This Paradox Took 17 Years To Solve. It's Still Debated.

[@ScienceAsylum]

The first 1,000 people to use the link will get a 1 month free trial of Skillshare: skl.sh/thescienceasylum07221

[@huepix]

Free? They want my credit card!

[@Secret_Moon]

Could the reason that we see the universe expanding is because our galaxy and other far away galaxies are moving at the speed of light? What what what?

[@nikhilsrivastava1931]

make a video on ALCUBIERRE DRIVE

[@ready1fire1aim1]

As Leibniz put it: “If an ontological theory implies the existence of two scenarios that are empirically indistinguishable in principle but ontologically distinct ... then the ontological theory should be rejected and replaced with one relative to which the two scenarios are ontologically identical.” In other words, if a theory describes two situations as being distinct, and yet also implies that there is no conceivable way, empirically, to tell them apart, then that theory contains some superfluous and arbitrary elements that ought to be removed. Leibniz’s prescription is, of course, widely accepted by most physicists today. The idea exerted a powerful influence over later thinkers, including Poincaré and Einstein, and helped lead to the theories of special and general relativity. And this idea, Spekkens suggests, may still hold further value for questions at the frontiers of today’s physics. Leibniz’s correspondent Clarke objected to his view, suggesting an exception. A man riding inside a boat, he argued, may not detect its motion, yet that motion is obviously real enough. Leibniz countered that such motion is real because it can be detected by someone, even if it isn’t actually detected in some particular case. “Motion does not indeed depend upon being observed,” he wrote, “but it does depend upon being possible to be observed ... when there is no change that can be observed, there is no change at all.” In this, Leibniz was arguing against prevailing ideas of the time, and against Newton, who conceived of space and time in absolute terms. “I have said more than once,” Leibniz wrote, “that I hold space to be something merely relative.” Einstein, of course, followed Leibniz’s principle when he noticed that the equations of electricity and magnetism make no reference to any absolute sense of motion, but only to relative motion. A conducting wire moving through the field of a magnet seems like a distinct situation from a magnet moving past a stationary wire. Yet the two situations are in fact empirically identical, and should, Einstein concluded, be considered as such. Demanding as much leads to the Lorentz transformation as the proper way to link descriptions in reference frames in relative motion. From this, one finds a host of highly counter-intuitive effects, including time dilation. Einstein again followed Leibniz on his way to general relativity. In this case, the indistinguishability of two distinct situations - a body at rest in the absence of a gravitational field, or in free fall within a field - implied the impossibility of referring to any concept of absolute acceleration. In a 1922 lecture, Einstein recalled the moment of his discovery: “The breakthrough came suddenly one day. I was sitting on a chair in my patent office in Bern. Suddenly the thought struck me: If a man falls freely, he would not feel his own weight. I was taken aback. This simple thought experiment made a deep impression on me. This led me to the theory of gravity.”

[@olmostgudinaf8100]

@@Secret_Moon We *are* moving at the speed of light. It's the universal constant. Everything moves at that speed through a four-dimensional space-time. It's just that objects with mass are moving mostly in the time direction. In the extreme case, they only move through time, whilst staying stationary in space. Objects without mass (such as photons) move only through space, whilst staying stationary in time. Relativistic effects that make it _appear_ that time is slowing down are caused by changing the _direction_ of the movement, not the speed.

[@BillDeWitt]

My initial thought was that the whole setup, both rockets and the string, are one unit, and it all shrinks as a unit. So the length from the tip of one rocket to the tail of the other looks shorter from a stationary observer.

[@olmostgudinaf8100]

Damn, I just posted the same thing! I should read other comments first 😇

[@ScienceAsylum]

The ships have their own thrust and the string is too weak to affect anything, so they act independently. They can't be one object.

[@BillDeWitt]

@@ScienceAsylum ok, well, at exactly what thickness does a string go from being a string and turn into a solid connection? Not trying to be pissy, I just wonder. Maybe when the string is stiff enough for the rear ship to be able to apply a little push to the front one? How about a bazillion strings where each one is floppity-floppity but together they are too strong to break?

[@olmostgudinaf8100]

@@BillDeWitt Until someone brings a _very_ strong argument to convince me otherwise, I would say that the thickness is zero. In other words, even a formation of ships with nothing but empty space between them should be treated as a unit, if it behaves like one.

[@Jim-uq1mc]

@@ScienceAsylum Not sure whether this holds true. The configuration space ship #1 + string + space ship #2 should be considered as ONE object. This object shrinks along with the space inbetween. There is no need to have any forces acting . . .

[@n4whhdb]

It's so refreshing to see the graphs and animations specific to what you are explaining. I know it takes a lot longer to produce an episode but it is totally worth it. So many people just use public domain looped graphics that become rather tiresome to watch after a while. Keep up the good work, it's outstanding!

[@ScienceAsylum]

Thanks! Yes, they're a lot of work. It's nice to know that work is appreciated 🙂

[@pinocleen]

He's got his twin to help... :P

[@kkarlsen_06]

@@ScienceAsylum I agree so much! Keep it up, it's awesome!

[@pwinsider007]

@@ScienceAsylum the distance between the rockets should decrease because when length contraction happens then the distance between two points on the body with same accelaration decreases ,consider two ends of the rocket as two rockets and imagine a shape string between material of two rockets then if rocket's length contracts then the material between two rockets(which were the ends of original rocket) also contracts and we had imagined a shape of string on that material between two end that means string also contracts and doesn't snap.

[@pwinsider007]

@@ScienceAsylum when we replace string with rod then the scenario shouldn't change because everything had same accelaration then first rocket can't apply pulling force on the rocket behind it because both rockets have same speed .

[@ScottLahteine]

I get the feeling Veritasium, ElectroBOOM, ActionLab, Steve Mould, Sabine Hossenfelder, and Don Lincoln are all going to need to post followups to this one! Everyone wants to know why the space between the ships isn't contracting in the *essentially shared frame of reference* that the two ships occupy.

[@wulf2121]

The frame of reference is only shared at the beginning, but as soon as they start accelerating, it's not longer shared. It's because from the point of view of a ship already moving (the MCIF mentioned in the middle) what happens in front happens earlier, so the ship in front actually must already have accelerated more than the one behind (it had more time to do so). So you get 2 ships moving at different speed - so different frames of reference. Now if you stitch the multiple MCIF together to see what that looks like from an accelarating observer, you get exactly the result as the video says (from the ship behind, the ship in front appears to accelerate too fast, from the ship in front, the one behind appears to accelerate too slowly). Also, minutephysics already has a video presenting this topic with the same conclusion.

[@ShionWinkler]

The ships aren't contracting and nether is the space, it is an "illusion" created by relativity. The truth is the two ships don't, and can't, accelerate at the same rate, space-time just won't allow it. Also this is a bit misleading, yes the two ships will start to move apart, but at such a slow rate at first that the string won't snap. The two ships will need to travel for a very long time, or accelerate rapidly to relativistic speeds before the gap would grow large enough to snap the string. At speeds we can relate too, like maybe mach 5, the gap would increases at a rate of like 1 atom every year.

[@gauravagrawal9265]

@@ShionWinkler why they can't accelerate at same rate?

[@robertcairone3619]

@@ShionWinkler Why won't space-time allow it? What makes space-time forbid it?

[@AnthonyFlack]

@@gauravagrawal9265 - Whether they are moving at the same rate or not depends on the observer. If they move at the same rate according to A and B, the rates will be different according to C. And vice versa.

[@emilio.occhipinti]

1. Why two ships and a string are not considered as a single entity? 2. Isn't the string accelerating too, contracting, and pulling the ships together? 3. What would happen if a somewhat long spaceship had two propellers one in front of the other?

[@swagdog100]

I want to know too

[@lokeshkalamalla]

exactly my doubts

[@brothermine2292]

Nick discussed this scenario near the end, where the weak string is replaced by a stronger string or a rigid rod.

[@Frankly7]

The answer to all three of your questions comes down to how strong the connecting material is. If it's strong enough to pull everything together, it's one entity. If not, it's multiple entities.

[@lamcho00]

@@Frankly7 well, technically it's always multiple entities on the atomic scale. The question is whether the electromagnetic bonds are strong enough to withstand the acceleration and not break the object apart. If the bonds are strong enough then the acceleration is propagated to the rest of the object, if not the object breaks and flies apart.

[@Bluhbear]

I'm glad you mentioned the thing about the rod, because I was going to ask, "What's the difference between two rockets connected by a string, and a single, long rocket?" I suppose this means, if you're planning on traveling at relativistic speeds, you'd better make sure you've built a tough enough ship to not snap apart.

[@Manuel-cx6ob]

If you get to relativistic speeds by accelerating very slowly for a long time, you don't need a very strong ship. It's the differential acceleration (along the length of the ship) that causes the ship to "snap", and the differential acceleration is proportional to the acceleration magnitude (and the size of the ship). So you have three variables you can affect, which are ship strength, ship acceleration and ship length. If for example your ship can take 1g of acceleration, then you can accelerate to relativistic speed in a few months.

[@eklhaft4531]

It wouldn't really happen with real ship. Real ship would just contract instead of trying to maintain the same length for some outside observer. Outside observer would see the front of the ship get closer to the back of the ship (and therefore not accelerating at the same rate) however the ship doesn't care. It's true though that for this effect to become noticeable you need huge acceleration and therefore the ship needs to be tough enough to withstand the force of the engine.

[@sly1024]

This is exactly what I still don't understand about relativity. well, maybe there's a lot I don't understand :) I've seen several explanations for the twin paradox, and I still can't believe they experience different timespans, because according to *relativity* the equations should give you the same result no matter where you observe it from. Whether I stay on Earth or fly with the astronaut, I see the same thing: the other guy accelerates away, then he comes back. So why is one older than the other? This guy just said that relativity can handle accelerated frames! Everything is relative, there's no absolute position in space, is there?

[@Bluhbear]

@@sly1024 I'm not an expert, but I think, even though there's no absolute position or motion, _acceleration_ is a different thing. It seems to be at the heart of all these paradoxes. 🤔

[@Manuel-cx6ob]

@@sly1024 special relativity is a special case of general relativity. Special relativity can handle acceleration when the general relativity effects that come with it are small enough that they can be neglected. Regarding the twin paradox, the acceleration is not even the important part. You can come up with a version of the twin paradox that does not have any acceleration, by using three observers. Observer A stays on Earth, observer B passes by next to the Earth at high speed without slowing down and synchronizes their clock with A to start counting. Another observer C is very far away on the same path of B, but coming from the opposite direction (going towards the Earth). When B and C meet (without stopping), B can tell C the value of his clock, and C can continue counting from that value. When C reaches A (again without stopping), they can compare the clocks, and they will find that A's clock has a measured a bigger time that C's clock (which includes the time experienced by B and C). None of tho three observers experienced acceleration at any point, but twins paradox effect happened regardless. So, how is the paradox resolved? When the "time keeping" is switched from the frame of B to the frame of C, their speed is the same, so time dilation does not play a role, but their direction is not the same, which changes the direction of the Lorentz transformation. This means that what B considers as "now" on Earth is not the same as what C considers as "now" on Earth. Imagine A and B met in 2010, with time dilation factor of 5, and then B and C met 5 years later in B's time. B would have seen Earth's time moving slower (because from his point of view the Earth is moving, so he would think that, when he meets C, the date on (far away) Earth is 2011, and is own time is 2015. C received the 2015 time from B, but does not agree that right now on Earth is 2011. In his frame of reference, on Earth it's 2059. Then, when C finally reaches A on Earth, he sees that his clock marks 2020 (B + C traveled for 10 years), and on Earth is now 2060 (his travel took 1 year of Earth's time from his point of view, because for him it's the Earth that is moving). Now you would argue, what about the Earth's perspective? Since the Earth's measurement of time was always in the same inertial frame, and Earth sees B's time going 5 times slower, then A would simply think that B met C in 2035 (2015 for B's time), and C then took the same 25 years to arrive at A in 2060. As you can see, the three reference frames do not agree on what time it is on Earth when B meets C. A thinks it's 2035, B thinks it's 2011 and C thinks it's 2059. This is relativity of simultaneity, which is as much real as length contraction and time dilation.

[@AlwinMao]

Most 'spaceships' we consider to be single entities because their tensile strengths are able to maintain a constant proper length by decelerating the front end and accelerating the back end: the tension of the craft pulls the front back and pulls the back forward. This is very unintuitive because the scale of this effect is L(a/c)^2 for length L, acceleration a, and speed of light c. For a typical acceleration 1 m/s/s and distance m, the acceleration difference to be overcome is only 10^-17 m/s/s. If we speed up to 10g (100x the accel) and 1000 m (1km) this is still only 10^-10 m/s/s. So even a relatively fragile 1 N string could hold together a 10^10 kg pair of spaceships at 10g acceleration and 1km distance. So it's easy for a bulk spaceship to be treated as one entity in many situations, and even easy for the string situation to be treated as one entity. But an infinitely fragile string would break apart.

[@burrahobbithalf]

I would treat the ships and string as one object, which experiences length contraction. Just like two ends of a ruler the ships would appear closer together, so the string doesn't snap in any reference frame.

[@richardb4313]

This is what I think as well. The 2 ships are in the same frame of reference. The distance between them does not change for them and there is no shrinkage from their own observations. To an observer in a relatively slower frame of reference the rockets and the space between them both shrink, as they act as one object. The thought experiment is trying to cast the string as somehow belonging in another frame of reference, which is nonsense,

[@user-pr6ed3ri2k]

the ships have separate accelerations + string =/= pole that maintains a "constant" distance therefore they are not the same

[@user-pr6ed3ri2k]

i mean 1 object

[@warchitect73]

@Retired Bore Exactly imo. just because people are in different places on this one "double-ship" shouldnt matter to space time or the universe. also, if the video is right, consider, a single ship is still a bunch of different components with length, so all the parts of the ship down to the plank length in the direction of travel would all snap apart everywhere while accelerating too right?

[@ScienceAsylum]

The ships have their own thrust and the string is too weak to affect anything, so they act independently. They can't be treated one object. There are technically 4 frames of reference here: Charles, Arthur, Bernard, and the string... but since the string isn't conscious, it isn't a very interesting point of view.

[@mikewagner2299]

New question: If Bernard sees Arthur accelerate too quickly, what if they choose not to accelerate at the same rate and Arthur calculates the lower acceleration (which may not be constant) such that Bernard sees Arthur always the same distance away. Would Charles see them getting closer together in a way that would exactly match length contraction so no one sees the string snap? I do understand peoples concern about Arthur and Bernard sharing a frame of reference. I think it would have been beneficial to explain why they see each other accelerate differently a little more in depth. I get that the curved axes would cause a changing distance but maybe explain the nature of why the axes curve

[@person1082]

if the distance from a spaceship with proper acceleration A to the rindler horizon (apparent black hole from acceleration) is D, then another spaceship a distance d from the rindler horizon needs to accelerate at AD/d

[@jergarmar]

Yes, this is exactly correct. Assuming there's a bit of "give" in the string, and something to measure the tension, you could imagine Arthur slowing his acceleration to keep the same distance with Bernard and keeping the string from breaking. In that case Charles straightforwardly sees them accelerating differently, moving closer to each other, and the string doesn't break.

[@RickSummon]

If the ships accelerate at different rates to keep the string from breaking, this is called Born rigidity.

[@robjames4160]

Length contraction is evident from the 3rd party observer, but it's not the length of the ships that contract, it's the apparent lengths of the ships relative to the OBSERVER'S frame of reference due to what light is doing during the ships' acceleration. The 3rd party observer is not seeing anything specific happen to the ships themselves, only to the light being emitted or reflected by them. The ships don't actually change in length relative to the space in their own refrence frame. Therefore, the string doesn't break because it too is accelerating at the same rate as the ships themselves, and its not the string that is contracting, only the appearance of the string that is contracting. The two ships and the string occupy the same inertial frame of reference and are ALWAYS the same object, regardless of the strength of the string. The appearance of BOTH ships AND the string will contract in unison to the 3rd party observer.

[@TheMonk72]

That was my immediate thought. People mistake length contraction as a physical change in the relativistic object. It's really not.

[@TheMonk72]

@@kylelochlann5053 why do the ships move further apart? If they're accelerating at precisely the same rate their instantaneous reference frame will remain consistent. In those frames they will not experience any length contraction either of themselves or the other entities in that frame. The rotating spacial co-ordinate is a way to translate between inertial reference frames isn't it?

[@TheMonk72]

@@kylelochlann5053 if acceleration is the same then instantaneous velocity is the same and distance travelled is the same. Whatever offset they had at the start is the same at all points along the curve. If length contraction is an effect that objects experience physically at relativistic speeds then that implies a universal inertial reference frame... which GR expressly forbids.

[@darkracer1252]

@@kylelochlann5053 the ships do not move further appart. stop being braindead

[@charleslivingston2256]

@@TheMonk72- The acceleration is only the same in Charles's reference frame. I'm the reference frame of each of the spaceships, the start happens at slightly different times.

[@andrewparker8636]

Fair warn: I'm definitely at the limits of what I recall about relativity here. I'm impressed by this paradox because I'm probably going to be unable to resolve it without getting the books out. Regardless, I'll state a couple of things that are confusing me about this. Much of this has been said below, but I wanted to highlight why I object to some of the rebuttals about these points. My initial reaction (like many below) is that we should be able to treat the two ships and the string as one "object". After all, what's the difference between a piece of string and the bulkheads of each ship. Much of the objections here come down to things like "we've got relativistic speeds so the acceleration is massive and thus the string is weak" or "there's time delay in the forces in the atoms and molecules". However, none of this is relevant as far as I can see. We can accelerate the ships as slowly as we like and thus with forces as weak as we like. Sure, it'll take a long time to get up to relativistic speeds, but for a thought experiment that's irrelevant. In fact, the starting speed is almost irrelevant. Why not just accelerate a blob of matter to 0.99C (relative to the stationary observer) then build the two ships and string from that blob of matter. In the blob's reference frame there's no issue with doing that - there will be no broken string. Now we just accelerate a little bit more. Apply a tiny force. The string won't break. The question is then what the observer would see. They should indeed see length contraction but that's a contraction of the whole system. It's not contracting about any specific point. It's contracting uniformly. This is similar to how the universe expands, but in reverse - it's doesn't expand from a point it expands everywhere. So the entire system contracts (as far as the observer sees it) and the string doesn't break. I feel like there's some erroneous thinking in the argument about the attachment points of the string. We said nothing about the attachment points of the string. We could repeat the experiment with the same ships, but attach the string at different points to the ships. Should this change the forces on the string? I don't believe so. It shouldn't matter.

[@JustMe-vz3wd]

wait wait wait. your brilliant post got NO replies. I scrolled a bit and this is by far the best post. I liked the "blob" explanation. First reach the near speed of ligjt and than construct the two ships and string, all while in THEIR reference frame. If this are three objects, two ships and a string, its just a matter of how you view it. It can be viewed as one object, or many objects, all the different parts of the spaceship. i feel these hypotheses get more and more just a funny trick with little essence to the awesome reality of relativity. I guess as a youtuber you gotta keep it up with the crazy stuff ^_^ Anyway, thanks for yr explanation.

[@ImHeadshotSniper]

@@JustMe-vz3wd they're setting entirely new conditions which oppose the ones set in the video, which is simply not at all the same problem anymore. firstly, their presumption that "applying a tiny force" to a string won't break it sets entirely new conditions, because the whole point of this paradox is to not just "not break" the string, but ultimately leave ZERO load between the rockets, so that the string between them is not being pulled one way or the other as the accelerate to the speed of light from a stop. secondly, building a string while you're already at the speed of light is ALSO setting entirely new conditions, because the point of this video is that you're starting from a speed of zero, and accelerating to the speed of light. building a string at the speed of light skips the MASSIVELY important step of the space time relation during acceleration, as was amazingly visualized at 8:06 if you simply start attaching the string together at the speed of light, then of course the space in between them doesn't change any more, because you've stopped accelerating... the takeaway from this video is that 2 things accelerating at the same rate leave a greater and greater distance of space in between them as they do so even if they're doing it at exactly the same rate, which is mind blowing to me.

[@JustMe-vz3wd]

@@ImHeadshotSniper the point some people including myself are making is, should we view it as two things accelerating, or as one thing accelarating.

[@ImHeadshotSniper]

@@JustMe-vz3wd i totally understand that, and i actually also thought as well at first, but i commented to someone else that at 9:01 he actually clarifies this paradox as requiring the ships to be stronger than the string, and that the string be strong, but not strong enough to withstand space expanding the distance between them as they accelerate as shown at 8:06 to be fair, this seems to limit the conditions of the paradox to the weakness of the "string", when of course as many people immedately said, if the string was as strong as both ships, that it would effectively be one load and the front load would be pulling the rest of it as one load because it's a theoretical unbreakable material that can withstand light scale speeds as well as the expansion of space which would make it one whole vehicle if i'm not mistaken (i could very well be :p)

[@Yambolic]

I may have missed something in all of this but "what the observer would see" depends on where the observer is in relation to the spaceships and string, doesn't it? If the observer is stationary, he ain't gonna see much at all as the ships and string are moving too fast. If he's in another spaceship travelling along with and at the same rate of acceleration as the spaceships and string what will he see then?

[@juzoli]

But why is there no length contraction for the cable? It is also moving fast, why it is excluded? It just joins them into one big spaceship. Because of length contraction, the 2 spaceships appears to be closer to each other, so it doesn’t snap… There is no paradox…

[@ScienceAsylum]

The string's _attempted_ length contraction is the whole reason it snaps for Charles. The string is _trying_ to contract but the ships prevent it by maintaining their distance. This lowers the "relaxed length" of the string without lowering it's actual length, thereby putting tension on the string. The distance does not contract.

[@juzoli]

@@ScienceAsylum But the apparent distance between the ship contracts too. Length contraction is not selective.

[@ScienceAsylum]

@@juzoli Ah, but distance and length are not the same thing. They may both contract sometimes, but usually not in the same frame of reference.

[@viralsheddingzombie5324]

@@ScienceAsylum The string has mass and it is accelerating, so it MUST also contract. Charles is an inertial observer, he does not interact with the ship-string-ship system at all. BOOM! We don't actually know if the two ships maintained their distance, that was merely an assumption or a possibility. Given Nick's logic, the string would snap ONLY if each ship moved away from each other, rather than move closer to each other. 2 x BOOM!!

[@yfakolh7154]

@@juzoli the distance from C's perspective doesn't change. The "distance" is not moving, just think about stationary ruler in the background. But the moving rope contracts and breaks. If the distance were to be contracting, the acceleration of trailing rocket would have to be greater than the heading one. Such acceleration difference could be caused e.g. by heading rocket dragging the second one when connected by stronger material. In classical one engine rocket at the rear it's the back that pushes the front creating the traveled distance difference equal to the length contraction.

[@christianosminroden7878]

6:22 When an accelerating ship A and a ship with constant speed B meet each other when they‘re at the same speed, the accelerating one did NOT catch up with the other, but the other way round. Before they meet, A was slower than B, so B caught up with A just to fall behind again immediately after the meeting.

[@KaktitsMartins]

Ok, but now im interested in the scenario where they are connected with thick metal bar - it wont break as easily, but will with enough acceleration/ speed? How does it differ from a single long spaceship?

[@tomkerruish2982]

It differs in that a single long spaceship is not usually thought of as having rockets at both ends, applying thrust in the same direction. With sufficient competing thrust, the ship will tear itself apart.

[@kurtpena5462]

@@tomkerruish2982 The premise was identical rockets with identical acceleration.

[@dj1NM3]

This should (in total) act identically to a single, lager ship with two identical rocket engines operating at the same time.

[@VOIP4ME]

@@dj1NM3 The difference being that the long ship could withstand a much greater amount of force (due to length contraction) before breaking.

[@dj1NM3]

@@VOIP4ME Considering that it would all accelerate together, an outside observer would see it all length contract together as a single object, because that's exactly what it is. The occupants at either end would see no contraction of their ships nor the rigid bar, because it's all travelling together and accelerating together. If that wasn't true, then the ships themselves would be breaking apart at the same rate as you're imaging the rigid bar would be and the whole lot would turn into fast-moving space junk at whatever velocity it all went "KABLOWIE!!" at.

[@Mysoi123]

The problem is again due to the fact that we don’t all agree on simultaneous events. while from one frame, the rockets accelerated at the same rate, the rocket behind were slightly in the future compared to the rocket at further ahead. that means the rocket behind is the one that has less acceleration in their individual frames where their clocks reads exact same time compared to each other, therefore it got left behind, cause a stress on the string. Great video as always!

[@ShawnHCorey]

Agreed. The rockets do not start accelerating at the same time, as seen from each of constant-velocity rockets. This is the same thing as described in the video but with different emphasis. That's the thing about relativity: there are often many ways to describe what's going on.

[@josephsalomone]

This is a bad explanation, as it would imply that rockets moving up instead of chasing would also have the string break, because they will not view each other moving at the same acceleration due to their distance.

[@ShawnHCorey]

@@josephsalomone Rockets that are side by side are different than rockets that are one in front of the other.

[@josephsalomone]

@@ShawnHCorey Not the way this is explained. The only proper way to explain this phenomenon is with a spacetime diagram, otherwise you end up having to explain why other cases are not paradoxes.

[@Mysoi123]

@@josephsalomone as I said, in their own frames, each rocket has different acceleration because their clocks reads the exact same time (ignoring the effect of a uniform gravitational field on flow of time). From the Charles’s perspective, the clock on the rocket at the left were slightly in the future compared to the rocket at the right in just the right amount, therefore both have the same acceleration in this unique frame. If in each individual frame, all the rockets were accelerated at the same time, from Charles’s frame, the rocket at the left will accelerate first, and the one at the right will accelerate last, and distance between them shrinks, also known as an increasing in length contraction.

[@gravitron12]

This is why I love this channel. I can finally understand another topic that I was not sure of.

[@Kazemahou]

Why is the string magical? It has to be, in order for any of this to be true. If both ships accelerate at the same speed, in tandem, then they are travelling together as a unit. They share the same frame of reference because they are stationary relative to each other. Putting a string between them is no different than putting a girder between them, or simply building more ship - a long tube of more ship - that connects them, making them a single, super-long ship. All share the same frame of reference! If this 'string breaking' business were the least bit true, then it would become impossible to build ships of any length. No super-long tube ships, no train-like ships composed of innumerable cars in sequence. Why? Because if the string breaks, so would the ship - the ship would snap in the middle. The string is effectively more ship between the ships. The string isn't somehow magically outside the observed contraction. What an observer not in the frame of reference should see is that the total unit of two ships and string between collectively contract. Uniformly, over the whole. Why? Because ships and attached strings all share the same frame of reference. The crew on the ships should see no contraction of themselves or the string, and no stress on the string whatsoever. Why? Because both ships and the string that binds them are one object, in effect, which shares the same frame of reference. All are moving at the same speed, all are accelerating in tandem, all are one single reference frame together. Again, replace the string with beams and girders and more ship. Now you have a really long jumbo-dog ship. The ship is a whole unit, it moves as one object because it is one object and the frame of reference aboard that ship is uniform throughout. Why? Because every part is accelerating uniformly. Just like the two ships bound by string. All is, in that frame of reference, accelerating uniformly. The string should not break, and there should be no paradox at all. Nobody should see the string break any more than they should see each ship snap in twain. The string is not magical, it is just sharing the same frame of reference as the ships. There should be no paradox, and no confusion about this whatsoever. The ships don't somehow magically contract separately from each other - this isn't about matter. This is about the apparent contraction of a frame of reference, and that includes anything between the ships that bridges them, so long as it is travelling within the same frame of reference of the ships. If this were not true, the ships would snap in two due to contraction, or, a very long ship would snap in two because of contraction. And if that happened, one would have to claim that, despite accelerating as a unit, together, somehow the front of the long ship is in a different frame of reference than the back. That should be testable even on a small ship. It should be universal that the front and back of any object exist in different reference frames - and I have never heard such a thing in my life. Tell me how I am wrong! Tell me how any of this is the least bit wrong! I need to know - because to my reasoning, all objects within the same frame of reference share that frame of reference. That is what I have understood for the past fifty years. How is that wrong?

[@TJTapia6]

It's a trick question. The ships are only in the same reference frame from the point of view of a third party observer. If they were in the same reference frame the distance between them would get smaller as they accelerate due to length contraction. They only appear to maintain a constant distance, even though they accelerate (and experience length contraction) because the ship in front is actually accelerating more. If you had an additional engine on the front of your long ship powerful enough to prevent relativistic length contraction it would tear your ship apart.

[@Kazemahou]

@@TJTapia6 The example given has both ships accelerating at the same speed at the same rate. This is clearly stated! That means that the front ship is - not - accelerating relative to the rear ship - they are moving at the same acceleration, at the same speed, together. It was specifically stated that they had the same acceleration and speed. If this is true, then from their frame of reference, they would be - stationary - with respect to each other. By definition, their frame of reference would be the same - that is literally what that means. If this is so, then the - space - between the two ships should appear to contract to an outside observer, which means the string should never break and no tension should be applied. The string links both ships, making them one object - it doesn't matter that it is a string, it could be a metal beam. They are one thing, one object, one frame of reference. They are attached, and all parts move at the same speed, the same acceleration. They are a single unit. My point stands. An outside observer should see the entire unit contract uniformly - because it is one single thing, one single frame of reference. It isn't 'Two Ships'. It is a single 'Supership' that just happens to have part of its structure made up of string instead of metal. If this were not true, all ships would rip themselves apart in the middle as they accelerated to relativistic speeds. The material - string versus metal - should not matter. The entire group is one ship, part metal, part string. It is one thing, one frame of reference. That is my point. That, I think, is the entire point. Unless, of course, it is true that a really long starship - say miles long, like one of the ships from Dune or Star Wars in scale - would, in fact, rip apart in the middle. I want Nick to address this. I think I am correct - that all things sharing the same reference frame must appear to contract uniformly because they share the same reference frame. It isn't about matter, it is about space. Thus no paradox. Unless I missed something and it - is - stated that the ships - are - accelerating at different velocities, in which case there still is no paradox since that would always break the string, even at conventional speeds.

[@wrOngplan3t]

"The string is not magical, it is just sharing the same frame of reference as the ships" This and the rest you wrote is my understanding as well. Sure I'm no particle physicist, and my head hurts from this video, but I'm not convinced. This continues to be my understanding.

[@jetison333]

Accelerating at the same rate only makes sense in one reference frame. Either the ships are accelerating at the same rate in a stationary frame and appear to accelerate differently in their own, or they accelerate the same in their own reference frame, and accelerate differently in a stationary one. In the former the string breaks, in the later it doesn't. If you wanted to build a long ship, you would just have to set the engines to not accelerate at the same rate as observed from a stationary reference frame, and rather set them to accelerate at the same rate in your own accelerating reference frame.

[@eyeOfAC]

I agree. The spacehip enters the frame of reference, not the other way around.

[@darenmiller2218]

I’ve actually seen this happen in a test. The string turns into a Jacob’s Ladder. Pretty incredible findings.

[@rjdruhan]

It's snaps because of the contraction for all three. But for a person in an accelerating ship, contraction "looks" like the other ship moving away a bit. The ship is still contracting, and that is what causes the snap.

[@jonathanhadden7157]

Not only was this really cool factually, but you explained it in such a way even laymen could follow. Thank you!

[@ScienceAsylum]

Glad I was able to deliver 🤓

[@rayzimmermin]

except length contraction is just an optical illusion that only the outside observer sees and is not a real thing phonically happening to the ships so how dose an optical illusion brake the string

[@ImHeadshotSniper]

@@rayzimmermin if i'm not mistaken, length contraction is very much not an illusion, but rather is the very real result of a lengthening of space between A and B being created by the acceleration of both rockets, which i thought was amazingly visualized at 8:06 again i could be mistaken as i don't know what i'm talking about, but that's just what i took from it

[@rayzimmermin]

@@ImHeadshotSniper if length contraction is real and not an illusion then why dose the third person observer not notice it also how is it that length contraction dose not effect the distance between sub atomic partials if acceleration resulted in space lengthening between A and B then why dose the space between sub atomic particle A and sub atomic particle B also lengthen and rip things apart as they accelerate remember solid objects are not really solid because they are made up of small things that are not touching or firmly connected to one another and have space in between them that should expand with acceleration yet they do not if the theory was correct you would expect that the faster an object moves the further apart the sub atomic particles that make it up will get from each other resulting in the object essentially expanding and disintegrating as all the sub atomic particles move further away from each other to the point they can no longer maintain the sebility of the object they make up

[@ImHeadshotSniper]

@@rayzimmermin 1. i would guess that it's because the contraction happens at a speed faster than the stationary third observer can possibly observe because they'd be moving faster than the light is reaching the observer. i don't believe there was actually an example of an observer that followed the rockets as they accelerated. i would imagine it was for the sake of demonstrating relativity because stationary things only receive light as fast as it travels to them, which i think was a really good idea by Science Asylum. 2. well he actually even showed shown in the video at 9:01 that with enough strength that the string would become part of the entire load and the 2 rockets would be connected, but it would be the front one dragging the back one because of the distance between them expanding as they accelerate faster than space can keep them in the same position in time as the were at at 0 speed. i believe the conditions of this paradox though is imagining that the string is strong, but not strong enough to withstand a certain amount of space stretching. 3. i would guess that this is because in this theoretical example, the rocketships forms are "infinitely" strong and can withstand a subatomic destruction, just for the sake of theory and thought experiment. 4. subatomic particles in solid objects don't expand with space because it's impossible for solid objects to move even close to the speed that it happens at a noticable level, unless you have an infinitely strong rocketship of course. as Science Asylum said in another video "The Speed of Light is Infinite", the equation for velocity has additional variables which Einstein determined, which relates a velocity to the speed of light, which of course are only practical for problems regarding objects moving at a large fraction of c or the speed of light. to the regular human experience and the fastest speeds we can possibly achieve, this variable relating to light speed effectively 1, meaning roughly zero influence in the additional velocity equation, because the additional equation relates to the speed of light, and even our fastest possible theoretical speeds for objects with mass are not even close to being able to influence the additional variable in the equation because things with mass can only move a very negligible fraction of the speed of light. (as far as we currently know)

[@cahdoge]

Wouldn't this scenario suggest, that with enough accelleration objects with finite bonding strength would eventually be torn apart? Yes and it dosen't even have to be relativistic. once the increase in distance between two points is faster, than the speed of sound in the material it should seperate.

[@cahdoge]

I never said anyhting abuot velocity. But I can answer my question now; If we accelerate by pushing, they would not. Since, length contraction and acceleration would smush them. If done by pulling, they would, for one reason or the other. But what it does notdo, is contradict spcial, or general relativity.

[@AnyVideo999]

Rewatching this one again months later, still great. Only thing I'm trying to wrap my head around is an equivalence principal twist. Imagine an observer in freefall over a uniform gravitational field, say an infinite planar planet. On this planet we have two rocket ships, each exhibiting enough thrust to hover perfectly. Attached to each rocket is a string. With this preamble, we should expect a similar outcome as the freefall observer should view the hovering rockets as accelerating. However, we should expect from real world experience that there should be no tension in the rope. We have an apparent break in the paradox. Thankfully, there is likewise no problems here. All I've done is changed who agrees that the distance never changes. And in this setup, we *finally* have the two rockets come together as their length is contracted together from the freefall observer's perspective. The two situations are rather quite different, and because one rocket is in a different vertical position than the other, the freefall observer actually does not see the two rockets accelerate at the same rate but rather that the back rocket has been accelerating faster.

[@brianm1143]

From Charles' point of view wouldn't the ship-string-ship contract equally as an entire unit? If so even from his perspective why would it place more tension on the string? Wouldn't this be analogous to drawing ship-string-ship on a stretched rubber sheet and de-stretching the whole sheet a bit? This scenario implies string would feel the effect of its own shape being distorted due to its (relative) speed regardless of frame of reference which seems to not make sense? I'm confused :)

[@dmitriy4708]

From Charles' frame of reference ship-string-ship contracts all the same, but the distance between ships does not contract, so the string is under tension and snaps. Only objects contract.

[@jimpinkerton1352]

@@dmitriy4708 Wait, so you are saying that objects traveling at relativistic speeds don't distort space/time?

[@dmitriy4708]

@@jimpinkerton1352 Speed is relative to the observer, so how can the speed of 2 objects influence the contraction of space between them if this space is not really moving in any frame of reference? Any moving objects contracts in space in the direction of movement, space is not moving.

[@paradox7358]

What if you replaced the string with a laser to measure the distance between the rockets? Would the measurement change? Would each rocket measure the same distance?

[@Manuel-cx6ob]

The laser would measure an increasing distance for all the observers. The inertial observer would see the "real" distance being constant, and would conclude that the "increase" in distance value measured by the laser is caused by the light having to "chase" one of the spaceships. Even if the internal mechanisms of the laser are time dilated, the roundtrip time measured by the laser will still be higher, since the time it takes for the roundtrip in the inertial frame increases by a factor of 1/(1 - v2/c2), while the time dilation has a factor of 1/sqrt(1 - v2/c2) [the first expression is always bigger than the second, because the square root of a number that is less than one is always bigger than the number]. For the two spaceship, they will see the same increase in distance measured by the laser, but they will say that it is a "real" distance increase, caused by the other spaceship actually lagging behind.

[@welrann]

@@Manuel-cx6ob The inertial observer wouldn't see anything since laser measurement is technique involving sending laser beam from one point to another to measure distance between them and both points are not this observer in this case. Also laser measurement would show increasing distance because of lower speed of light in accelerated system or it could become infinite depending of acceleration.

[@Manuel-cx6ob]

@@welrann there is nothing preventing the inertial observer from looking at the result of the laser measurement, even if the device is on the ship. For example, if the laser measurement device has a screen and the ship has a window, the inertial observer can just look at the screen as the spaceship is passing by. The value they will see will be the same as the value that the observer on the ship will see, but the explaination they will have for the value will be different. Also, while it is true that acceleration causes an additional lag, it's not really the main effect as long as the acceleration is not very big. The laser would measure a bigger distance than the inertial observer sees even if the ships stopped accelerating, as long as they are at relativistic speeds. That's caused by the light having to chase one of the spaceships, increasing the distance it has to travel to do a roundtrip. The distance that the light travels for a round trip in the inertial frame is cL/(c-v) + cL/(c+v), which is 2c2 L/(c2-v2), or rearranged: 2L 1/(1-v2/c2). As you can see, the factor by which the light travel path has increased is greater than the factor of time dilation for the laser (which is the square root of that factor), so the inertial observer will see the (moving) laser measuring a longer distance than the distance between the two spaceships in the inertial frame, and they will conclude that the measurement is "wrong" because the light has to travel longer to catch up with one of the two ships.

[@jetison333]

@@welrann they wouldn't see the laser, but they could still model and figure out what the laser would measure.

[@Manuel-cx6ob]

@@classicalmechanic8914 But I'm the one saying that the effect of the acceleration on the laser measurement is negligible. :D The whole idea of the laser measuring a bigger distance than the inertial observer measures from its frame depends only on the fact that the ships are moving at relativistic speed, no acceleration or general relativity required.

[@voiceoftreason9212]

Hmm, no. The stationary observer does not observe any Lorentz contraction. That is just the spatial aspect of the Lorentz transformation which applies to a frame in motion relative to the observer we take to be 'at rest'. It is the observers in the spaceships who will measure contracted distances in the direction of travel (relative to the observer at rest). Lorentz contraction always has to be relative to some 'rest' frame. Incidentally, this is also why if you travel to Alpha Centauri at almost the speed of light, it will seem to take virtually no time at all on your clock: it's because the distance to Alpha Centauri is contracted in your frame. It's NOT due to time dilation. However, the observer at rest will think your clock is running slow, and THAT'S why they think your journey took almost no time (on your clock). So you end up agreeing on what happens, but for different reasons. The string does not snap! Try the experiment...

[@joepierson3859]

No, relativity means motion is relative. The string must contract relative to a stationary Observer. Spaceship will see the Observer Contracting but that's incidental and irrelevant to the thought experiment. But in any case contraction always occurs both ways it's symmetrical.

[@kerryhaycock9446]

Thanks Nick , great explanation ! On another topic , I agree with Joe Scott that KZbin is increasingly filling with sites offering science content from opportunists cobbling together snippets of others work with inane voice overs and comments . This is why I turn to you for sensible discussion about sometimes controversial subjects . JWST and it’s achievements so far is something I’d love to hear your summary of , as there is so much “ science “ rubbish and click bait out there. Always look forward to your videos ! Respects !

[@ScienceAsylum]

I hate that those "cobbled together snippets" perform so well. It's so frustrating. Whenever my content shows up in them, I have them taken down, but it's like wack-a-mole. They just keep coming.

[@Antifag1977]

Nick really does have a gift for explaining the most nuanced complex concepts in a way that even people of average intellect like Yours Truly can understand and conceptualize. In times past those who were gifted in teaching were afforded the greatest of respect and highest of honor. . Its such a shame...and irony...that humanity doesn't seem to value the imparting of knowledge now as much as it did in the past even though now we have much more knowledge to impart.

[@glenncaughey5044]

Finn McMissle, British Intelligence Tow Mater, average intelligence -Cars2

[@DEtchells]

Ok, so here’s the super-extra bonus points question: ==> What is the *stress* in the string as a function of acceleration, velocity, masses of the spaceships and the separation distance? I’m having a hard time thinking of the ships and string being in different reference frames, but I think it comes down to the distance between the ships divided by the speed of light somehow; that seems like it would be the key to the amount of difference between their reference frames. The way the problem is drawn leads us to think of a separation of maybe a few hundred meters, and I think over that sort of distance, anything other than an unimaginably weak string would keep the two rockets in the same reference frame. OTOH, a thousand KM separation would result in a greater effect. I think I can accept that the string needs to exert a force on the rockets to keep everything in the same frame, it’s just that the effect is going to be pretty tiny unless the acceleration gets really large. So here’s the question: Can we calculate how much force the string needs to exert as a function of distance, the masses, the velocity and the acceleration? I don’t remotely have the math or physics ability to do that calculation, but I, sure someone does, and I think the answer would be very interesting. (!)

[@themcchuck8400]

Nicely done, sir. Informative and entertaining as usual. Thank you for your efforts!

[@Mr-Pulse]

If the ships accelerate at the same rate and began at the same time, then why are they both not in the same interial frame, like the passing rockets going the same velocity. Also if the string is taught and attached to both ships why isn't the string accelerating and moving at the same speed? Why doesn't the whole conjoined setup contract together as one?

Exactly this. If they really accelerate equally they should act as a single object and thus contract as whole, including the distances. The explanation makes it look like the strength of the string affects spacetime, which doesn't seem right.

[@ScienceAsylum]

They _start_ in the same frame of reference, but that's it. Once they're accelerating and acting under their own thrust, they're in separate frames.

[@Mysoi123]

@ The distance between the them stays the same in the Charles’s frame, but not in their own frames. remember, in Charles’s frame, the clock on the left rocket ticks ahead of time compared to the rocket at the right due the the fact that simultaneity is relative. on each rocket’s frame, the rocket at the left has slightly less acceleration, because comparing to each other, their clock reads the same time, (ignoring equivalence principle gravitational field effect on time). in fact, if both rocket accelerated at the same time, Charles will see distance between them shrinks, because from his frame, the rocket at the left were accelerated first, because it was in the future compared to the rocket at the right. Welcome back to length contraction:-)

I think it'd help to define "same acceleration" relative to what and how we measure "started at the same time". Anyway, if it's correct the string snaps but a durable rod doesn't then there should be a force that can be calculated. Also the engine cones should feel this force trying to tear them apart in forward-backward direction.

[@axelperezmachado3500]

not gonna lie at 8:41 I genuinely thought that both clones broke character. Took me like 2 seconds to realize that didn't made sense and it was hilarious.

[@petarmisic281]

When I first heard about this my initial thought was that since accelerating objects towards the speed of light actually increases their relativistic mass (assuming if affects the spacetime curvature), that at some point the mass of the two spaceships would be so great the gravitational pull (curvature) they both exert would pull on the string until it snaps.

[@ophidahlia1464]

"Relativistic mass" isn't actually a real thing, it's more of a misguided teaching aid for beginning relativity (since it causes exactly this confusion), it's more correct to say that the momentum of an object increases with velocity. Rather than do a poor job of explaining it myself here's a few videos that dive into it deeper: kzbin.info/www/bejne/goWtkqiXmsuHkK8 kzbin.info/www/bejne/pJDcqZJ4p8mFfJo

[@coloradoing9172]

No.

[@14arma]

I actually did this experiment, 2 model spaceships with string taught between each, both accelerating at the same rate away from CBR at the edge of the observable universe. String stayed in tact and no change.

[@fredk4745]

Yeah but i dont think you accelerated them to the speed of light...

[@14arma]

@@fredk4745 That depends on where you're measuring from. From the edge of the observable universe they were going the speed of light. But really you can accelerate forever in 1 direction and never get to the speed of light.

[@MrPaxio]

where? on roblox?

[@SD-vd3mh]

Not sure if its a joke, but this is actually valid. Any gravitational field should be indistinguishable from acceleration!

[@TheMrJizzus]

That blooper with the clone is pure genius and mastery of story telling. Keep it up Nick!

[@dcterr1]

Wow, that's a neat paradox! It's much more subtle than the twin paradox or the car in the garage (aka the pole and the barn) paradox, which is probably why it wasn't resolved until much later. Relativity is great, and this was a great explanation! Keep up the good work, and stay a bit crazy like myself!

[@thegwangster9097]

I see. I guess this illuminates the fact that acceleration at relativistic speeds stresses an object, due to each part of the object existing in a different momentary inertial reference frames from each other. In order for the object to stay together at all, it’ll have to be strong enough hold together, otherwise it’ll break just like the weak string.

[@levyroth]

Duh, that's why ships in Star Trek had a magnetic bubble around them to allow them to go at warp speeds.

[@darkracer1252]

no this illuminates the fact that this guy has no fucking clue what he is talking about and shouldn't be allowed to post vids about science.

[@pulsar22]

The theory of relativity is about observational illusion. An observer does not create any force on the objects they observe. That is why an accelerating observer feels no different than a stationary one though the inertial frame observes the accelerating frame to be shrinking (but it really does not). Thus all three observer Charles, Arthur and Bernard will not see the string break. Charles will see everything shrink even the space between the rockets. Arthur and Bernard will see that they maintain their distance to each other. Imagine this, two very tiny balls simultaneously falls into orbit around a black hole near the event horizon then both accelerates to near the speed of light at exactly the same time. Will they be maintaining their distance to each other or will one lag and the other catch up?

[@yoshtg]

there is no paradox. this video shows what happens when physicists spend to much time in theoretical physics. what we are asking is: "if an object contracts due to length contraction, will it cause stress to the ships fuselage?" thats what we are asking here because the line itself gets accelerated, too making both ships technically 1 ship with a very weak fuselage

[@bierrollerful]

The paradox arises from having *two* objects accelerating independently. We _falsely_ (hence the paradox) assume that a) because they are connected by a string and b) because they are accelerating at the same time & rate, that we can treat them as one object, and so we expect the string to not snap in the rockets' frame of reference. However, in their frame of reference, the rockets do _not_ accelerate at the same time & rate: the string then snaps as the distance between the rockets does not remain constant, resolving the paradox. Edit: Maybe it helps to think that the paradox isn't about the string, but rather about preserving distance between the rockets. The inertial observer sees the rockets contract, so the space and therefore the distance between them must be become "bigger" - whereas the accelerating rockets do not observe any contraction, so the distance should remain constant.

[@hillarysemails1615]

@@bierrollerful I agree with the 1st guy. If the string breaks, then the rockets must shatter into billions of pieces as they too are collections of many parts and also must undergo the acceleration stresses. The nose and the tail must be considered as separate Frames from the Observer.

[@erhard_gesagt]

I agree. And thx for not being alone. In fact I immediately saw why the strings break in the view of the space ships, but it wasn’t clear to me why it would break in the perspective of the observe. I’ve thought all behaves like one single object. I can now understand that it breaks if it really cannot stand any stress. For the space ships it is assumed, of course, that they can stand the stress.

[@erhard_gesagt]

Yet I wonder whether this is due to length contraction. Let’s assume it can stand some stress. Just that it survives the start. Will it break later? Will there be more stress? Observers perspective: ships are closer to each other, rope is shorter. However, as they move, they cannot maintain a path such that both ships are equidistant to the observe, which is crucial to this problem

[@aniksamiurrahman6365]

Imagine a college professor torturing students with this.

[@davidedrich6985]

I liked the “visual approximations” of the two other scientists. It’s like an Easter Egg.

[@ScienceAsylum]

😂

[@ludvercz]

I was just about to comment that.

[@lolerskates876]

Before watching it my take is, if you feel angular acceleration spinning in a circle then I want to change it to 2 people holding a string spinning around on a merry-go-round. As the wheel spins close to speed of light the people experience the effects of special relativity. But it seems like the circle wouldn't disform nonuniformly because the acceleration is the same everywhere. So I say string wouldn't break

[@gravitationalvelocity1905]

Interesting idea.

[@adamreynolds3863]

if the metal rod was as strong as the ships, would the two ships be considered one ship? and would that ship break apart? why would only the string or rod break and not the material that makes up the ships? i would argue that each ship is a mini "ship-string-ship" entity.

[@ScienceAsylum]

8:43

[@olmostgudinaf8100]

Conversely, wouldn't the entire system with the string be considered one big ship, no matter how thin the string is, including a string of zero thickness (also known as empty space)?

[@lamcho00]

Yes if the rod is strong enough, then it can be considered as single object. But your question was answered in the video. It would be like watching just one of the ships. The stationary observer would see the length of the rod and the ships contracting.

[@fewwiggle]

@@ScienceAsylum I believe that AR is asking what does it mean when the ships become one ship, IOW, what (if anything) is happening or trying to happen to the integrity of the spaceship?

[@Blox117]

because the string is weaker duh

[@BrianMelancon]

How is this scenario different from one in which the string goes from the front to the back of a single ship? Also it would seem to be the same as a single ship with one engine in front and one engine in back. In those cases one would think the string does not break - the system as a whole would contract. You seemed to briefly touch on it by mentioning the strong bar connecting the two. Where is the dividing line between three separate systems acting in unison and a single system with three parts?

[@theoj608]

I think the main issue is how he failed to define "same acceleration" if the 2 spaceships were in fact accelerating at the same exact speed, they would both be moving relative to the external observer, which is equivalent to the external observer accelerating while both ships stay stationary the observer would then observe a contraction of the ships yes, but also of the space between them regardless of contraction, saying the string must break when the ships accelerate would imply that the string would break between static ships when the observer moves, which is absurd the false assumption is that the space between the ships must stay the same to the external observer

[@jeffwells641]

This is my thought, he's treating the ships and string as three different frames of reference, but since they're all going the same speed shouldn't it be a single frame of reference? When the frame of reference contracts, shouldn't the ships shrink toward each other? I don't see how an arbitrary strength of connection is what makes the ships have a different frame of reference.

[@welrann]

It isn't but acceleration from one drive is moved through a ship by a ship material and it can't stand an accelerations in which relativistic effects take places. The drive will just move through the ship and flew away. Leaving the ship behind with a hole from relativistic drive in it. In absolutely hard indestructible ship string would snap.

[@Manuel-cx6ob]

@@welrann reaching relativistic speed does not require "relativistic" acceleration. You can reach values very close to the speed of light in a few months of simple 1 G acceleration. To answer the original question, if the string is attached to the start and end of the same spaceship, it will contract together with the spaceship because both ends of the string are accelerating at the same rate in the ship's reference frame, so there is no tension in the string (let's say the mass of the string is negligible). The acceleration being the same is enforced by the fact that the ship is a solid object. When you attach the string to two different spaceships, the string will experience a tension due to the two spaceships not accelerating at the same rate in their frames, meaning that the two ends of the string will accelerate at different rates, stretching the string. In the external inertial frame, where the two accelerations are the same, the tension is intead caused by the string contracting. If the two spaceships were correcting their thrust so that they always accelerate at the same rate in the moving frame, then the string would not snap and the inertial observer would see the string contracting while the two ships will be getting closer to each other to not break the string.

[@eklhaft4531]

@@jeffwells641 It's not the problem. The acceleration being the same in the inertial refference frame is the problem. If an object contracts progressively more and more then the front and back of it don't accelerate at the same rate. The observer accelerating with the object won't see any contraction but the inertial one does. If you have two identical rockets they should have the same acceleration for the co moving observer not for the inertial observer.

[@jussayinmipeece1069]

So with out watching the actual video here is my understanding. Two things accelerating one behind the other at the same rate of acceleration means that eventually the front object will be opening a bigger gap between them so snap goes the string

[@lamcho00]

This was a good one. My intuition really makes me think classically (Newtonian mechanics) and my first thought was the string won't snap. But after a while I came to the conclusion, if an object accelerating to relativistic speeds undergoes length contraction, if we are not seeing both spaceships coming closer to each other, then it means one of them accelerated first or is accelerating at a higher rate. So the string should definitely snap. If the string does not snap, then the stationary observer should see the string contract in length, like it's a stretched rubber band pulling both spacecraft closer as they accelerate.

[@enomoto-kudamono]

The video caused me to raise some questions: Will the spaceship break apart eventually during the acceleration? Or will the spaceships break apart in pieces of one atom thick slices if the EM force between atoms assumed to be infinitive weak?

[@Bela12351]

I have the same question. After this video I assume that the rocket breaks apart as well.

[@brothermine2292]

Nick answered this question when he discussed what would happen if the string is strong enough not to snap or is replaced by a metal rod. Spaceships don't break apart from the Lorentz contraction because spaceships are strong enough not to snap.

[@ScienceAsylum]

Whether or not your ship would survive would depend on the structural/material properties of the ship, where the thrusters are located, and how large the acceleration is.

[@brothermine2292]

@@ScienceAsylum : The temperature of the spaceship's o-rings could also affect whether it would survive.

[@abrahamvivas9540]

@@ScienceAsylum the real question is: the ship is at tension or at compression?. Every point in the ship drift away as in the non-inertial frame looks like for the rope... ergo the ship is in tension. Or the engines pushes the structure and the ship is in compression?

[@chrismcgarry3160]

6:05 - 8:05 Momentarily Comoving Inertial Frames : What an elegant tool for accelerating Frames! Very nice job on the graphs btw! I understood everything on first watch! Plus I'm a big fan of playing with SpaceTime diagrams and black holes, so there's that too!

[@ScienceAsylum]

Glad you liked it 🤓

[@pwinsider007]

​@@ScienceAsylumwhat force causes the snapping of string,length contraction is not a force and everything in the universe must be explained by four fundamental forces.which fundamental force snaps the string if length contraction is not a force ?

[@DavidPysnik]

This brought up an interesting question for me: When you observe length contraction, from where does the contraction take place? The object is going to shrink toward a line perpendicular to the direction of motion, but is that line located in the middle of the object, the front, the back, or elsewhere? Wherever it is, why is it there and not somewhere else? This mattered for me in the problem as the separate items (each ship and the rope) would distort differently if they were separate items that shrunk toward their own centers (or fronts, or backs, or elsewhere) than if they were one item with a common center they would shrink towards. This seems to reflect acceleration causing internal stresses on the items which would lead the rope to break, but could then break other things if the item is not built to handle the acceleration.

[@Manuel-cx6ob]

It depends on the constraints your are putting on your experiment. If you want that the inertial observer will keep on seeing the same separation between the ships (let's say they have to accelerate at the same rate and start at the same moment in the inertial frame), then they shrink separately around each own center of mass. It could not be any other way, because we are requiring them to accelerate at the same rate in that frame and start at the same moment from zero velocity. Their distance in the ships' frames will not be constant though (will be increasing, as shown in the video). If instead we require that they maintain the same distance in their own frames, then they will appear to get closer to each other in the inertial frame, which would look like a contraction around a middle point between the ships.

[@torgeirtheodorsen1301]

This is caused by the time light from this object takes to get to you. it only Looks like they are futher apart and so on. due to how an observer viewes the light comming from them. There is no actual change to the object in Question. Only how it looks to be for the observer.

[@Manuel-cx6ob]

@@torgeirtheodorsen1301 that is not correct. Length contraction does reduce the real length of an object, as measured in a certain frame of reference. If it were only an optical illusion, muons generated by cosmic rays wouldn't reach the earth, electromagnetism wouldn't work (a magnetic field generated by a current in a wire can turn into an electric field in a different frame of reference because of length contraction on the metal's lattice), just to name a few.

[@newtypealpha]

From nowhere. It's not the objects that contract, it's the literal value of all measurable lengths. Wavelengths are shorter, rulers are shorter, atoms are shorter. A perfect sphere appears to be an ellipse, etc. Not only would the spacecraft appear shorter but the distances between them also shortens. The effect appears instantly for every single object at that velocity.

[@Manuel-cx6ob]

@@newtypealpha the distance between them in the in inertial frame cannot reduce if we define that they have the same acceleration and they start from the same velocity at the same moment in the inertial frame. Did you watch the video? If the distance between them was reducing, it would mean that the string would not snap, and they would not have the same distance in the external observer rest frame. It would be as you said only if the setup of the experiment was that the two spaceships have to maintain the same distance in their own frame, which would imply they would not have the same distance in the external rest frame, so the distance between them would contract (i.e. they would move closer to eachother at the same rate as the string contracts).

[@skeltek7487]

I remember explaining this to people on my forum many years ago. Essentially it is a shift of the plane of simultaneousness during acceleration.

[@3obby]

I'm having trouble understanding this, why wouldn't any shift in space/time apply to every part of the accelerating body? I mean, it is all the same accelerating body, right? There should be no difference in behaviour between the string between the ships or any part of the ships.

[@skeltek7487]

@@3obby Accelerating and changing velocity changes the metric of the object or observer in comparison to everything else in the universe. It is like tilting the accelerated observer into one direction. One may imagine it as being an angle being orthogonal when standing still and being tilted in a direction when moving relatively. While from the observers view things happening at his front and back may be simultaneous, another relatvively moved person would consider them not happening at the same time. Is is similar to having two rockets start accelerating simultaneously with one of them being 1km in front of the other. From the perspective of a non-moving observer, their distances would remain the same, but from the perspective of the front rocket, the other rocket would have lower acceleration lagging behind(and increasing the distance). If they both start decelerating after a preset time of 1 minute, the front rocket would think the other having reached their max speed later and then decelerating faster. Such phenomena also apply within the same rocket.

[@3obby]

@@skeltek7487 but wouldn't that orthogonal angle be exactly the same between both bodies? If the two bodies start at a set distance apart, simultaneously accelerate, then simultaneously decelerate, their frame of reference for each other was fixed independent of the surrounding spacetime.

[@skeltek7487]

​@@3obby Their distance would stay the same in the eyes of a non-accelerated observer. From their own viewpoints though, that distance experiences a lorenz-contraction. Imagine an observer, who was already moving at the rockets maximum speed when they were still standing still: From that observers view, the rockets did not start simultaneously. From the very beginning, he was in the rockets final (accelerated) frame of reference. That is the frame of reference the rockets will arrive after the acceleration phase is finished. By accelerating, distances change (contraction or lengthening) and so does the plane of what is perceived simultaneous. There is another example, where rockets fly parallel: Before starting, they see each other next to each other (looking left and right 90° angle from flight direction). When they start accelerating, they will see each other slightly in front (89° for example), since the light travels a curve from their perspective and hits their eyes slightly from direction of travel. Accelerating shifts the angles and distances of the whole universe towards the direction of travel - or from a non-accelerated observers viewpoint: The universe stays the same while the accelerated object experiences being warped.

[@3obby]

@@skeltek7487 in your example of the two rockets beside each other, wouldn't they each seem to view themselves as slightly ahead, as the spacetime between them expands? Even though this is happening, the spacetime distortion is proportional to the speed, and I don't think that impacts matter that would be strung between the two crafts. The spacetime should simply become thinner, stretched out, right?

[@Smitology]

Also when you said the problem I immediately questioned the statement that the rockets see the other rocket the same distance away. If from an outside perspective they appear to have the same velocity at the same time, that doesn't translate to the rocket POV as seeing the other rocket being stationary at the same time because simultaneous events aren't absolute.

[@Pseudo___]

Why does the space between the ships not contract? Like if I just put a hypothetical box around the whole thing that accelerates with the ships…. The box would contract, right?

[@SyDatNguyen-r4j]

Because the string is weak, so they can’t be one object

[@Pseudo___]

@@SyDatNguyen-r4j Dont think you understood the question

[@narfwhals7843]

Because we set the condition that the ships remain a fixed distance apart in Charles' reference frame. It's like saying "set the box moving but make sure it stays the same size for Charles". Since the box moves and lengths contract, it would have to grow in its own reference frame to accomplish this.

[@Pseudo___]

@@narfwhals7843 "Because we set the condition that the ships remain a fixed distance apart" I must have missed this?? I believe the question was two space ships, exact acceleration and timing. Not that they had to have fixed distance. 3:23 he says the string cant contract.. Im saying the whole system contracts string ncluded

[@narfwhals7843]

@@Pseudo___ That is the definition we use for "equal acceleration". (I suppose it is only implied by the animation at the start, but that is the scenario of the "paradox") Because in relativity, we have to be careful with what we mean. We want the ships to start accelerating at the same time and at the same rate _according to Charles_ . Because that is the frame they all start out at rest with respect to each other. Only in this frame can all these conditions be true and we call them "equal acceleration". The string does contract. It just can't keep spanning the distance between the rockets, which is constant according to Charles, and if it's attached to them that means it can't keep connecting them. The rockets contract. The string contracts. The distance between the rockets contracts. But since we _demand_ by "equal acceleration" that the contracting distance between the rockets remains constant for Charles, it has to _grow_ in the rockets' frames.

[@KungKras]

What if the string is considered as being part of both spaceships? Or if you accelerate a long spaceship with the same length as the string? Why are the two spaceships and the string not considered the same system and contract as one single block? EDIT: Extra question, what implications does this have for objects that are longer than a planck length?

[@javiej]

it is explained in the video, if it is strong enough then the front ship is towing the back ship and they act as single object (no snap)

[@ScienceAsylum]

If the ships operate under their own thrust, then they act independently. You can't treat them as one object, even though they're connected by the string.

[@judgeomega]

@@ScienceAsylum so any ship with multiple thrusters needs to be considered multiple ships?

[@ScienceAsylum]

@@judgeomega In that case, I could imagine an acceleration that would tear that ship to pieces. Technically, yes, you should treat the one ship as multiple objects... but whether or not you could get away with still treating it as one ship would depend on the material properties of the ship (and where the thrusters are located).

[@judgeomega]

@@ScienceAsylum so for the very same reason the string would snap, any ship without just a point source engine would also snap. this seems like this is important information. if the string would snap there would be no ships left, right?

[@jdmayfield88]

The string wouldn't break, because all of it's space-time, along with both rockets, would be stretching all together. Since it's spacetime stretches along it's length, to it nothing changes. The string does not stretch. Space stretches around it. So, to the string, it's all the same.

[@gaberowe2]

Great video - i think an additional conclusion might be that while the Charles frame would be able to justify that the string breaks due to length contraction - which would be visible during both the acceleration and the constant velocity phase, the string would not break during the constant velocity phase for the moving rockets, but would during the accelerating phase (I think?).. so a 1km long string would break during the accelerating phase but a 1km long string would not break if strung between the two rockets after reaching a constant velocity

[@ScienceAsylum]

If the velocity were constant, then the length contraction is whatever it is. It won't increase. In that case, the question is "Can the string survive being stretched that specific amount?" If the rockets continuously accelerate, then it's likely you'll reach a "too much stretch" point.

[@KeirRice]

I would like to hear more about joining the ships with different materials. Where is the boundy between two ships with independently length contraction and one big ship that contracts as a single unit?

[@michaelcherokee8906]

I was wondering that myself.

[@Manuel-cx6ob]

The deciding factor is the strength of the material. In their frames, the ships will have different accelerations, so the two sides of the rope will have different acceleration, thus the rope will experience tension. If the material is not strong enough to sustain that tension, then the rope will snap. If the material is strong enough, then the rope will exert a force on the two ships so that their acceleration (and their distance) is forced to stay constant in their frame of reference. If their distance is constant in the ships reference, then it cannot be constant in the external observer reference. The external observer will see the rope pulling the spaceships closer to eachother by contracting, so in his reference the ships will not have the same acceleration. In all the frames the tension of the rope is the same, but the "explaination" for that tension is different.

[@darkracer1252]

nowhere. there could be nothing in between them and they would still be concidered one single "object" or better said. one single frame of reffrence.

[@darkracer1252]

@@Manuel-cx6ob their frames? I FUCKING HATE THE DUNNING CRUGER EFFECT

[@Manuel-cx6ob]

@@darkracer1252 what do you mean? "Their frames" -> "Their reference frames" Do you think they don't have reference frames? :D

[@joshandmomma]

Would this be the same outcome in completely empty space?

[@narfwhals7843]

Define completely empty. If there are ships and strings it isn't empty. Are you asking if this is due to curved spacetime? It is not. This is due to the acceleration of the ends of the string. Spacetime is completely flat for this scenario.

[@TheHumanHades]

"Now we're cooking with gas" I don't know why I thought it was a bad joke but I laughed hard😂. I hope of understanding relativity completely after my NEET exam day after tomorrow. Thank you for making me understand many physics concepts in these 2.5 years of my preparation for the exam😀.

[@SR-se6ii]

All the best for the results.....🤗

[@hairlessape5107]

I laughed out loud at that too. Most of his jokes would be categorized as 'bad' by many (even though they laugh) but this was particularly good as both of him laughed afterward and the timing was perfect.

[@hairlessape5107]

[Edited to add] And yes, good luck for the exam. [Edited to add] Lol at myself, nice editing.

[@jensphiliphohmann1876]

I didn't really know this idiom before. However, the humor in it is quite a lot darker to me since I'm German...

[@Blox117]

i dont get it, the meaning of the phrase is not a joke

[@SaniFakhouri]

Here's an interesting question. Assuming a similar setup, and assuming the string between the ships is extremely fragile to the smallest of tensions, is there any situation of rocket accelerations (aside from the obvious a_A = a_B = 0 m/s^2) that results in the the string not snapping at all?

[@erumaaro6060]

I think so, if the difference in acceleration counterbalances the speed at which the distance changes due to length contraction. We know that length contract to 0 at the speed of light, so the two curves in the space-time diagram (from clone C's perspective!) must touch at infinity (both time and space) (=>asymptotic), meaning the front rocket must accelerate slower than the back, and acceleration cannot be constant for both, unless the distance is zero. So the difference in acceleration needed is a function of time (#canOfWorms) and the initial length of the string.

[@AlexBesogonov]

Not with constant acceleration. If ships can change the acceleration, then yes.

[@jameelarosetafoya2058]

NO

[@lamcho00]

Yes if both rockets accelerate at the same time in their frame of reference. Then the stationary observer would see the string contract in length and the rockets come closer to each other. It would be the same as accelerating a metal rod to relativistic speeds (or just one of the rockets). Length contraction will be observed, because light (and information in general) travel at the speed of light until it reaches the stationary observer. Even at the speed of light it takes time for light to reach the observer.

[@mitymi]

Good question. I'd like to see the plots of that in comparison. I think it would make it more clear that the setup of the problem is from the 3rd person perspective. Maybe even add 4th moving observer to really drive home the different cases...

[@mattpietz5179]

A great video, but I don't get why A sees B lagging behind and B sees A getting farther ahead. I guess the explanation is referenced quickly with the world lines diagram, but it doesn't seem to immediately clear to me what the real - world reason is.

[@feynstein1004]

This thought experiment has the same vibe as "when you heat a metal plate with a hole, does the hole expand or contract?" 😂

[@ScienceAsylum]

I loved discussing that metal plate example back when I was teaching in the classroom.

[@dj1NM3]

I know from experience that if the metal "plate" is only a few times bigger than the hole (ie: a typical bushing, pulley or gear), then the hole will get bigger because its not constrained, as the outside perimeter will also get bigger. If the plate is effectively an infinite plane (eg: a 12mm hole drilled in the hull of a container ship) and the heated metal is constrained by the rest of metal around it, then the most likely thing to happen is the hole to get a little smaller and the plate to buckle and pucker to give the expanded metal somewhere to go, the extent of both dependent on how much of the plate around the hole is heated.

[@jameelarosetafoya2058]

So what's the answer? Do the electrons closest to heat source move faster and expand. Is it quick cook or slow roast? Is it iron or copper? Ok, what if we take a basic element like Hydrogen to its plasma state, then with electric charge and Lazer cooling immediately harder said plasma in the shape of a plate wit ah hole n it, could I still eat a donut off it when places in the center and thrown at light speed 2 my buddy?

[@feynstein1004]

@@jameelarosetafoya2058 I don't know 😅

[@givrally]

New question : If, in both moving frames of reference, Bernard is lagging behind, couldn't we design an experiment in which Bernard doesn't have the same acceleration rate, but instead the acceleration needed to make the distance stay the same in Albert's point of view ?

[@ooloncolluphid9975]

excellent question! the string should still snap (since Charlie will observe the distance between the ships increases), but now Arthur and Bernard do not have an explanation for why it snapped.

[@narfwhals7843]

@@ooloncolluphid9975 This is not accurate. If Bernard accelerates in such a way as to keep the distance constant in the rockets frames of reference then Charles sees that distance shrink just as the string contracts. The string remains taut and unbroken for all observers. Charles will now, however, not see the ships accelerate equally.

[@BartvandenDonk]

@@narfwhals7843 The string will always snap. It is like the bang while using a wip. Or like stretching when getting near a black hole 🕳️. The forces are so immense and different behind the first rocket and in front of the second rocket. The rope will change and get the shape like a wip and break.

[@BartvandenDonk]

The acceleration itself should accelerate. This is (aside that reaching speed of light is impossible) completely out of reach.

[@BartvandenDonk]

@@ooloncolluphid9975 Charlie is not important here. 😏

[@juliavixen176]

Ok, it's been a year since these comments were written, but nobody has mentioned Rindler Coordinates or Born Rigid Body Motion... ok, so here's the thing about proper acceleration of rigid bodies... the material gets compressed if you push on one side of it. In an inertial reference frame everything is _weightless_ Change in velocity is acceleration, change in linear momentum is force. When a spaceship starts throwing reaction mass out of it's engine, the engine changes velocity relative to the rest of the spaceship... this "acceleration" propgates as a "force" through the metal (or whatever) at the speed of sound in the material. This compression wave is either higher than the elastic electrostatic forces in the material and the spaceship gets crushed into a pancake, or the material pushed itself back apart with *_force_* i.e. acceleration. Stuff inside the spaceship "falls down" towards the engine. The reason why any rigid object is feeling a *force* is because one side of the object is moving at a different velocity relative to the other side, and so the two sides of the object are either moving away from each other or moving towards each other. Inside one spaceship, the engine end, on the left in this diagram in the video, is moving right at a greater velocity than the front end of the rocket (where the string is attached). If this was the only engine accelerating, it would compress the string and then compress the right spaceship, pushing it to the right just like everything else. Everything would reach an equilibrium feeling the same compression force and the "same" time-delayed acceleration... Because proper acceleration is only meaningful between two parts of a material object. It's only "absolute" because it's local to the material itself, not the rest of the universe. If you have two rocket engines, both throwing exhaust out, and changing velocity relative to the exhaust i.e. accelerating. Rather than compression between the two engines, there is a tension force pulling them apart from each other. In this video, the string is the part feeling a streching force... because the two ends of it are moving at different velocities and the left side is slower than the right side. If you have one long spaceship with only one single engine in the middle. It will push on the front of the spaceship and *_pull_* on the back of the spaceship. If the spaceship is made from metal with enough tensile strength, it can change the velocity of the back end to catch up with the engine in the middle. If there's too much acceleration (force) from the engine then the back end of the spaceship will break off, just like as if it was held on by string. I you _don't_ want to break this string in this video, the rear (left side) engine must apply more thrust (force) than the engine of the right side spaceship. So... yes, there's time dilation between the ends of each spaceship, and each other spaceship, and everything in front of the spaceships and behind the spaceships. In fact, very far to the left side of this diagram is an event horizon, beyond which no light will ever reach these spaceships for as long as they are experiencing proper acceleration. This is also how gravity works, but the acceleration isn't the compression forces within the rigid object itself. (It's the inertial coordinate systems.)

[@olmostgudinaf8100]

But wait, Charles can see a _single system,_ consisting of two spaceships and a string. The _entire system_ is accelerating as a unit, therefore the length contraction applies to the _entire system,_ including the string. So it does not snap.

[@ScienceAsylum]

The ships have their own thrust and the string is too weak to affect anything, so they act independently. They can't be one object.

[@olmostgudinaf8100]

@@ScienceAsylum Not physically. But they move as one object, so to the outside, or even inside, observer, they _appear_ as one.

[@huepix]

But wouldn't the space between the rockets contract? I mean, the two ships and the string are all part of the same "space" arent they? Does this mean that at a certain length, rockets will split in half, then quarters eights etc as the get closer to SOL?

[@ralphwishart]

Absolutely! The rope wouldn't snap at all just like in the similar example of a stationary observer watching a train accelerate by. the train will contract but the wagons will never come apart. if both objects accelerates in unison, it must be regarded as a system and not as individual bodies with discreet paths the space-time lines will be equivalent.

[@Frankly7]

@@ralphwishart The train scenario is not the same. The links are not fragile like the string is between the rockets, and so the train is strong enough to stay together. The problem is that you are treating the accelerating objects and the space around them as the same thing, when I fact what they do is completely opposite to each other. Any section of train will contract relative to the space around it (i.e. space expands), and any section of space will contract relative to the train (i.e. the train expands). The only reason the parts stay together is because they are strong enough to do so, which is exactly the reason why the spaceship paradox here uses a fragile string, so that we can ask what happens when the accelerating matter does in fact break apart.

[@Frankly7]

The space doesn't contract, the matter within it does. See my above comment.

[@olmostgudinaf8100]

@@Frankly7 I disagree. Relativistic equations do not care about _what_ moves, only how fast. An empty space contracts too, if it moves fast enough relative to the observer - or vice versa. There is even a well known example: a photon moves through empty space at speed of light. From the photon's perspective, space and time contract to zero length. The photon observes leaving the star an reaching your retina millions of light years away as one event that happens at the same time and place.

[@Frankly7]

@@olmostgudinaf8100 So I messed up the example in my explanation originally but it's edited now, and my point still stands. The problem is you are in fact the one treating the object as a discrete whole, whereas I'm pointing out that it's a collection of continuous matter. Any section of matter you take will want to contract, pulling it away from the sections around it. So uniformly the matter of the train will experience outward forces parallel to the direction of acceleration. If you claim a weak rope won't snap, you are implying that the ships move toward each other in the acceleration frame, but they have no reason to do that unless they are only edges of a discrete object and not simply collections of matter that can form countless configurations of objects.

[@HD-fy2wu]

Many people are asking why couldn't we treat the whole system as a body. From my understanding, length contration happens in any scale, so each and every molecule is contracting. Even if you focus on just one single ship, it does not contract as a whole, but every point on the ship is contracting. So if the ship is weak, for example it is made up of toilet paper, then when each and every point of the ship contracts, it will be torn apart due to the extra tention pulling the head and the tail of the ship towards each other (since all the material in between them are contracting). The reason why the ships in the video can contract as a whole is that they are made up of stronger material, so the head and the tail of the ship can be pulled towards each other when the material in between them contracts. That's why you can't treat the system shown in the video as one body. You can only do so if the string connecting the ships is changed to a rigid rod that doesn't break.

[@AnthonyFlack]

The length contraction occurs only in the perception of an outside observer. The ship doesn't experience being squashed. It's a core principle of relativity that it won't feel anything.

[@HD-fy2wu]

@@AnthonyFlack What I've said is indeed in the perception of an outside observer. The ship made of toilet paper would be torn apart because of contraction in the perception of an outside observer. If you want the perception of the ship itself, the ship will be torn apart because different parts of the ship are accelerating at a different rate in the perspection of each molecule of the toilet paper that makes up the ship, which is similar to what's explained in the video.

[@thedeemon]

Love those "visual approximations" of scientists! Great episode, although how exactly length contraction leads to rope snapping may still raise questions. ("is rope's length reduced or is it the distance between ships reduced? then the rope may get too long for it instead of snapping...")

[@jasonremy1627]

Under what conditions would the string not break? That is, how would each ship need to accelerate so that in all reference frames, the string remains taut but unbroken?

[@apirx]

The front rocket would have to accelerate slower than the rear one.

[@jasonremy1627]

Yes, but how much slower? What's the relationship?

[@Manuel-cx6ob]

@@jasonremy1627 the distance between the two spaceships if they didn't have a rope connecting them would be the initial distance D multiplied by the lorentz factor: d = D (1/sqrt(1-v2/c2)) If you find the rate of change of the rate of change of this distance, that would be the acceleration that the ends of the string would be subjected to. Then if you know the masses of the ships, you can compute the tension on the string, to know when it will break.

[@ShellShocks14]

8:44 holy moly this has bothered me for a long time, thanks for clearing it up. As i watch these i pause and think, and just before the timestamp i mentioned i paused and worked out in my head this issue and im so happy to see you explain why we’re using a string specifically. Also, this thought problem reminds me of the observers in the train/lightning thought experiment. It seems like ultimately, the string breaks for the same reason the observer in the train sees the bolts strike in succession (in the direction if the train) but the outside observer sees both bolts strike at the same time. W video good job!!

[@ScienceAsylum]

Thanks!

[@ghost9199]

I love how you make logical sense out of relativity. So many people think it's a mystical thing but it's not and your graphs show it. I love your videos 👍

[@Nulley0]

Imagine this way, Let the string length be 1 light second The forward ship accelerates and pulls the string, and backward ship compresses the string (if the string stays in a straight line), the tension caused due to forward ship accelerating, travels as waves to the other end of string and it takes time, depending on the amount of stress the string can put through and the amount of acceleration, the string can break I think, All observers notice the string broken for the same reason, even if they don't agree on when the string got broken

[@flexeos]

to add to my confusion, the scenario is that the 2 space ships accelerate the same, which means that they start to accelerate at the same moment. But what does at the same moment means? it has a different meaning for the 3 observers. What you describe is the case where Charles equally distant from Artur and Bernard is the one defining the start for both space ships. If Bernard sees the 2 ships starting at the same time his speed will always be lower than Arthur so the rope snaps, but if Arthur sees he 2 ships starting at the same time his speed will always be lower than Bernard?

[@david_porthouse]

When you do your analysis of the elastic string, you need to adopt a framework where all parts of the string have the same time co-ordinate. A line of constant time will be rotating as the rockets accelerate. It’s only a paradox if you don’t know that.

[@ninehundreddollarluxuryyac5958]

The rotation part is subtle. I don't think anyone got that part except you.

[@shawng4896]

I don’t understand how the unbreakable rod connecting the ships changes things. If the rear ship is being towed it is still accelerating at the same rate as the forward ship, just the force causing the acceleration has changed. Fundamentally what does it mean for the 2 ships to be “no longer operating independently?” If the 2 ships were on a giant space barge everything else stays the same, and the space barge started accelerating would the string snap?

[@omargoodman2999]

The string isn't a physical string; it is a metaphor for relative distance between the two ships or, more abstractly, two discrete points. Think of it as one of those math problems where you assume friction and wind resistance aren't real and cows are spheres and mass doesn't matter. Even in the context of a single rocket accelerating, there will be internal stresses on the materials; rockets aren't made of solid "rocket", they're made of mechanically connected parts which are made of bonded molecules and atoms. For the sake of simplicity, we consider it as being a single lump of solid, contiguous "rocketanium" and internal stresses be damned. Likewise, the "string" is an abstract representation for how, for each accelerating ship, their acceleration will appear different, thus the distance between the two changes, but for the stationary observer the distance remains the same and it's the lengths of the ships that change. If there were a single long rocket with propulsion near the front, again made of solid "rocketanium" so we can disregard internal stress, then what happens depends on the specific properties of "rocketanium". If it is cohesive enough to overcome even relativistic effects and allow the entire rocket, front to back, to move at a consistent acceleration as observed by a passenger, then a stationary external observer would actually see the back moving *faster* to catch up with with the front. In other words, it would be impossible for the front and back end of the ship to appear to accelerate at the same rate *because* the entire ship is contracting in length as a single unit; just the "contraction" is a result of the back apparantly accelerating faster than the front. Of course, this would require "rocketanium" to be able to allow all points along the length of the rocket to accelerate simultaneously relative to one another which would require violating causality as force would have to propogate at infinite speed. On the other hand, if "rocketanium" *can't* keep the ship in pace with itself, then a passenger riding at the front of the rocket would see the entire thing stretch out as the back end lags gradually behind the front end. If there were also a string woven of Metaphex fibers, a synthetic polymer made of pure metaphor, stretched between any two points along the length of the rocket (behind the pointnof propulsion), then the string would break at some arbitrary point because of the lengthening of the rocket. And, for an outside observer, they would also see the rocket stretch out and the string snap because the front end actually appeared to have started moving before the back end would have due to differences in light/causality propogation. A fundamental issue with the setup of the model is the assumption that an external observer is even *capable* of seeing both ships start accelerating simultaneously. If the observer sees this, then it means that the further-away ship actually started accelerating first and it just took time for the light to reach the observer, meaning *of course the string snaps.* But if the launch were carefully timed in such a way that they actually *did* start moving at the same moment, the external observer would not observe this; he would first observe the closer ship start moving, followed by the further ship start moving. Thus, there is never any breaking tension in the string in the first place. But if the further ship started moving first, the tension would propagate through the string at the speed of causality (c) and, for the sake of example, let's say it snaps exactly in the middle. The observer would first see a ripple of tension start at the closer ship and propagate up the string towards the farther ship. Then, the string would snap in the middle. Last, the two ships would *appear* to launch at the same time, just as the tension ripple finally appears to reach the further ship.

[@EpicMathTime]

The ships are accelerating at the same rate in the wrong reference frame. It isn't moving like a single ship.

[@brunocardin4935]

There's an extra force if there's a rod

[@tomaszkisiel9915]

Many years ago, when I was studying astrophysics, our natural philosophy professor asked us this exact question. We couldn't agree on the answer and he never explained it. Up until now, I wasn't sure of the answer but your explanation makes total sense. Thanks so much! :D

[@Martin-kn1cn]

If you want to finally get closure you can look at this paper from two Japanese scientists about this exact problem. Just search the following on google and click on the first link: Takuya Matsuda and Atsuya Kinoshita “A Paradox of Two Space Ships in Special Relativity” AAPPS Bulletin February 2004

[@alaridkonz]

what about, inside one ship you have a piece of string connected to two pipes within the ship no need for another ship if that ship starts accelerating, will the string snap or loosen? because I saw in the video that the ship as a whole contracts, while the space between ships would increment. So I ask now what happens within one ship?

[@snowthemegaabsol6819]

Inside the ship, the string remains intact

[@darkracer1252]

the space between the ship also contracts. because they are accelerating at the exact same acceleration they can be treated as a single object (along with the string and the space in between them)

[@Manuel-cx6ob]

@@darkracer1252 they only have the same acceleration from the point of view of the external observer. In special relativity, three-acceleration is relative and not necessarily the same as proper acceleration.

[@darkracer1252]

@@Manuel-cx6ob no he said they had the same acceleration. he said nothing about it being the same acceleration from an outside perspective. the distance between the 2 ships would ****SEEM**** to shrink along with the string. that is what would happen. nothing else. (well the string would burn from the rocket exhaust but that's not part of the thaught experiment)

[@Manuel-cx6ob]

@@darkracer1252 he said that they have the same acceleration when they start (and they are at rest with the external observer). And he said that they stay at the same distance in the external observer's frame (which is at rest with the frame where the ships started from). As they accelerate, they cannot possibly mantain the same distance, if we define that they have to maintain the same distance in the external reference. To keep the same distance, they would have to tweak their engines to adjust their relative acceleration (or be connected by a strong string that pulls them together), to account for the fact that they did not start at the same time in their current instantaneously comoving inertial frames.

[@joeturn4130]

Can you take this to the next level and figure out what the acceleration differential would have to be for the sting not to snap between the two ships?

[@eigenchris]

Science Asylum mentioned that an accelerating spaceship travels on a hyperbolic worldline on a spacetime diagram. You can draw a series of accelerating-ship hyperbolas on the spacetime diagram, with each one being a bit "less curved" (less accelerated) than the one behind it. You can do this in just the right way to make sure strings tied between all the ships never break. This series of hyperbolas are called "Rindler coordinates" (shown very quickly at 5:37 in the video). Wikipedia gives the equation for each of these hyperbolas with given a starting position "x": the formula is x*cosh(αt) where "α" is the acceleration of the first ship at the back. Source: en.wikipedia.org/wiki/Rindler_coordinates

[@joeturn4130]

@@eigenchris Thanks for the info, but that doesn't answer the question. Sorry to say I am not a super Mathematician who could calculate things on that level. From the video I did gather either one ship would need to travel a certain percentage faster than the other for the string not to break, or if the were to excelerate at the exact same speed one ship would need to start a certain fractional time frame before the other. Just wondering what that difference would have to be.

[@ScienceAsylum]

@@joeturn4130 If you know the material properties of the "string," then sure you can. How rigid is it? What's it's breaking limit? That sort of thing.

[@ToxicityAssured]

@@ScienceAsylum Why don't you go do an in depth analysis of string, collecting is material properties and how rigid it is. You can then "do that sort of thing". I disagree with your conclusion and have been reading comments for an hour trying to see it your way. This one pathetic attempt and sounding smart sends me right back to thinking you tried to understand a cool topic for the views. Seriously, read his question and your reply. You sound like Pelosi being asked if she should be able to trade specific stocks.

[@joeturn4130]

@@ScienceAsylum Thanks, but I would think that there would be a solution to the problem where the material properties of the string would have no bearing. I would imagine that there would be a perfect offset with either speed, time or both where the stress on the string would be zero then the properties wouldn't matter. After that the properties of the string would only have an effect in widening those variables to give enough play to make it possible without there having to be perfection to make it happen.

[@laxminarayanbhandari855]

I remember reading about this paradox on Physics Stack Exchange. Also, Sciencephile the AI had also made a video about special relativistic paradoxes in which he mentioned this. Nice video.

[@BartvandenDonk]

Can you place a link to that one?

[@arabidllama]

The "string" represents how we think of distance, when really distance is entirely a property having to do with the travelling of light. The "strong metal rod" version is connecting the ships via electromagnetism - aka light!

[@petrowi]

a weak string is also electromagnetically bound

[@davidhand9721]

This would seem really straightforward to me. The two ships and the string are all in one reference frame. They don't length contract relative to one another. If you're in a "stationary" frame, then why wouldn't the string contract right along with everything else in its frame, but that's not a physical strain any more than squishing the metal ships is. I guess that's just when they're already travelling, not accelerating, though.

[@krshna77]

That is exactly right. The OP (TSA) is deeply confused in his reference frame. Keeping the same distance as measured from Charlie's pov is entirely artificial. Side note, a real string would absolutely break as it's being pulled to near _c_ (tensile limitation, inertia etc) but the string is simply a material representation of the distance.

[@darkracer1252]

no. accelerating or not, it's the same. they are one single frame of reffrence. because the rules of the thought experiment says they are. they velocity is at all times exactly the same, those are the rules.

[@nocare]

I will start by saying the following explanation relies on accepting that in the two seperate ship scenarios when they are not attached, the lead ship sees the rear ship falling behind. -Now if the second ship accelerates harder it can maintain its distance relative to the first ship from the perspective of the first ship. So we know greater acceleration from behind maintains the two ships as "one object" from the perspective of the lead ship. -Well if you replace the second ship with a long pole behind the first ship and each atom in that pole is like the second ship. The space between those atoms is the space between the first and second ship. The acceleration/thrust of all these atoms is the bonds to the atoms in front of them instead of an engine. -These atoms in the pole are maintained as "one object" because there bonds allow them to accelerate faster than the engine and spaceship itself. Once the acceleration required to keep pace with the ship exceeds the strength of the atomic bonds the end of the rod breaks off. From an outside observer this looks like the rods stretching as the rocket length contracts but the rear of the rod fails to keep up and upon failing to length contract to stay with the ship breaks off.

[@darkracer1252]

@@nocare come the fuck on don't you people just freaking understand it? THERE IS NO ACTUAL STRETCHING INVOLVED. that's all part of the (for lack of a better word) illusion that the outside observer sees. the rear ship doesn't have to accelerate faster then the front one. because they both have the same frame of reffrence because their distance is close. and they are moving in the exact same direction at the exact same speed. for them. there is ZERO contraction of space going on. the other ship wouldn't even move visually because by their own frame of reffrence, everything else is moving and they are standing still. for the outside observer. it would seem like they are one SINGLE entity. your explaination of the atoms also further brings home my point. because the idiots here seem to be so sure that the rear ship would lag behind because it's not attached. well NOTHING IS ATTACHED TO ANYTHING. (except for inside a blackhole or a nutron star where atoms are actually touching) the only reason things SEEM to stretch or contract for an outside observer is because of an illusion. not just optical, their speed introduces time dilation and their time is moving at a diffrent speed then our time. for us they change size. and for them WE change size.

[@sohamtapse]

My answer is Let the back rocket be A and front rocket be B Now as shown in the graph of both rocket be hyperboles and they are identical to observer. Let the observer be Q Now if the rocket A has constant increase in speed So that the graph of A is more curved towards B then it will not break the string apart length contracts. As length contracts the rocket A will increase the speed and will match up the contraction. Thuse paradox is solved ... In my opinion (Soham Tapse)

[@rogerahier4750]

it does not snap. Moving near c doesn't actually make the rockets shorter, it only makes it look like they are shorter. With the string attached, they become 1 object, so the ships look shorter, the string looks shorter, so the entier object is shorter meaning the space between them is (looks) shorter.

[@nocare]

I will start by saying the following explanation relies on accepting that in the two seperate ship scenarios when they are not attached, the lead ship sees the rear ship falling behind. -Now if the second ship accelerates harder it can maintain its distance relative to the first ship from the perspective of the first ship. So we know greater acceleration from behind maintains the two ships as "one object" from the perspective of the lead ship. -Well if you replace the second ship with a long pole behind the first ship and each atom in that pole is like the second ship. The space between those atoms is the space between the first and second ship. The acceleration/thrust of all these atoms is the bonds to the atoms in front of them instead of an engine. -These atoms in the pole are maintained as "one object" because there bonds allow them to accelerate faster than the engine and spaceship itself. Once the acceleration required to keep pace with the ship exceeds the strength of the atomic bonds the end of the rod breaks off. From an outside observer this looks like the rods stretching as the rocket length contracts but the rear of the rod fails to keep up and upon failing to length contract to stay with the ship breaks off.

[@rogerahier4750]

@@nocare If that were true, the ships would also split for the very same reason. They don't actually change size. They only look like it.

[@nocare]

​@@rogerahier4750 Yes at a high enough fraction of light speed any ship will break apart. I said nothing about the ship actually changing size. Edit: Also the ship is actually shorter from the perspective of an inertial frame. That's how relativity works each reference frame is correct about what they see because all interactions with world are governed by the speed of light. That's how the ladder and the barn are able to work where the length contracted ladder fits inside and both doors close simultaneously but at rest the ladder is longer than the barn and so sticks out both sides. The entire premise of relativity of simultaneity is that viewers can disagree on the timing of events and both are actually in reality correct. Since the waves responsible for physical interactions also propagate at the speed of light a bending of the visual representation of an object also must bend all the waves propagating the physical interactions.

[@rogerahier4750]

@@nocare It's not actually shorter, it only appears shorter. The reason it looks shorter is because of time dialation. It doesn't change the actual dimensions of an object, only how it appears. Since there is no difference in the relative speed of the 2 ships, they will be in the same time frame so they will notice no difference in each other. Relativity is just what it says. It works on the relative speed of an object in question. If there is no difference, there is no effect.

[@SyDatNguyen-r4j]

@@rogerahier4750It’s NOT an illusion

[@religiouslyscientific4371]

No paradox. It is a connected system like an assembly, so, length contraction happens on the whole. If your paradox is correct then rocket engine should contract not the rocket body because engine is accelerating body is just connected to it.

[@williamdowling7718]

The 2 ships and the string can be thought of as one "observer" or reference frame. It wouldn't make sense that the two ships would contract at their center of mass independently of string that is moving equally as fast. All 3 bodies together should experience the same contraction together, not each individually. The whole "system" of the ships and string are moving together..

[@SpectralRedshift]

Lenght contraction also affects the space between the spaceships. For an outside observer it would look like the spaceships were getting shorter AND moving closer together. For anyone travelling with the spaceships there is no change in length or distance observable.

[@gravitationalvelocity1905]

The distance between which two points on the ships? Front, center, back? Of the ships just get shorter around the center then string from center to center should not break.

[@phillipcoetzer8186]

I visualized it like this The view is like tilting an image ... the whole image shrinks so no snap. Also take into account if those rocket people measure the rope they'll find no change in length.

[@narfwhals7843]

But the whole reflection _doesn't_ shrink here. It is a condition of the experiment that the distance between the ships remains constant in the frame of the "stationary" observer.

[@phillipcoetzer8186]

@@narfwhals7843 I realised that error in thinking and edited it ... thx

[@phillipcoetzer8186]

@narfwhals7843 the stationary observer sees the shrinkage not the travellers

[@narfwhals7843]

@@phillipcoetzer8186 The stationary observer sees the length of the rope shrink because it moves. But by definition of the experimental setup, he sees the distance between the rockets to be constant. The shrinking rope can not span the constant distance. So it snaps. The travelers see the rope have constant length, but since the stationary observer sees the distance between them as constant(again, by definition of the experiment) it must _grow_ in their own reference frames. The constant rope can not span the growing distance. So it snaps. Two rockets who are seen to have "constant acceleration" in some inertial reference frame can not have the same acceleration in their own frames. They must drift apart.

[@misteratoz]

I lost you when you said that the rockets ships see each other accelerate at different speeds despite being the same speed.

[@juliavixen176]

Accelerating reference frames do weird stuff... mostly everything in the forward direction will appear to be speed up, blue shifted, and the speed of light will appear to propagate faster with increasing distance ahead. (This is a measurement thing in the accelerating frame itself, the rest of the universe is not actually doing anything. ) Everything behind the accelerating frame will appear red-shifted, clocks run slower, and light slows down. At a far enough distance behind, there is an event horizon beyond which light will never catch up to the accelerating frame for as long as it is accelerating. The instant proper acceleration stops, all of these effects will stop.

[@bluestarwars]

Interesting. My thought was that the string itself undergoes length contraction, so if (to Charles) distance between spaceships is constant, sting snaps due to contracting … Consider this: in the final example, the string is replaced by a strong enough bar to make the chips a single object - that object, bar included,used will length contract for Charles. By extension, so it will as we make it weaker and weaker. Ignoring the spaceships, we have a string that accelerates at near C speeds - and a stationary observers will see it contract.

[@Iohannis42]

It only appears to contract. It will never snap, because the distance between the ships does not change. Otherwise, by the same thinking, both ships and the string would disintegrate as every atom is torn apart.

[@narfwhals7843]

"the distance between the ships does not change" Exactly. But the rope is shorter. If it just looks shorter, will there be a gap? No. Length contraction has real, physical consequences. The rope length contracts, but the distance remains constant. It can't span that distance so it snaps. The difference to the individual ships is that the rope is individually accelerated at each end. That is not the case for each ship. Each ship is accelerated at one point and the rest is pushed by a compression wave that moves through the ship at the speed of sound. The ships do not tear themselves apart.

[@martinhofman80]

If you consider the rope to be something unbreakable it would create a unit that is equivalent to one solid ship. What happens then? A ship breaks apart? In other words, if you apply this chain of thought on any very 'left' and very 'right' point of one ship, it would eventually break apart. Which doesn't happen, so my conclusion is that the rope (moving also near the speed od light) doesn't snap, the whole system shrinks and the distance between the ships too.

[@juliavixen176]

It will tow the spaceship on the left-side. If the difference in acceleration is great enough it can pull the left-side spaceship apart, but this will happen whether or not the left-side spaceship has its engine on or not. (The spaceship gets pulled more than it gets pushed.) Assuming that the spaceship engines don't apply enough thrust to crush the rest of the spaceship into a flat pancake, it's probably not going to pull the towed spaceship apart.

[@deathpyre]

I feel like you could say that the STRING also has a different frame of reference. The string itself will have a frame of reference spanning the two ships, which will also change the size of the string along the length, maybe making it staying connected

[@Martin-kn1cn]

I don’t think you can say that the string has its own frame of reference here, because it has no propulsion of its own which is important here

[@daviddavids2884]

8:12 WHAT.?!? horshett relative to one another, the two ships are Stationary, the WHOLE time.!!!!

[@hughshaw6109]

So, all the great explanations about space-time, and what catches my eye? The fact that the closed captions at 4:30 read "Besides, this is the Asylum."

[@brianhiles8164]

_“The string breaks because of [Lorentz] length contraction.“_ This is not the way spatial (and time) contraction works, due to relativistic velocities approaching the speed of light! To say this affects the spaceships without affecting the string is like saying that a magnifying glass can magnify in its view one thing without affecting the view of anything else in the same image. The third “viewer“ is not a third spaceship, but the string itself. How about this consideration: It is my informal assertion that since the string is a material object of _any_ tensile strength (or length, by the way), it still has to endure the accelerations imposed upon it by the leading spaceship pulling it against the inertia it will always have. The “string“ can never be so light _or_ so strong (as addressed by Dr. Lucid) that there will _ever_ be one that will not eventually break -- _even though technically the two spaceships stay exactly the same distance apart from each other._ So what do you think?

[@snowthemegaabsol6819]

We could address that consideration by defining the rate of acceleration for both rockets as an infinitesimal, reducing the physical effects caused by a change in motion to 0. If the string were to still break, the only force that could have acted upon it is the relativistic tension. The string is also its own 4th observer. We can have as many as we want, but don't need all of them to understand why this happens at all.

[@juliavixen176]

The spaceships don't stay the exact same distance apart from each other; The distance between each spaceship increases with time in their own reference frames... They're _not_ in an inertial reference frame. Acceleration is litterally defined as anything not inertial (constant velocity in a straight line). You need to use a Rindler Coordinate transformation, not a Lorentz transformation.

[@kendrickmcelfish2805]

This didn't go as I was expecting, but it made plenty of sense. I was very suspicious when you said the rockets accelerated "at the same time." Because I have been exposed to these relativity paradoxes before, I was expecting the observers to disagree about the order of the events. Also, how strong is the pull due to the space stretching? Like how tough of a string do we need here?

[@punitsolanki5744]

Doubt - I think we don't jump into general relativity when we shift to accelerating reference's frame(also called as rindler coordinates) there could be curved axis but it's not means that spacetime is also cuvred because riemann curvature tensor is still be zero. Right !

[@henriedelbauer3955]

It's amazing, how linked, conceptually interwoven, Lorentz-contraction [LC] & relativity of simultaneity [RoS] are! Both together give rise to a lot of seemingly paradox gedanken experiments within the frame of STR. Like hyperbolic twins. Two circumstances are mind-doping: 1) LC (inverse gamma-multiplication) is an effect of 2nd order; RoS - not to confuse with time dilatation - of 1st order. 2) There are robustly physical & apparent asymmetrical consequences, hiding in the depths of - at first glance - simple and evident M4-formalism. The principle of special relativity seems not at all to induce just a change of perpective! Greetings from Vienna Woods. My comment, I guess, would come a year to late - if time were more absolute:-)

[@ryuzakilawlight]

Start: I understand perfectly. Middle: BLACK 🕳️ End: aah I understand the conclusion perfectly! This is my 🎢 every time I watch your videos. And I keep coming back.

[@tylerhamlin166]

Nice VFR sectional bookmark! It’s rare to see pilots show such restraint ;)

[@SiqueScarface]

My first intuition was, that the second spaceship is not only accelerating from its booster, it is additionally pulled by the first spaceship, and that the string itself also has to accelerate, so it would experience a force, causing it to snap. Apparently, I was non-relativistically right. Special Relativity to me was just some fancy setting for a well known plot.

[@WernerBeroux]

7:56 Now I finally understand why I always feel the bicycle in front of me seems to go faster than me.

[@ManyHeavens42]

I noticed throughout history These debates have done one thing slowed progress down, even to a Holt,Everytime, a closed mind" should mean close the door on your way out 👁️