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Quantum Entanglements, Part 3  Lecture 6
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Title  Quantum Entanglements, Part 3  Lecture 6 
Title of Series  Lecture Collection  Quantum Entanglements: Part 3 
Part Number  6 
Number of Parts  9 
Author 
Susskind, Leonard

License 
CC Attribution 3.0 Germany: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/15127 
Publisher  Stanford University 
Release Date  2008 
Language  English 
Production Year  2007 
Content Metadata
Subject Area  Physics 
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full down her
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real this program is brought to you by Stanford University please visit us at Stanford died EDU through all the talk a little bit
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about the laws of nature and former who will take the bomb exam over possible war nature which is not a war Richard but the discuss what wrong so there's a lot nature articles collide Maxeys XMY take then permanent on the black of fixed and permanent and the role is particles will call and go often we're direction 1 ever x coordinate is saying whenever the 2 particles whenever there could X coordinates become the same they suddenly turn around the war orphans from other direction sold for example of this particles moving this way Aswan in this 8 when they get the same value of X rulers something happens we specify precisely what happens maybe they get turned around maybe they go off from some other actions if they're moving bike the particles in moving in some other direction not horizontally but they're moving this way of moving this way and this was moving this way again Ruiz whenever there X court has become a saint a I'd go off and do something else while possible on nature and good it could it could be like that there's nothing mathematically inconsistent but we're but this doesn't seem like it seems the Violet something deeply ingrained our both in physics and the laws of physics and growing tuition were what's the rules BY what not only did they know it's OK but it's not almost maybe says that you could translate things and those stay the same laws of nature are independent of where you put the center of your accord and this it is independent where you put in your the center of the accord now the gates that yes I specify the exodus told your next becomes equal a light so we could say it this way if we have to particles and they have coordinates 1 called X 1 and this is not a component of the coordinators is just labels which particle with talking about when x 1 minus X of 2 will become equal to 0 that collide bets on the difference between 2 axes recalled 0 collide while I don't specify what particular value of Mexico BX where here could be X way over there the alkali it if we lift everything up vertically collided removed everything down vertically so it but it doesn't violate homered unity of space spaces everywhere is the same as guy not all the I always collide and if In all their axes are equal the violates the isotropy of space violates the idea that when you rotate coordinates the laws of physics status but the laws of physics are the same and independent of the orientation of Iraq sees not the place where you put your but they all eradication of Barak's sell for example if I really only entered my taxis so they look like the freely and then so that Our this becomes a white coordinate y coordinate hours over here the x coordinate here simply Ridder Babel and it is a rotation of Michael admits a rotation by 90 degrees than exactly the same physics will reap IRS say that particles collide when the Hawaii's people different layoff different forms if we entered the taxis in some awkward and arbitrary direction than law will become much more complicated some combination they would the lonely interactive who only collide with some combination of the exit is and why worry will sold what would be violated here as the law that says a lot laws of physics on the scene in every coordinate system independent on the angles independent of the angle of the Xaxis however we don't have to ensure that the coordinator perpendicular to each other off things in the rules of physics which require us to use perpendicular coordinates to express the way we can express the laws of physics in accordance with light but they take a special and simple my coordinates for which the perpendicular to each other but the rule that says it doesn't matter which way we are coordinates the amount of this way this way this way this way the laws of physics should be the same odd that violated here the UN example of law physics which is not by abated which is not violate the rules that our that was as it should be the same rotational various is called rotational variants and variants on the rotation of XT let's require that particles only interact if both the X coordinates of the 2 particles and the white coordinates of the particles are equal are equal words effects 1 equals X. tool and why 1 equals White what does that say that says that the particles collide with each other or whatever it it is they do maybe they turn around them specify exactly what but room now would be they can only collide sir if they had the same point and the rewards of both the X coordinate and the White would that they collide and fact they don't have to be moving along the same line of a particle comes along this axis article comms along this axis and they find themselves at exactly the same point Nikolai now would be a lot of nature that would be the same no matter how we Orient Barak sees if they collide when I come to the same point that it doesn't matter whether we use this axis from the centers and so this law of physics would be 1 of which was invariant under rotation saying the matter what orientation widows for taxis I am the but for the moment than just ignoring time Vermont your full of course we do but I wasn't about to come back as if there's a Secord began to coordinate right cell exactly self and if spaces threedimensional it sea of 1 physical fuzzy of if and only if these conditions are satisfied Our then to particle COCOM to time we can really express this in a single equation using vectors we think the position of a particle as a vector of course but position are through our on topped indicate a vector but now I mean the threedimensional vectors in ordinary space not time for moment will come back to time we write this as 1 later
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because I have the particle collider only all I want is equal tied to our 1 miners are of 2 physical to 0 and this equation implies all components of our should be equal to 0 it's a vector equation back to acquire unit has 3 components and potentially variant it was a factor is that there is you are an effective 0 goodness of the Royal particles collide when a certain Vector 0 Italy Vector 0 0 8 0 in every coordinate system of all components of a vector being 0 means all of its components of being are equal to 0 if all of the components of the vector your that's true in every quarter system if 1 or 2 components of the vector is equal to 0 and the other 1 is not meant that is not something which is Chaudhary coordinate system for example it is a vector was X component is equal to 0 In some other coordinate system the x component of that victory is not equal to 0 so when I would say this or that happens when a certain X component of a vector is equal to 0 that we're not in general be rotation on the other hand we have a vector which is equal to 0 at means there is no link that all of us just but vector of 0 length just origin then that 0 in every coordinate system the matter how we rotated that I we express laws of nature in terms of actors as we chose to express in terms of vectors and the advantage of a vector extra a description that it is transformed certain way they transform when you rotate coordinates the components of vectors change they transform but although vectors transform the same way you when you rotate so vector is equal to 0 1 frame its equal to 0 another friend if 2 vectors are equal In 1 frame they are equal In any frame and that's why we expressed that is the reason that we express physics in terms of vectors because we know we express a properly in terms of vectors than the rules are independent of the intention a bike use them but comfort time about time we could Webster letter the horizontal axis X again but now that the vertical axis be time Ex and now again I could have a rule which says particles I win enough votes a viable collisions take place only when the time of the 2 particles as the saying what would that mean that would mean it's possible but 2 particles collide this indicates a particle going forward and then going backward of moving along the xaxis moving on the back along the xaxis it's possible for particles collide like back when the 2 times are equal it's possible for them to collide at a later time as long as the 2 times a recall collide means both of their velocities simultaneously change from collide velocity simultaneously change but always had the same time that's a perfectly good law of physics In the coming in physics and the fact it is even the fool or physics and physics are while it is agenda is a true it's as a possible war 6 particles collide only their velocities change simultaneously if they change at all that's not possible loss in the special theory of relativity and the reason is because of the idea of is not in area so if I had a lot of nature which said the 2 particles collide in 1 particular framing you offering only when the times are equal there some of the frame of reference this frame of reference here the Times of collusion would not be the same come to the conclusion that the relativity principle was not correct the laws of his are not the same in all reference frames of 1 particular reference frame in which particle collisions take place our an equal times and other frames of reference it's somewhat complicated story in which some combination of fees have to be all right that's not the way it works the way physics works is Lorenzen very Lorenzen variant means again that the laws of physics are the same in all reference frames moving 1 with respect to the other what does that entail about simple particle collisions very simple particle collisions while offer very simple particle collisions the simplest particle collisions particles collide don't produce any other particles not much happens they just bang into whichever it off that implies bad collisions can only take place when the 2 particles find themselves at the same pace time point another words not only when yeah times are equal but when their position also recalled Mr. particles only collide when they're at the same event at the same point of space part then it doesn't matter which would to use fear the same space time point is invariant can't now so that a good laugh physics from the point of of view of relativity theory that particles collide not just when they're the same position space but also only at the same time our that also convene expressed in a kind of vector notation if we represent the location of a particle but look a scallop point along a particle trajectory but for vector before that now has only components x y and z but also has a component he then we would alien 1 more equation here that the particles can only collide and event and that event pasta have the same location space and time they could say much the same words about space that just about space if a vector in space Rebecca on has 1 or 2 or 3 components which are equal saga which are equal to 0 vector has space component equal to 0 that doesn't mean 0 it can have a time component is a factor which has a time component but no space if I have some law of physics that says nothing more than something or other happens when the space components of a vector are equal to 0 that's not Lorentz invariant and the reason is because in some other frame of reference the exodus of these points are not the same the axes of these 2 points and not the same in some other friend reference and sold are saying that something special happens with X components of a Y components of Izzy components of a hectare equal
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0 that can be learned that the only Levinson bearing statement that you could make concerning a rule about that Is that all of its components have to be equal to 0 1 if if in some frame of reference the law nature is that some components equal 0 had better be all of that White because if you rotate or rents transform you coordinates you will love inconsistency on unless you require all components to be 0 so as as I said a lot of physics could be that all 4 components of these location of 2 events where the 2 events the 2 events are the sudden turning around the 1 particle and the sudden turning around of another particles both this law of physics would tell us this event 2 events could only happen if they work the same spacetime point so we express the laws of physics In terms are 4 directors or objects which transformed in a particular coherent way what electors transformed the components transform into each other we've seen for example that under Lorentz transformations X T for example get mixed up with each other but they get mixed up a mother with a with each other in such a way that of all components are equal to 0 1 frame and all 0 in another fresh so we express laws of physics in terms of objects which transformed we always require either we transfer Trans pulls everything for the left hand side and stay Parliament's art for vector transformers for vector must be equal to 0 all are sometimes might express these laws of physics as an equation left inside the right hand side X of 2 why 2 is you have to of till we express this particular form is expressing research ruled by saying that tool for vectors every quarter which are not of some for vectors Yukos you certain former rector associated with 1 event is equal the 4 vector associated with the other events so sometimes we express them by setting some 0 sometimes expressed something by expressing a lefthand side right inside but the equations only makes sense in this particular kind of case if their whole expressions for higher for vectors not just the pieces for vector for vectors of course things which transform under Lorentz transformations the same way that X Y Z he left but the final 4 vector it's an object but opponents From at 1 for each direction based 1 kind that transformed doesn't need to transform it means its value as measured in different reference frames transforms fight node in 1 reference frame I can give you the rules for computing it in another reference price for example position Bossidy and so forth transformed from 1 Lawrence friend to another position when or how that transformed let's take the case of 1st Lorenzo transformation was x equals X miners see our why it is X Prize was at my square of what might be squared incidentally the notation McCabe writings square 1 might be square dominated here the standard notation gamma is equal to 1 although the square root of 1 might be squared Gamma Gamma Read saves writing nothing deep about just every writing another later writers is that gamma times X minors the for well so why Prime said again the AIDS ago the 1 everyone see back into a form was put down here I see is eagled 1 why try musical Hawaii prime physical dizzy as a prime physical 50 miles X the everybody what might be squared or gamma times team miners VX visited the transformation properties of coordinates of the event other transformation properties involved rotations of space right down transformations associated rotations space replaces a space together and is an example many maniacs easy diva rotate about the rotate about zaxis vertically is zaxis erupted my coordinates X and Y get mixed up with each other Our Zeke does not get mixed into it and neither the time for all I do is take my year record taxis then the equations would look like not totally dissimilar today as well so totally similar data but they would look like best X Prize is equal the X times cosigner angle plus Y times sign of an angle West putting he opposes Eurocurrency plus 0 times he just remembered that is that tear also part of the game why Prime is equal to minus minus X very minus X cosigned data when sex figure sex sighing freighter plus wiII cosigned later plus 0 z plus 0 fee prime is equal to the GE stacking things Of the Z backing things up so that the X always appears in the same place wire with appears saying Glazier both the place arrive 0 x 0 0 why plus plus 0 t being pedantic but putting in always your beer and finally Our cheap his eagle to cheat for all the others being 0 this tells me how things and former rotation about the world about the zaxis his tells me how they transform underwent transformation that similar kinds of equations in each case they linear equations of the form of a particular form a matrix is involved a lot like this in matrix form I think I wrote that last time matrix form in each case the linear equations which have the form but X find you know you've got 1 2 3 0 1 2 3 0 1 means X Cummings white creeping Zee and 0 0 means time there of the former X Prize it is equal to some make matrix mutual funds X Nova for example let me right down the matrix got will rent and remission in that case the Matrix looked so it makes on
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patient 1st column will stand for 2nd column Hawaii 3rd column z and full of column feet of zeroth columns and with Rose X Y just labeling now entries OK the entries here well from here you could X Prize is equal to gamma X was a gamble over here our desire minus V Jamil a year and a 0 0 that indicates that why don't come into this equation but again X and again we see with a minus sign and so on but see why Prime that's equal to 0 times X plus 1 times y 0 times EPA's Europe times saying z 1 0 and TV time that's equal letters or minus V gamma here 0 0 and gamma that's the Matrix which governs the simple Lieurance transformation he matrix which would govern our rotation of Spain will also be a matrix was followed by 4 structure but he likened just rewrite off here because funding matrix rotation about zaxis a rotation about the axis would look like Our cosine citing later 0 0 hour might decide later cos I'm afraid at 0 0 0 0 0 1 0 0 0 0 1 that would be the big tricks which represents protection now you could combine and nation's together in a wide variety of ways by mixing up a rotation moment transformation for example if I wanted to do a Lorentz transformation along some different angles works in Angola the y plane the 1st I would make a coordinate transformation that turned Xaxis new xaxis I would use the rotation matrix defying who coordinates along which the new Xaxis is along the direction that I wanted to learn transformation then if the having gone I would do a 2nd transformation which would be a Lorentz transformation of between time and don't tell multiplying these 2 matrices together basically that's what that's what matrix for the full attention would be as it would be the product the of Florence transformational combination of looked out of the lower rents transformation a rotation would involve a product of the relay it's enough to know that physics is invariant under rotations and know how to write the rotation matrices and invariant Lieurance long call Lawrence boost whose means to speed up the velocity it's enough to know those who form here and not makes a mop all kinds of ways to through do more general transformation or general transformation Will you get enters new different answers yup if you 1st Lorentz transformation transformed mystery why letting you 1st Lawrence transformed so that brings you to cordoned frame in which this is the new space axis and if you do you rotation your rotation a rotation about this Plane here not this so it depends on the altar if you 1st transformed and then rotated you would be a rotating about different surface you be rotating different surface then if you 1st rotated and inland transport and the order of action does matter but it's enough to know these 2 0 and have a combined to work to know how any transformation of a kind that applies to vectors bite so the more general statement that doesn't matrix of some store trips as to satisfy some will also have basically the rules are out of various kinds of products of these matrices and we In a more general form that the prime coordinates are some matrix times prior court and 3 each specific situation but we want we wanna why transformed from 1 coordinate system to another we have to know what the Matrix is but usually written 3 about we can also like this in the form of combat the next prime is equal to matrix just figure of ticks now plans a combat do the matrix multiplication all are manipulation of indices remember remind you want when you have repeated fourdimensional indices like it's particularly when 1 of them is exhibited as an upper index in 1 of them was exhibitors a lower next week PT the indices season means some overtheair some overly indices it's the same summation that you would do if you multiply the Matrix loyalist but X 1 X X 1 X 2 X 3 x 0 right when you work out the components of the product you have disarmed gamma transit time 0 sorry gamma times X 1 0 0 6 2 0 0 and extreme so full that sustained some simplicity here and some over the repeated index but so this is just a notational device for riding out generally what a Lorentz transformation as a linear operation like that that's a special case this is a special case of the transformation of vectors with a vector in this case happens to be space time position of a point there are other factors that either for vectors which are interesting for example before vector associated with velocity will come back to it the full vector associated with proper velocity remember what Travelocity years the proper velocity is not the X by X by tea but it's X you by How was appetite 1st of all before even go ahead even before we talk about 4 vectors to talk about scalar scale of things which are missing in every reference for example if we're talking about only rotation us forget for the moment Lorentz transformation just rotation the temperature at this point right over here it is the same no matter which way we already are actually so temperature is a thing which doesn't transformable rotations of case temperature it is a scalar scalar quantity which means it does not component to just 1 number characterizing it and the temperature of the point in space doesn't depend on what what taxis are used to describe them about the distance between 2 points the distance between 2 points I'm not thinking about based time thinking only about space the distance between
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talk points it doesn't matter which way I turned my head the distance between them the number of inches between these 2 points is they had no matter which way I already my taxi so distance is also a scalar quantity components of vectors on upscale quantities they they change from 1 reference French or rubber but the distance between 2 2 points is a skier or quantity I might have a dialog nature so much but a simple claim I might tell you for example that the distance between 2 points measured meters happens tends to be the same as the temperature measured Fahrenheit right to stop the train but they want big might be that was the temperature Fahrenheit above 70 degrees yeah I so I can imagine took points separated by 70 centimeters and that I would tell you it just so happens that the distance between my fists measured in centimeters happens to be the same as the temperature in the room but that depend on what taxis I now because the distance between the points doesn't depend on the ax eased the temperature doesn't depend on taxis that both scalar quantities and sold the statement that the temperature recalls a distant bats AT rotationally invariant statement and a strong 1 framer which when every frame in this case a statement about scalar quantity I should have talk about Scalia's even before I mention vectors vectors of more complicated offer vectors thing is true it better be true for all components of the vector it is drew with every friend reference or how to the border meeting of saying it's a former sector of is 1st of all has 4 components but let's call it works just call it general for and not specific 1 of the general for rector let's call but for for the moment 18 new this 1 is usually called muumuu but With is generalized and think about the general emotional for vector and let's call it a view could return its components article 81 . 8 2 0 8 3 and by law start the definitional for record and it does apply for velocity or the beat proper Milosevic the definition of 4 vectors of thing which transforms again transforms means that if you know what 1 frame you could determine the the definition is that transforms exactly the same way as acts as a component of that song if I know a 1 frame and another frame the kind frame the components 8 trying new are equal a L times their values in the 1st exactly the same as that for example before all components of a if I just did a ordinary laments transform our transformation will vary in exactly the same way as x y z take about writing it down the yo transformation properties of former rector of the same as XYC you take that the definition of 4 record if they follow a vector 0 in 1 frame of reference this is 0 1 frame of reference it's obviously 0 and every frame of reference that every component 0 here every component will be 0 here if 2 vectors are equal and warm friendly reference than they would be equal in every frame of reference assuming all 4 components are equal souls backs the value of thinking in terms of 4 directors of things which are true what truly in every frame of reference this makes them useful the describing the laws of physics which is supposed to be true remembered firm reference now represent think of that they're in this form here is called the Contra variant form factor it was another form of the same sector which is called the color barrier for political covariance components that really imagine to be the same geometric object just described in a different way and the describe it we put the new downstairs upstairs downstairs and 1 of is the scale collection of components exactly the same except the space school 1 with 3 are changed inside a lot too we invite Linus a nor wire I want to do that will 1st before what I want to direct religious described mathematically but will be differently is a mathematical connection between 2 80 mule is equal to a certain matrix all right down matrix in a moment our new no found a new stood now before we do anything else let's look at this equation before you say what he has a nice feature that its 1st of all summer with the new index always will use some of new and index it had better be 1 downstairs with 1 upstairs otherwise you're writing something was Kim private property with me so whether makes sense from a from this telling you now for future purposes if you're writing out equations like this and you discover that you some kind of a indices both of which are upstairs or both of which are downstairs you made some mistakes this is a nice form that Lorenzen very equation should take some of industry should always be 1 up went down but you do understand why in due time but let's see what this means that exactly what this means is 1st of all what is Ada a dozen matrix and the Matrix minus 1 0 0 0 0 0 minus 1 0 0 0 0 0 my this 1 0 0 0 0 plus 1 what does it do whatever multiplies call is a column so much for the column of multiply aided by new by eBay AD Acer Times AT 4 of them make fixed on the comic aII that would just get by writing down 81 82 83 and 84 or get the western inside this would look at the proper it gives us my USA 1 In the 1st entry minus a 1 minus 8 2 0 minus 8 3 plus aII not beat up minus 1 and they want possible zeros minus 1 times 82 branches euros and so forth so you go between the call their former Contra variant form by multiplying but matrix aid but convenient notational but bites so why we do it not only us all way to us are urging that the creative original level of over the
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disease on but but that we could identify were 81 8 2 8 3 and a 0 0 where lower actually right I was not consistent up thank you look at why would will do this we wanted to this because it's a useful trick informing various things which don't change on the coordinate transformation for example supposing that I take a new and I multiplied by a new downstairs again some indices which are repeated what this means this means we can read it directly off here 81 upstairs might say 1 upstairs services just 81 squared plus or minus a 1 square might to square I might say 3 square plus 84 square well this is very much like picking the X vector and computing the properly the late for the proper time associated with the expected if we go back to the X Media will remember that difference of the Times Square components and the space squared components we called how how Square squared difference of these components and the main feature of it was that it didn't change from reference frame of reference frame it wasn't very well since they a New Year's by definition transform exactly the the same way as Exs it follows that this quantity does not change the Lorentz transformation although rotation or any other kind of transformation of reviewing this is an invariant Paris gala so just as in the occasional the light and car back our 81 square club might say 1 square mile say 2 square mile 3 square plus a foursquare we invent the notation with lower index and opera index where the lawyer index changes designed all 3 space components and leaves the time component that keep ready for Burnley 0 hour so the trick to keep track Of this might Saeed in multiplying vectors on the form rents invariant skewers for example and this language we would just right that the proper length other vector next Is X view X meals and this would stand for he squared minus X square miles 1 square mile busy square so let's leave notational trick to keep track of my and this Haider matrix is part of that additional trekked it's just another compact Lady of writing that they were the law components is thing the upper components with your design fifthlargest ordered up with them . follow us I actors over wherever long Carlos this is there is a lot of folk that the both to be thought of as cold snap this is not going from columns arose the well and some center could think of it that way but I would not now think of them rose as another later right new a new which is invariant Mr. Right they don't know they yield a that because of her multiplied immune known by a mule 4 of the exchange you you know when you some index doesn't matter what you call it so this lowered the index of AT and then multiply that by the year the contrary right so they have various ways to write the same things you know you're doesn't matter not Europe doesn't and that was with its symmetric yeah Villalobos bloody order would have the really does not matter how do we set the required Tottenham might never do this OK pro something else suppose you'll have to force vectors but Quelimane beat the top 8 EU being yield is of course exactly the same thing incidentally as sub super why is that will both of them are just 18 not being not minus 81 B 1 minus 8 to be to minus the same thing for rate it doesn't matter if you put the opera index on 1 a lowering make on B or swap index on me aII upper index on be saying things get the same effect so basically the product we just think of this as a new kind of product of 8 times but 2 4 vectors Sabah thought to number is the product of 2 4 vectors and the product of 2 4 vectors are the only thing I know about it is that this 8 is in the the study signed business otherwise it would be very much like the of ordinary vectors are now what about this subject here is this Top variant would you have 2 different vectors of you just have 1 vector and transforms on the Lorentz transformations and this particular combination doesn't change as you go from friend preferring well above yet to vectors a times that also was variant when program is 80 is a B is a form of active don't a Class B a bit it might be on both for record that means that a Class B kind a Class B above 4 actors a paper B of 4 actors a plus because of 4 and now when I write about me in this 1 copper and 1 but it doesn't matter which one's upper which was well this must be invariant because of a and B R for vectors was a plus base so this is an area about the difference miners B a minus that's also invariant because if you have to work for that is the difference between them is another 4 vector well this is a square This is a candidate for speedy times cost twice 8 times beat this is a kind a Busby times B minus twice a tons of both of these are invariant than it had better be that 8 times B is also a very white because if you subtract we used to subtract BAA squares OK over be squares would cancel you just get twice
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a times B twice 8 times B is also an area so anything with 1 up index like this in 1 lawyer index combine them together you've gone an invariant quantity which is the same in every reference for course good defying things which are the same every reference frame because I love nature supposed to be the same every reference for measurement and measurement of 80 of the state quantity should be the same every 4 temperature distance between particles whatever all right but let's let's go want to consider some specific for vector velocity for vector which we do last time but now armed with some powerful rotation find some things that would have allowed 1 of article and we have to points then the separation between them is a little in the past more of week DX meal time it has was Israel components are its components are D XTY Desi and dt equivalent 3 X 1 B X 2 wouldbe 6 3 in BX not that's a compound that's a 4 vector it's infinitesimal small if we divide that by D. Talwin what's good for with detailed Square Square a how is he squared minus Theatre Square miners BY square mile disease quarters but that's just the X New X meal squared minus DX whereby why squared with his town as Tao square I think I was just a square room the extramural divided by D Tal so this is the rate of change of the X of the X quarts of VX new coordinator particle but not rate of change with ordinary time but rate of change with proper the rate at which the coordinates of the particle are changing but as measured by Clark moving with the particle STX new detail and that is called the proper form lost you but its components well what we can before I talk about the components its now before velocity None as far components that's weird you think of you know the 3 components of velocity that you know everything about particles molding that's true reason is because of 4th component of our city is determined In terms of the 1st 3 components why is that that's because his velocity here satisfied a constraint satisfies and new relationship which allows you to compute the 4th component from the 1st 3 let's see what that constraint that I maintain that this subject here is a unit vector was even that mean for a it means that it is a product of itself or that you you knew is equal to 1 it's some like a unit vector In ordinary space if you have a unit vector an ordinary space and you know what's X component but you also know what's why component up was signed as an ambiguous why because the sum of the squares from most people 1 well before Milosevic is a unit vector how we see that but just check out but just check those work out DX muses equal to D X annually by detail VAX New by detail the upper component law component but look here be count Square is just BX BX so we have in the numerator so the numerator and the denominator of the same thing BX BX Mueller's deep how squared in nominating you also house for and so the flow velocity is a kind of unit full of vectors a floor components of a full velocity are not independent 4th component parts from an ambiguous sign is completely determined Greek a bleak he stated less than he year for a loss of vessels that let them last I remind you With tell us exactly what was about to do so would know let's you want you to 3 writers X and Y Kinsey components let's just call OUX were thinking about space for some reason I rather think about exwife is even want to and 3 but use of X will is equal to the X but details but that's the same as X by D. C and D T by so we need to know when there's of course this digital ordinary Milosevic agrees to the same thing for y component this is Eagleton D Y by GTE T by Didao saying thing for Uzi physical easy by BT but detail so space components here we can just write a simple equation we can write warning equation we can write that love as a ordinary threedimensional vector is equal top order velocity of ordinary velocity time deeply but did tell the truth all 3 components not to worry about the 4th component to the country 4th component of you in a minute but let's just take the 1st 3 components the relation between order a velocity and the proper velocity is a simple factor T but let's work out what that is what like Elwood P. by powers we tell is equal amount arrive about explicitly decay squared minus X square minus D Y squared minus square aid if because glared sorry that detail squared so it's the square root of its but with this way because now what we want is seen by now the race obediently how that they divide us by DTC square Tsquare bt squared 50 square feet scored or is as easy as just 1 might squared is equal to 1 Martinez velocity squared the velocity of the particles or not talking about velocities of 2 frames of reference necessarily were really just talking about the velocity of the particles and they indicate that the of threedimensional velocity ordinary velocity Opel on top of it indicate ordinary 1 miners velocity squared it is DT by Tao squared so did thousand dt so all I have to do in order to compute bt by how is turned upside down and take it square root of team by town that's equal to 1 the but the square root of 1 might be squared recalled again after them it's just a demo the sitting down but now it's associated with the velocity of the particles as the 1st velocity all 3 components of the square answer
1:01:59
now we know the relationship between before velocities on the 1st of the 3 components of space before velocity and the ordinary velocity but just write down what is is school for 1 divided by square root of 1 these glared could words How about course dead death South Park where the this is not square a day after having lost their last loss the losses where other slow we calculated her through again this is definition because square definition because court here's G he squared my this DX my STY squared minus disease square that definition of detail square now we just providing it and it's a square of something not the 2nd derivative of anything it's just a square d Tower square we divide that IDT square by squares just 1 the expo led by DP squared is the VX component of velocity squared Rice's writers VX squared the well that was definition I actually leaving you guess but they new use of just the definition of what I meant by Didao squared about it just waters as you said the year about products were yacht now where we are we have this these basic components philosophy and all those multiplied the ordinary velocity which it would buy and number which is very very close to wonder if the velocity Is which wore a because you go fi loss this sort out what's a Thibaut Newport In some particular Frank the get disdained at a similar Mathematical Expression but never that yet it attracted spew out of that saying it's not the velocity of of friend right but could not so killer again like this again year it is a particular function of the magnitude of the velocity any velocity it happens to be the velocity of a particle last article efforts velocity of a moving train velocity will framed by definition here's 1 divided by the square root of 1 miners squared that definition with these square here means the squalor of the magnitude of the velocity nearly velocity squared seemed velocity the tuatara right Connecticut and you would be 1 half and squared saying Milosevic so that a particle would put the velocity all 3 components we them square them and that defines score beat it also happens to be the same Mathematical Expression because the Lorentz transformation that the it's not completely accidental but at this point it's just a notation for a particular function OK services Best it the 1st 3 components of you what about the 4th component of your of component of you is by by definition DT by D. Kyle well the X by detail which is his by now it's and that is just 1 provided by square of 1 mind is peaceful and oversee square as TV Tower by so we now know what the floor components of you are in terms we ordinary velocity is 1st of all they're bigger than the ordinary velocities 1 might be squared overseas squared with less 1 1 over it is greater than 1 so so the space components of the velocity of always bigger then component of ordinary velocity is something very strange happens when the velocity gets to be up near the speed of light ordinary velocity when ordinary velocity approaches the speed of light you grows and 1 so what velocity gets hugely big went through when the ordinary threedimensional velocity gets near the speed of light what about what component before a component of the velocity is small as essentially 1 the 4th component is just the number 1 in the limit of velocity is small when the velocity gets big or gets up near the speed of light thing you not also appears set on what is that the speed of light equal toward our soul we've lost that particular fact we could multiply by velocity the speed of light and give birth give a components of the velocity if we wanted it have sorry we want units of velocity when the speed of light was not equal to 1 we would multiply it by sea Our that's fine you could do that if you like it does receptor remember what you're talking about it more usual to think of a 4th component of velocity is something you want Robert you could think of as being near seat now that brings us before components moment but if you kid they were so I thought I will report erkle Ernie object affairs a certain rest mass call arrest then it has 4 components it's also a vector momentum is Standard Notation Forrest Court P that's true in the Newtonian mechanics through a relativistic mechanics momentum the symbol is Pete now in nonrelativistic physics is just in time be mass times velocity but that's not a very good for vector rest times ordinary velocity it's not for of Roanoke of Slovak they're out of it because we don't want to write equations which makes sense in every reference frame then we want to take it to be advanced kind you meal that of course may I'm really new divided by square root of 1 might be squared overseas square but Polaroid 6 square so a definition if you like this is the definition Of the momentum of an object of mass I am moving with velocity you were doing with 4 velocity you and has 4 components the firstrate components are very close to the ordinary momentum when the velocity is small last component is the energy I showed you that last time it's essentially antsy squared plus so 50 . 1 here FM be squared Markham backtoback local could visit will come back about but let's just know that this is the definition of a 4 components of velocity got momentum by the
1:11:15
laws of physics study think we may actually get to it tonight effective a good match Rimbey about 2 start anyway from nobody is not for right I did the does for components fell 1 of which is just wonders Top Of Westchester just Riverside equation Maramec ourselves confused Kirk but what do now is take a bit of a vacation from relatives write a lot of emotions ordinary laws of motion even before relativity the a day that I was removed Ma 0 to with the sauce back he knew of not only do is pick a lot of physics but offers makes think about it from the point of view freely and then trying to figure out how we would on redesigned changes modified so that it would become a relativistically invariant law motion it's a lot of emotions are charged particles in electric and magnetic fields now tell you right now decomposition and electric and magnetic it is not an invariant decomposition what in 1 frame of reference it is a magnetic fields may become a combination of electric and magnetic and another frame and vise versa souls I want to write 1st the novel feed treelined Stein long motion of particles charged particle in an electromagnetic field and then we can try to think how modified in order to make it a legitimate relativistic equation that will be the same in every reference right as I write it it will not be the same in every reference because it will be based on 3 Einstein rejects the fact that it's not the same everywhere from whole world of things are really led lines I'm His along with with ordinary kinematics with ordinary space type emotional right it is loss 1st of all articles MA forces massed times acceleration that we haven't even talked about acceleration but I think we can get rule for relativistic acceleration if ordinary accelerated his time rate of change of velocity then I don't think it would surprise you very much at this point if I told you that the poor dimensional acceleration it is the proper time rate of change our before that Bigelow velocity remember that these 1st 3 components of you are very close to to the ordinary velocity and all is very core was ordinary kind of our office slowly moving objects of a slowly moving objects the 1st 3 compartments of very close to the acceleration ordinary acceleration but will come back Abbas for the time being let's just think about ordinary acceleration the oldfashioned kind articles and they always need to know in order to write down the names equations for the motion of a particle is what the former Sunnyvale the forests of an electric particle of electrically charged particles force on an electrically charged particles is made up Out of the electric field just which is 3 vector their meaning the has 3 components X Y and Z some good about that and also the magnetic field is which also has 3 components something good about that worry because after all we want the right things in terms of fourdimensional objects not threedimensional objects so as something quite none in not right about right down at least until we finally learnt understand better the relations between electric and magnetic fields but for the time being but just write down before and here it is before low rents is forced discovered by the rents sometimes 1895 because was always known was proportional to the electric charge it was not bad sir acceleration everything's an ordinary threedimensional here was charged times electric field that goes back where who was the 1st one wrote down charged under electric field that Barry a 1st had the idea electric fields charges and electric fields of forces exerted on event plus also charged times yet velocity crossed the magnetic field OK now I have to tell you what the cost proper there's somebody anybody not know what the core of product the moment
1:18:06
anyway this place over forests is called velocity independence is not velocity and there is a velocity in here but remember velocity was in 1 frame of reference a particle may have let suppose there was no electric field altogether OK but because the particles woman with a velocity in a magnetic field a force on that's interesting because in some other frame of reference the velocity ladies year the velocities yearold than they would be no contribution to the force from here but the cost on we just see the on the force or makes except accepted accelerates in 1 frame it accelerates in every time on Wednesday at doesn't accelerate 1 frame doesn't accelerate another friend and sold such as funny if there was only a magnetic field of all the magnetic field there is some other frame of reference that would have to be an electric field In a frame of reference in which the velocity of the particle was but frame of the particle that could only come from the electric field so that immediately tells us that electric and magnetic fields have the mixed up with each other when you do a wrench transformation of what Is is the connection between electric and magnetic fields and the collection but the 2 of them together form a new kind of object which we haven't we have written down yet it's called depends here is an object which has sex components altogether issue only most scalar components x y z b also has components exwife Missouri Our it it doesn't have work components that has 6 components altogether so it's not what it something no it's gotta be something new and what all over it should these things transformed into each other on the various kinds of Lorentz transformations that's what we know so the next question is what kind of other objects are weakened defined besides galas and vectors and the next class of objects is called cancers in order to write down what be electromagnetic field from a point of view of relativity we have To understand and considers arm greedy well I tell you 1st of all the simplest way to think about it and From recall can choose capital city a scalar is concerned it's called the cancer of 0 rank it has no indices why had no indices used scalar has no plans to use its gala doesn't have any components so scalar is the simplest kind of cancers a vector a vector of 4 electorate is also Lee Kansas but tensor of rank 1 which means it's an object which has only caught 8 not aII new has only 1 index it as what components but has only 1 index that index could run over 4 different value was a cancer the thing which has 2 industries the simplest cancerous take 2 vectors let's call the main B B does not stand for the magnetic field may be a sequel to copy take another vector and put them next to each other and this object the object which has finally components altogether it's got 16 components altogether right 16 component because 80 has 4 components BA has 4 components Nuuanu each could be anything from 1 4 or from 0 3 as altogether there are 16 components we give this display play these 16 Kahan as a matrix as a floor by 4 matrix and the 1st place would put a easier what's 81 B 1 then I think it's a 1 B to a 1 B 4 E 81 B 0 and then 8 2 will be 1 and so forth we could display that is all for composed 16 components just lay them out on a grade and draw matrix for that's the simplest form of can't just take 2 vectors but and juxtapose them together take all 16 components and all 16 components Foreman object which is called a 2nd rank cancer you could take 3 vectors figure 3rd vectors now we would have to make a Cuomo represent a
1:23:38
stake of 1st of all this is a 2nd rank vector 2nd rank cancer is but about whitecollar cancer nervous about not a fractured does come from the word can a resort Marisa justice to indices secondranked concerned because we can construct thirdranked cancers of bank cancer will be made out of 3 vectors New New similar saying that would be a thirdranked concerned many components doesn't have 4 times for 16 components would want to display them we might display them on a cell fuel and thus it got us we can have fourthranked enters fifthranked tend to know what is it that characterizes Is it just a product of 2 vectors not its any object which transforms the same way as a product of 2 vectors so let's let's ask Howell a product of 2 vectors transforms the proud also 16 components supposing we know the components of the 80 kept the components of a prime and another frame of reference definition L knew the 80 that's the transformation property it's also the transformation property of babies GE kind these new but use different variable said they doubt or say how old 8 Powell I remember it doesn't matter what you call the summation index this 1 new and quote was 1 town left inside I haven't opened Sydney index which means really does correspond to some particular component on the side what how does what's the transformation properties of aII time new be time that's the crying component the news component of the I can't climb means the components and the prime reference for a while so easy just tell you know yes 8 With detail I'm so whatever I can Is it transforms with an object which has 2 elves it was great to next year's it next year the stigma index noticed that the law component of this goes with aII the upper component of this also goes with a prior to the local parliament can goes would be the upper components also goes would be prime now I can tell you what the general off 2nd rank cancer transformation general law is take attend shore In the kind of reference is given by in exactly the same kind of structures L you Nu Tea Co New professor Israel if not component of in 1 frame the on time frame than you take the Lawrence transformation matrices and you simply apply them the new index transforms into the new index but the new no matrix the stigma and index becomes the sentiment index by El Cid Tuesday the pattern of his pal repeats itself 3rd ranked cancer there is just 1 more making it transformed just as if all the product of 3 vectors this is what is called the answers you can best me why that called concern you can best be while Rhode or you could say is useful as it used to define it is very useful to define but he is a definite it's a thing which transforms as if it were a 16 components the father of 2 vectors now there are other things it doesn't have to be the product of 2 vectors for example here some bring take a new these stigma that's not the same you're not the same as a stigma menial is not the same thing if for
1:29:16
example I take the the water 3 matrix supplement a 1 3 matrix all of this will be a 1 B 3 1 3 matrix element of this over 1 3 component of this will be a 3 D B 1 serve and not the same thing a newbie soup where a Rabin meals but both of them transformers cancers you could end the result is still cancer but it is not an object which is just a product 2 vectors in and all other things you could some 3rd vector and you could add some 3rd and fought back an Arab it seemed doomed DEC admire see nude these segment is concerned with you and some previous still transform so not every cancer is just a 2 sectors but every cancer transformed the same weight as a proper tent the directors got a mild Toulouse special qualities at centuries of rank rank to to a special kinds of called In isometric and symmetric and a symmetric tensor is 1 that has a property team you know equals team meal this symmetric sorry that's a magic at says that she won truly cost c 2 1 symmetric it doesn't in this case it doesn't matter which ordered you put be indices that's called a symmetric tensor rank to an end the isometric Frank 0 is 1 which changes sign you change your work Internet matrices they correspond symmetric in a nearby symmetric matrices so we take a symmetric tensor and we display hats all 16 components of it that it will have the property that it will be symmetric the 1 to element over here will be the same as the key did to 1 element of the he's the 1 1 element will be whatever it is the choochoo Alimento be whatever it is that they're C130 will equal T3 1 team on free will be the same matrix Ohio a symmetry biggest symmetric matrix which means that the components off diagonal will reflect the whichever that's a symmetric matrix Vermont just thing for us today the next time is going to be in 3 matrix What about the investor that the property will be reflected changes sign he won 3 is minus T3 1 what about T1 1 it's gotta be minus T 1 1 2 1 1 must be might be 1 1 on his mind he won more than a year die symmetric matrix must have 0 I was on that at all zeros on the bed but now and then off diagonal he want to see 1 3 he warned 0 stoic nicely don't make a nice big matrix subdivided for subdivided it before and its components isn't isometric cancer diagonal elements off the B 0 that's because he won 1 must equal minus or more and the other hand T 1 0 0 has equal minus to what she wanted to appear and he want to over here this is my 1 3 miners he 1 3 and finally see 1 0 minus 1 0 about over here here we have T 2 1 and here must be my who won t truly 0 ish the latest T 2 0 0 and finally down hero ahead of key c 3 0 my mystique 3 0 Source and the isometric which means that when you reflect the about the bag with changes and symmetric also means that changes signed when you interchanges to industries browser called symmetric Annie and isometric cancers now I'm really how only interesting components does a symmetric cancer have looked counter symmetric cancer as for diagonal components 1 2 3 4 1 and then harmony often components works of art there are of course 16 components altogether but the component over here is not independent of the component over here so really the independent components number of numbers that you need to know is 1 2 3 4 5 6 7 8 9 right now 1 2 0 3 4 5 6 7 8 9 10 attained independent components determine asymmetric cancer so what nothing but how many and Annie and isometric cancer but while entice magic cancer bag so we could track only this minute this this and this zeros on the diagonal and the other one's or from the other half of the matrix yo determined in terms of companies that 6 sex as the number of components of the electric and magnetic field that's a number of components of electric and magnetic fields this is only object that you could construct be enticed to metric tensor it is in fact he only an object that has 6 components which transforms while you could take 6 6 are full of vector to scale areas but there were also have 6 years but none of those Our correspond to the electric and magnetic field electric and magnetic field form a but it cancer symmetric cancer however that can't symmetric consider we want understand this equation here and there How will it transforms the Lorentz transformation now it has transformed the complicated what that means of electric and magnetic fields of transformed To richer like elegant finished now I'm good putts go fast I'm right down the next time right down the precise relationship over the riders will derive on the right now right down the precise relationship between me and isometric cancer and opponents of electric and magnetic field how the components of electric and magnetic field combine themselves together make a name that metric tensor we take our for all but 4 matrix Dawson guidelines of nowhere elements are zeros on the diagonal the off diagonal I don't need to specify both above and below 0 once I specify above you could 3 or 4 Pelotas changing the site a repair more visitors Our 1 2 3 0 1 2 3 0 X Y Z T this is the 1 to component In the until component we play my the 3rd component of the magnetic field B Z but say that I put a 3rd component of
1:39:03
want to place we put the 3rd component of the magnetic field and the 1 Place we but the 2nd component of the magnetic field white component magnetic field guess guesswork goes down here then he goes minus the 1st component of magnetic field so we don't have time to read in that time we only had 1 2 and 3 then be enticed symmetric cancel only have 3 components and mostly components would be identified with the 3 components of the magnetic field we we'll discuss this further prefer time being this it is we set up for the way that the magnetic field enters into a and symmetric tensor that was not my class minus will see White who did up this could be this could be busy component this could be a white component for wouldbe missile BVX component and then we were right x y z time so too meant young workers right now we have 3 trees over here 3 more entries obviously those entries must have something to do with electric field of course they do he X U Y easy the son yacht but just put a more profound here just about where plus busy here penis BY here my EXE a plus BXC let's see if my interest you and easy so it's now that 6 independent components fills out a but won't make example symmetric by former Turks with electric fields in the time slot here and magnetic fields in the space spacebased slots the Space Space components of magnetic field is based on component of electric fields now why is this house I know this has anybody know it will come to back next are and what they the knowledge base fiddling around and trying to find a sensible Hey Our relativistic generalization of this formula here so the next time we work that out will work out at equals an 80 foot particle electric and magnetic field in relativistic notation incidentally this followed by 4 matrix here is not usually called team you know it's called afternoon I don't know the history of whatever Y F is called a half but I suspect Barrett I suspect that stood the cancer yeah it's and isometric and it's composed of the magnetic and electric fields I'm going to use it as a threat that life is components of the field the components of a field they'll sell the here what's left just concentrate on a magnetic Putthividhya but serves as both events around the Times staff and get rid of it for a moment this now just constitutes a threedimensional members of the potential of a house it now much like the equation is equations that F 1 2 0 is equal to mine is busy all miners B 3 3 said 3 z next F 1 3 In whose busy plaster finally F what's the last 1 Baroney unless 1 is 2 3 2 3 is equal to be X factor for me this is ready yes minus B 1 be too OK now I can write this In nice of former former you'll see the pattern let's right this 1 as F 1 3 but where have as my years at 3 1 at a time like this is my S 3 1 that I have that f 3 1 in his beat you an F 2 3 years might be 1 by what see in that former alike about it now works let's let's start with the equation F 1 truly Eagles miners being arranged and lets cycle from 1 2 2 2 3 in each spot a about the next equation gold wanted to hold 2 2 3 3 2 1 Clark Clark goes around the circle 1 2 3 and let's proceed clockwise right next equation down Everything motorcycle word so what would happen to anyone because ritual the rate will happen 3 miners 1 Watson X 1 1 to the next would be 3 3 2 3 1 2 equals minus 3 1 2 well with any luck at all that will be exactly what we out fear of of . 2 with my 3 3 1 the minors 2 former upstairs downstairs forget that left 2 3 as miners B 1 so this has a very nice structure the 1 to component of the cancer is the 3rd component of the magnetic field to 3 component of the cancer is 1 component and cycles 1 2 3 2 3 1 3 1 2 so could always remember all you have to remember is 1 of them and the others will follow just by cycling media members wanted to 3 other electric field different the electric fields he what it is equal F 1 0 0 2 0 0 is F 2 0 83 is F 3 0 0 0 so the elected Olds on vectors where you just ignore the time component and you see what the index structure is you want is F 1 0 0 2 is at 2 0 0 3 3 0 magnetic field a little more complicated the someday might notice to magnetic field seems have to identity it has an identity president and isometric cancer was composed and recalled half and has another identity as a kind of that order we threedimensional vectors Islamist bigger vectors speaking a threedimensional and has an identity as an investor isolated cancer in 3 dimensions miners b b b assigned here has a 2nd identity as a vector that
1:47:36
connection between enticement cancers and vectors only true threedimensional it is not something which is true other dimension it's just an accident but not exactly an accident but think of it as roughly accident tho that if you take a 3 by 3 and symmetric matrix it happens the have 3 independent components those 3 independent components also have been performed an ordinary threedimensional vector but if you have a format for matrix and isometric it cannot be identified with the 4 so we see the the magnetic field sort of has dual identities you want to be the 1st so indices and the other identity it has only 1 6 Brown With next time out of this by symmetric cancer and far velocities and accelerations fourdimensional accelerations were below the equation which were recognized as having transformation properties that make certain that it's the same every reference that's our goal to rewrite this so that it's same in every reference frame have modify not quite right modified it will form a equation which is the same in every reference Ford got so act there was of war over the Senate has not yet he have a suspicion ripples of nature to be the same in every reference frame this was a matter of wish to hammer out of conviction and of course also had a good deal of experimental support the fact the speed of light was the same in every reference frame from were measured the nobody could detect using light the motion of through ah through a space but Joe Ali I don't think he knew that animal feed which he said he did not return often were not armed but he himself according was owned description was more influenced by the principle of relativity and he could not make these equations b the same every not without doing something that makes even be up and even that wasn't enough he had the mix up the components of a velocity eventually here the figure out special relativity he did not actually right his equations and for vector notation our just put in all the various extra Gamez and functions of velocity what it was doubling was basically making sure the equations were the same in every reference frame by making them before their request was Murkowski who really together before their idea and rewrote the equations of Einstein was more clumsily somewhat clumsy clumsily going about letting these equations in a way that they would go from 1 frame toward next by some systematic transformation properties they would always be the same reference so young it was it was a century Europe conceptual guests on his part that Maxwell's equations together with blue and was should be Herbert Hoover or or Cecilia lost a lot of traffic deaths at 1st he I don't know a hell of a lot about the history of the you know history is always in the eye of the beholder and what people say that we will have no but I I accord was writing if 1st did not like Minkowski is ideas he didn't like the idea fourdimensional geometry at 1st but it didn't take me long to catch on the 1st East simply wrote everything out they sought clumsy brutal her I have a hard time adjusting we need a notation 40 said you if you have a lot of money was 2 years was 1907 young women things most Florida leading the protest well I think was today would have much faster well I'm not sure the people necessarily distinguished that our arms and I don't know I don't know if he was he contributed much else the physics of an octopus was enough the Huge amount but I don't know what else to do it with the long history of famous papers this was a century Zurich are not I saw her I was detected by were lower rents but I think probably yes was in effect a buyers Fitzgerald who euros Hey 1 thing I know is you don't trust people narrowing answers as Christians they don't remember that out of yes we a lot was of Einstein's paper 1905 yet but he just had been alarmed no written out I hear Maxwell of his equations he wrote every single equation X component the white component busy component company of the law there of together but they eat doctors 1 equation he got 3 equations 6 equations 7 Haiti quake aid equations 80 equations altogether now but I you can then we are not so this was 3 Asia's 6 paid equations thank you wrote about all the x component of its X 4 . 1 2 3 4 5 6 but that all 80 equations and never put them together and some symbolic a former wants that was going into exactly the same thing in space time you which is writing everything out and so doing it by trial and error you get the feeling when you read the paper that he did it by trial and error rather than anything is systematic just for vectors was Jankowski who understood the before vector idea so hidden this with some good intuition about OK the next time will come back and work out EU Iran's for so long as modified by Einstein yeah end down her
1:56:03
room the preceding program is copyrighted by Stanford University please visit us at stanford . edu
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1:01:56
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1:11:12
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1:23:36
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1:29:12
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White
Suitcase
Source (album)
FACTS (newspaper)
Yacht
Weight
Missile
Cartridge (firearms)
Electronic media
Membrane potential
Level staff
Miner
1:47:33
Weapon
Matrix (printing)
Effects unit
Formation flying
Speed of light
Schubvektorsteuerung
Year
Brown
Lecture/Conference
Cardinal direction
Steckverbinder
Spare part
Transformer
Measurement
Magnetism
Buick Century
Refractive index
Electronic component
Single (music)
White
Paper
FACTS (newspaper)
Mixing (process engineering)
Spant
Computer animation
Fahrgeschwindigkeit
Mintmade errors
Throttle
Noise figure
1:56:03
Computer animation