[Music] okay this is 11.2 time dilation so this is where relativity is starting to get pretty interesting time dilation is the idea that the faster something moves relative to us the slower it experiences time and this is true this is not really I mean it has been confirmed it has been there have been experiments that have shown that this is in fact what happens So the faster something is moving relative to us the slower it experiences time you can think about this um in terms of how it was described in 11.0 where we are always moving at the speed of light and so the more that speed is in the space component the less it goes through the time component um there are a few other ways to think about it and in this lesson we're going to be looking at Einstein's original thought experiment that led to this idea of time dilation so this thought experiment we're going to say imagine someone moving horizontally with speed V someone moving horizontally with speed V and they are measuring time with a light clock so this is going to be a clock that measures time using light our light clock here is a device you can see a picture of it down below right here that's our light clock and the way it works is light balce es up and down between two mirrors taking time delta T equals 2D over C for one cycle 2D over C so we can take a look at this picture of the mirror you can see that the light moves Upward at speed C and back down at speed C and we have distance is speed over or um distance is speed times I guess time is distance over speed so delta T is 2D over C because it's traveling twice this distance between the two mirrors so you can see our roundt trip time there is 2D over C hopefully that is clear enough that for it to go back and forth it needs to um take that much time okay so that's our light clock clock we've got somebody who has their light clock on um on some surface here so this is our Observer one he's watching his light clock he sees the light go up go back down it takes delta T to do that so that's Observer one Observer one is stationary relative to the clock so from his perspective delta T and we're going to call this Delta Ts for stationary so the T stationary delta T is 2 2 D over C Observer two in this picture we have an observer somewhere outside watching this person go by and from their perspective the light bounces back up up and back down and so it travels the same vertical distance up and down you can see it also travels this horizontal distance so that um after some time the whole cart here has moved that distance that means that the light wasn't just bouncing up and down it sort of followed this path for it to end up there still on the mirror okay so the light has actually taken a different path here for Observer 2 Observer 2 sees the clock moving horizontally and the light travels the same vertical distance which is 2D but also a horizontal distance of V delta T and remember the light still moves at speed C it doesn't speed up or slow down it's still moving at the same speed but now it has a longer distance to travel if it's moving the same speed further distance that means it must be taking more time to do that which is the whole result here that it's taking more time from Observer 2's perspective than Observer one's perspective means that time seems to be moving slower in observer one's uh frame than in observer 2's so uh we have an equation for this we can say that Delta TM which is the the time from the moving uh frame of reference so from Observer 2's frame of reference because Observer 2 is moving relative to the clock or the clock is moving relative to The Observer okay so Delta TM is equal to Delta TS the stationary time over the sare OT of 1us V ^2 c^2 and the whole derivation for that is in your textbook if you want to take check it out um it's a cool derivation but there it is there's our relationship so from The Observer 2's perspective the time seems to take this longer amount Delta TS over < TK 1us uh v^2 over c^2 so you can see that this bottom term here the largest this bottom term can be is well this is going to be less than or equal to 1 because v^2 over c^2 um um we're we're always going to be subtracting that from one so that bottom term is less than or equal to one which means that this ratio is always going to be at least one okay so um proper time is what we call Delta TS Delta TS is the time that the stationary Observer was measuring so the one that wasn't moving relative to the clock so Delta TS which is the time measured by an observer at rest relative to a clock and Delta TS is less than or equal to Delta TM always okay there's our relationship we have an equation for Delta TM and um we're going to do a problem on the next page you can see there's a little graph here showing Delta TM over Delta TS you can see that when um V / C is zero so when the uh when the clock isn't moving at all you can see that the ratio here is one you can see that as the clock is moving faster and faster and faster relative to the the second Observer our ratio goes up and up where approaches Infinity as the as the clock approaches the speed of light all right so on this next page here we have on Earth an astronaut has a pulse of 75.0 beats per minute he travels into space in a spacecraft capable of reaching very high speeds determine the astronaut's pulse with respect to a clock on Earth when the spacecraft travels at a speed of 0.10 okay so we can use our equation for this we have Delta um okay delta T M is equal to Delta TS overun 1us V ^2 over c^2 so to get my delta T first I'm going to convert 75 beats per minute into the period for one beat so I'm going to say here here Delta TS is equal to 1 over um we'll just say 1 over 75 and we're going to get a value here of 1.34 * 10 -2 minutes I didn't convert it into seconds because we're going to convert this back in uh just a minute anyway so 1.34 * 10 -2 minutes that's our Delta TS okay so I can get my Delta TM here is equal to 1.34 * 10^ -2 minutes over sare < TK of 1 - V ^2 over c^2 well we have um basically this is going to be well I'll do this here 0.10 C squar over c^2 and you can see that the C's are going to cancel out here so we get c^2 over c^2 so we get 1.34 * 10 -2 over < TK 1us 0.01 and this gives us a value of one second here I'm sorry I did make a bit of a mistake here so Delta TS was 1.33 here 1 Point oops 1.33 which means that here I also need to have used 1.33 and here 1.33 1.33 1.33 and that gives us an answer of 1.34 * 10 -2 so sorry about that mistake here so that's that's how many minutes it takes for one beat and now we can get the pulse again so pulse you can see the the delta T has changed a bit so we're going to get the pulse is going to be um 1/ Delta TM so 1 over 1.34 * 10 -2 which is equal to that's equal to 74.6 beats per minute so you can see that it's a bit slower but it's still that's approximately 75 if we're using our Sig fakes okay now we'll do the exact same thing for Part B determine the astronaut's pulse with respect to a clock on Earth when the space spacecraft travels at a speed of 0.90 C so we just looked at 0 one of the speed of light now we're looking at0 n of the speed of light same idea we have Delta Delta TM is equal to Delta TS over < TK 1 - V ^2 c^2 and we had our Delta TS from the last part 1.33 * 10 -2 minutes over sare < TK of 1 - 0.9 2ar um or 0.9 c^2 I'll fix that there 0.9 c^ 2 over c^2 and of course the C squs cancel out so we get 1.33 * 10 -2 over the square < TK of 1 minus so 0.9 squar is going to be 0.81 and this gives us a result of Delta TM equals 3.05 time 10us 2 minutes okay so you can see now one from 1.33 to 3.05 we've more than doubled the duration of the P of the pulse so now we need to get the frequency of the pulse pulse it's going to be 1 over Delta TM 1 over 3.05 * 10 -2 this is equal to 33 beats per minute so you can see now at 0.9 of the speed of light this astronaut's pulse has changed quite a bit okay so that's how we do these sorts of problems there was the added sort of trickiness with this of um of having to take the reciprocal to go from frequency to period because Delta TM had to work with a Time measure of period all right so that's time dilation it's a very real thing that does exist we can measure it it happens all the time one of the places where you do notice it is things like halflife so we have um particles that have a certain halflife we know that they Decay at a certain rate and if we take those sorts of particles and send them at very high speeds we'll see that they Decay slower relative to us than they than they should normally because they're experiencing time at a slower speed than we are and so it is something that is very measurable and has been confirmed now along with this whole idea is the concept of the twin paradox and any textbook the twin par Paradox is covered in the next lesson um we're going to cover it here instead cuz I thought it goes well with this lesson because it's all about time dilation so our twin paradox imagine two Rockets or sorry imagine two twins sorry twin one leaves Earth in a highspeed rocket a high speed rocket then returns home and what does he find when he returns home finding that Twin Two has aged more than twin one okay so this is the twin paradox that is what would happen if if a twin jumped in a highspeed rocket and flew away from Earth and then came back well everybody on Earth would have aged more than than this person in the spaceship would have and at first that makes sense cuz you say oh time dilation well he he experiences time more slowly than we do but I'm going to say here why does this happen sure it seems like time dilation should explain it but actually for both twins time should seem slowed down for the other twin so the one that's hopped in the spaceship from his perspective he's not moving at all and everything else is moving around him so he flies away from Earth uh so which means that the Earth flies away from him him and then Earth flies back towards him so his twin back on Earth was moving at this relativistic speed he was experiencing time slower than uh than than the twin in the spaceship so it doesn't seem to make sense because really from both perspectives time should have been slowed down for the other person and it shouldn't um it shouldn't really work there's two ways of viewing the solution to this solution one this is using using what I talked about in lesson 11.0 the idea that we're always moving at the speed of C through space and time so solution one for paths from A to B the shortest path the shortest path through space is also the longest path through time for paths from A to B the shortest path through space is also the longest path through time so if I'm here sitting on Earth and I'm not really doing much to leave Earth I'm staying here now Earth isn't a perfectly um inertial frame either it's rotating it's it's accelerating in a slight bit but mostly we're just sort of we're not accelerating too much on Earth and this is sort of an important idea that we are basically in an inertial frame that is not accelerating here on Earth the person who jumps in their spaceship and uh drives away and comes back well they're taking a much longer path to go from point A to point B through space so that means that they must be going less through time okay so that's one way of looking at this and again it gets into those some trickiness of saying well is is it really a shorter path what we're experiencing on Earth who's to say that that's shorter and again it's sort of relative so solution to is maybe a better way of looking at it so let's see when twin one changes Direction so twin one is the one on the spaceship when twin one changes direction to come back home he switches between two inertial frames two different inertial frames that line up differently with twin two so we can say twin one is driving away at in an initial frame away from Earth and then he drives back towards Earth in an inertial frame but at some point those two frames are different at some point he had to accelerate to change directions he had to um switch inertial frames and that switch is what causes this mismatch in their in their time alignment so you can see the picture to the right here of of what's happening so we have the stationary twin is just going straight here not moving from from their perspective the other person is going away and then coming back okay and so going away and coming back even though going away is an inertial frame and coming back is inertial that switch here is where things get sort of changed so you can see we have these lines where time lines up so we can say that the stationary twin here at that point in time lines up with this point in time for the moving twin that point in time lines up with this point in time that lines up with this then when we switch inertial frames now those two times are lined up in this other location so that at that change there's a large span of time that's sort of all of a sudden um gapped bridged over okay so that's um that's what's going on that's why uh things don't line up is because we're not staying in one inertial frame now for the for the twin that's driving away from Earth to stay in a single inertial frame they'd have to keep on going away from Earth and they could never come back okay now this whole situation has been verified time dilation has been verified the twin paradox has been verified in the 1970s time dilation was verified using atomic clocks on passenger Jets atomic clocks are clocks based off of the Decay rate of some radioactive substance and they put these clocks onto passenger jets that flew around the Earth a couple of times in opposite directions um and also they compared that to one that was sitting on Earth and took into account all the you know different effects going on but they did find that the time on these clocks ended up being mismatched by a very measurable amount it's been replicated over and over again this is definitely something that happens in the real world um so that's it's a pretty cool phenomenon give those homework problems a try and we'll see you in the the next lesson so