Relativity Paradox - Sixty Symbols

so I'm gonna talk about gamma the Lorentz factor which we have already done the video about but there's enough to say about it that I thought we'd do a kind of gamma reloaded video and have another go at it gamma reloaded yeah gamma is a factor that comes up in special relativity it's it's kind of appears in a lot of equations in special relativity and one of the places it occurs in this thing is this thing called time dilation you see something going past you so something that's traveling relative to you the clocks in that reference frame seem to be going slow relative to your reference frame so that time you know the clocks tick more slowly and everything goes more slowly biological process go more slowly radioactive decay goes more slowly and so on so really time really is slowed down when something's moving relative to you okay so in our last video about this we talked about one of the proofs that this actually occurs which is that there are these particles called muons created when high-energy particles crash into the top of the atmosphere and the muons are then detected down on the earth and the reason why this is a bit of a surprise is because muons don't live very long they have very short decay times and actually if you just figure it out you know even traveling at the speed of light the fastest light speed they could travel at the muons don't actually have enough time to make it to the earth so most of the muons should have decayed before they get to the earth but actually if you put a muon detector on the earth you find plenty of them are getting through and the reason is is because these muons are traveling at high speeds close to the speed of light that means that as measured from our reference frame their clocks are going more slowly which means that they don't decay so quickly then and so they have actually got time to get all the way down to the earth without decay as measured in our reference frame suddenly their lifetimes become maybe ten or a hundred times longer than they were and the factor by which they become longer is actually this factor this gamma factor that we're talking about so as I say there was lots of other phenomena that are sort of associated with this factor of gamma and one that we can get out quite easily is by thinking about what happens from the nuan's point of view right so from a muons point of view it comes into existence it suddenly born somewhere at the top of the atmosphere and it sees the earth rushing up to meet it okay so it's just sitting there and sees the earth coming towards it because that's you know that's relativity that's what it's all about and the question is well in its own reference frame it's just going to live as long as it usually does what there's nothing special going on as far as this muons concerned it's just sitting there stationary and so you know if it looks as if watch everything you know times ago in the normal kind of rate and so it's going to live for the normal lifetime that anyone's going to live for and so in its reference frame if it had the same distance to travel it wouldn't live long enough you know the earth rushing up towards it wouldn't it wouldn't have the wouldn't be time for the earth to travel at fifty or hundred kilometres from from where it was when it was first created to smash into it and so there's got to be some other effect that's going on that from the muons perspective actually allows it to reach the earth because it does reach the earth we can detect them on the earth and so the muon has to also agree that it reaches the earth and so this other factor is this thing called Lorentz contraction that as well as time being affected by motion distances are affected by motion as well and so when something's moving along the line of sight in the direction it's moving distances get squashed so instead of it having to travel 50 kilometers from being created down to the earth it might only have to travel a few hundred meters because in its reference frame it sees that distance greatly compressed and so all the distances are squashed in that direction so as far as its concerned the Earth's atmosphere is only about 50 meters thick and it only has to travel that 50 meters so it actually has time to get all the way from the top of the atmosphere to the earth because even though it's clocks going at the normal kind of speed it sees all the distances squashed remember it's actually only contracted in the direction that the muons moving so actually everything gets really distorted so your tree has the normal width but it's suddenly a few millimeters high maybe so actually ya know you things could look very very weird from your perspective when you're traveling at close to the speed of light because all these distances get contracted but as a famous thought experiment involving a train in a tunnel the idea is you've got a train and you've got a tunnel which is exactly the same length as the Train and so if you're a train spotto standing there beside the railway watching the trains go by you see this train approaching and because this trains moving relative to you you see it Lorentz contracted it's a bit shorter than it should be and that means that as it goes into the tunnel it disappears completely in the tunnel because it's shorter than that unlike the tunnels not moving relative to you you're standing next to it so therefore the tunnel is its normal length but because this train is moving relative to you it becomes a little bit shorter so it disappears into the tunnel fine so it actually is briefly kind of hidden at both ends of the tunnel but now if we think about somebody sitting on the trains perspective somebody sitting on the train they think they're stationary you know the table there in front of them they can drink their cup of coffee when they put it down it just stays there so they were in their reference frame everything's normal and they're seeing the countryside rushing by and so they see this tunnel approaching them at high speed and of course because they're seeing this tunnel moving the tunnel should be contracted and so from their perspective they should never completely disappear into the tunnel because they see a tunnel which is shorter than the Train so therefore the Train never completely disappears by the time the back end of the trains disappeared in the tunnel the front ends come out the other entrance and that's all well and good and okay maybe they just agree to differ about whether the Train is shorter than the tunnel or not but unfortunately you can then do another experiment which is you can say okay so if this train spotter actually is you know has a rather a lot of power and bizarrely someone has installed two large guillotines one or either end of the tunnel and at the moment the Train disappears into the toll he pulls a lever which briefly makes these guillotines come down and then go back up again then from his perspective that's fine because the trains disappeared into the tunnel you can do that without doing any damage okay from the point of view of somebody approaching the tunnel if this happens it's not going to be such good news because they're from their perspective the Train is longer than the tunnel which means either the front of the train gets chopped off or the back of the Train gets chopped off or both okay and this is something that everyone has to agree on like this if you happy doing experiment either you end up with a smash train or you don't and ultimately that's going to be something that everyone has to agree about it's not something that you can just say well that's just a matter of perspective that's all just relative right even the train gets broken or it doesn't so what happens so there is a resolution to this paradox and the resolution to this paradox turns out to be something very fundamental in in relativity and it's called the relativity of simultaneity which basically means which things which are simultaneous in one reference frame and not necessarily simultaneous in another reference frame so in fact both of these people agree that the Train doesn't get smashed but the reason why they agree the Train doesn't get smashed he's different right the guy who's pulling the lever says well the Train obviously didn't get smashed because it was shorter than the tunnel so that was fine the guy in the Train see something completely different in that what he says is that Ashley wore hands is the train was approaching before it went right through the tunnel the front guillotine came down and went back up again then after it had pretty much passed through the back guillotine came down and back up again and so actually they kind of missed the Train as it went through and and so this is what both of them see they both agree that actually the Train is fine but the reason is that because things which was simultaneous in one reference frame the guy was just standing next to the tunnel said these things happen both at the same time the guy was in the Train said actually no one happened first end the other and so that's you know that's the resolution to the paradox is that this whole concept of simultaneity kind of goes away in special relativity that things which one person says both these things happens at the same time somebody else will say no actually that one happened first and that one happens second because whether things happen at the same time really depends on what reference frame you're you're measuring them in what's the fact did the Train disappear in the tunnel or did it fall out the ends what actually happened both a bit it just depends on who the rest will train you in right in one reference frame that's exactly what happens them you know if you think about it the Train sticking out of both ends of the tunnel relies on somebody's simultaneous measurements right the front was sticking out at the same time that the back was sticking out and because different people disagree about simultaneity it's quite possible for them to disagree about whether both ends of the Train is sticking out of the tunnel at the same time how does this go down when you teach this to students the students love it I have to say I mean one of the really nice things is although we avoid all the mass in these videos actually the maths you need to do this properly isn't very hard the most difficult mass you need to do this properly is trigonometry and just basically Pythagoras's theorem you know the square on the hypotenuse is equal to the sum of the squares on the other two sides so the nice thing about it is this is something which is actually mathematically not that difficult but which is conceptually mind-blowing and so it's what actually one of the things we teach the students quite early it's in the first semester of the first year here just because it's something where you can do the maths but actually end up with a result that has you really scratching your head and actually one of the things I always warn the students when we're starting to do this is that usually in physics what we tell them repeatedly is if you get something that looks like a stupid answer at the end you've probably done something wrong so go back check make sure you haven't messed up this is one of those cases where if you get something that looks like a stupid answer at the end you've probably done it right because the results that come out special relativity is so bizarre that actually you do end up with these apparent paradoxes all over the place you

Channel: Sixty Symbols

Views: 892,335

Rating: 4.959538 out of 5

Keywords: sixtysymbols, relativity, lorentz contraction, time dilation, train, tunnel, muon, ladder, barn

Id: kGsbBw1I0Rg

Length: 8min 36sec (516 seconds)

Published: Mon Mar 04 2013

Reddit Comments

[u/IthinktherforeIthink]

Wait a minute.. In the perspective where the tunnel is shorter than the train, and the guillotine goes down, the front one opens again to allow the train to pass through before closing the back one.

What if the guillotine was never set to raise again after it came down? Would the back one cut off the train? I would guess that train would stop and then simultaneously the tunnel would widen and the back one does down, trapping the train inside.

[u/Sparhawk7892]

That's my Quantum Physics Professor.

[u/Tendow]

Wouldn't the tunnel expand behind the observer as soon as they entered the tunnel?

[u/Tomcatjones]

Sooooo Han did shoot first?

[u/staser9er]

Wilhelm scream!

[u/[deleted]]

What the fuuuuuuck

[u/BoobsThatStareBack]

It's unbelievable how different our middle ground of speed and size is from the extremes

[u/rasa1]

wtffff I think this changed the way I look at speed. Really everything just shortens distances in the direction its travelling? The more energy in that direction, the more the distance is shortened???

[u/johnnyb3141]

Shouldn't he have mentioned that simultaneous events do not exist in the 2 reference frames because the events were happening at 2 different locations. As I understand it, 2 simultaneous events can occur in both reference frames if the events occurred at the same position along the path of motion. Is that correct?