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.
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.
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.
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.
2 года назад
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?
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?
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.
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
2 года назад
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...)
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!
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.
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 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”
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
Nick is the best science communicator on RU-vid. I understand the concept immediately, instead of having to watch over and over again. Bad for Nick, good for me!
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.
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.
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!
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.
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.
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.
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.
A few years back on a physics video somewhere on RU-vid, 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.
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
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.
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.
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
The fact that most people dont clarify that when we say "The speed of light is constant for all at any speed" is talking about causality and not the ray of photons. Good job Nick!
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.
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.
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.
"For velocities greater than that of light our deliberations become meaningless; we shall, however, find in what follows, that the velocity of light in our theory plays the part, physically, of an infinitely great velocity." - Albert Einstein, "On the Electrodynamics of Moving Bodies" section 4, 1905
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.
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
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!
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
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.
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!
I think its interesting that this video skims over that we only observed the 2 way speed of light, never the 1 way. Veritasium has a video on it, which boils down to: we dont *quite* know the 1 way speed, but we accept the convention that it is the same in all directions for the sake of simultaneity.
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 was just caught up in an argument about someone saying that there was experimental evidence that the speed of light is not constant. I tried to explain that no matter what speed you measure the particle, it will never tell you about the value of the speed of light, which is a stable or metastable constant that if it changes, would causally result an entropic cascade (unless firewalled by the internal geometry of a particle or similar) which would propagate at whichever following maximum speed from it's location.
@@JamesDavy2009 Yes, but it is important to acknowledge that C is actually a constant defined by an asymptotic limit of relative speed, and is not dependent on the particles speed. Rather the photon anneals to that speed under certain circumstances such as non-interaction. I know it's nitpicky, but there are interstellar observations that seem to indicate that even in a relative vacuum, photons can travel at different speeds through macro space, including perhaps even faster. One possible explanation for this is that something bends space allowing it to travel quicker. (these differences are too small to confirm) However, if we want to look out for photons that are in regions of space where the C is not constant, we should be looking for expanding balls of entropy, not photons getting here at odd times.
So THAT is why everyone says you need to deform spacetime to "move" faster than light. (I used "move" in quotes because you aren't actually moving, your frame is the one moving, you are actually just travelling through your frame.)
3:40 "Science: do whatever works as long as your measurements are logically consistent" this is exactly why I had no trouble with Liebniz calculus in physics but failed Newtonian calculus 2 the same semester.
Science, do whatever works!! My new catch phrase. If the spacetime math works only in hyperbolical space perhaps our universe is actually negativity curved as well.
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. :)
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!
Basically, the speed of light is kind of like a carrot that you dangle in front of a donkey as you ride the donkey. It doesn't matter the donkey's speed; the carrot will always be out of the donkey's reach by a fixed amount. So it is with the speed of light. It doesn't matter how fast you're going relative to anything or anyone else. Whatever your frame of reference is, time dilation makes it so that you will always measure the speed of light as 299,792,458 m/s relative to you. If you're going 90% c _relative to earth_, and you shoot a projectile at 11% c, someone on earth will see that projectile traveling at 92% c. However, from your frame of reference, you will see the projectile, indeed, traveling at 11% c _relative to you_.
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
What blows my mind is that even if we discovered FTL travel our observation of events would still be limited by the speed of light. If we warped to Alpha Centauri, blew it up with some kind of super weapon, and then warped back to Earth, we would still see it unchanged for over four years.
Einstein :- Nothing can travel at the speed of light , it can be slower or faster . Nick :- Nope , the speed of light is the limit at which speed can travel .
The last part makes sense in layman terms. A billion miles an hour doesn't make sense in reality therefore it doesn't exist on the hyperbolic graph. I'm glad you explained it that way
