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Quantum Entanglements, Part 3  Lecture 1
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Oh my honor
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wrote this program is brought to you by Stanford University please visit us at Stanford died EDU relativity remains
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special relativity mostly and the end a little bit a little bit of general relativity but general relativity done our with some not rigorous so much as well conceptual that's not mathematical regret the before I go ahead let me just tell you 1 thing may be gone for 2 weeks I have to be in Israel had to get some lectures proved to be an of somebody Our some organizations of philosophers and those so I will be gone for 2 weeks that means I think there's anybody ever calendar and they hit I'm coming back on the 24th but I don't know what they yup yup now that we will continue we will we will gets 10 classes even if we have to go into the euro into June a little bit will get classes but I will ask necessarily have to stagger them a bit yacht old yes that's
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also on Monday isn't it Yao will be I'll be gone for the next 2 weeks well you know life is enough if this flexibility of this flexibility about the day of the week we could just won a thing like that happens they were well we will meet on Wednesday instead of on Monday or something like that Our I know some people arrange their schedule world where another and doesn't fit our butts we will have 1 class before the House that elected and then we can decide what I think so no 1 I think the 20 I'll wait wait wait wait wait while her half if nothing else the hostile overture I think begins at 8 that is a problem there I may have to be available on the well it 1 way or another what they do other gun next to Monday's and back Let's assume will leave here on for Monday after I get back at the 30th as the hostile lectures I will be I'll be now I am also the hosts the Hofstad Overture Services so it'll probably means will go to the House that overture but I want and that dog account that as a regular collapsed if we do that I will see at that time will see if we could figure out a prime tool of some way to make adjustments with the fact that a shot of us so we never do that we're know you don't Wendell that all mean specifically young now we could do that that OK good good good laps but still that yup yup so let's let's assume that on the week after I returned that we meet here Monday after returned we meet here at 6 was at work promoting talk a good 20 the 30th Young but then but that I would like to find the maybe some ways of I'm always in a situation where I have to be someplace else I have to be 2 places at the same time it's hard to do so if we can somehow double kind there but those held yeah 1 more than once that most of the time here know not none summer ahead summarize eyes I want to be a financial usually here but I need their need to stop and do nothing for the summer and the think about things but have let's Oh let's begin with special relativity that's really do the mathematics of special relativity carefully was work out our work out the details of energy momentum will press were going to work out the details of space and time and space time be details and symmetries Lynch transformations our what it means for things to be synchronous and so forth to work these things out with some care and some precision and then move on to bigger collect kinematics kinematics space and time how things move but not why gabled then comes the question why things move things moves because of forces and forces of course subject of dynamics and so will study a little bit about the dynamics of special relativity energy momentum be analog of Newton's equations not Newton's gravitational equations but Newton's equations of motion for special relativity equals M C squared all that stuff word means and I think that should probably take us about 5 lectures 5 or 6 or something like that then I would move on to some general relativity the general relativity that will do we will not be correct it will be absolutely correct but what I will teach a less of is the Einstein field equations and more on what happens to particles and objects when they're in a giving gravitational field that's always a gravitational field curvature of space time back and that kind of thing without trying to get into the harder it much more difficult subject of solving Einstein's field equations that heart is hard for me anyway so that's although this quarter basic relativity special and the general will probably get will probably get to the point where we can talk about black or a little bit maybe also get to a point where we can talk about expansion of the universe some of these some of these interesting things the flow out of a special general theory of relativity right now Thomas let's begin 1st of all what we mean by relativity specially Loblolly mean by road general would be my relativity is the idea that Of course all of you know that the laws of physics on independent of the frame of reference which is studied and frame of reference simply for our purposes here means a state of motion of set of coordinates if you like the way to think about it is still imagined based being filled up with a lattice in an imaginary lattice of meter sticks so that every point and space is labeled by a our X Hawaii and v Mr. meter sticks and at each the vertex of the lattice o'clock and the clock measures time now in Newtonian physics Newtonian physics Newton imagined that time was very very universal imagine guards time God's time with the same for everybody are there was no question of what the time wars a particular Clark clerks was synchronized
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it didn't matter how they moved it didn't matter how you'll move it did no matter how you move your watch without you watch everybody would agree and every incident at every place and the world on what the time mayors that of course had to be given up our way Einstein moved the special relativity but the idea of relativity was already there before Einstein long before I'm started basically goes back to Galileo so images remind you would Galilee in relativity is how how and why it failed on way I like to think about relativity as I like to imagine I was like to have a picture in my head father that I can our think about intuitively the picture that I have on my head and I here it is saying on Einstein North or about knows along a railroad train our with in a railroad which extremely well made so well made attracts a perfectly smooth so small that train doesn't rattle the dozen shake and they just moves with absolute uniform velocity what are the laws of physics well my favorite description of the laws of physics in this train is wellliked magic the laws of job juggling his a year's good physics incidentally you it to be a good job where you have to have very very good sense of timing you have to have a good sense of gravity now that the good stands on a lot of different aspects of physics but you don't be mean that consciously be using those unconsciously all kinds of physics comes into juggling down there are some models you throw the ball up and you wait a certain amount of time you ketchup transferred and to do whatever it is that you're supposed to do when you call the laws the laws of physics of laws job now it's a fact of nature that in this train which is moving with absolute uniform velocity relative tossed stalled trainer train that happens to be rest on the surface of years for example and surface 1 train moving 1 train standing still the laws of juggling are exactly the same this this in fact before Galileo this was something of a surprise I'm not surprised people didn't understand that they actually imagined the costly that trains to think about but you call them about trains and move along with the train the average scientists probably would have thought that when you throw a ball in the air and the train you probably have to compensate for the loss of the trying to buy the train is moving that way you probably have to blow your arm or whatever right in order to catch the ball because the trains moving will leads to you moving out from under the wrong ball of course is wrong are the laws of juggling are the same in the stationary train in the moving train identical exactly the same so father names that was something that Galileo realized go it was the 1st to realize it and Newton spelled out in great detail in his laws of motion so let's begin with Galileo Galilei relativity like all series of relativity its a theory of transformations transformations of coordinates and transformations of phenomena from 1 frame of reference to up again a frame of reference means as I said our lattice meters sticks with clocks which have been synchronized will have to discuss what we mean by synchronized but clerks synchronized at each corner of the lattice in Italy imagine having to such lattices 1 is moving relative to the other 1 is moving relative to the other In each frame of reference the clerks of the synchronized and the appropriate way and so we have the transformed information from 1 frame of reference to another of various kinds of information was the energy of an object or is a temperature of an object are I don't know any number of different kinds of quantities we could ask if you knowledge in 1 frame of reference what is it and the other Framers reference number simplest the simplest of quantities has to do with location location in space location and time the things which are recorded by the meter sticks and the clocks so how we transform from 1 frame of reference to another it when a frame of reference less 1st of all draw a frame of reference 1st thing to to do when you start a relatively problem was to draw up a picture picture is a picture of space time prime vertically as always space horizontally and at least in the beginning Our were mainly going to be interested in problems involving motion along 1 particular axis so what caught the Xaxis Hawaii in Zeit directions for the moment will be rather passive they won't play any great role for a while the commenter of things and so every point in space and time our every point on the blackboard stands for its place in space and time in time so it called based time veteran enough no that's and if you if I wanted to draw meter sticks that system engine drawing meterstick here the origin physiology Ikea tying equals 0 here the clerks of all adjusted so that time is equal to 0 all along the horizontal axis and as we floor the clocks tick off time seconds years whatever happens to be ironed out of removed their online a little bit we come to the end of the 1st meter meterstick so he is the end of the 1st meterstick its trajectory is a trajectory in the 2nd meterstick is only the stickers are here and the sequence of Swedish sticks laid out along the horizontal axis as time goes on the fire a series of stripes like this which represents locations in space a few light x equals 0 x equals 1 needless to say we can also go backward Texaco's minus 1 x equals might to insult for our there's no reason to stop Mexico 0 x equals 1 x still lexical Soviet sickle forms of their the clocks and saw clock which reads visited there is no time that we call 0 is that we never hear of talking about 0 amenities aroundtheclock Clark time zeros usually we call 12 midnight against her but anyway let's use term time equals 0 people 0 was over here Siegel's 1 1 1 1 2nd or whatever 2 seconds 3 seconds for seconds 5 seconds and draw some horizontal lines those represent lines of constant time so of space and time becomes all lattice our real Iverson imaginary lattice and you could make that lattices flying as you like but doesn't have to be meters Jacoby centimeters are microns plant points were smaller than that makes any sense we can we can make finds we like and that way every point in space time represents a precise time and a precise place according to the latter are now that you are a reference frame you are the people in the station's already the book bound train which is moving you're sitting there waiting at rest at the surface of the earth but my train is moving and my train is moving with velocity V right I in the caboose of the train at the tail end and let's say a kind equals 0 the caboose right over here but eyes was on as time goes
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ahead my train goes forward Our was across a more backward and saute Nova caboose address the my next car might look like this tragedy cause 1 need along and so my series of armed constant position coordinates on chill did relative to orders the space time plot like right now that were not up to to I signed it we're doing Galileo and noted Galileo in Newton Caressa questioner enough new people in the station give a certain set of coordinates let's say X and is an event of the year the vent means a point of space time if you label the point of space time extant Tapie in will call the stationary reference frame and what are the admits that I assigned to it in my moving reference for well as Galileo he he would look at this and say What kind this time I believe like Newton will believe in a few years that kind is universal everybody agrees about for a time and in in fact he could thought shop that they want to a good watches at that time we could do Checketts with the pendulum or something like that could take the pendulum choose carefully to put the pendulum moving train it would not trains that when a watches but he could imagine taking this pendulum moving along and bringing it back and checking it against the pendulum at arrest and if he did it well they did it carefully you would find that the 2 pendulum is agreed by the way the pendulum move would not in any way influenced however how rapidly fickle often such the pendulum until the pendulum stop moving too fast so Galileo 1st of all Lewis said he kind and that's the time in my frame of reference to moving reference frame is the same as your time but my coordinate ex prime but as Michael warned me much stake his point of India the warned that the x coordinate their assigned to this point over here it is a distance not or origin coordinates not to your origins space but the my space smile bases of he so I will assign a smaller X X Prize will be smaller than X and will be will be XUR X minus the amount that you move was sorry the amount that I moved in time T and that's going to be minus the velocity times 4 people in recalls tea exporter physical DEC's Warren's Vt where are we Is the relative velocity of the 2 frames of reference laugh we can go back and what we can of solve this equation this gives me find Courtnage in terms of the on time I could solve this and rewrite it as T equals steep trying x equals X. prime plus VAT nice I would Have I just use the 2nd equation I will write plus but since t is this a misty prime I'll just try to keep prime here the top equation would be how the more would be how our stationery observer converts his information into the information of removing observer and inverse equation would be how you go back and this is a basic mathematics of Galilee relativity let's check something 1st of all what kind of things are invariant the idea of invariants it is very important the idea of invariant or in the or invariant variant is both an adjective and now Our equations can be invariant end invariant means of things which everybody agrees about the quantity that everybody agrees about what we give you an example of a quantity that everybody agrees about Everybody meanings these 2 observers or any other ones that we also throwing question saying yes but I I was I was a young now fuzzy that go drop where beer but at the top of Garcia Entergy doesn't but you do it with ordinary watch with good Birmingham have Galileo would not only that Galileo just guessed it How would do it as you would take a highquality Rolex watch and you would just go and only 1 person would stay in the sea and the other person would march around noon up 100 times and then come back and check their watches now the answer of course that we know because of hindsight and because we're living in the 21st century and Einstein was a hundred years ago is it will be a trainee discrepancy but that discrepancy is so small that ordinary purposes but when nearly daily experience we would simply say the rate at which clocks kickoff time doesn't depend on emotion and that's an experimental fact experimental factor in the sense that they know that if we actually did it With ordinary clerks who would work who worked at very high precision because at some level it fails but that's what special relativity is about problem that but they are while the or if you might sped up as your 1 way in the we all want the bed might have happened Our that would mean that it was basically a difference between left emotional right marchers so we could try this expanded going the other way while may be the same thing would happen but that we did with its eastern West who tried North and South we might also take my watch him go around you and circles for a long period of time all kinds of experiments you will be all kinds of all kinds of ways we try all sorts of obvious things will work work right arm of in the Yukos abused telephone 3rd communicate the information back our satellite signals back and forth to to check our watches and selfless and would work very well there were very well until we start to move too fast we then the bricks that public the point is that Galloway notably was based on guests who frankly based on a guest and the mathematical translation of the guests disease equation right now there are some obvious invariants for example something that's in area and it instead of time to look at that distance between 2 can't switch are synchronous or simultaneous simultaneous being used to mean at the same time they truly events at the same time the distance between them will be invariant Why is that let's take the tour events would stick the tour events such call them X 1 at a time 1 an X tools at also at the same time feel it's not even bother labeling the time at the same time these 2 events at the same time as the distance
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between them is simply X 2 minus X 1 and that we call distance about the current quarter the prime coordinates 1 is equal to X 1 minors X climbed to his X 2 miners VT notice that the actual coordinates are not in area they don't assign the same values of addicts but they do assigned the same values if we now subtract to or subtract and the bottom 1 the top 1 from the bottom 1 next time minus X prime 1 is equal to X to minus X 1 and then the canceled so what separation between 2 point at the scene instant of time I emphasize at the same instant times would be true if we good different instant of turned out all right now that With an event X 1 t and another event that X this is city over here Chuck can't resist acts 2 years Ex warm the separation between 2 points is in an area that story withdrew due lets legend that we have a particle trajectory a particle trajectory meetings and X T X as a function of TV would be a trajectory of some particle probity trajectory of some particle is an ex at each a function of time 1st of all is the position invariant not XNX primer not the same thing but we can write next of physical tax miners bt IXR final I know the trajectory acts of tea memory of tea but times the final the trajectory of a particle what you'll know the trajectory of a particle acts of Ucon communicated to me and the communication will be in the form of telling me what X kind of 2 years or I will say is an ex prime of treaty which will be your acts of T minus but the coordinator the particle itself is not a very warm out the velocity the velocity is just the kind a river of the position price of a velocity which would just be be a car the ex prime might seem at to DX by C minus V E so not velocities are not in area what you'll see is the velocity I see the velocity being diminished or the leak of course be negative fearing narrows the finger intrinsic go about assigned every but armed I see a velocity which has shifted by that's obvious of course I'm moving along you should demand I see the arrow going slower than you see the arrow going for obvious reasons what about acceleration personal position and velocity are not relativistic invariant Galileo's theory of relativity but differentiating didn't get the acceleration of these 2nd acts prime by a pretty 2nd X Prize by deeply squared that's the acceleration that I see is equal to the acceleration AUC minus the time derivative me but he is a constant I am assuming that the relative velocity between the 2 frames of reference is constant at Sequoyah weren't saying that these tilted lying a straight line so don't be acceleration the difference he doesn't make any difference the time derivative of a constant 0 so acceleration is in an area the acceleration of icy of an object but my acceleration and that accelerator and the caboose train and moving along with uniform velocity but I see objects moving back and forth in front of me with less the right sometimes accelerate you see the same objects you also see them accelerate and we will agree about the acceleration of acceleration is invariant now I know I shouldn't and that stretches Marshall which is cause and effect here dreary but let's say it this way the fact that acceleration is itself an invariant is a strong suggestion that the laws of physics should be in terms of acceleration not velocity not position the laws of physics the laws of motion if they are to be invariant if they ought to be invariant the laws of physics should involve acceleration since we all agree on the acceleration well of course that's exactly what sequels and said Forest physical demands fans acceleration acceleration we all agree about acceleration well let's analyzes so little more closely surprising we're talking about tool objects which exert forces on each other and for the moment everything is taking place along the xaxis With we usually a magic correctly for most cases that the 1st between 2 objects only depends on the distance between them only depends on the distance between them between 2 objects gravitational force electrostatic force only depends on the distance between 1 bits of the particles themselves are not variant but the distance between them is invariant bats what we win we write Obama top everybody agrees about the distance of the distances in invariant and if the force only depends on the this and it says that the force between 2 objects Is it not meeting to say that everybody agrees about it so their fuel in your framework to measure the force between these 2 objects which are moving some and I my frame whizzing by it measure before us according to the Newton and Galileo and everybody who came after them motto Einstein is a invariant be cause the distance between 2 objects is very badly is all the more similar in this equation massed in order for Newton's theory to make any sound it was necessary that masses of area if all the terms of the equation are invariant then so must be the mass and so according to every object has amassed and everybody agrees about that there is no discrepancy because of object is moving our about 1 kilogram object is 1 kilogram object is 1 kilogram object even it's moving past your railroad train so no use law motion is In variant his language no long motion is In invariant under Galilei and coordinate transformation is a gallon coordinate transformation and Newton's laws of motion are invariant the idea of relativity is all laws of nature wherever they happen to be are invariant under the appropriate analog of of egalitarian and none of the going transformation now the problem is that that clashed with electromagnetism at the
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moment we have learned anything about electromagnetism but you all know that electromagnetic waves a called light and the right moves with certain velocity Maxwell's equations predict what that velocity years and you think of Maxwell's equations as lawyers of physics that everybody should agree on the speed of light must be the same for everybody that's really all there was to it Einstein really use no more than that because he was guessing he was guessing that Maxwell's equations were the same every reference frame virtues guessing Maxwell's equations which will will laws of nature which work so compelling that they would have to be the same in every reference frame and that of course led to a big problem it led to the problem of the speed of light that if Maxwell's equations of the same in all reference frames than the speed of light is the same in all reference frames but how could the speed of light be the same in all reference frames 1 frame is moving relative to the other we work it out work it out using Galilaean relativity Our let's let's take a library a light gray will Blair the butt of a lightly moving along in along with x equals see kind please that's what we see is a speed of light arms that's what Maxwell's equations said well galileo would then have had to say let's ask however light rate appears to move from the point of view of the molding reference for him and that will be X Prize which is equal to X's might Vt so far this is just a just be a gallery and representation of the transformation that is equal to CT Exs eagled the CT miners bt said but same thing as saying C minus they say so Galileo would have not choice but to say in removing reference frame the speed of light differs from the speed of light in the stationary reference frame by the relative velocity Our that of course make good intuitive sense it is always salmwood work it is the way you water waves would work if water waves move which awards typical velocity waterways Kiepper meters 2nd or something like that much that EU law yes yes yes yes we possess a 1 meter per 2nd but I'm flying over and a half a meter per 2nd from fly all the way that I see the way it will be half a meter per 2nd so it's completely unreasonable throughout to expect that waves Have a universal velocity that all observers see now most physicist of course at the turn of the art and the 19th century simply came to the conclusion that well 1850 18 sixties when Maxwell was it was active simply came to the conclusion that there some finds a reference better than other frames of reference that there is no relativity father theory exists AT frame of reference God's frame of reference don't tell the ombudsman said that what do we tell the ombudsman guards frame of reference where Maxwell's equations are correct in either of frames of reference Maxwell's equations must be somewhat different because other frames of reference obviously like those with different velocity and saw as you well know various physicist tried study b dependents of loss b of light on the state of motion of the observer and systematically they found no dependents will go through media various experiments several different kinds of experiments to try to detect the fact that our moving observer would see a speed of light which would depend on the state of motion and as you know no such effect was discovered after result being the physicists had no choice but to go and review the basic ideas that went into this definition no I himself claimed have even knew about the experiments I'm inclined to believe them not bidding call Einstein Einstein for nothing he is a 5th so that give credit them here he said of his 60 years on but about along with a lightweight politely of didn't did oscillate didn't move to do anything just look flows in and he somehow know enough to know that Maxwell's equations were inconsistent with that and his choice was to believe that Maxwell's equations were basic laws of nature basic laws of nature which should be the same in every reference so something was wrong had to go back to some very very 1st principles In fact the very 1st principles which this lattice represents the idea of coordinate the idea of kind that distinct times and so forth and went back and said let me begin all over again with 1 assumptions about to assumption the 1st assumption is a principle of relativity namely that in every frame of reference the laws of physics are identical the 2nd assumption is that in every frame of reference the speed of light is always the same let me take those as posture works and see what I can derive the 1st thing he derived it was something that is implicit in this diagram which a partly spelled out partly didn't spell out but let me spell out now that if true events are simultaneous as seen in 1 reference frame and they will be simultaneous in every reference frame it's exactly the statement that keep prime equals if 2 events car at the same time in 1 reference frame they occur at the same time in any reference Einstein realized very quickly that he could not maintain that his visit is roughly the way thought about it so let's let's go way back to the beginning and even start with the whole idea of synchronizing clocks we can lay down on leaders sticks clear enough where we can do we could lay out meter sticks with my own eyes fixed on the ground but now you have your clock you have your clock you of your clock you of your clock over their enormous synchronizer clocks Mitchellville read the same at the same instant at time what does that mean it was various ways you could try the check with your clocks were were synchronized but Einstein decided to use a light you want to use light but for the obvious reason that his basic postulates was about life so we wanted to use light to synchronize clocks so he is years away he synchronized clocks facility but somebody in the middle now the middle Maine but stick you in your to end points was a means to say somebody little well you could measure with sticks and just the leader sticks and Colette person middle is exactly halfway between you now had exactly whom you're going to eat a flashbulb you know life a flashbulb our like signal was go abroad rule out all around you but particularly the goal from our from your position I'm not sure you know exactly the same thing when you are Clark is also at time people 0 5 people 0 0 1 or whatever 2 I had known you will also send out flesh if the person who was in the middle sees the flashes simultaneous 40 members of the left flash right flashed go right by him simultaneously then we will
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agree that your clocks a synchronized if he sees few like your flesh before your flesh misses you overlook the you'd better readjust to clock a little bit back try again set the clock back a little bit and keep doing it the person in the middle Our sees the flashes of light instantaneously saying that everybody in the room does the same thing with respect everybody else until everybody's clocks a synchronized by the same exactly the same definition that somebody halfway between them our will see flashes of light emitted at the same Coetzer saying time will see those flashes of light at the same time located then I Einstein realized it was a problem that that much is OK that much is OK we can synchronize all of your clocks but know about me moving past your honor now so let's meet moving past you ends with the should work I have my recorded frame laid out hands final past you Oh I look as your life Ray Araujo a light ready but because I'm moving by the kind your lightly gets to me has had to go further than wiry said quite obvious that I'm going to see your library before a to lightly even if you omitted them at exactly the same time even if you emitted them exactly the same time according to all of you reckoning full happened to be moving this way will wrecking your life ready as being needed later because I see it later because it has a greater distance to go so I say I realized that the idea of sidewalk and was a relative idea that if everybody synchronizes their clocks by means of light means of light it will be a discrepancy in what in different frames of reference cares counted as as simultaneous 1 a work that out I wanna worked out in detail just using light How the simultaneous how this open 80 break down another way to it is if these 2 points counted in the stationary reference frame as having the same time which killed points In the moving reference frame will be said to have exactly the same time that something or work out X will pick 5 minutes and work that out in detail and that'll give us the Lorentz transformation grunts transformations which implicitly have in and that's I'm open AD is a relative concept show that will be our next challenge to replace this equation but 1 that consistent with the idea that light travels through the same velocity all using what that meant you will find then you will find that the law that says that the speed of sound is what is it about a thousand feet per 2nd if that's not a lot at all and morning observers do detect different velocities of sound you if you do and ever arrest in a frame of reference would the at rest that's a perfectly good way to synchronize is about to synchronize clocks but then somebody moving our you'll do all these things you discover in the and that it's not consistent to say that the speed of sound the same in every reference or 2 of the abuse you lose the idea that like this there's nothing wrong with synchronizing clocks that way nothing at all you wouldn't want to do it so you wouldn't wanna synchronized clocks on a moving train is the train was an open air train and and you get the point it was an openair train and the atmosphere was arrest and you are moving through the atmosphere that would be a better way desynchronize supervisory require require Yup yup yup exactly exactly uncensored using care he is going to simply suing the speed of light is same and see where it takes an hour would leads lead to something consistent but so has said his 1st observation was something wrong with the idea of but sigh Multan maybe is universal or that the Symington maybe 2 we is an invariant Sally began years I had acted on how we gives away it's been too long since I've read newspaper but again we begin with the drawn and now we have to we this is the rest frame and we have 2 0 0 points laid out a distant they apart this distance L and this distance else there now they could be 1 meter but let's keep a general distant L is measured off another distant Elise measure off and here are the trajectories of the points which corresponds to Lula being a part of this could be your seat and that could be your seat and around the world since equally distance between the 2 of them is a person Rapids to be vacant but was sitting in the middle of a between the 2 of your OK so that that's refer set up now if the clocks of synchronized wasn't mean exactly it means that if at times he equals 0 0 most observers at 2 wins here send a light signal the light signal light signals will intersect at the same place like best Forbes list the right signal from this side it it sent down at the same instant another words synchronous in this frame of reference will light signals sent from this instead that the light signals will cost the intermediate observer here at exactly the same time now right of course very fast that means the drawing is a bit the well it's a little funny because like moving so fast that would mean that like the the Raiders a practically horizontal they move Lola's distance in a short amount of time but we can fix that we can fix that by working units in which the speed of light is equal to 1 let's do that later on will cut back the speed of light but for the moment let's set the speed of light equaled the 1 how do you do that you work in units of space which are adjusted the right way so that in 1 2nd of time our light goes 1
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unit of distance alike set for use in light years or whatever whatever you like and so the vertical axis in the horizontal axis here are measured in units in which the speed of light is equal to 1 of the speed of light is equal to 1 that all light satisfies x equals t not CT but just TV and that would be a 45 degree line that's a lightweight moving to right a lightweight moving to the right satisfied politely moving to the right From the origin over here satisfies x equals wants a light moving right from some of the positions as well then it satisfies X eagles he plus some constant may be offset of like raised any library moving to the right satisfies x equaled keep plus something since thank you b Colorado removing the left but X equals might tea or minus steep last summer but constant prime so it is that these are the emotions of light rays and they cast on the 45 degree angles 45 degree angles slope of 1 a slope of 1 corresponds to a velocity of the speed of light things moving slower than the speed of light mover move more toward the vertical if it makes sense to think about things moving faster than the speed of light which it doesn't they would be moving on trajectories which or more horizontal selloff these Diller had these 2 clocks are synchronized because if at the same instant of time or another way of saying it it is 2 points here must be at the same instant of time in this reference frame the stationary reference frame be caused polite raised will intersect intermediate observer here right at the same instant of time now let's straw in a 2nd observer 2nd observer of words a 2nd reference and the 2nd reference frame is moving supposed to be a straight line and its moving with most of them ex it equals v he is now the velocity of removing observer and this year is now x equals VT from might From newer perspective I'm moving along with velocity you watch me and you say each incident of time my position moves and moves with velocity read this is infinitely the definition of velocity This is the definition of the velocity of you see me move I think so this immoral Ikea let's try some more lines he is here's a 2nd I wanted those good parallel to the 1st seller Palo close enough resources also pretty peril bird back OK we wandered now must look at the moving frame of reference again this guy is halfway between these rule we started out over here halfway between the 2 of them only with uniform velocity he stays halfway between these 2 0 but noticed that at his location the 2 libraries do not arrive at his position at the the same time and his location 1 of the light rays as much later than the other 1 so according to this woman observer he must conclude that these 2 points were not synchronous the light rays emitted from here here could not amend synchronous from his point of view because from his point of view they show alive at his at the intermediate position at the same time in fact he can reverse the logic he can say where are all along this white ready does it correspondent for Tse however say this right arm was forget this library frenetic and let's follow this 1 Grant Hill and actually hits the observer this is the person the where donor life away half to come from all along this trajectory in order that the guy in the middle he could would conclude that the 2 libraries were admitted simultaneously it's easy just faces a backward like also at 45 degrees for that point tracer library backward along 45 degrees point of view and in the moving frame of reference for like is in no way here and I like isn't over here this is a bit of an exaggeration I think this line 2 were too shallow but more at exactly the same place he added if at all right they were emitted from here the left analyze they were remain from he had the right they would hit the central observe all here at exactly the same time so the motor observer will conclude that this point and this point us inquiry was a stationary observer concludes that this point in this point us agree that's obvious from the beginning that the idea of synchronous cannot be the same in the 2 frame but can we do it quantitatively really figure out quantitatively exactly where this point lies news and true we figure out all the point that the moving observer will call synchronous with the origin over here and that's not too hard just requires a little bit of simple algebra geometry now let's work it out the 1st thing to do the 1st thing to do is to make some simple equations for line drawn he reasoned simple quantitative I wanna right summer equations for the areas line this line x equals was a slide over here this line is X X equals V class AL white because the shifted over by distance so for it's X Is people assailed the acts means from the stationary reference frame the stationary reference frame seized the black
1:01:57
lines shifted up by distant Al at x equals beekeeper sell about this 1 bt plus 2 at Texaco's weekly post to we won locate disappoint the object is to locate this point right over here why because it's the 1 that the moving observers says a synchronous will be origin like a final word what's the equation for this line Mexico's take that equation as x equals to eat less 1st locate this point at this point is at the intersection of 2 lines that we not 1 of them is is Mexico's cheat and the other is X equals people cells but Frankel's 2 equations and find out where those 2 lines intersect so religious Texaco's tea and the other 1 6 Eagles VT plus L I guess easy will UCX but yet TV Eagles fee plus or 1 minus V terms TV equals 1 more step T equals Aiello divided by 1 might be glad right this jacket yes but this point right over here it closed in the rest frame at a time which is V which is divided by 1 might about acts would IPX about point did Siegel the tea all along this line the same thing X equals L although 1 might that's this point of ideological appointed capital baby this is a point capital now don't we know appoint capital 88 is let's see if we can trace all backward on till it hits here and figure out with this point it at all gold defined that with this point but what about this line early about this line we know about this alliance that's a backward going like ready so it's sort of a former ex my he is equal to or constant Posada X equals mowing the people's a constant the backward Gordon library his ex equals minus see plus something but it is give us something our Kobe Bryant accord be part of later right it is that X plus T Eagles beat widowed is yet come identifying beef this is X plus equals be qualified be Ligon idea that right now we don't know is we use the fact is lying goes to this point and this aligned goes through this following at this point X equal normally equal to very equal to L Over 1 is so at this point over here X plus she's just twice L divided by 1 minus V and that's equal debate so we found we found out what these His is that goes into this equation was backward moving Light Railway here and try them next plus is equal to L. writer by warmer and his date so now I know this line but I wanna find its intersection with this 1 all right right underneath this equation all right the equation for this line that X equals VT plus 2 this is right have to equations that county or where the intersection pointers that's a show we solve them a bit of a nuisance to solve them but let's do it Our OK not hard let's subtract the 2 equations that will get rid of X for us subtracted 2 equations that will be rid of X that'll give me on the left here and on the righthand side I will have equals tool L 1 miners very might this looks like the guerrillas right minus Vt minus 2 well but plus not my tour minus rout let's move all the key staff that gives us she plans 1 minus V equals plus 3 eagles layout of these turns out to Alan M. and then 1 of them has 101 miners V and the other 1 just as minus 1 Tourello it is common to both of them out here and it's just below is just 101 miners minus warned OK what's 101 miners B minus 1 hour over warm as ministry yup 101 miners B minus 1 might feel warmer and as we want runs were cancels be over murders rewrite the organizers and finally Herrera now located where teachers key is frenzy delighted by 1 minus V times 1 plus G 1 Minors NYTimes rain it is just 1 might be squared so 1st of all I found is equal to over 1 might be square that's the kind of this point over here what about it acts what's easier defined the confined different to ways weakening the Finder but clothing into this allowing off but into this which 1 simple Alexy II looks to me like it's easier to plug into this 1 OK so as this 1 that says that X is equal to 2 L provided by oneliners might this he might QLV over 1 might be squared
1:10:02
later than they had now bill that allow left we will be that really do we better multiply top and bottom by 1 close beasts of earlier the same with the same amount Saudi multiply by 1 plus well what to accept a break 2 at all times 1 plus provided by 1 might be squared minus 2 L beat it looks like to me like it's just too well divided by 1 might be squared Her sorts x equals to over 1 might be squared no I'm actually not so interested in Munich he or acts what I'm interested in is the the slope of the line uninterested in the slope of this line which is the ratio of FedEx I'm actually interested in what the this is interested in the current year this curve has the former x equals something kind key bowler occurring off this now it's is the the ratio of expertly what the ratio of extricate the ratio of expertise is being so this curve is Key equals VX was checked that T equals VX yes she has an extra factor review unit from X T equals NYTimes Air or scanner interesting we did this whole big construction and what we found out is that with the more things observer corn synchronous this stationary observer recalls equals VX we could always everything else was just keep track here annually observer he also with x equals Vt but his X is that his climax x equals V but they Xaxis is people's VX all these points along here are the same position from a point of view of the moving observer all these points along here are at the same time from the point of view of removing observer all these points a synchronous from moving observers point of view not from the stationary point field not 1 of them says X equals VT the other assist Eagles reacts very symmetrically related to each other very symmetrically related rights as abstract from the whole thing now what we learned because Siegel 1 so let's actually put some units into it but Skippy units right when you don't feel like it sets equal to 1 then of course links in times have exactly the same units and velocities of dimensionless velocity is a ratio link to a time if he is 1 velocities a dimensionless but let's try to get some dimensions back into would restore some correct mentioned till it always have to do is multiplied by the appropriate powers Of the speed of light in order to get the dimensions to be consistent this is already consisted X is equal to velocity which is a length over time of time but this 1 does not consistent with the do it to make it consistent the answer is I think divided by the square of the speed of light plans alive is set equal to what this cursed equals VX is a unique way to restore the seas so that you get the same insolvency is equal to 1 but so that dimensions a restored but check the dimensions this is a time maritime equals now a velocity is only over time and X is a link and dominate us we have a late squared off over it times Square so let's see that is that is in fact dimensionally correct yacht a C in the demand made us squared the nominator restores the image of every include the speed of light that would be formula in fact including the speed of light you see something interesting you see that when the speed of light is very very large the slope of this line is very very small Key equals V times context divided by enormously big number the speed of light square metric units the 17th or something so this an enormously big number downstairs and that's says that this is a very very shallow line so shallow that the most practical purposes signed alternate any really is the same in all reference frame but when things are moving with anything like the speed of light we have to account for a road for these discrepancies and so but we still feel like the 1 in the diagram has a lovely symmetry the slope of this line relative to a horizontal is the same as a slope of this line relative to the vertical pituitary drug purchase extract that we have your coordinates ex ante and we have my coordinates our key prime Annex prime Our white signal what's the right signal and they all like signal moving to the right is a 45 degrees at AlliedSignal Melinda letters of 135 degrees C facility these no coordinates are so symmetric relative a library here but we surface which is the surface of Symington it's given name is the surface of simultaneous when moving reference frame set of points that the observers says of all at the same time till did relative to the surface of some of them may be in the rest frame and they still did exactly the same way meant so that the axis of the time axis is tilted so some very simple construction now from this construction we can make some guesses not guesses we can actually derived former transformation wraps a less work out urine strands with no good deal about the relation between X T Annex prime and keep prices we good deal but we don't know everything so we need to we need more input but saw back Galileo Galileo said X prime is equal tax miners VT all this said is all this says it is that when X is equal to V T X prime is equal to 0 but that's should be true that should continue to be true it should be continued continued that as far as your concerns my position satisfies x equals VT so X is equal to Vt you're talking about my origin of court so must still be true not next primacy Quebec's might but that when X is equal to E ex prime must be 0 my center of coordinates is what you call x equals bt that continues to be true that's just a statement that this wine is X equals Vt and this 1 of courses also ex prime equal 0 it's also like prime equal 0 it's the center of cool admits Of the moving prime observer so of this equation may be wrong but it must be correct that 1 x equals VTX Prime is equal to 0 how could this be wrong but still be true that when Exs equaled the VTX Prime is equal to 0 is only 1 way but they could be some factor in Gila call and factor could depend on velocity that's a possibility Mrs. still consistent with the statement that when X is equal to VTX Prime is equal to 0 but it's not quite the same as saying its prime Quebec's weekly it's a little different than were Delaware were written depending on what after views here so Our this
1:19:42
is about although you could do with the equation with Galileo's 1st equation that still allows X primed to be 0 when X is equal to VT perhaps some miracle multiple there and then the medical multiple can't principal depend on the velocity and as a say all this is telling us here is that X equals VT is the same as ex prime equals Iraq now let's look at this 2nd careers the 2nd car here is keep time equals 0 why is it keep prime equals 0 because all long this the prime observer moving observer says that the time was saying if we just she was it at this point to be 0 0 here they keep time is equal to 0 all along the curve source says for every equation for keep violators it must have performed key miners VX possibly from some other function of velocity what is it being model OMI saying this all this it is he equals VX that deep prime is equal to 0 . 2 eagles VXT Prime is equal to 0 this is about all that you could have and still be consistent with saying that this is the access X Prize equals 0 and this is the axis steep Prime musical 0 so I signed notice down but 1 more step this is not quite enough because we don't know what this africans G or they could be anything they could be anything these 2 equations he tell us about the tilt of these acts he used but they don't tell us what F and G R B 1 more step up there with 1 more step the 1 more step is that whatever the equations that you used to convey to me information should be essentially exactly the same as the equations I would use to convey information back to you with 1 exception the sign of a velocity the of lost city another words other words if we were to solve these equations for Exon In terms of export keep prior the equation should have exactly the same war accept the sign of the velocity should change so much so that let's solve these equations off for X and TV in terms of ex prime prior to the inverse relation if this is what removing coordinates are in terms of the stationary coordinates we solve for ex ante will know what the stationary coordinates are in terms of moving toward and the pattern should be essentially exactly the same if you see me moving with velocity v that I see you moving in the opposite direction with velocity and the only difference in the family relationship would be the site of the velocity acts on all these equations with this is a nuisance very straightforward but let's do it anyway Clark no but proved something courage let's 1st proved that Athens GA have to be the same why the weather and you have
1:23:37
to be the same that is statement that everybody agrees about the speed of light the words of 1 observer causes lying x equals T 3 other observer must call X prime Nichols if 1 of their says the speed of light is 1 that means x equals the other observer most also said the speed of light is warning that was Einstein's basic principle that we all agree about the speed of light life race satisfies x equals and must also satisfy X barnacles P prime let's see if we can see what that says if X equals T ready then X Prize here is just equal to T E times oneliners we all I just use that X is equal to carry on the other hand seemed time is equal to T Cunha 1 minus V apology me again just used that X is equal to T along like rate so alike were 80 x primers given by this time is given by this but it must also be true that X Prize must equal T prime otherwise the speed of light would not be the same 2 frame while at that time was must be called deep prime it means and G have to be the same function choices about it so Einstein's assumption that the speed of light is the same in all reference frames required that F and G of the same so we learned a little more With Jan reviews the idea of sidewalk maybe as being defined by a light rays coming from the left and the right and hooking up at the same instant of time for the observing halfway between that told us what this curve wars and now we've used the statement that the speed of light it is the same in all reference price and other words x equals along here then ex prime mistake will keep trying and that tells us that the functions it was a spot not quite everything we cannot walk we still don't know what FOV years he has revealed valid that is just some unknown function that maybe you will find some reason to set equal to something particular must be some which tells us where is Ilex now solve their X and he in terms of X time end deep price that's just solve these relationships of OK let's just let's do it multiply this by effort the FOV plans X prime is equal to X's miners VT F RV tires keep Prime is equal to T miners VX Nike let's see what we do with 2 I want to software X In terms of the prime variables so let's X in terms of prime variables and the best way to do that is to divide this equation by let's see some 18 hours multiply this equation by each year why I do that I did that referred to subtract them or get rid of the BT now have an equation for acts Philips Europe's says just a Ed them so I get X times 1 might be squared Armour right V please canceled and that's equal to move on f viii X prime plus V F of V T prior religious right that were Meister effort crimes I think it was just X close VT Prime and now divided by 1 might be squared that's the equations that acts as an equation for tea which is very similar the equations the team looks very similar in looks like FOR divided by 1 might be squared times cheap closes X Prize that's the inverse relationship with you go from T X X climbed through 1 7 equations than later you go back is this equation well you expect if there is symmetry between the 2 frames of reference that the 2 sets of equations should look very parallel each other In fact X and he should be related tax Prize T Prime in exactly the the same way that expired you keep prime related to accident scene with 1 exception with the exception of the sign of the velocity should be opposite well that doesn't look so bad we have acts Prime is X minus V T time something Healy have X is equal DEC's prime plus VT prime time something if the somethings all the same after aren't we all 1 might be squared with the same as 1 over the relationship would be completely reciprocal relationship would be completely reciprocal saying here the relation between the prime variables when you In verdict simply involves changing the sign of a velocity from T minus V X could keep prime plus BX that will be true FOR V delivered by 1 minus V squared is the same as 1 over half of these so far I haven't told you ever years now I'm
1:31:00
telling you it's the thing which satisfies this equation for another later right it is that is equal to the square root of 1 mind is a square multiplied by effluvium both sides squares this multiply by 1 might be squared 1 Masri squared and take a square what follows is what I do what I did was simply to make sure that the relation between the kind in neon prime coordinator for reciprocal mad 50 on observer tells a primed observer that his coordinates are related the accord a related a certain way inverse relationship should be of exactly the same form except with the velocity having the opposite side all art says if you see me move velocity v I see you move with velocity minors they so now we have the complete set of relationships between the coordinates of a point of spacetime in 1 frame and recorded in another frame and here they are X prime is equal X minus V T 5 by square root of 1 might be squared he Prime is equal to those key minus V X all square of oneliners square we can solve them just didn't work right now with the solution is solving it meaning to relate right equations that and annexes the private people the investor relations are X is equal tax time Class Prime by the square root of 1 mind as we squared and key is equal to keep prime plus VAT at Prime of a square of 1 might be squared why has 1 have a possibly in the other speak for exactly the reasons I said 1 velocity is the opposite on the other you see me with 1 0 loss the ICU with the other velocity these are the relationships between the coordinates of 2 reference frames which is sure that everybody sees the the same velocity of light that life has same Milosevic everybody and that the relationship between coordinates a reciprocal you never know I email reciprocal but they have the same form no matter which way you go from 1 side to the other and the quite unique so that's it and new thing he ever knew bizarre thing is that truly is not just times that what's synchronous in 1 frame of reference in other words he can't there is not the same as a prime equals a constant as new things that synchronicity is not the same in both reference right now I left the speed of light by now I should put back the speed of light see not equal to 1 and all I have to do that is to on and tomato the equations have dimensional consistently X In VT have the same dimensions that they have the same dimensions as prime next time Ex and NVT all have the same dimensions well the only thing that's out of kilter here is the velocity does not have the same dimensions as 1 you cannot subtract the thing with 1 set of dimensions from beam with another set of dimensions 1 has no dimensions of dimensionless Bossidy linked over time you always have to do to fix this Is divide square by square 1 might we square oversee square if he is set equal to 1 reproduces original if is not equal to 1 its dimensionally consist of about this equation he'll sell this 1 it is a little bit tougher art tea and VX do not have the same units keep climbing have the same units but VX does not have the units of time we have divided by the square of the speed of light C squared and again puts square dance George P. minus V overseas Quarterdeck's provided by a squirt of warmer as we squared overseas quarters the those Jacinto look at what happens when the speed of light get you work the speed of light ordinary units is huge that means that this is negligible and the usual situation the usual situations where the speed of light tend be 3 times and meters per 2nd square of the speed of light is 3 per 9 canned 10th of the 17th this is a huge number in the nominator so far almost all circumstances this is negligible also be square overseas squared it is also negligible the velocity of an ordinary object is very much smaller than the speed of light so this practically reads keep Prime is equal to take Galileo's equation is only begins the failed when the velocity gets up near the speed of light only when the speed the relative velocities of the observers and near the speed of light does this differ appreciably from Galileo's keep prior Nichols please same thing here when the velocity is much smaller than see you can ignore this and it's just good old ex prime equal tax miners bt only when velocities anew the speed of light that the collections from road Trivedi become important power but there are especially important for light which does more with the speed of light Yelets even in see from these equations just directly are better like moves with the same velocity alive the reference frame we're ready dated but overtures do again column Yao looks spurred him if X is equal to see a light rail I race satisfies x equals T effects eagles x equals see this just becomes 1 minus V times t divided by square root of 1 might be square oversee squares that's 6 5 what about
1:38:28
T prime again X is able to T T prime TV my guess is that they are making a mistake I can't both have x equaled and have the speed of light about equal to 1 I think I want sickle CTU young Mexico City so this is C minus V see my bt all of the squared 1 oneliners but overseas quartered and this is see priorities are prime is a corporate TV Miners overseas squared forms X which is CT that's C T provided by square of which is equal to 3 times oneliners Viola CE wherever square XOX Prime is this key Prime is this what is ex prime over tea but got all the works are nowhere to be seen as a B C but multiplied by by Celia C minus V oversee but yacht this Steelers NYTimes dealers squared 1 oneliners be squared but we have an extra factor see downstairs so see climb on our X prime is equal C prime whenever X is equal to CT magic the speed of light was the same in all reference frame with this transformation on end product here it is important that you cannot Trista stomps or whereabouts or as X equals we just proved that next time is equal to seek a prime put exercise is to show that x equals of minors CT lightly going in the other direction that X prime is able to minors CT price that's something else you could check from it also true so that means a light really going in either direction will move with the speed of light in either of the 2 frames what's special about these transformations namely that like most of the same velocity and the frame and the relationship between the 2 of us at coordinates is reciprocal has exactly the same form except for the velocity having changed as special about them and visa Kosovo transformation is not the way that learned the right there and that a new node Lawrence did the right arm souls that special relativity or at least a portion of special relativity that has to do with the X he scored armed and they were met finished now what I was going to do which food next time is to derive the contraction of late the below the violation of time and some various a special relativity effects of violation time be good most interesting 1 and then move on to more about the kinematics a special relativity were proper time is what momentum is what energy is so full of our any questions thus far about worried about us the love of the boxer dropped up any questions by the physics of the young it to is not that time that he is not what you're saying OK but stood but the but they do Isidra picture is a forget this but our yeah let's talk about somebody else so happens the pesto at this point over here these moving but he sees little flashes of light at the same instant of time right as we are talking about but he's not midway between the 2 observers he doesn't satisfy the criteria that he's midway between the 2 observers the yeah right right so there's no reason for he always easier the city's to flashes were the same instant he sees the light get him at the same instant but he's not midway between them so he again you will not call this point simultaneous this is a point which is midway between them and this is the test point for deciding signed with the maybe from moving frame yes a yacht right right right right because at the Rose was further away from him the network this this is a nice construction to go through is not very hard all you do is you take x equals tea which is a light gray passes the point x equals beat people settled and Mercury backward with a backward like moving like our backward library until it hits the point x Eagles people too well and that tells you exactly where the point is that tells you what count as simultaneous moving reference for that's key that have the key as have the key to Lorentz transformation of other half is a statement that relationship between the 2 frames should be symmetrical a pod From the change of sign velocity Bozo the 2 key ingredients of Einstein's theory Our in 1 form or another these were the equations these will be principles of you know what our digital for sold were finished that foreign book him a room in the
1:45:49
preceding program is copyrighted by Stanford University please visit us at stanford . edu
00:00
Juni
Schiffsklassifikation
Relativistische Mechanik
Jacht
Computeranimation
Woche
01:30
Erder
Summer
Magnetisches Dipolmoment
Kugelblitz
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Schulflugzeug
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Heft
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17:54
Magnetisches Dipolmoment
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Front <Meteorologie>
Waffentechnik
Windrose
Bett
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Druckkraft
Großtransformator
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Schalter
36:25
Erder
Atomphysiker
Fahrzeugsitz
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Magnetisches Dipolmoment
Lichtgeschwindigkeit
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53:51
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1:10:01
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1:19:42
Sommerreifen
Relativistische Mechanik
Zugangsnetz
Pulsationsveränderlicher
Rollsteig
Kraftfahrzeugexport
Bergmann
Lichtgeschwindigkeit
Wetter
Schnittmuster
Satz <Drucktechnik>
Grundierung <Technik>
Gleichstrom
Januar
Bildfrequenz
Jahr
Quecksilberschalter
Source <Elektronik>
Modellbauer
Fahrgeschwindigkeit
Passfeder
Vorlesung/Konferenz
Multiplizität
Stunde
Unterwasserfahrzeug
1:31:00
Magic <Funkaufklärung>
Penumbra
Direkte Messung
Kraftfahrzeughandwerk
Bergmann
Drachenpunkt
Stromschiene
Lichtgeschwindigkeit
Satz <Drucktechnik>
Korporale
Jacht
Leistungssteuerung
Nanometerbereich
Propellergondel
Bildfrequenz
Messschieber
Fahrgeschwindigkeit
Reziprokes Gitter
Passfeder
Vorlesung/Konferenz
Klangeffekt
Störlichtbogen
Kaltumformen
Relativistische Mechanik
Waffentechnik
Licht
GaußBündel
Standardzelle
Rootsgebläse
Schiffsklassifikation
Glimmlampe
Satzspiegel
Gleichstrom
Großtransformator
Ford Mercury
Boxermotor
1:45:49
Computeranimation
Metadaten
Formale Metadaten
Titel  Quantum Entanglements, Part 3  Lecture 1 
Serientitel  Lecture Collection  Quantum Entanglements: Part 3 
Teil  1 
Anzahl der Teile  9 
Autor 
Susskind, Leonard

Lizenz 
CCNamensnennung 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  10.5446/15133 
Herausgeber  Stanford University 
Erscheinungsjahr  2008 
Sprache  Englisch 
Produktionsjahr  2007 
Inhaltliche Metadaten
Fachgebiet  Physik 