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

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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?

make a video on ALCUBIERRE DRIVE

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.”

@@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.

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

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.

@@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?

@@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.

@@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 . . .

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

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

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

@@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.

@@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 .

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.

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.

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

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

@@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.

I want to know too

exactly my doubts

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

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.

@@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.

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,

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

i mean 1 object

@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?

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.

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

@@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?

@@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.

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

@@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.

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.

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.

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?

@@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. 🤔

@@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.

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.

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

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

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

@@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.

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.

@@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.

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

@@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)

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?

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.

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

@@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.

@@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!!

@@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.

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.

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.

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

@@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.

@@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.

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.

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.

@ 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.

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.

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

@@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.

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

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.

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

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.

@@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.

@@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.

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

@@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.

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.

@@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.

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

@@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.

@@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.

8:43

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)?

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.

@@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?

because the string is weaker duh

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

@@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.

where? on roblox?

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

😂

I was just about to comment that.

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.

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

NO

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.

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...

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.

@@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.

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.

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

Imagine a college professor torturing students with this.

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.

@@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.

I was wondering that myself.

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.

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.

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

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

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

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.

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.

@@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.

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.

@@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).

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

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

@@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.

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

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.

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.

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

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.

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.

@@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.

@@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.

​@@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.

@@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?

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.

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

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

Inside the ship, the string remains intact

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)

@@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.

@@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)

@@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.

a weak string is also electromagnetically bound

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.

@@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.

@@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.

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

@@ooloncolluphid9975 Charlie is not important here. 😏

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

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.

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.

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.

@@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.

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.

@@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.

​@@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.

@@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.

@@rogerahier4750It’s NOT an illusion

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.

Thanks!

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

It doesn't matter whether the spaceship is a single object or not. It matters whether there are several points of acceleration that we insist remain a constant distance apart in Charles' frame of reference. You can encase the whole thing in a hull of titanium. If the condition is that two points remain at a certain distance for Charles, then the hull between them will break.

Interesting idea.

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.

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.

😂