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Quantum Entanglements, Part 3 | Lecture 4

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Roy this program is brought to you by Stanford University please visit us at stanford . EDU let's review now I am
not going to be consistent about choice of the speed of light sometimes I would put it in in quality other times I will just call the 1 I will try to tell you when I do and I don't see that's just formulas get a dupe complicated when you put this season and you all should be able to figure out if I give you a formula which the speed of light is equal to 1 you should be able to figure out by dimensional analysis where the speed of light with Lucy's go our but here and they're all overseas buckle when it's important when I think that's a relevant and interesting to reduce our otherwise I will choose the speed of light to be 1 which means we working units like light years three-year-old like seconds per 2nd quarter hour or such Sojourner artillery review quickly we just start with Galileo relativity be Galileo in 3 Einstein no Newton Galileo relationship between reference frames moving stationary reference frames it is given by the simple equations which it called Galileo and both were going transformation Our for example from motion along a one-dimensional lined a year are relationship between the call admits in moving railroad trains and the stationary stations are acts prime is equal to X minus the velocity of the train times AT & T it is equal to take time is the same for everybody and the difference between X coordinates is just Vera because we move the origin of the molding frame is moving with velocity v relative to the stationary train to draw up a picture of the drug picture of what these transformations look like if this is often go myself as a moving train think of you with the stationary our our observers arrest or speak of media the moving as big of you that's arrest of course I think I'm at rest and I think you are moving as of course what relativity is all about but let's a wicked Galileo refers Galway relativity if I were the plot the kind access and the X-axis and then on the same Gaurav draw he prime acts as the steep rise in taxes which stands for the location the locus of points of origin Of coordinates for the prime moving observer would look at an angle like self and the slope of Bangalore angle would be controlled by the velocity of the moving reference be that blind would be given by X just given by X equals city which is the same as X prime equals 0 sewn back so much Our moving frame boats but then ballot the plot leader lines of constant time as seen in the reference frame their exactly the same as the lines of constant time in Kneeland Prime reference for so they look like that you don't have to
draw to actually a space but you do have to draw on the key axes and this will become the tee time Texas after the moving reference reference removing court so if we now remain agreed at this would be the grade for the moving observer an agreement for the stationary observer would be similar except airlines review vertical it is the basic equations that govern be relationship between observers America our Delaware relativity notice that if you solve a few and the equations they simply look like this they like X equals X. prime plus V E but I can also write that is meeting Prime TNT same thing and he is equal to T the relationship between the prime prime coordinates is exactly the same as the relationship the inverted relations between on prime the prime Courtnage with 1 exception you change the sign of the velocity that represents the fact that used moving down the track with velocity plus VAT ICU moving backward in the opposite direction with velocity might so that's how Galileo if you written the equations would have thought about relativity it is awaiting Newton for about it it's the way everybody else did until the advent of Our new discovery that the speed of light is the the same in every reference frame those equations are not consistent with the speed of light being the same every reference frame they have to be corrected and the last time we were about what these rules about life in imply about the corrections to those equations of so we write them down for you to remind you of course the low rents transformations and they look like X primacy Quebec's miners bt exactly as no and Galileo might've written it but then the square root of 1 minus V square downstairs now that if you want to put back the speed of light see equals 1 sometimes you want to put back the speed of light this equation doesn't make good dimensional sense of velocity has units of length per unit time you want 1 has no units at all so what you really want to do is survived by a speed of light Square squared oversee squared and of course if the velocity of the moving reference frame the train in this case is ordinary velocities 50 miles an hour 100 miles an hour thousand miles an hour or even a million miles an hour that's quite
small by comparison with the speed of light and v square overseas squared is very negligible even from million miles an hour I'll be square overseas where is a very small number and sulfur most practical purposes terrestrial physics physics not involving got things moving with anywhere near the speed of light this just becomes the same as Galileo and scored 1 of 1 exactly the same as God about a price that was the new thing this was a really knows saying that Prime is just equal to take another words that kind it is not as universal as Newton had thought Newton had believed in the universal kind got time equal to that of a court that that everybody has exactly the same time keeping and therefore chief privacy with seat well that was exactly what was wrong with Galloway relativity that may be inconsistent with the rules of the speed of light is the same in every reference frame to correct it Einstein that way then my ears X divided by that score saying the square root of 1 one-liners square square but this is also not correct together to be dimensionally correct after right velocity divided by sea squared current X again if we velocity is small by comparison with the speed of light so that overseers a very small number then you can ignore this terms and you can ignore these square overseas quit the nominate at and it just reproduced he equals to prime time but this is a correct of formula end it differs significantly from the Gallo and formula when the velocities get up near the speed of light in fact on the velocity gets to the speed of light something crazy happens here because equal the overseas Eagle Cole 1 1 minus 1 as 0 and U.S. to divide by 0 that's not a good thing our mathematically inconsistent facility tells you is that a frame of reference can never quite be moving with the speed of light relative to another 1 but when it starts to get close to the speed of light we start to see significant deviations from New Delhi where they might ask What about XMY there 2 other space baton besides X is Iowa about y and z New Mexico worry about wanting to other components of a moving down the train track then disease corn the space along the track but there's also the recorded so space perpendicular to the track was a white V and the Lorentz transformations for them very very simple there simply that why Prime is the same as Y N Z crime is the same z coordinates along the direction of motion get mixed up with the time this complicated way but the perpendicular to the motion they are the same for both frames of reference that your Hawaii this imagine having a meterstick moving on the train and his stick standing still and sticks past each other like so now does I'm observer and they pass each other we just cut the instant of time and the question is whether the ends of the meter stick are the same place if their exactly the same place then the 2 meters sticks will look the the same to every observer the question is will actually passage of the argument that says that they will stay exactly the same size goes like this and then only itself and a reference frame halfway between the moving train and the stationary train stationary station when halfway means an observer moving along who sees mean moving with the velocity and sees you will be with exactly the same velocity backwards some frame of reference not moving as fast as a the train not standing still Close to Use Somebody riding a bicycle hold is midway between 2 reference frames and that they see me going with velocity something you going back with the same velocity that but see that entirely symmetric situation entirely symmetric situation and what you'll say it would seem to me fix approaching each other with exactly the same velocity To me the sticks in the perpendicular direction approaching each other with exactly the same velocity and sister situation is entirely symmetric between the left and the right because no expectation that 1 of the sticks will pass the other 1 looking short period Bell pass each other looking exactly the same way and sell for that reason our there's no funny business about Exon sorry y z but ex ante get mixed up with each other OK let me show you another way to write to Iran's transformation which might be illuminating gives you a little bit of a geometric picture about how these equations work Komodo was black for over here after get rid of the speeds of life because Sarah nuisance but just cervical 1 simplicity and now let's and these 2 equations on both sides will get on left-hand side is X prime plus keep prime that's the left side will we get on the right hand side we're gonna get 1st of all something of the same the nominator we get is going to have a denominator square root of 1 mind square and then the Noriega it looks like will get X past take a but they going to get another which is a B minus V times X plus t so that will be X plus figure times 1 might some stuff at plus see and in mind is be experts to
101 miners 1 might be divided by square warmer squares what's that worker over here one-liners upstairs and then the square root of 1 mind square downstairs that's the same as the square root of 1 minus V times 1 minus that is what writers Drury row warmer divided by the square root of 1 mind this plans 1 plus former lenders we squared as 1 might be times 1 plus the 1 canceled and all of this is just equal to the square root of 1 minus V of 1 class OK so 1st of all we rewrite this equation says that X prime plus 2 Rory is equal to X plus Tracy time something which depends on the velocity depends on the velocity as 1 might be all 1 plus nuts look at X prime prime must look at X prime might Steve prime instead of X plus he keep called prime plus deprived Matsuo we get X Prize Money a steep price subtract them within a given X minus from here we're going to get a plus vii because was subtracting arms expert nor do I just for lawmaker small changes I wanna right says he prime plus X prime same thing I haven't changed anything plus it just because it agrees notes in our like this 1 keep prime Maringa 6 Brian keep price minus X Prize Texas track this from that's where we get there and we get team ideas X times 1 plus B divided by the square root of 1 mind as square 1 plus V because of the opposite side of appears to get T minus X and they will get miners V Comdex might that all a up 2 201 1 plus divided by squared away might be squared but if you go through the same exercise here 1 class 1 plus 1 plus with you get is square of 1 Class B over 1 it looks almost the same as the the equation fact prime plus prime but with a slight difference the square root that appears here the square root of 1 plus or former industry now but stop Maryland look at what we have is that I'm working with ex ante I've worked with no coordinates no coordinates X plus and X minus t no didn't think of those as sheriff Mrs. CTN this is acts and 45 degrees remember represents the motion of libraries and ex- prime plus elect Class C is a coordinate along that axis X might be is accorded along this axis This is the X plus T coordinate and this is the X other T minus X coordinate what this is telling us is that X plots tea when it's Lorraine's just get multiplied by by something that depends on the velocity and at the same time he might miss X also just gets multiplied by something but that something is the inverse the something here is 1 over the something here so 1 picture of what is mathematical transformations light is that keep plots X depending on whether these positive or negative he X gets stretched its workers is begets stretched or squeeze depending on what it is these positive and one-liners B is less than 1 Brooks I would say that this transformation squeezes keep prospects people aren't perfect prime p plus X number less than 1 so it squeezes 1 axis and stretches the other axis axes by the inverse amount so for example Lorentz transformation would be appropriate velocity would squeeze this access by a factor of 2 0 and double this axis by a factor of 2 that's the nature of Lorentz transformations there really quite simple their stretching along 1 axis 1 45 deg axes and squeezing along the other 45 degree at notice important thing is they don't change the 45 sees the 45th 5 degree taxis Stacy 45 degrees wise that important because that you that light rays continued to move with the speed of light in any reference frame city learns transformations are really quite simple there stretching along 1 axis squeezing along the other taxes but they preserve the 45 degrees taxi they don't preserved other lines here other lines will get stretch this way and squeeze the other direction but the ones which are along 45 degree taxis do are meeting so that's nature of Lorentz transformations squashing a 1 like like direction and expansion of the other like like their actions are along the direction of motion of the relic of relative motion to reference for bribery questions about the structure of Lorentz transformations before we go on at they are now route OK said but there was to make some assumptions but the assumption is but if To what objects quit and instead of space time if 2 events are at the same instant of space time and now
that would mean for example released 2 6 cross each other a flash of light go off from each from the end of each 1 the moment that they had each other we call that an event if an event occurs at a particular space time point our or if tool events a car at the scene space time point in 1 reference frame and they occur car at the scene space time point of reference so that's what that's an added ingredient the sort of you really traced every possible assumptions sure there's an infinite number of assumptions and things like that but they also easy to believe that there about to Evanston space time and now yes it till events in space car at the same space time location according to 1 observer and they occur at Sing space another observer notion of the event smoke point of space time where which thing occur that's invariant concept so you could bring it back the question of these events of the crossing of these 2 meters 6 gap so that His all mm while bicyclist of course stands for a whole moving frame of reference but what did that well war he could see the flashes of light come out From there when the 2 were sticks pressured to each other he concedes flashes of light come out and indeed do so flashes of light came out from the same 2 . 4 the same points so I don't think he really has to be right but it has to be able to deduce what was going on right here by looking at libraries O'Shaughnessy 2 meters sticks contest each other and then seized a pair of flashes of light emanating from exactly the same point to flesh what clashes altogether 1 from the right in this the 1 from the left and the meter stick 1 from the right and this sweetest they 1 from left and sees that these till rays of light emanate from the same point in space time and the other 2 Back on my shoulders here which I can't squeeze together and as flexible there used to be OK so that's Lorentz transformations when I forgot to do which will do next time was to show you how a compound velocities our will do that next time bomb but for the next thing I tell you about proper time I don't remember I don't think we talked about proper time last time many of you know what proper time this summer you don't but let's spell it out as I said no didn't believe that time was the same for everybody moving stationery accelerated whatever you do kind will agree Fuchs synchronized to Clark's where do you do with them after they will stay synchronized if you have a whole collection a synchronized clocks they will stay synchronized that has to say he is not true for moving clocks In the a special theory of relativity and an ocean of time which is proper Tyler Hill a proper time as a new article walking walk around your walk around a high-velocity up near the speed of light but you carry a wristwatch with your everybody sitting still has the wrong kind peace and sitting still people all synchronize their prime pieces sir they all agreed that synchronize the kind pieces would light raised according to the way the Einstein described synchronizing so you people all have perfectly synchronized light rays are perfectly synchronized clocks and I have my clock walking around the first one I started synchronized clock with viewers and then I walk on and I take a walk around the table a few times at close to the speed of light the end I look at my time peace and you can compare it with the the prime piece of a person standing next to or here in general there is no reason for them to to agree and they won't agree the proper time of mind moving Clark it is the time of read by removing clerk of the court when the time is definition for Quarter NYTimes time measured by your stationary clocks so proper time is something which follow the observer around and kicks off intervals of proper time our as a clock following a wristwatch would would kick off surprise 5 give you the formula for a proper time initially the proper time is indeed concept the proper time between 2 we've been were talking about to events we could think of them as Chile events along a trajectory of we wanted the trajectory could be the trajectory of removing Clark but for the moment let's just think of them as 2 events and 1 of them was simplicity let's take it the origin of the origin of coordinates so this point over here is x equals 0 people 0 prime equals 0 T. prime because 0 not incidentally was always seemed lost we're at like Olivia 5 x Eagles X mines V X Prize was excellent we see the square root of 1 might be squared keep Prime it is equal to T minus V acts of Route 1 might be squared noticed that when it acts and TO both equal to 0 x prime Primeau also equal to 0 so we set up in such a way that the origin of space-time coordinates is the same for both observers OK so here's our warming observer or let is not even be moving observer is system only particle or moving person it doesn't have to correspond to work to radioed a reference frame it just some object moving and it could be moving with an accelerated velocity but let's just take a small piece of its trajectory along which we can approximated by a straight line OK now if we replaced time but would never be space the free replaced by nearly space forget this just think about ordinary space not ex ante but X and Y we could also have different coordinates describing the same next prime and why Prime nailing coordinates with Georgia's rotated relative to the original coordinates what kinds of things are the same for both I'll call them observers but I simply mean rotating coordinate frames well 1 very invariant quantity is distance from the origin of the distance from the origin of any Point necessarily is the same in both coordinate frame that's clear your distance from me he doesn't depend on which way armorial oriented on my wife doesn't which way I'm oriented you're still sitting 3 meters off 3 of 4 meters from the matter which way a terror cell that
implies that the distance from the origin of the squalor of sicker square of the distance from the origin Claude Esquerre that's equal to X squared plus Y squared it must also be equal to X's prime squared plus Y prime squared another words the quantity X squared plus Y squared is reined in any coordinates system ordinary coordinate system involving the rotation of court something like that is truly its transformation something slightly different a little slightly different mathematical expression is X she and this is some point here With coordinates XT all our X Prize keep prime depending on which frame of reference I'm looking at that he said the statement which would work out from this equation here I would do it with Stewart let's take the point X which could also be labeled by the prime coordinates worked out the quantity X prime squared not people square plus sex prime squared but keep prime squared minus X Prize square a little bit of a glitch from what we would do just rotating quark gay pride squared minus X prime square let's right right here 1st of all the denominator will get 1 might be squared because we've square mile note the next prime square that has X squared it has Class B squared he squared and that it has money this tool X we see bats all X X prime square have that's that comes in with a minus sign but then let's and keep prime square feet high squared has he squared plus squared x squared minus 2 x also provided by 1 squares I know this is T prime squared this is next prime squared and we subtracted the 1st thing that we Norris is that these things these crossed terms cancel we subtract beast To the crust terms canceled now be X squares don't cancel because Seeley have these square X squared minus X squared so of minus 1 might be squared x squared them what about 50 square Terrence C squared here my ears are square T squares so that's he squared times 1 might be squared all over 1 might be square to make a long story short bus right about keep squared minus X prime squared equal 50 squared minus X square just work out if you didn't follow algebra he does work out square this square this subtract in in your way and you'll find out that keep prime squared minus X prime squared is the same squared exit manner 6 quarters very very much like when you rotate coordinator the square of the distance stays the same square of the distance from the origins stays the same now a physicist would say that means that the quantity he squared minus X squared is invariant it's the same no matter whether you look at it from the point of view of the prime frame or the young prawns and has a name because it invariant did support anything which is invariant which is the same for all observers must they have a special report it's something which everybody will agree upon and so we ought to give it a name the name for it is proper time the proper time between this point and this point proper time the proper time between the origin and the point X is by definition a proper time equal to the square root he squared minus 6 square 1 square root well EDS's is very similar to definition of course it's a definition we could work with definition with the proper time was just uh D squared minus X squared but it's a definition the square root but is very much like looking at a distance between 2 points in using Pythagoras around the Yout Pythagorean theorem says the distance between the point is the square root of X squared plus Y squared the parallel with that we we call this a proper time between origin and the point X T and that says definition but a highly useful definition will see why it's useful go along the main thing about it is that everybody agrees about it OK 1st of all let's look at some possible values that the proper time can have various . 68 if he is bigger annex where the mean 52 annex In magnitude of T squares bulimic square 50 square is bigger than X quarters that means we're talking about points which is a wedge here with time with time component is larger than the x component T-square is larger than x squared all in here but let the particular stay in the upper half from space these squared is begin next quarter if you within this wedge the his alike are why should it it in essential if something and if we're working in 3 dimensions not sorry 3 spatial dimensions not 1 spatial dimension that we should also a minus why square minors a square he miners why Prime square miners prime square the a that makes sure that our definitions don't depend on direction space as long as we know there were only talking about Lorentz transformations along a given access we don't really have the worry about wine but wanted the transformations along some other axis 1 day and y z we would note that this whole quantity keep prime squared minus 6 women's wife women zinc prime squared is invariant why is that incidentally well remember the zinc prime why prime don't transform where we go from 1 frame of reference along the x-axis torn up so that doesn't make any difference but in prime is a prime abuts a convenience store edit and then we have an expression which doesn't depend on directions space
but if you only 1 think more simply about directions of along of single axis that you don't have to worry about wine is OK so here we are these this point here he squared is greater than heck squared and that means that squared minus X quit as positive and the square root of a positive numbers as another hour another real number so in this I'm like region this is called a time like region because the coordinates are more in the direction of time than they are in space and time like region be proper time is just a positive number square root of a positive number we take a positive square root of course that perfectly normal positive proper turned up in here what about along the light from lot the light call to that's exactly who were Exs either equal to tea or X is equal to minus along here Exs inculpatory belong here Exs eagle from minus X is a quote from in either case he squared is equal to X square so although like comments along a place for light rays can travel the proper distance between the proper time between the origin and that point is 0 now this is a big difference between ordinary gift distance and proper time ordinary distance the distance between took . 0 both points of the same point is no way that the distance between 2 points can be Zero and not be sitting on top of each other in Minkowski space in the space describing the special theory of relativity space time 2 points can have 0 proper distance between them was 0 proper time between and and still not be the same pouring hearing here war issue for is the 2 points separated by 0 proper time then a light gray single from 1 to the other so the trajectories of light rays correspond pay of the which have essentially 0 proper time along with real particles asserted that real particles but mess but particles moving with less than the speed of light have trajectories which are in the time like Correction like that moving slower than the speed of light and the proper time along those is positive now drove a simple fact property time measures ordinary time as seen by a clerk moving with the object for example supposing we have a clock which is at rest of clock which is at rest and are frame just has a vertical world line is perfectly at rest and for such a point of point along here what's the proper time from the origin of the square root Of he squared minus X squared but axes euro for this point was just a square squared which is just feet so the proper time along the trajectory which is standing still it is just be ordinary time was that tell you that tells you that the proper time is nothing but the time read by a wristwatch standing still what about what about proper time along a trajectory which has a finite velocity examples of posing a fire velocity like that and we look at the proper time for particular point over here well the principle of relativity the equivalence between different reference for years tells us the following things we could always think of this particle object observer as being arrests with the always dual arrange transformation so this guy is a guy arrests and then use the same logic that we use a 2nd goal in the reference frame with this is it rests this point he'll have X prime equals 0 but keep trying not equal to 0 why is it next prime equal to 0 because in the moving reference frame moving with this guy over here this whole trajectory corresponds to his origin of what next so X Prime would be 0 for somebody moving along this trajectory here but he Prime would not be 0 over here we could work out exactly the same argument the proper time to this point over here would be keep kind squared minus X Prize squared but time square 0 and so the proper time along here in the same way is along he and others along the vertical trajectory measures the kind as red by moving watch moving Clark moving clocks take off proper time our that's the message that meaning a proper time is nothing but meaning of of the time as read by watch which is moving with the moving the herd object so that's the most of a proper time it's kind of funny in these regions over here in these regions over here X is bigger than CD so T-square minus X squared is negative and the proper time is imaginary points up in here are called time like wrote of the origin points along the light cone here are called like light and points out here course space like space like points have an imaginary proper distance imaginary proper time wrote the origin but points and within all right call have a real proper time brook any questions about the notion of proper time it is very similar to distance along the trajectory except for the funny property that is a minus sign it throwing here a minus sign it you throwing instead of a plus sign peace squared minus X squared but because many similar properties and in particular it's the same in every reference frame the proper time between 2 distant 2 distinct points is very that we can use that fact use that fact Fairfax similar to 8 well that the time interval as related vii at Clark that
moves between those 2 supports the view that the star face Fernand Alberta are your soul the product another way if there are 2 points in space time and a clock a physical real Clark can move from 1 to the other along a straight trajectory that means a trajectory with constant velocity then the difference of time readings at the 2 ends of the trajectory over here will be the proper time Federer was visited the difference between the time rate at this point and that point will be proper time along a trajectory you could calculate their forget it to points of space-time we can calculate breezily how long it takes to move between these 2 points as red by removing Clark not lured by stationary clerks and different as read by stationary clocks the time difference is just whatever the time was at this point that whatever the time was this point where the times we just today but the molding Clark moving Clark reads its own time kind that's the message of special relativity that the rate and which clocks moved depends on how the rules that we can work out in some detail the relationship between moving Clarkson stationary clerks would do that and the tool is to use proper time so let's take a clock which started the origin and moves Shalala trajectory which takes it to a point Let's thanks to Torpoint on the origin of the moving coordinates I'm sorry that the line is not reversed rate was intended to be a straight line do we have harmed moving coordinate frame Clark moves from the origin to point of view what is that point of view that point over here is X prime equals 0 it's the origin of spatial coordinates for the moving observer and its summer point prime it also has coordinates X T and other words in the stationary reference frame and has coordinates x seat in the moving frame has coordinates X Prize equals 0 as some tea prime I would like to know if I know X and I know to be what is steep Prime incidentally acts of class is equal to 80 really all I need to know is T and the velocity of the moving observer is a moving observer His trajectory is is x equals bt his and here's x equals weekly what is it kind by the clock moving from year to here was crime was now find out keep prime squared minus X prime squared must equal he squared minus X squared for a few hours should be prime keep times squared minus X prime squared should equal squared minus exported from this point because the expression for proper time is varied same expression neither reference frame for the beef proper time between this point the origin of what I know about this point I know that it's X prime is equal to 0 0 so let's cancel out I can sort out to through right keep prime equals squared personnel at this point X is equal to leaky so that minus V squared he squared less interesting we now have a simple formula for tea Prime 1st of all it's right that says 1 minus V times T-square 1 might be squared times to squared but we just take a square wrote prime time raid but removing Clark is equal to that kind red by the stationary Clark remote stationary Clark is just this time here times 1 one-liners square root 1 might be squared that's the thing this formula of time violation that when you see a moving Clark that moving Clark will appear to move a little bit slower Weisel the what might be squared is less warned but Everybody will agree about the value of the proper time between these 2 points although they will disagree about the actual core wouldn't times in particular the people of equality time she prime seen the movie reference frames will be the Accord NYTimes as the young stationary reference frame from square 1 Missouri square so what that's says is at the moment observer Rupp the stationary stationary observers seen removing clock run a little bit slow we really sit at this point but in the speed of light C squared but b square overseas queries typically are very very small number of services almost keep prime Eagles but with a slight discrepancy this is the kind of ovation of removing Clark now how come how could possibly be that we have relativity which says is a complete parallel between your talk completes symmetry between moving observer and the stationary observer utterly then account for the fact that the stationary observer sees the bowling observer having having his time slowed down what what does it mean if we look at the other way what does the moving observers see about the stationary observer as he sees the stationing of observers time sped up no also sees the time slowed them so I moved past you with my clock my clock seems to slow down as far as your concern the other hand if I watch 1 of you'll I have all of my fellow lined up with me forming a reference frame they will also see your clock gross World Cup and that possibly be that sounds contradictory well it they're talking about 2 different things so we warming observer remember that removing observer his
Torre or his lines along which kind up current data are not these horizontal lines here lines which look like this this is a prime equals a constant This is a prime equals a constant sorrow for so long on which he Prime constant are different than the lines of which he can't do this is this relativity are of synchronicity the relativity of being synchronous different observers call different things synchronous now let's take the mood observer here the only observer looks at the stationary clerks and instead of his time and asked what does the stationary Clark really well an incident of his time works like this so is really asking not what this stationary Clark reads at this point but where the stationary Clark reads at that point so they're 2 different questions were U.S. what 1 observer seas of the other ones clock fog there is a complete symmetry but to have a say in this thing Avery said it but on Stratus query comparing different things these surfaces along which he is constant them along which T prime comes different surfaces and so when you asked whether whether clocks a synchronous from up synchronous so full you have to account for the different from what you mean by synchronous talk so at least that's the kind violations the fact that moving clocks repute stationary observers to go slow and stationary clocks appeared to moving observers also 2 well we couldn't certain from the beginning that whatever 1 observer sees the others Clark that they will be reciprocal because of the reciprocal nature of Lorentz transformations OK that slowing down the clock let's talk about the twin paradox somebody asked me about the twin paradox we're now in a good position to do the twin paradox Stewart says force worker Aussie the 20 paradox twins are born at the origin of coordinates 1 stays home police stays arrest doesn't move doesn't accelerate doesn't move off and just moves along a trajectory like that were like that are the other 1 is shot out a high-velocity not quite the velocity of light a high-velocity for a certain period and then at the end of that period turns around and comes back and meets the twins at that point there they compare that clerks will all kinds of Clark's sick compare the clocks watches might be their aging made them metabolic a time differences between hearing here on how long beards or whatever whatever corresponds still good Clark they compare their clocks at this point and ask what the relationship between the kind grid by Clark moving here and by Park moving here well overall set to calculated now because we have to do is to calculate the proper time between this point in this point along the vertical trajectory and the proper time our trajectory which consists of 2 different pieces with a lot top proper time from here to here the proper time for beauty he said and that tell us what the clocks measures clocks measure proper time I 1st of all this triangle assuming that observe that these 2nd twins comes back with the same velocity that he was shot out ways with some velocity v then this triangle is a nice isosceles triangle and weakened the vibe vertical lines and have called the side he is have side it takes there for time to see her every move from stationary person here it takes time to to go from this world point of this world porn and therefore the personal stays home says it took amount of time twice capital chief for my twin to return as pets twice capital please that's what the observer who stays home says the time interval between birth and meeting between now what about the twin who goes out and comes back but said calculate toggle proper time along this trajectory here he goes out with velocity v How far out to sea get he gets out a distance times Fish NYTimes distance along the x-axis the moving twin gets out of this point then has coordinates x Eagles VT and time sees let's calculate the proper time from year to year the proper time the square of the proper time he squared minus X squared so the proper time on the 1st leg of the journey here is he squared minus V squared he squared which is equal to 1 might be squared times squares that's the square of the proper time and we have to take the square root of it to find out how much the clock change from year to year so the time change the proper time along with trajectory here square root of 1 square times what about on the 2nd leg of the journey the 2nd leg of the journey is exactly parallel to 1st 3rd of the 1st leg of the journey except going backward the proper time between here and here is exactly the same as the proper time between here and he the same next coordinator and the same feat song answer for these observer going along this bank trajectory here His twice this twice times Square might be squared as opposed to just twice team which won a smaller the smaller 1 of the smaller proper time is associated With the Twins less child and came back because 1 might be squared more than 1 so the amount of time read off by moving clocks here is smaller by factors square root of 1 might be square than the time measured along the trajectory Of the twin who stayed home all kinds of times as I said the length of beer a number of heart beats all sorts of things be time red along as moving trajectory will be less than the consequences that these moving moving twin comes back looking younger feeling younger looking younger being younger than
Wednesday home is a paradox that paradox because nothing inconsistent about the look inconsistent is nothing particularly inconsistent sometimes people ask How do you tell which twin moved in which 2 Indian didn't but not hard this 20 reverses trajectory of accelerated this point over here and so if there were a rocket ship they would feel a rocket ship accelerate and there is no symmetry between the 2 2 will twin 1 twin stays at home experiences no acceleration experiences no tendency to be thrust back and forth by by the acceleration of whatever owed me sitting in the other twins suddenly experiences a sharp acceleration point until there's no symmetry between our is no more of a contradiction that asking about the size of the triangle dissidents or longer the sign of the triangle is not the same as the distance all won't be broken path composed of the other 2 sides is no not want nothing a paradoxical and the proper distance along the sides of these 2 try was broken trajectory here is just different than the proper distance along here now know would not have understood that it probably would have understood was going to him smart enough but With his all of you of kind of his own view of space this would have been a shock to discover that when goes out comes back comes back younger our between who stayed home gap question look at that but Dick well between feel something quite different what no you don't worry avatars exhilaration but you fell out of say OK on the assumption drizzle you could smooth this out is nothing there was nothing new power there was nothing very special or about making this a sharp turner prior to save lives without just says that finite acceleration rather than an instantaneous In the acceleration of the assumption is that the clocks that we have their way the case is insensitive to acceleration from a cinch full of real clocks real clocks a few accelerated harm the bright new accelerator Clark real hard you break it now it also depends on the size of a clerk the bigger the clock it if you give it a hard acceleration the easier it is to break it to take up a trainee kind kind little Clark and you accelerated it's hard to break thinking in Adams an accelerating it 10 meters per 2nd per 2nd Soviet carmaker clock that's composed of electron going around nucleus is not going to be affected by that meters per 2nd per 2nd on the other hand a great big Clark a hundred miles wide and so forth that's that you try to accelerate get Our it's gonna be very very delicate to accelerate it without changing the way that it fixed so the assumption is that the clock kickoff proper time that's assumption that good faithful small clocks which are very well below and built to withstand acceleration that they simply take proper time the takeoff proper time all we have to do is measure the proper time along now they said we can do this experiment a way that minimizes the amount of acceleration here we don't need a lot of acceleration and we can have we began have doing this period he had the acceleration be very modest it could be a small fraction of the 10 meters per 2nd per 2nd that we feel this room here will normally that to that 10 meters per 2nd per 2nd has these any very big significance and big effect on our wristwatches restrain armourer little bit we could swing around the more than 10 meters per 2nd per 2nd if the clock is built to withstand shock and so forth acceleration won't have much effect on the assumption is that the acceleration doesn't have much effect on the clock in the meaning of that is the clock always takes off shoppers time so it ticks off the number of units a of proper time along the trajectory now that is assumption that assumption that assumption of an ideal Clark is what you mean by an ideal clock is set up but now they b but health here is pressured jab all of the gasses that yeah that acceleration yes yes yes yes so bright let but the analysis the announces the logic of the analysis the logic of the analysis has to be supplemented with some additional physical facts are that I don't think you could arrived just from what we said but we derived from the experience of knowing that if we have a small Clark well-made built to resist shocked and so forth that as a matter of experience we know that that clocks rate of kicking will not be sensitive to modest acceleration for rugby particulars got the of your class now you get a lot of ground yeah but would you cannot 19th seed it's not gonna help say general Motors because then littered the began asking is what is the effect of grab the gravitational field on the way the clock here and I can't answer that unless I know more about the clock I have to know the details of how the clock works and so full of again if the clock is a big sloppy Clark made out of very big components and their components which are very rigid and so them it be very sensitive to acceleration on the 1 hand but also gonna be if I put on the table here it's also barely squash down by year by the gravitational field and so little also fail to learn to be a good faithful Clark I take a small hard clock um of my heart I mean hardened against acceleration then No. 1 it will report faithful time keeping a proposal on the table in the gravitational field the tool it will report faithful timekeeping interests women around my head on a roll but I can't I can't prove that from the things that we've done from mathematics with done I say that's a matter of experience in a matter of physics and a matter of how Clarkson made and what made out of and so forth but the talk about yet you have the right then at the top
there is right OK right right so Of the extent we talk about the fabric of space what we will be talking about here would be proper time measurements along with along with trajectory the fabric of space whatever the right word for it geometry of spacetime geometry of spacetime is a geometry of proper times and proper distances but more rubles yes 0 yes absolutely absolutely out of that was implicit that was implicit I should make that exploit experts yes absolutely indeed yes we do and yes I version of an explicit right we divide the trajectory interests more segments each of which has approximately of vary constant velocity along rebels say we calculate the proper time along that segment we ate at the proper time along the the next segment we had at the proper time along next segment but it's very very similar mathematics to the calculate the lane along a curve the calculator link along the curve by dividing it up until a lot of little intervals and the each in double calculating the linked by 80 x where plus Y squared back square pass built the wise scored square root them all up and that gives us going along curve very much the same way we take this current trajectory now we divided into small little intervals each interval can be thought of as almost a straight line and therefore it each interval our can be analyzed by thinking of it as having a constant velocity along those that trajectory we end up or constant and Zeller the question was that while we were wrong no I know exactly what you're asking the way can analyze that is to imagine actually sending like Ray is back in full of our let's tick off some time intervals here slips imagine ticking off 1 second intervals along both trajectories 1 second intervals of proper time or both trajectories so here would want second intervals might look something approximately like this for the stationary person this is very many seconds so we gotta go over a long period and 1 second intervals would be longer here because some moving clocks appeared to go slow to the stationary observer Michael Ljubicic faster In this work like that OK now we could start sending light signals back and forth every instinct these stationary observer could send the or light signal to the molding observer now what will happen in the 1st set of light signals a wasn't barely would come the moving observers stretched out this is Doppler shift this is Doppler shift in fact but we have figure of Doppler shift this guy's moving fast he sees the light coming a series of light pulses coming from the stationary observer coming gone delayed varied delayed 1st and then all lumped up not delayed but squeezed together In these 2nd part of the trajectory here so it sees all the light signals a 1 2 3 4 5 6 605 light signals but sees the 1st set of them come very slowly and the last of them come very quickly now what about the stationary observer looking and the moving observer the moment observer may also send out like signals and I also there would be kind of Doppler shifts but the number of light signals the stationary observer sees would equal the number of light signals that the moving observers sent out and the number of light signals of the observers would equal the number of light signals of the stationary observers both cases they would see something taking place our somewhat regularly in time at 1st they would see things stretched out was slowed down and then they would see things speeded up in both cases but at slightly different rates and rates which would which words we which you could calculate but they won't any missing frame by missing frames you would mean missing light signals they would see although each 1 would see all light signals of the other 1 and doubt but coming at rates which would depend on their state wrote state emotion eventually they thought a lot Rios and father but the saying the bourse where he that the this person transmits light of low frequency yup young right so that more wheat show we OK before can answer that we have to know more about energy so let's wait on that kind of question something and at the moment all the talking about is axes but that's a reasonable question whether what how's the energetics appeared to both observers and the answer is that energy really only makes sense in the usual sense to an observable was not accelerated to talk about the energy of the moving of the accelerated observers very tricky the today talk aloud I was born here is but noted that the we have of Asia those that gays the threat level but after week of its steady at last rather than the what 0 0 0 yes but they would nevertheless be an asymmetry between the 1 they went around these direction enormous still well a woman observer would discover surprise he would discover that he kept repetitively passing through the same places on it or if we were Will the Circle will you and we would discover that repetitively we keep a passing the same place where I have to think about that that's a good
question but the answer is definitely there asymmetric their asymmetric the person moving around proposed and the 1 standing still young missed at no no no no that us you to quite right let let's back were to blame I need to think about the good hands it's a nice question or give me some time to think about it but I wanted to do I don't wanna give you a on thought out answer so we could back to its limit is still a question wants the questions of interest more young supposing you live in a world of true dimensions of space to a dimensions as they are this was only aware of the one-dimensional space and your world was a cylinder a cylinder of space now this is bases not space time you could imagine such a world special relativity doesn't actually quite makes sense for it is a problem with special relativity here our but but suppose perfect special relativity sense then an observer could stay home over here and another 1 good nor around on a trajectory that a well sorry this is time vertical is time vertical space 1 observer could stand at a standstill and the other thing go around and a large velocity and come back that observed was moving around will large velocity would be unnecessary from 2 experience any acceleration Laura around here just more the same velocity and would leave and come back without ever having experienced an acceleration however it is not true that they would experience the fact that they were on me on the course can't where they were young she I'm sorry a moving around them the scores curfew no no no no no no on this no talking image asking no you not living in a world which is a solid cylinders in which the radius means anything you just living in a world which is just a pure surface pure surfaces no notion of distance from the center opposes nothing in them let's comeback direct this is this is way way way beyond what I want to discuss political declined theory this is not the misses is a a somewhat more advanced question where do I get to before we quit his link contraction we did time dilation by doing proper time with a link contraction in a very similar way so that let's but by but the 1st thing when you want to understand the special relativity problem draw pictures always drew a picture and see if you can cast your question and diagrammatic language our which Richard then a cancer by using laments transformations and that sort effect so let's begin but I start that way always on coreference we put in a light coat that's the speed of light nothing woes restrooms light now we have are moving observer a stationary observer we have are moving observer he is the trajectory of the moving observer here Is the surplus of constant time this is T equals VX Mrs. X equals Vt 1 of these is service along which the moving observers clocks a synchronized this motion here is the origin of coordinates the origin of spatial coordinates of the woman observer and now let's could amend a meterstick but sticker meterstick which is arrests in the rest frame so new car a mere stick which is at rest has tool in let's take the left end meterstick left then the sticker might start over here and it's at rest of left and the leaders stickers over here the right the meter stick His at x equals 1 and it's Anatolia the meter stick sweeps out a space time stretch he changed at time the meter stick is 1 meter longer but as time moves on meterstick sweeps out a kind rejected a ribbon through space time I know I know how long the moving observer says that the meeting meterstick how long is meterstick according to the Observer What does moving observer me the moving observer means by the length and this of course the definition of summer expand the definition of a link of meter stick In the woman frame of reference is the ball or you take the true what ends meter stick at the same instant time buckles time the moving observers time removing observer takes it to win the looks of it joins of the meterstick at a synchronous time his own reference frame and asked his own reference frame how far apart is the front-end thick in the rear and the stick after he gives instant of time while a given instant of time in them and they are moving reference frame means T equals VX trajectory over he was equals VX surface over here sell here is 1 of the media stared at that instant of time here is the other and of meter stayed at the same instant of keep trying fuel but the American stationary reference these 2 points in the moving the reference frame these 2 points of sucrose sophomore Mulhall meterstick looks immediately moving reference all we want to know is not what these acts of this point theaters but we want to know what the ex prime of that pointers who want to know what the ex prime of that point which would be what the molding observer would call the length of that now is an easy way to find out what 1 later find out just compute with this point is and the dual Arens transformation defined our X primers but there's an easier way
and it's just use proper distance proper distance means he squared minus X squared the square of the proper distance Peace Corps X women's tees with because now 1 of the other and you the fact that did the same in the 2 reference frames so let's think this point over here what is its proper distance of all call a proper time because it's not time like but instead of looking at he squared minus X Quebecor of excrement these where the what is invariant distance four-dimensional space time invariant distance between his point in that point well on the warning and if I work it out in the current frame of reference it's just keep trying squared minus X prime squares that's a square of this proper interval here where this Point has coordinates T prime what what he primary this point the prime of this point 0 0 This is the surface keep prime equal 0 this whole surface corresponds to to the origin of time coordinates for the moving observer so long hair prime is equal to 0 and the proper distance from the origin is just what square the proper distance is just minus X prime square on the other hand let's find the on Prime's coordinates of the sport this was supposed to be an 1 meter long so this corresponds to X equals 1 equals 1 and stationary reference frame the stationery stake in the stationary reference frame is exactly 1 needle of this point corresponds to X equals 1 this fall trajectory here what about that way about the height of this point facts take right that's T equals X. all along this trajectory so long hair TV is equal to 0 does was the expert lexical 1 and so on here this point 4 years x equals 1 and T equals I'm storm left inside here we have the proper distance expressed in terms of prime Court lets expressed it now in terms of the aren't prime coordinates that's going to be he squared which is B squared minus X squared which is what so I find out we discovered that kind of space coordinator of this point over here is X Prize is equal to the square root of 1 mind is a square I just Jesus sorry and take a square root Watts that removing the meterstick which is arrested you offering reference in mind frame of reference appears to have waned square root of 1 mind square another words coordinate distance between the 2 ends of the meter stick in the mold reference frame areas square root of 1 might be square next prime this . 2 0 x prime this point square warmer be squares so you see what that says that says that stationery meterstick is seen in the moving reference frame as having a shorter life again and application of these variants of a proper distance he prime square mile sex prime square is equal to T-square minus X of fraud you just go through it again this point or a 1st offered 1st point of the left here everything is equal to 0 XTX Brian primarily 0 and that's trivial but this point on the 1 hand he Prime is equal to 0 so across this out on the other hand X is equal to 1 and was equal to V so that gives someone might be square over here and tells us the next prime is equal to squared warmer squares so the comparison of the weight of the meters stakes in the tow reference frames says that when meters thick it at rest a moment observer being shorter request a is also Treblinka could be worked out a similar way removing what I we said stationery meterstick looks short the moving observer need moldings meterstick looks short the stationary observer that's the reciprocal character of a of Lorentz transformations OK any questions about the length contraction kind their waste in our Lorentz transformations or any of the geometry that we the and not boat yes revenue we not only do we know it world we certainly know it within the framework of the general theory of relativity yet yes within the framework of the general theory of relativity yes do we know as experimental facts we really have never directly seen gravitational wave would like to see gravitational waves but what we have seen is that the effects of gravitational waves coming from binary pulses ours and what's known about that tends to suggest not terribly strongly the polite thing gravity travel with the speed so now has no direct experimental evidence that gravitational waves travel would be electromagnetic waves on the other hand Einstein would have had to be very wrong about 4 not a threat to be untrue the problem but they I'm worried about not every accelerate fasteners be like Bosnia but that's a lot here are others yes From there it a lesser for years a stroke yes that's true and the latest that OK so here's a moving observer it is woman observer he is star No. 1 year's star No. 2 and that's for light away or we wanna find out how long's not in power stationary reference frame it takes to get here but how long those these moving observers say that it takes and that we have to calculate the proper time between here and here we the point is that the proper time at this point is point maybe she an
Ex the point is the proper time which is he squared the square the proper time is T-square minus X square meters only smaller than just T-square by itself so the proper time between here and here is less than the actual coordinate time so for that reason yes you've actually write this personal but he would say that it takes less time than then former light years away several would be at the stationary observer would the robbers that but the lawyer start fast well what what the only observer would say Is that he would order mowing observer would save for himself was on frame of reference and then you would look at the 2 starters and he was ceded to 2 stars were a lot closer together than what a stationary observers said so you would say it took a short amount of time to go from 1 to the other another words the moving observer would see the low rents contracted distance between the stars and so it would conclude that it took a small amount of time to go from 1 start to the other it would be a good buy you know nor Norway the stars would not appeared to be travel faster than the speed of light it what appeared to be traveling with exactly the opposite velocity the stars see the astronaut traveling with a certain velocity and the astronauts star traveling with exactly the negative velocity vector the yes but after S 1 is moving How far apart disease stars moving with close to the speed of light that see the stars as being not 5 right use of 4 light years over with said before a light years Times Square 1 might be square and that his reference frame would be the description Sarwar difficult fighters but right is With sorry set for the hour OK like like earlier this year you're correct but then we have to analyze a more complicated problem complicated problem and starts with an observer which is arrests and may again be observer starts to accelerate and has to accelerate up to the structure is velocity and we have to take into account the effects of acceleration the simple a calculation to do is to imagine an astronaut there was always forever and ever moving with uniform velocity remember the special theory of relativity just like Newton's theory only makes sense inertial reference to find out what happens among commercial reference from we can answer the question we can answer the question but the question will not be intrigued by application of exactly the same formulas that would reduce the social let yet laughter a guy with a limit talking about it an inertial reference the study none the reference frames we have to begin all over again and as you correctly point out there was obviously don't be some effects of acceleration which we have to take into account for that there is little right but Rafael at yeah not erased sorry yes the show violence yeah yup yup native was born Jew was born moving with a larger losses if West that right revenue still the same figure still calculated total proper time along trajectory so the proper time have contributions from different velocities only those out there so blue workout but if this right now we still want when twins born with the huge velocity in Ireland has left off the the poor the poor woman could be it works a mail order all Rock said but that well we that time is not a special prize our yet we tell EU right in the back of our minds we can we can't isolate time special but the lesson the special relativity is time is not that special power there were over the real question is can we recover all of the physics of slow velocities that we normally known learned in love from our Darwinian circuits that we that we inherited from from all the years that nothing ever moved faster than 50 miles an hour recovered by taking the formulas from special relativity and armed specializing in the case of very slow velocities of course we can but I also got but where as far as they that's as well yes as necessary not only unnecessary Einstein never thought about that he said time was the 4th dimension is not necessary to think of times the 4th dimension but once you start mixing it up with space where are Lorentz transformations here with an awful lot Of the physics which looks very symmetric between space and time for SunOS symmetric key squared minus X square were square mile squared they don't get mixed up with each other under a under Lorentz transformations in a way that they wouldn't under these Delhi and transferred so he's seeing Galileo kind really just his time whereas in Einstein or low-rent time really does a period of time really does get mixed up with space In a way that makes you not want tool our think of it as are fundamentally separate them and be couples geometrically part of a whole lot which is bigger than either of her young at each other but that not think of the money needed nearly million plant I wish down her so the preceding
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Titel Quantum Entanglements, Part 3 | Lecture 4
Serientitel Lecture Collection | Quantum Entanglements: Part 3
Teil 4
Anzahl der Teile 9
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/15112
Herausgeber Stanford University
Erscheinungsjahr 2008
Sprache Englisch
Produktionsjahr 2007

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