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Special Relativity Paradoxes
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0:00
That Einstein guy had some pretty wacky ideas. Black holes, gravitational waves, he was even the first to realize that friggin lasers could be a thing. But that all came later. Way back in 1905, when Einstein was just getting started - he was already rocking our understanding of the universe with his special theory of relativity. The theory started with the simple assumption that the speed of light was the fastest speed
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possible, and that all observers should measure the same speed of light, regardless of their velocities. But from that can the inevitable conclusion that space and time themselves were relative - depended on the velocity of the observer. The result? A cascade of seeming paradoxes.
0:43
Today we’re going to delve into a couple of the most famous paradoxes of special relativity and see why, against our intuition, the universe really does work in this seemingly nonsensical way. But the point of this episode is to go much further - we’re going to try to break the universe by pushing these paradoxes beyond the limit. Let’s lay some relativistic groundwork. Say we have a spaceship traveling from Earth to a nearby star at a good fraction of the speed of light.
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Special relativity tells us that the clocks on the spaceship will appear to tick more slowly from the point of view of a stationary observer back on the Earth. This is called time dilation. And the spaceship would also appear squished in the direction of its motion in what we call length contraction. But in relativity, every non-accelerating frame of reference should be considered equal. The spaceship can think of itself as stationary - it perceives the Earth as racing away from
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it and its destination star racing towards it. That means it sees clocks back on the Earth ticking more slowly, and the Earth and the distance traveled being squished. These seeming contradictions only become paradoxes if the different observers - on the spaceship and on Earth - can compare the results of an experiment and get unresolvable conflicts. For example, there IS a disagreement between the astronaut and an observer back on Earth
1:57
about the relative passage of time and the distance traveled - but those aren’t paradoxes because both agree about when the astronaut arrives at their destination, and how much they age on that journey. The observer on Earth thinks the astronaut’s clock ticked slow, but the astronaut thinks they traveled a shorter distance - and the two conspire to give the same age for the astronaut on arrival. Let me give you an example of a trickier apparent paradox.
2:23
Imagine our spaceship can travel very, very close to the speed of light. So close that the distance across the entire universe contracts to a size smaller than the spaceship itself. Let’s also imagine that we live in a closed universe - one that loops back on itself, so by traveling far enough you can get back to where you started. In the frame of a spaceship moving at near light-speed, the universe could contract to
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the point that the spaceship wraps all the way around and its nose smashes into its tail, presumably puncturing the matter-antimatter tanks and destroying the ship. But in the frame of a stationary observer, the ship is just moving ridiculously fast, and doesn’t destroy itself. That would be an unresolvable conflict in the observations - a paradox. This paradox was actually presented to me by one of our Patreon supporters - and it stumped me for a while. But I think I figured it out.
3:13
However the resolution requires us to think about two much more famous paradoxes in relativity - the twin paradox and the ladder paradox. Let’s start with the twin paradox. One of a pair of twin hops in a spaceship and travels at a good fraction of the speed of light to a nearby star and then turns around and heads back to Earth. According to the twin back on Earth, his sister’s clock has been ticking slower than his own.
3:36
But according to the traveling twin, Earth appears to have raced away and then raced back again. And during that time she sees her brother’s clock ticking slower. So if both perceive the other’s clocks ticking slow, who has experienced less time by the time they reunite? The answer is - the astronaut has aged less. The resolution to the paradox lies in the fact that the traveling twin hasn’t been in a single moving frame of reference, but rather in two separate moving frames
4:05
- one moving away and one moving towards the Earth. The best way to see this is on a space-time diagram. We have time on the y-axis and just one dimension of space on the x. Let’s say we’re in Earth’s reference frame so the Earth doesn’t move in space - just straight up, which means forward in time. The spaceship moves in both space and time - first away from the earth and then back towards it. One of the consequences of special relativity is that different observers give different
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accounts of what events are simultaneous. On the spacetime diagram, the set of simultaneous events for a motionless observer lie on a horizontal line - all events corresponding to your notion of a given tick on the time axis. But someone moving relative to you has a totally different sense of “now”. Their lines of simultaneous events are tilted. For the details on why this is the case, check out this episode. We can use these lines of simultaneity to solve the twin paradox because they allow
5:01
us to track the apparent passage of time back on Earth from the point of view of the astronaut. She counts every time a new years day happens on Earth according to her calculations. On the spacetime diagram, that’s whenever one of these lines of constant time extending from January 1st on earth crosses the spaceship’s path - its worldline. But those lines are tilted due to that motion, and they tilt one way on the outward journey and the opposite way on the return.
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In total, the traveler counts fewer Earth years because she misses some time in the middle corresponding to the turn-around point. Those years do happen back on earth, but they don’t correspond to any time point that happens to the astronaut. Ergo, the twin returns younger. Our ultimate question is about a spaceship crashing into itself in a closed universe - and trust me we’re getting there. But baby steps. Let’s first look at the twin paradox in a closed universe.
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In that case, the traveling twin would not need to turn around in order to get back to Earth to compare ages. She could keep traveling in a straight line all the way around the universe. And if she travels in a straight line she sticks to a single reference frame. Both twins still perceive the other’s clock to be ticking slower - so which twin is older when they reunite? Again, it’s the traveling twin. But to see why we need a much weirder version of the spacetime diagram.
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We’re still just doing one dimension of space, but now that dimension loops back on itself. We end up with a spacetime cylinder. The earth-bound twin moves straight up as usual, but the traveling twin now does a single loop of a helix to intercept her brother’s upward path. A line of constant time for the former is just a circle around the cylinder. But for the traveling twin “now” takes on much weirdness. Again, we tilt the line, but on a cylinder that leads to a line of simultaneous events
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that spirals up and down the cylinder as a helix. The traveling twin declares that multiple points in the stationary twin’s past and future are all happening in her present. So by traveling around the universe, she’s actually traveling towards one of those future versions. And those future versions are older - by just enough to agree with the time dilation that the stationary twin would calculate. The traveler returns younger than the stay-at-home twin. There appears to be a little issue here.
7:25
It seems that in this closed universe there IS a fundamental difference between frames of reference. There’s a stationary frame of reference where lines of constant time are closed loops, while in every other frame they are endless helices. Doesn’t that violate the fundamental tenet of relativity? Actually no. Relativity tells us that there’s not a preferred LOCAL frame of reference. But the universe as a whole can have a special frame.
7:54
In the case of the closed universe, that closed topology DOES pick out a special frame - it’s the one with these closed constant time loops. Our own universe also has a special frame - whether it’s closed or infinite. That’s the frame of reference in which the cosmic microwave background appears still - or un-Doppler-shifted. Ok, enough with the twin paradox. We need one more paradox before we can wrap this up. Let’s zoom in from the whole universe to just a simple barn and a simple ladder.
8:22
This is the famous ladder paradox. Our ladder is slightly longer than our barn, so if we move the ladder through the barn at least one end always sticks out. Remember, length contraction causes moving objects to appear foreshortened in the direction of motion. So if you move the ladder through the barn at a significant fraction of the speed of light, then to someone standing still relative to the barn the ladder may be length-contracted enough to fit in the barn.
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We’ll also open and close the barn doors, timed so that the ladder has just enough time to pass through each. At some point, the ladder appears to be inside the barn with both doors closed. Let’s switch to the point of view of someone moving with the ladder. They perceive the barn to be moving so the barn is shortened. If the ladder didn’t fit in the barn to start with, it most certainly doesn’t fit now. One observer says the ladder fits, the other says no.
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The solution to the ladder paradox also relies on the relativity of simultaneity. In this case, the different observers disagree on when the ends of the ladder enter and exit the barn. In the frame of the barn, there’s this single instant when the base of the ladder has just entered the barn and the front of the ladder is just about to exit, but the door hasn’t opened yet. But in the ladder’s frame, those two events are NOT simultaneous.
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The ladder’s observer perceives the front of the ladder exiting the barn before the base enters. Again, the best way to see this is on a spacetime diagram. This is in the barn’s frame. The doors of the barn are a certain separation apart in space - solid lines in the diagram. We’ll open gaps to indicate the period when the doors are open. The ladder also has a certain length in the spatial direction - initially longer than
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the barn, but when the ladder starts moving it shortens so that it neatly fits inside the barn before emerging again. But the ladder sees things differently. The ladder’s sense of “simultaneous” is represented by these sloped lines. Those lines are the x-axis in the ladder’s frame of reference, which means the ladder sees its present state as this line. So the ladder sees itself moving through the barn like this. At no point is it entirely inside the barn.
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So does the ladder fit in the barn or not? Yes AND no. The answer is entirely relative. OK - we’re finally at the point where we can answer the money question. Does a near-light-speed spaceship smash into it’s own rear-end in a closed universe? This is a variant of the ladder paradox, but here the doors of the barn map to each other, pacman style. What happens to a ladder in a pacman barn, or a spaceship in a closed universe, when
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the barn or universe are length contracted to be smaller than the ladder or spaceship? The answer is in our looped spacetime diagram. Remember that the line of constant time is a helix. So that’s also the line on which we lay down points along the length of our spaceship. That’s the “current” spaceship as it sees itself. And look - the spaceship DOES cross this point in space - our seam - and sort of exists there simultaneously.
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But there’s no collision, because the nose of the spaceship exists in the future, and the tail in the past. They exist at the same point in space at different points in time. And no matter how fast the spaceship travels, it will only elongate along this helix - safe from collision even if it wraps around the universe multiple times. Long story short -near-light-speed spaceships in closed universes or ladders in pacman barns are safe from colliding with their own asses.
12:01
Relativity is weird, but its seeming paradoxes always have resolutions. You just need to follow the logic and the paradoxes evaporate, and so we make sense of this deeply strange, but unfailingly self-consistent theory of Einstein’s spacetime. I want to thank two of our Patreon supporters - Hank and Mark - who actually asked the very similar questions regarding length-contracting closed geometries that inspired this episode - guys, i hope this helped clarify matters a bit.
12:28
And in general thanks to all of our Patreon supporters - for the support and for the really active engagement, especially on our discord channel. But today's extra special thank you goes to Clinton Robinson who's supporting us at the big bang level. Clinton, we took your contributions and accelerated them to near the speed of light, causing them to wrap around the universe several times and multiply. Then we bought bitcoin with the money and forgot the password - but we learned a lot
12:52
in the process! Thanks you, we couldn't engage in so much beautiful frivolous science without you. See you at pineapple motel. And speaking of crypto, today's comment responses are for two episodes - how cryptochromes - the protein not the browser bitcoin wallet - may give birds quantum magnetovision. And also our episode on what happens during quantum jumps. We'll start with the latter. Regarding the experiment which showed that quantum jumps could be predicted, tracked,
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and even reversed in an artificial atom. David Robinson and Drakinite asked the same question: how can we be sure that the artificial atom is really a useful analogy to a real atom. So the answer I think is that we don't yet know - but it's important to point out that the system used in the experiment IS a genuine quantum system. The energy levels are represented by a very small number of photons in a cavity - 0 to 5 - so quite quantum.
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Transitions involve quantum tunneling in a superconducting circuit - so entirely quantum. But I take your point - can we really be sure that the same underlying complexity exists in real atoms? This experiment should be taken as motivation to look deeper, rather than a final answer. Gerry R asks the most insightful possible question: Could it be that asking which interpretation of quantum mechanics is "true" is like asking if light is made up of particles or if light is a wave?
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The more we probe the fundamentals of reality the more we realize that the stories we tell to make sense of the math are just models to help our intuition, and that strikingly different-seeming interpretations map to the same theoretical frameworks. This doesn't mean the models and the stories and the interpretations are meaningless because they guide us deeper - but I don't think anyone has given a satisfactory answer as to WHY contrary-seeming interpretations can be simultaneously "correct".
14:58
This could be one of the weirdest things about the universe. And a question from Mishel from Hamburg, Jörn, will you marry me? ... hmm, welll I know what my answer would be, but I don't think I'm actually qualified to answer this one. And on to quantum magneto reception in birds. Actually, I have research update here. Right after the release of that episode some researchers from the university of Tokyo reported on a new experiment that helped validate this theory.
15:23
They managed to get a culture of cells containing cryptochromes to autoflouresce -to glow - in a way that changed depending on the direction and strength of a very weak magnetic field. This was done with human cells, but the mechanism is exactly that proposed one for bird magnetoreception. This is the first time the flourescence has been directly observed, so it lends credence to the theory. Does it also mean that humans could develop this superpower?
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Maybe, but my cellphone has a compass, and I choose to think of that as cyborg magnetoreception, which is just as cool. I wish I had more questions on bird magnetoreception - but there were more bird puns than bird questions in the comments. I don'thave time to go through all of them, so I'll just give Matteo Alessandro's summary: Even though all these bird puns just fly over my head, toucan play at this game! It'd try to wing in with my own put, but
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instead I'll swallow my pride and instead congratulate your impeckable and emu-sing humor. You all quack me up.
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