Twins Paradox Explained: Special Relativity Solution

[00:00:03]This episode was supported by Brilliant.org Hey Crazies.

[00:00:07]It’s time to tackle my favorite paradox of all time.

[00:00:10]The Twins Paradox!

[00:00:12]Unfortunately, I don’t have any twins, even though everyone keeps asking.

[00:00:15]But I do have clones.

[00:00:17]I’d like to introduce you to Rocket Clone.

[00:00:19]Hi!!

[00:00:20]He’s super useful for this sort of thing.

[00:00:22]To infinity and beyond!

[00:00:24]Yeah yeah, whatever. Just get in the rocket.

[00:00:26]Ok, let’s do this!

[00:00:27]Rocket Clone zips off at 55 percent the speed of light to, oh, I don’t know.

[00:00:32]Let’s say Wolf 359, which is a modest 8 light years away.

[00:00:36]By the way, for those of you who don’t know, a light year is not a time.

[00:00:40]It’s a speed multiplied by a time.

[00:00:42]A light year is a distance.

[00:00:44]Anyway, Rocket Clone gets to Wolf 359 and immediately turns around.

[00:00:48]When he finally gets back to the station, 29 years have gone by for me, but only 24 years have gone by for him.

[00:00:55]If you’re familiar with relativity, this isn’t that surprising.

[00:00:59]Rule #2 for Relativity says: Every observer carries their own set of coordinates and their own clock.

[00:01:06]I get a clock.

[00:01:07]You get a clock.

[00:01:08]Everybody gets a clock!!!

[00:01:11]Different observers traveling different paths through space-time are going to measure different amounts of time.

[00:01:16]There’s no paradox there.

[00:01:17]A quick way to calculate the difference is by using something called the gamma factor.

[00:01:22]It’s a number that shows up all over relativity, so get used to seeing it.

[00:01:26]Using that factor, I expected Rocket Clone to be only 24 years older, even though I’m 29 years older.

[00:01:33]There’s no surprise on my end.

[00:01:35]Rocket Clone is the one that’s surprised.

[00:01:37]Rule #4 says: It will always look to an observer as though they are stationary in space.

[00:01:43]As far as Rocket Clone is concerned, he stayed in one place and I’m the one that moved.

[00:01:48]We also have Rule #1: No point of view has preference over any other point of view.

[00:01:53]Rocket Clone is the one with the paradox.

[00:01:56]He aged 24 years and if he does the same calculation I did, he expects to find me only 20 years older, not 29 years older.

[00:02:05]He expects to be the older one.

[00:02:07]Paradox!!

[00:02:08]Ahhhhh!!!!!!!!

[00:02:12]Huh... Not sure why that happened.

[00:02:14]I’ve said it before and I’ll say it again.

[00:02:16]Paradoxes are not a problem of the universe.

[00:02:19]They’re a problem of the human mind.

[00:02:21]The universe doesn’t have a problem.

[00:02:23]We do.

[00:02:24]When it comes to paradoxes, there are only two ways we could have made a mistake: Option One!

[00:02:29]We’ve made an assumption we shouldn’t have made.

[00:02:31]Meaning one of the rules we used isn’t actually a rule.

[00:02:34]It’s difficult to argue with Rule #4 since perception is pretty self-centered and Rule #2 can’t be the problem since that’s what relativity is all about.

[00:02:42]That leaves only Rule #1, so let’s take a closer look at that.

[00:02:46]Throwing out rule #1 is a very common way people try to solve this paradox, but that unravels the entirety of relativity.

[00:02:54]I mean, do you think I would have spent an entire video giving you a list of rules for relativity that could break in relativity?

[00:03:00]Don’t be ridiculous!

[00:03:02]It’s rule #1 for crying out loud!

[00:03:04]Anyway, here’s how the typical explanation goes: Since Rocket Clone accelerates, he feels a force toward the back of his rocket.

[00:03:11]A force of uniform gravity to be specific.

[00:03:13]This breaks the symmetry between his clock and mine somehow making mine the correct clock.

[00:03:19]But here’s the problem with that solution.

[00:03:21]There never was any symmetry between him and me to begin with.

[00:03:24]Like we said earlier: Different observers traveling different paths through space-time are going to measure different amounts of time.

[00:03:30]And that’s not just true of time.

[00:03:32]It’s also true of distance, length, and a whole list of other measurements.

[00:03:37]Having one of the observers accelerate just adds another measurement to the list of things they disagree on.

[00:03:42]Forces are now on the list.

[00:03:43]That’s not that significant of a change.

[00:03:45]It certainly doesn’t make one person right and another person wrong.

[00:03:49]It just makes them more relative.

[00:03:51]But don’t accelerated points of view require general relativity?

[00:03:54]Yes, but general relativity fixed that by strengthening rule #1, not by throwing it away.

[00:04:01]Rule #1 holds true in general relativity just as much as it does in special relativity.

[00:04:06]The fact that Rocket Clone accelerates isn’t the problem.

[00:04:10]Then what other mistake could we have made?

[00:04:11]I’m so glad you asked!

[00:04:13]Option Two!

[00:04:14]We didn’t consider something we should have considered.

[00:04:16]I feel like there’s something missing here.

[00:04:20]1 2 4 Oh! It’s Rule #3. We forgot Rule #3.

[00:04:27]We have to use the correct coordinate transformation.

[00:04:30]Remember this quick calculation Rocket Clone did?

[00:04:33]It’s wrong.

[00:04:34]And this quick calculation I did?

[00:04:35]Also wrong!

[00:04:37]But didn’t you get the right answer?

[00:04:39]Yeah, but that’s, well, OK, maybe “wrong” was a bit harsh.

[00:04:42]Let’s go with “sloppy”?

[00:04:44]When we do this, we get no perspective on what we’re calculating.

[00:04:49]The number Rocket Clone calculated is valid.

[00:04:52]It’s just not the answer to the question he was asking.

[00:04:55]You’re not always calculating what you think you’re calculating.

[00:04:58]It’s safer to use the full transformations or, better yet, written in matrix form because you transform both space and time together.

[00:05:06]It gives you a better idea of what you’re looking at.

[00:05:08]But you know what works even better?

[00:05:10]A space-time diagram!

[00:05:11]If you’re not familiar with those, you should go watch this video first.

[00:05:14]It’ll also help with all these rules I’ve been giving you.

[00:05:17]So, in the Twins Paradox, the two paths look like this.

[00:05:21]Rocket Clone accelerates away from the space station reaches 55 percent the speed of light and stops accelerating.

[00:05:28]He eventually gets to Wolf 359, turns around, and the whole thing happens in reverse.

[00:05:33]The straight paths are constant velocity.

[00:05:36]The curved paths are the accelerations.

[00:05:38]So, yes, there are three sections where Rocket Clone accelerates.

[00:05:43]But like I said before: The fact that Rocket Clone accelerates isn’t the problem.

[00:05:48]This paradox does not require accelerations.

[00:05:51]It persists even if Rocket Clone goes the same velocity the whole time.

[00:05:56]These are the lengths of the two space-time paths: 29 years and 24 years If we’re going to remove all the accelerations though, then we can’t even let Rocket Clone turn around.

[00:06:07]That means these are the numbers we expect to see: 14.5 years and 12 years because we’re only looking at half the trip.

[00:06:14]OK, now let’s look at the simplified diagram.

[00:06:17]Say Rocket Clone is already going at 55% the speed of light when he flies past my space station.

[00:06:22]Sometime later, he passes Wolf 359.

[00:06:26]He just keeps going because no accelerations.

[00:06:28]I say Rocket Clone’s path is about 14 and half years tall, but he says it’s about 12 years tall.

[00:06:35]Exactly half the time differences from earlier because this is half the trip.

[00:06:39]So this is totally consistent with everything we got before.

[00:06:42]The paradox arose when tried to look the other way.

[00:06:45]Starting from Rocket Clone’s coordinates, we can look back at me and Wait a minute!!

[00:06:50]We’re not looking at Rocket Clone’s path anymore.

[00:06:52]We’re looking at mine.

[00:06:53]That’s a completely different space-time path.

[00:06:56]Hmm.

[00:06:57]Rocket Clone’s path measured in Rocket Clone’s coordinates is 12 years tall, but that only includes a portion of my path.

[00:07:05]A portion that’s about 10 years tall in my coordinates.

[00:07:08]Exactly half the time difference from earlier because this is half the trip.

[00:07:12]But that doesn’t actually say anything about my path through space-time.

[00:07:15]It’s only saying things about his path, which isn’t the question he was asking.

[00:07:19]The event of Rocket Clone passing Wolf 359 isn’t on my path.

[00:07:24]It’s on his.

[00:07:25]To line that up with the event at the end of my path, Rocket Clone has to wait over 5 more years because my entire path is almost 17 and half years tall in his coordinates.

[00:07:37]These "horizontal" gridlines are lines of equal time.

[00:07:41]That 17 and half years is the number Rocket Clone should have used to calculate my time using that sloppy calculation from earlier.

[00:07:48]Except, he wouldn’t have known to do that.

[00:07:51]That extra 5 years doesn’t even exist in the full Twins Paradox because, by then Rocket Clone would have turned around.

[00:07:57]He wouldn’t even be on that path anymore.

[00:07:59]So, here’s the real problem.

[00:08:01]Both of us are trying to calculate the location and time of an event on someone else’s path in someone else’s coordinates.

[00:08:08]There are four different measurements here.

[00:08:11]Two different paths measured in two different coordinates.

[00:08:14]Everyone agrees if those measurements are at the same place and the same time because we just take the measurement together.

[00:08:21]But, if they’re at different places, we have a problem.

[00:08:23]At distant locations, those calculations are just predictions, not measurements.

[00:08:29]Each of us has to take a measurement of ourselves, maybe one selfie every day, and then send it to the other person.

[00:08:35]And that’s a whole other can of worms that I’d rather not open.

[00:08:38]My point is this: The acceleration doesn’t make one of time measurements incorrect.

[00:08:42]It’s just that, without the acceleration, we only get one shared event.

[00:08:47]The acceleration gives us a second shared event.

[00:08:50]If we don’t want to deal with sending selfies to each other we need two shared events to measure the passage of time.

[00:08:56]One at the beginning and one at the end.

[00:08:59]We can’t agree on any other events we might want to include, so we can’t compare measurements between them.

[00:09:04]The acceleration doesn’t give anyone priority.

[00:09:07]It just fixes a measurement problem.

[00:09:09]You don’t have to throw out rule #1 if you do rule #3 correctly.

[00:09:13]The Twins Paradox disappears entirely if you remember all the rules of relativity.

[00:09:19]So, are you satisfied with this solution?

[00:09:21]Please discuss in the comments.

[00:09:22]Thanks for liking and sharing this video.

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[00:09:29]And until next time, remember, it’s OK to be a little crazy.

[00:09:33]Still feeling like you need some practice?

[00:09:35]Then check out Brilliant.org.

[00:09:36]Brilliant is a problem solving website designed to train you to think like a scientist on a wide variety of topics.

[00:09:42]If you liked this video, then you’ll probably like their course on relativity.

[00:09:46]Anyone who is new to relativity might want to check out “Reference Frames and Observers.” It gives you that traditional introduction using Lorentz Transformations.

[00:09:54]You know, these equations here.

[00:09:56]Once you feel comfortable, you can check out “Space-time and Paradoxes,” where you hone your skills using space-time diagrams.

[00:10:02]That’s where the true power of relativity lies.

[00:10:04]They even have a quiz on the Twins Paradox.

[00:10:07]They do talk about acceleration in there, but we’ll forgive them.

[00:10:10]Brilliant was nice enough to sponsor this video and crazy enough to offer you a discount.

[00:10:14]The first 72 people to visit brilliant.org/ScienceAsylum will get 20 percent off an annual premium subscription.

[00:10:20]It’s a really good offer if you’re interested in getting some extra practice.

[00:10:24]Several of you were disappointed I didn’t do a space-time diagram for the two ships moving away from each other.

[00:10:29]There’s a reason for that.

[00:10:30]You can calculate relative speeds using angles in a space-time diagram, but they’re hyperbolic angles, which means you need hyperbolic trig.

[00:10:38]That’s a kind of math I wasn’t interested in explaining that video.

[00:10:41]Anyway, thanks for watching!