that there are more explanations of rapidity on YT, but this is the first one I encountered.
To summarize: speed = c * tanh(rapidity)
gives max speed = c * tanh(infinity) = c
Awesome animation on the trig functions there! Glad to see how this channel has grown in the last couple of years... Keep it up!!
7:08 casually as an aside, lays out the greatest explanation of trigonometry I have ever seen in my life.
This is the power of intuitive instruction. Teachers like you are the future of education and I am truly glad to be living in the tomorrow
Thanks again for making another muddled math/physics concept *Lucid* once more
Clicked on this video 11 seconds after the upload. Light can eat my dust.
it took me 13
😂
Fast fast
9:52 _"... traditional units suck."_
Did you just endorse *Planck Units?* Ma man! I also see that you're using a lot more SI than Imperial / US Customary - and that too is a huge improvement IMO. Absolutely fabulous 👍
Not Planck units, but I'm definitely a fan of c = h = G = 1
@@ScienceAsylum Yes please!
@@ScienceAsylum Your preferences exactly match mine! I've always dreamed of our civilization redefining all the physical quantities as multiples of these fundamental constants creating modern physical units.
Like if intelligent aliens exist in our sphere of observation, exchanging information would be much easier since these constants are assumed to be the same everywhere in the observable universe. It'd be only a matter of modifying just the mathematical base and learning their numerical symbols then done.
As soon as I saw the title, i was like, oh boy, here we go again.
Dude! _"Do Whatever Works"_ is every *_Engineer's_* mantra 🤣! You're welcome to use it though-eventually, you might even become one of us. 😜
It is a strange notion that there even should be some kind of infinite speed. Great video.
Another great video! Where was this video 5 years ago when I was struggling to teach space-time graphs to young Physic students and trying to explain to them why the speed of light is a maximum!
I never had the privileged of teaching relativity in a formal classroom, but I did give an informal mini lecture to a few grad students once.
technically, this video didn't explain why the speed of light is a maximum. Showing a graph and formula that are based on the speed of light to be the maximum to begin with, doesn't actually explain why it's the maximum.
It's the maximum because it's the speed at which time stops. It's actually very simple. It seems like teachers don't really understand this simple explanation. I had to figure that out myself.
@@Dragon-Believer why does time stop at the speed of light? All the explanations I've seen of it just doesn't seem logical, all the explanations resolve around "because the light of some other object can never reach you, then that means time stopped". Which is an illogical statement.
@@resresres1 well, we don't know why time stops at light speed. Which shouldn't be surprising because we have no idea what time actually is. So yes I agree those are dumb explanations. Time is a property of space? So moving through space affects the passage of time? I mean, we could be moving really fast or really slow through time. How would we know? Time affects all of our perceptions equally. We could be moving backwards.
What I wonder is this... If you were moving at 90 percent light speed it change the speeds of light around you? Would light you were traveling towards slow down? My thinking is that it would not. It would still be moving at light speed in every direction relative to you. Because it's actually accelerating to infinity but it just appears to stop at light speed from our perspective. But the stars and planets would seem to move faster because they are at a relatively fixed speed. Light is accelerating at infinite rate to infinity but appears to stop at light speed. Because of time dilation. Light just interacts with time much differently than we do.
Been wondering about this for a while now. I'll watch this about 20 more times to get this down, but it's all there. Thanks for your efforts TSA!
A short summary of me watching this video today:
1) A new video, nice!!
2) Oh, alright, but what does that mean?
3) Amazing, Nick always explains exactly what I have just been asking myself a second ago.
4) Alright, alright. Now I am lost.
5) Okay, back on track. I get it!
6) WHAT? That is amazing! *Mind has been blown* Again!
Thank you for another great video. :)
Excellent way to bring many slightly complex topics together to teach us something new! Nice job
This is hands down one of your best explainers Nick!👏 Your the greatest science communicator of our generation!
That slide at 3:46 is so funny 🤣..
"SCIENCE, DO WHATEVER WORKS 😂!!"
I can relate to it so much being a science enthusiast!!
Where's the T-shirt?
Yeah, I can't do Physics and shit like that too well. It makes no sense.
It's why I use magic in my cartoon, and break what science in the "Outside of consistent results" form
Your acting abilities have improved so much, I love it!
Once again a really nice video! You blew my mind when you said that small velocities' addition is just small angle approximation!
I'm still a bit puzzled by what it means for spacetime to be hyperbolic, I think it would make a great video too.
I love these visuals. Thanks for the great explanations as always.
You're welcome 🤓
Hyperbolic trig functions + Nick getting whacked = Amazingly insightful lesson + Crazy entertainment
Interesting. On earth and at low speeds we use the “small angle approximation” of trig functions. Love the relationship between math concepts and the real world
10:05 Wait, how does that even work mathematically? Aren't angles by definition bounded between 0 and 2 pi? Or what does the angle represent if it doesn't represent the angle our triangle makes with the horizontal axis?
I'm glad you gave me simple, practical usage for geometry, which school was unable to do over several years!
I think this is one of your very best videos. Good work!
Thanks! 🤓
Another splendid video ! I ❤️ the whacking clone ! He keeps you on your toes ! 🙋♀️
What I particularly like is how you can easily see that Euclidean spacetime wouldn't make sense, since if you add angles in Euclidean space you can add two angles in the first quadrant and end up in second, for example, meaning you could run and throw object so fast that it ends up going in wrong direction or back in time.
Thanks
This is by far my favorite channel on TH-cam. Always amazing videos that I can fully understand. Thank you for your content!
Great video, love that you used the TARDIS for infinite speed 😊
The TARDIS just felt right 🤓
I consider the speed of light to actually be slow in the grand scheme of things, while Daddy Nick Lucid is over here claiming it's infinite! :)
Incidentally, this is the first explanation that truly clicked for me on why relativistic speeds don't "add up" as expected, and I've been consuming this information for 20+ years.
I think the same way. It's actually what frustrates me most about the way relativity describes frames of reference in the passage of time. objectively speaking if light is a historical record of information then we should only describe time as the defined points of objects minus the delay that is the time light takes to travel. If light was instantaneous this would be a different conversation.
Light:look how fast I am!
Expansion of the Universe:Very good Light , one day you might catch up ha!
it's infinite in the same way an asymptote is infinite. the number is not infinite, it just represents a wall you cannot go pass
@@knightsljx Yes, I saw the video, as well.
Slow?
Just like distances are vast, so is time.
Ok that was amazing. Thank you very much.
I've encountered Lorenz transformations and all the stuff around adding velocities but the concept of rapidity works wonderfully to clarify it.
Truly wonderful stuff. You're an excellent person!
Amazing video! as always... thank you Nick! 🔥🔥
I had a thought about doing this after i saw a video explaining that "Everything is moving at the speed of light, but it's a combination of doing it through space and time. light moves through space at C and not through time, and a theoretical stationary object would move through time at C but not through space." and then i thought, huh, would could measure speed as an angle. now i want to do it, but i'm certain it would confuse too many people.
Been imagining of “piggybacking” light, past, present, future would be meaningless… kinda the definition of “infinite”
but Never thought of this way.. very logical & easy to understand.. thanks Nick!
The universe would be so contracted, that you would be everywhere at once and frozen completely
@@AverageAlien true and that too
This is exactly what I wanted to know, other videos treat light in a euclidean way, which doesnt fully explain why the speed of light is the upper limit of speed.
Glad I could deliver 🤓
@@ScienceAsylum A hearth and answer from The Sciense Asylum? what is this, the best day of my life? (Answer is yes)
This doesn't really "explain" it either. It can't be explained. It can only be observed. And once it has been observed, it needs a mathematical framework to live in.
If we want to cram it into geometry we intuitively understand, the mathematical constructs need to be contrived. If we want the mathematical constructs to be nice, we need unintuitive geometry. Neither is better or more explanatory. They are just different frameworks to use on the same physics.
@@MasterHigure I was about to say the same thing, that the video doesn't explain why there's an upper limit of speed. You mentioned "it can't be explained", but I don't believe this either. Everything can be explained, just that we may not understand it.
@@resresres1 Ultimately, there is no way to explain why our universe chose to be the exact way it is. We can only observe that it turned out to land on the particular rules it happens to have. This goes for the particular values of physical constants, like the speed of light, and in my opinion it goes for more big-picture stuff, like special relativity as a whole.
this is a really interesting way of putting it! never seen the speed of light represented geometrically, let alone as a hyperbola.
When I saw the title I was expecting something about "relativistic momentum" and how that value approaches infinity as you approach the speed of light, or how at that speed any nonzero mass has infinite kinetic energy
The reasoning that always helped me wrap my head around this was remembering that time slows down as you speed up. helps with both the "why you can't go faster" and the "why it's the same in all reference frames".
and actually, I think this "time slowing down" is directly linked to the hyperbolic geometry you mentioned.
would love to see a follow up relating this to time dilation and/or momentum. great video man!
Time dilation and momentum? Already made that video: th-cam.com/video/HU6t8QvGZmA/w-d-xo.html
@@ScienceAsylum woah, just watched it, that's exactly what I was thinking of when I wrote that comment haha . Another awesome video!
I love your perspective on things.. It bends my mind but helps me to see things in a new way.. So lucky to have your perspective..!! Very grateful and thankfull
You are a Real Genius! I mean you do all the work yourself. I definitely want to meet you once in my life!
This is first time I’ve seen hyperbolic trig functions explained in a way that doesn’t involve exponential functions and it’s so much more intuitive!
Ultimately, hyperbolic trig is very closely linked to spherical trig, and both are also very closely linked to the exponential function. Hyperbolic trig is like the mirror image of spherical trig, and both are different forms of the exponential function, just split into symmetric and antisymmetric parts. There is technically a version of trig that lies on the mirror line between hyperbolic and spherical trig, which I sometimes call "euclidean trig," but it's significantly more boring (or perhaps interesting specifically _because_ it's so boring).
@Mr Fl0v That's exactly why I call complex numbers "spherical numbers," split-complex numbers "hyperbolic numbers," and dual numbers "euclidean/flat numbers." In every case, the "angle" is twice the area of a sector.
Nick, you are so darn creative and insightful we are lucky to have you here. Thank you. Your last name is too descriptive.
Nice, explaining Rindler coords in a new and refreshing way. TY
Great video, It helped it enlightened a corner in my understanding that was bothering me.
Looking for your next ray.
Huh, I always thought of it like time dilation. The faster you go you slow down time and at the speed of light you stop time. So technically light speed is infinite because in the photons perspective, any two points that are any distance apart is the one and the same. Because it takes 0 time to get there and back.
I picked up your book a month ago and it’s one of my favorite things. Thank you for making tensor math less awful!
Glad I can help! I had to teach tensors to myself during grad school. Awful is a good word.
"As long as your definitions are logically consistent, you're fine" it's the best phrase ever
If you replace "logically consistent" with "reasonable" you can then apply this phrase to any aspect of life. Logic is reason in a closed system where all variables are accounted for. Reason is an open system where evidence and axioms dictate one's synthesis. The only difference is the amount that perception must to change before we reevaluate our paradigm.
*cgs units enter the chat
A new Nick!
And all about C!
I'm personally thrilled :)
Thanks!
Thank _you_ !
This episode has this “something something” that makes Your channel stand out.
If You ever feel You’re losing this special thing, go back here.
Yes, i want rapidity now instead of speed.....adding the rapidities would just help intuit the universe, and then measuring "speed" without a unit, just radians, how radical!
Well....you do still have to measure the speed.
I know how to measure a car's distance/time
I don't know how to measure a car's micro-radians
@@tomc.5704 1 km/h~1.07nrad 1 mph~1.49nrad 1 m/s~3.33nrad
Finished the vid and im proud to say i actually understood everything
I assume you're a little crazy.
10:14 "...because a billion meters per second doesn't exist..."
Those speeds... they simply don't exist in this reality... BOOOOM my mind was blown completely.
I've been working on Relativity for years and somehow this just made this concept click for me.
Glad I could help 🤓
3:47 Oh my gosh, the expression on the clone's face has me laughing so hard. Don't think I noticed it when I watched this video previously. It's especially hilarious that there two of you, side-by-side, one of you looking absolutely _flabergasted_ and _horrified,_ while the other is just friendly and enthusiasticly explaining. If I didn't know better, I'd say you must have a twin, because otherwise doing that shot must have been tricky. (Though I suppose you've had a lot of practice with the clone shots over the years -- it's still one of my favorite edutainment gags.)
Yeah, there was a lot of practice. My early clone shots were quicker because I wasn't as comfortable. The quickness made the issues less obvious.
If Light can only be Light Speed, does this mean the Fine Structure Constant can only be that number as well. Why is the FSC such a mystery.
In my admittedly limited understanding, the FSC is an observed value that our models do not yet constrain.
The main difference is that the so-called 'speed of light', as this video states, is a result of hyperbolic geometry, and its 'value' is a result of the permittivity and permissivity of free space, which are 'more fundamental' in a sense, in that they act as the interpretation of that infinite rapidity into a measurable amount of space and time, given a reference frame. The speed of light can *only* be what those underlying values allow it to be.
In contrast, the FSC is a dimensionless constant, a factor that 'scales' some part of the equations in the model, and can only be determined through matching observations to the model - there is nothing in the model that makes a specific value of the FSC 'fall out naturally'. There are a fair handful of these 'free parameters' in the current model, which is why the hunt is ongoing to find a model which is both effective at predicting observation, and which naturally produces values for some or all of these free parameters.
When the 'fine tuning' problem is brought up in the contest of physics, this is what it refers to, the idea that these parameters appear to be free to vary, and yet have very specific values for as-yet-unknown reasons; this leads into things like the Anthropic Principle ("The values are what they are because we wouldn't exist to observe them in any universe in which they weren't those values"), which is unsatisfying to say the least.
There is no reason that we know for the speed of light or the FSC to have the exact values that they have. Same with Planck's constant, Newton's gravitational constant and a few others. Our universe just so happened to land where it landed.
@@HeavyMetalMouse nice comment.. or satisfying, perhaps? Given that we exist because we can, I don't see a problem with the Anthropic principle. I might be missing something?
Broke : v = s / t
Woke : v = p / m
But seriously, the more I study physics the more it feels like, momentum is a much more fundamental quantity than velocity. (Same goes for angular momentum)
The problem is light doesn't build momentum.
@@DrakeFellwing That's the neat part, it does.
i haven't watched this vid yet but its probably cuz of the fact that photons can warp space-time lol
And, as I understand it, photons do not experience subjective time.
Like as soon as they are formed in, say, like the core of a sun-- bouncing around for however many millions of years to finally emerge triumphantly out of said Sun --and then come down to, say Earth, and help give a person some sunburn, or maybe knock off of a few free radicals, the whole journey would seem instantaneous to the photon if it was conscious... I think that's how it goes? 🤔
Physics is so weird! Cheers 🙂🙂🙂
@@munstrumridcully lol true to that last statement... but i get what u mean XD
@@TheHyprBeastX 🙂
That was really good. Straight forward and easy to understand. Thanks m8.
I love your channel! It’s my favorite on here. I always walk away feeling like I get what was being talked about instead of feeling confused. You’re a great science and math communicator
Glad I can help 🤓
I couldn’t get this intuitively until I thought about it like “things can’t go faster that the speed of light in the same way that things can’t be more stationary than stationary”
Oh, I like that a lot!
Wow! Another amazing one! Just like the earlier video about superposition, this one knocked it out of the park! So much clearer than the standard explanations! Thanks! (!!!)
Glad you liked it! 🤓
Something I've been trying to relate for decades! What you named "rapidity," I've been calling, "the velocity parameter," and using the Greek letter alpha for it.
β = v/c = tanh α
γ = 1/√[1-(v/c)²] = cosh α
βγ = sinh α
I think I like your name better. But the key thing is, the rapidity is what adds when compounding velocities (such as your basketball thrown from a moving car).
And so, it's what goes to ∞ with ever-increasing speeds; e.g., with acceleration that's constant in the co-moving frame.
Thanks, Nick, for explaining & illustrating it superbly! I guess it took another physics geek to independently come up with all that.
Fred
it is actually named rapidity
I have found to be extremely helpful this explanation highlighting the contrast between the High School Vs Hyperbolic geometry for understanding c, as well as another analogy of Poisson's equation for gravity & Einstein's equation, used in another video of yours, which talked about how elastic spacetime was.
This is probably my all time favourite video of yours ever. Thank you.
Why didn't you spin up one your mind-blow clones for this video? You totally blew it out of the park with the woah factor in this one. Keep up the great work! this video is definitely in my top 3 favorite videos from your channel because it makes me completely rethink some of the ways I view the world.
Thanks for making difficult subjects so much more... lucid. You probably hear that pun a lot lol. Your videos are enjoyable, high quality, and your teaching/editing style keeps me engaged and paying attention 100% of the time.
Now if you'll excuse me, I now have to try to re-learn Trig again now that you've made the subject feel so fascinating and approachable.
Very nice overview of rapidity from simple concepts! Great work! I’m to far from intro level to know if it’s helpful. 😆 Nice that you mentioned hyperbolic geometry. I suspect viewers who are not familiar with that will remain very confused about what that is and how it works exactly. At least they now now a new word to google to find out more.
Yeah, this video wasn't meant to be a comprehensive explanation of hyperbolic geometry. It's just something I use in explanations occasionally. I leave the deep dive to the math channels.
@@ScienceAsylum And I appreciate, that choice of scope. Once again, great work!
Fantastic video. I'll admit, as a fellow physicist, I started crafting an emotional, strongly worded comment about the angle addition at the 4 minute mark with a long polemic about "you need to learn about rapidity and hyperbolic angles!" Lessons: a) watch the whole video b) Nick Lucid knows his stuff.
The number of comments I get that are obviously from people who didn't watch the entire video is staggering. Thanks for being patient.
Absolutely love learning from Professor Nick. Thanks for your videos, makes me appreciate the complex world we live in
Canadian eh
Glad you like them! 🤓
At 5:12, the words « unitless unit » were uttered….made me smile!😊
One of your best videos. Thank you! This filled in some gaps in my knowledge that I never even really knew were there. You have a special gift for presenting teaching in a way that cuts through all the extraneous noise and gets to the heart of the information.
Yo.. the way you desibed the angle of a slope, and how the higher the angle then the greater the output difference between two adjacent angles.. dude you just took the derivative of relativity and made me intuitively understand the 'resistance' (for lack of a better word) of spacetime with a perfect descption a dimension (or two) lower. Basically you just gave a very good model of 4d space in 2d. Wow thank you, that was just elegant in its simplicity and profoundness. 😲👏👏👏
You are genuinely the best science educator I've ever had the pleasure of learning from. You'll go so far.
Thanks for this explanation! This would have helped tremendously in High School Trig and Physics. This gets me closer to understanding the math BUT I’m still having trouble with the cosmological constant and relativity, and finding it hard to just accept that C is C, and C is the speed limit because physicists say it is. I sure wish I had a better physics teacher in 11th grade. I still love this stuff though. Thanks again!
Hi Nick! I can answer the apparent confusion as to why infinite angle looks like 45 degrees near the end. The most natural definition of angles in both Euclidean and hyperbolic geometry is made using the arc length parametrization of the unit circle/hyperbola. For the circle, that's why radians are the natural unit. On the unit hyperbola, If you use arc length from a fixed point (say (0,1)) on the unit hyperbola, then the angle of another point P on the unit hyperbola is defined to be the length of the path from the reference point to P. That way, rapidities corresponding the light-lines intersect the unit hyperbola at "the point at infinity" which is infinitely far away from the reference point!
This is a great video! Definitely gave me a different way to think about it. Keep up the good work!
Love your channel. It's a pleasure to watch, listen and learn!