In this video, you’re going to delve into a couple of the most famous paradoxes of special relativity: the Twin Paradox, The Ladder Paradox (aka the Barn-Pole Paradox), and a paradox suggested by our very own viewers, which asks whether a spaceship could wrap around the universe & destroy itself. You’ll explore these paradoxes 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 – you’re going to try to break the universe by pushing these paradoxes beyond the limit.
Transcript:
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 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 – depending on the observer’s velocity. The result? A cascade of seeming paradoxes.
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. 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 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 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 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. 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, and you can get back to where you started.
In the frame of a spaceship moving at near light speed, the universe could contract to 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. 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 pair of twins 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. 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 – one moving away and 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 accounts of what events are simultaneous.
On the space-time 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 us to track the apparent passage of time back on Earth from the astronaut’s point of view.
She counts every time a new years day happens on Earth according to her calculations. On the space-time 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.
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. 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 space-time diagram.
We’re still just doing one dimension of space, but now that dimension loops back on itself. We end up with a space-time 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 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. 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. 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. 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. 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, and the other says no. The solution to the ladder paradox also relies on the relativity of simultaneity.
In this case, the different observers disagree 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. 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 space-time 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 the barn, but it shortens to fit inside the barn before emerging again neatly when the ladder starts moving. But the ladder sees things differently. These sloped lines represent the ladder’s sense of “simultaneous.” 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. 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 its 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 the barn or universe is length contracted to be smaller than the ladder or spaceship? The answer is in our looped space-time 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. 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 asses. 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 space-time. I want to thank two of our Patreon supporters – Hank and Mark – who asked very similar questions regarding length-contracting closed geometries that inspired this episode – guys,
I hope this helped clarify matters a bit.
