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Special Relativity  Lecture 6
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it this has seen a university where
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we ought to talk about electrodynamics tonight with talked about scale of theories scalar fields we talked about how particles couple scalar fields are scalar fields influence the motion of particles and particulary receive grungy and same I with charm Cal's feel for how to influence the particle also tells the particle outward words of CEO William ever do that again tonight we electromagnetic field worries part but before we do I really want to nail and place the notational ideas a war was that will Sequoia mechanics for me what do at don't any McCarrick summer White for tonight just wants noir briefly I want to go over the notations of Florida says In this car use and how you manipulate them now just by way of Rome Prosser Pfizer Inc of good notation can be extraordinary with powerful good notations in mathematics might sign Zero signed the equals sign for goodness sake extremely powerful more modern vector notation began extremely powerful Our were going to be talking about tried a little bit his cancer rotation he heard track that showed you last time off upper indices and Laura indices that actually is blind completely dude starting operand this season Lauren disease which means nothing more than just changing the sign of the kind component of vectors of
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vector with this season vector with lower indices are really no different except assigned time component of course at a special case of something much broader it's a special case of manipulating draw vectors together with metric tensor but not the metric tensor yet so for purposes now goes just conventional our conventions end notations notations which are quite power forwards you will Street let's just go over them again quickly just to remind ourselves our work 2 of those notations or or about all the other rotation which is truly brilliant again good Ornstein is astonishing convention bark the summation convention is something that only to to be used in the right way it's to be used when your half when you have Hey 2 0 indices which are say 1 of them upstairs In 1 of them downstairs the only time you use summation convention with proper index of OR index were nervous saying you could set or use at the nickel to each other and some of them are about series starring summation convention or use it all the time but will only use it in a special form which Einstein invented it if we have summations the duo and not Of that special form all right summation OK so let's begin with 4 vectors again for vectors have 4 components 3 of which are spaced components 1 of which is prime component and when interested in the fourdimensional geometry we write components 18 mule Boeing can also remember that they consist a time component which is usually called a Norwalk end From release base components which are usually called AT a & & from want a free new goes from 0 to 3 5 now on Ovrette Maria vector with upstairs and next because of called upon for various next the Contra variant index is a sort of thing that you work catch DX meal and the fact that you put upstairs and downstairs is purely arbitrary but there are you have to put the index someplace and Einstein chose to put these Index associated With differential displacement like this in the upstairs slot and ideas who 1st call that a country area Mexico why it's called kind of areas that exporters and the goal from a Contra variant notation called area notation this is pure definition of a new call variant counterpart of as a saying that it's just a mother notation Ford on our way of describing it and that is equal to 0 8 Muna a new problem are you were a them you know it is a is a collection of numbers forms and make trucks are floor by floor matrix because we would go from 1 before Peter it is just a matrix of components a 0 1 1 0 0 1 ever are just a components minus 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 that's almost EU it matrix and in this instance with his relativistic geometry with funny money aside from time to time they really does play the role of a kind of Arab identity matrix but is Where's minus has 1 1 slot warnings of all the other slots are purely diagonal what this complicated formula while what's kind heated it's a nice need them simple set up but really all says is that these kind component of the Co variant vector it just might miss the time component of a Contra variance vector minus 1 over here and all the other components of the same all the other components of the same so we can write a sub naught minors a super nor a and plus a super and and that's all the formula meetings but In written nearly a year ahead they kind of neat white will you find the real or not at this point is not relevant the point is that as you start doing things with it you will find you begin by very soul butts take it for granted that it's a useful thing to do White Sock 1st thing the formation of scalar 24 that there used to talk about this at least twice a day drive found blackboard if you take a tool for vectors that could be the same for vector different for this and you take 1 of them to be kind of call they are that the Contra variants now this thing that I've written on the blackboard automatically songs over you I do not have the right some overview here by Einstein's convention this means 80 someone be super ones start Norwalk warned to 3 so forth this it a scalar another words this is a thing which is a quantity which are doesn't change from friend friend another example cement this came another example lead derivatives signed with just a year from the directive signed theme IDX new it is a collection of form differential symbols derivatives symbols derivative with respect X knock which means derivative respect kind and the derivatives with respect to the other cordoned off is his often just written as you'll pay another brilliant patient get rid of the sub x just right decent meal this symbol by itself of course doesn't mean anything it's got back something or whatever acts on it adds another index relax on a scalar it creates a thing with index meal that is next of the Debye DX mule or being used is a core value index another
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words for example of this would act on a scalar field it would give you the collection of derivatives which formerly called Marion vector award to remain as just ended its its name is Dean you 5 parts of derivative symbol yet so far you can fly flu is bound to toughen the fractured bar LYE indeed our at last time I explain why explained why if you take defined by IDX meal and you multiply by DX orders at the river of fire with respect to time Times d time and so forth and so on and this is just a change in 5 as you grow from 1 point to another by differential displacement BX this is nothing but the cost . 2 1 . 1 This is just justified tool minus 5 1 by divided by a differential separation but it is just a difference of 2 scalar is and therefore itself as a scalar stop by fuel I didn't pull but I told you about it you have a Contra various sectors and you multiplied by a collection of symbols With a script and if the result as a scalar theorems shows the things that other piece of it he is call variant so there is a steel COM which says what ever you hit anything where the army in that particular KOR but both produce more examples a moment scalar would be by DX Newell it gives you would for vector everything I'm saying insolently work also be true about sectors the only thing they got about 3 sectors is if we're thinking about Pulido threedimensional language not fourdimensional everything is the same except for every is seeing laws and rules put in In news a matrix it's just unit maker books is just the components of a data which said In New year special components for that reason because a There is a unit matrix is no difference between opera components so you don't have to say if you're talking about ordinary 3 dimensions it's unnecessary to say whether nexus covary controversy the book saying OK another example of for me gala out of vector well 1 example forming scare was our vectors is just to take a new baby but nothing you could do is let's suppose we have a vector quantity which happens to depend on positions and also kind X knew it brought about right way it's a form that their field depends on space and car and are just indicate that arriving next year it depends on space time and awful can be differentiated as it stands it's a former rector of victory each point of space differs from 1 place to another could also differentiate 8 with respect to X and for example you could differentiated with respect the X and somewhat over the index knew this means derivative with respect the time of the time component B plus the river respect the X Of the x component of B and so forth and so warned this is also a scalar price that's another example of forming a scalar is From vectors body were the call index contraction index contraction is the the same as identifying operate that's what the Lauren and summary as call index contraction and that's contraction in this kind of situation makes you from better while picture from a quantity which has all sorts of other components and leads to a scalar OK now considered are for Skelos and for vectors transformed and that's it that's the defining property that defining property is away transformed Skelos for example or just transforming themselves as transform under Lorentz transformations unwilling to give a a a broader definition aware mean by Lorentz transformations with views of Lieurance transformation as we talk about or along the xaxis we can't cost talked about Lawrence transformational axis zaxis and so forth but there's another class of transformations which are also considered to be part of the collection of Lorentz transformations rotations of space rotations of space is plight the collection of Lorentz transformations now want to say that then you could say that a rents transformation along the Y axis is simply a low rents others simply aII rotation of the Lorentz transformation along the xaxis you could compel rotations together with Lorentz transformations the make Lorentz transformations in any direction rotations about any taxis and just the general set of transformations where Shah visitors and there are not so sure about that some important to us right now the important thing is just to keep in mind the physics is to be invariant not only of the Lorentz transformations along the xaxis or the y axis or zaxis but more complicated things we wrote takes place transformed the rotate back again and our errors compound rotations with they that I thought but my favorite places I mean effective or exactly storm What's a nice notation for Lorentz transformations well let's take work said the transformation of the current Romario vectors for example just X meal has X new transform X new kind can't protect tour X naught prime at the time
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component that equal to at nor miners vii X 1 either by square 1 V square and so forth and so are the Segura Lorentz transformations except time next marked and excess called X 1 we can always write those transformations in the form of a matrix acting on the components of the vector archery Wyoming are we right 8 new crime as components of a certain form vector in my frame of reference in terms of the components In your frame of reference are given by some kind of matrix matrix has increased with dual where are you recognize was operating next year from 1 of pro or next down here it has disease that makes a matrix has a value for every meal and every it's a 4 by 4 matrix and narrow multiply a new is a properly formed the equation although he had not left hand side has an index New which could be anyone wants for the right side in the next meal and index nobody index New was some Nova and that's no was summed over its not the exquisite variable in these equations and so yes is properly formed equation amazing convey a convention are true and may give you an example are the matrix L. just for the simplest Lorentz transformation with written about your cup or after 4 of the Lorentz transformation let's say along the X access right there we will put it here we would Putin he this is the kind exwife Izzy time in the 1st column in the 1st 4 0 x y z so a onehour square 1 might be square he added a mine is and I think I better make just matrix a little bit bigger OK make it bigger With the upper corner here we have won over Route 1 might be squared and then we have my his over Route 1 might be squared about next 1 where y guest 0 Xerox wherever here but how my his the square root of 1 mind might be squared various Amax place here 1 or squared away squared 0 0 0 0 0 0 0 what down here 1 and war 1 0 0 1 This is a stay there Lorentz transformation along the xaxis and every write a column vector she x y z which occur also right X Nortek for exports 3 but Volkswriter way he X Y Z all right this is equal to keep time prime why prime is a prime 1st receiver that's correct complex here this is a car wreck was say it's as keep Prime is equal to 1 of about block terms these miners over blah blah blah times X transformation Lafferty prime X prime is my his lead over square root and he lost or or square root Prime's X sets a standard Lorentz transformation on X and then why and see do not next with and X Y and Z unit expand he shows you that white primers equal Kauai prime musical go so this is an example are a low rents matrix matrix along the xaxis if we want help transform along the Y axis will just shuffle these around a little bit I'll leave it to you to figure out where Lorentz transformation along the yaxis looks like it was matrix location but let's let's consider instead of a different operations a rotation in the White easy plane in which Annex completely left along misses 1 of rotations which is somebody said don't involved all but that would look like this is a different object now different transformation we would put ones here 0 1 that sales see an ex doing nothing Borja put down here this law block Perahia ago suggestion I wanna rotate by will favor and the wise Wiesiek planes are you will be call sign later signed Minor signed favor I figure and this would just say why primers cosine favor times White sign Hey primers my assigned later why cosigned and I'll let you could take these matrices start multiplying and combining them make much more complicated transformations which for example are partly rotations along some axis partly rotate properly Lorentz transformations along some other axes but this is the basic of buildings the building started pouring guard they form York PARIS fight so this is a transformation property army for With Contra variant index I'm going Kaleva to your 0 the compute the transformation property of a former actor with a lorry index tell you the answer now but you work out if he'll have they a call variant vector you wanna know what looks like in my frame given what looks like in your friend then there is another
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matrix this Lake Success have all index mule because we have come out of our Lauren next year and operating that's new and a Saab meal were to tell you right now what the Matrix O thank your noble if the left hand side has Mueller the right hand side must also have on and on some over a new so that's correct OK now we're talking about the same Lorentz transformation L & M represent the same physical transformation between quarter
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of frames and sought in In a L must be connected there must be a connection between them and Allen is really very simple and has just given by a death a 0 8 Pollitt you work that our to prove that is incidentally is only in just like the unit matrix is onerous saw hated summer minus 1 is it's only in inverse and that's because the entries here are such that they are their only verses the universe of Warner's 1 numerous minus 1 of my sworn wiser Yukon prove this that there are forgiven Lorentz transformation hello connected I'm not going to use this very much the main point is lawyer for vector Riaz call variably or contraband we wanted stands for is an object which transforms from 1 frame for another in a particular special wary of a special way is our parallel with the way the cordon that's themselves transformed biological the cancers were to make heavy use of cancer patient What is it can attest it is simply a thing with more industries actually a scalar and vector are special cases of cancers are scalar would be called a transfer of rancour 0 0 which means it has no index vectors of cancer rank 1 and they can serve rank tool would be a thing with To so I'll give you the simplest example of a cancer that 2 vectors impact factor would be enough to have 1 vector boats take to vectors With a and B now we can make a subtly can make a proper burial beat Marty at isoscalar but now I wanna consider a more general kind of product the more general kind the product has How Bureau New are works begins with these contrary version of it was just take put next to each other 8 and be now this is a larger company components is 16 4 times for it is a naught being knocked a nor B 1 a be too and not be free 81 Norwalk and so forth so onerous 16 This is a symbol here stands for a complex of 16 different objects numbers off it's just that sound numbers you get by multiplying a component of a with any component the beat cut to indices has called attempts abroad the label just generically younger right he for cancer a particular transiting you know not full cancers are of this form not all cancers or simply constructive 2 vectors this way but 2 vectors defined a cancer that way how such an object transform houses subject transform team you know well if we know how 8 transforms going all out beat transforms which is the same way we can immediately figure out what but scored 8 find Bracken around here knew would be crime New always have after Dover is transformed the a in the lead but we know how they'll be transformed let's rewrite this using the transformation here Otto York roared change and and a symbol here instead of calling mule Newell said remember it doesn't matter what you call a summation index as long as you're consisted of summing over it so it's just a finger a sort of index OK after and let's do it here also Segarra said West slug here this is just a quarter L mule said Monday for over here a a sick but now we also have to put beta R & B Prime has the same sort of thing El New unless court how a cow L music where firms a say Well that's a prime hello can't say sorry skill may be sick lack basic will be a primary that shows new Sirois said Mark Wright broke Carol found of corporate say 1 goes with a towel goes would be and how the equations consistent so helium and know more about kind of object with tolling disease with which tells us how it transforms it transformed With the actions of Lorenzo matrix on each 1 of these this season I could extract from that saying more generally the way a cancer transforms scores the tensor find concerned the quantity the set our quantities that look pricey are related to a set of course is that you'll see y el Hussein l cow trial he said
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filed a terms beat that's seasick now so this is a rule for example for the transformation properties of a simple consider withdrawing has this can be off will be you read it all burden that he also had good year 8 character there's only 1 matrix you're held every matrix mutual itself of his Norway go ashore a a commutation here talk on you could rent more complicated turns of cancers cancers worth 3 indices mural known How would this transformed an you could think of it by just thinking of it as the a product of 3 vectors within index new new and warned that I will write it down but the way that transforms a straightforward for each index a transformer of reached index Howell and there we 1 more land TD Sigma Tau Kappa that would be a general transformation proper course to generalize the hell out of this and a number of industries and that's the way that basically the definition of the definition of the cancer is a thing which transforms like now not as a ship not every cancer was formed from a product of 2 vectors for example supposedly what other factors suppose other factors see in these being you a new bureau war and a CD 0 0 D No Dale cancers by assumption now Eddings cancers gives other cancers selected cancer 8 times being with your index a new index of airports and they are something which cannot be Gerald written as the part of 2 vectors buzzer cancer with defined by its transformation properties not by the fact that it may or may not be associated with just a pair of actors North cancerous promised transformation properties or there this month layoff Beijing by world her 0 at it worked exactly this way this is sorry this way right here His a matrix L Iraq to you right I would right out of what this means in detail resort of components for your for example teacher for Reedy warned Clark were set that equal to 0 l Meris's 3 said Mark L. 1 cow he said Powell is like going through the Ark signal could be 0 Powell could be 0 the 16 possibilities 0 0 you make that with T 0 0 so 1 firm would be L 3 0 0 1 0 c 0 0 there would be another turn a L 3 0 L 1 1 T 0 1 and so forth and so on will be 16 such terms 1 for each index here each index here and a UN not sure that's the idea of the transformation properly with cancer nothing about cancers vectors scalar cancers thing about cancers is if they are equal in 1 frame they are equal in every frame that's easy to proof but to say that their equal means of all components equal if all components of the cancer Our equal all the components of some other cancers cost in the same cancer by them are another way of saying it His if all the components of a cancer Uh 0 hour shift everything from left side all the components of cancer or 0 0 in every reference I sought to say that cancer is 0 invariant statement said it's not enough to look at some components and say that comport to 0 the whole thing is a cancerous or must be 0 on every frame not if all the components 0 0 every frame and that's the power of cancers that allows you to make statements of allows you to write their equations which if they truly want frame will be true another frame after transformation just the basic power of it all right now I've told you how to transform a With all of its embassies upstairs I could start writing down the rules for canned foods with some indices upstairs summoned the seas downstairs but I think I want and instead I would just tell you that once you know how what cancer transforms you could immediately do how what other variants transformed the other variants for example the other versions of the same cancer saying cancers saying geometric quantity but with some indices upstairs and downstairs for example but AU be that would be some cancer with 1 index upstairs in 1 index downstairs how it
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transformed never mind you don't need to to worry about it cause I will tell you I'm immediately that this cancers here is given by T. Newell said Nora which you'll understand not about power transformer at times New Sigma R&D indexed to take cancer and taking the index from Contra variants the Cove area and you multiply you do exactly the same way that you do it for a veteran take a index and you will lauded by the operation of a song again this is a wellformed equation wellformed assembled stigma some Nova where's who this here object is he knew wife but there's another way to think about an easy way to think about it given cancer with all of its indices upstairs what do you do propose some of the indices downstairs and the answer is very simple if the index which appalling downstairs is a time index you multiply by minus 1 of space index you don't multiplier auto aided so for example here is the cancer Nought Nought which is exactly the same as see naught naught because I've lowered to win this use Loring toll the city's is to my side as like the relation between 18 or what b North and a sub B sub not to minus signs in going from a super naught a not going from these super b sub North each 1 has a minus sign this is equal to its for about this 1 may not be worn and how that compare with 18 North be 1 sorry your BUS but 1 that works both of them downstairs these super ones and be someone are the same but a super North In a sub not differ by my side so he 1 would be my not 1 because only 1 kind component was lowered every time you log or erased a crime control and you get a sign and that's all there is to it but having matrix notation is a fast summaries of that fact that the relations between call variant the controversy is just every time a kind indexes goes from upstairs to downstairs my beside yard while easily at all go if circle right things like that how else would rewrite 18 Aug being minus a 1 B 1 2 minus a will be Brooke owed it was startup Aswan but there was a gap this complicated thing which requires for firms are part of signed here for plus 1 minus signed free plus signs and for terms altogether Hi how is just do do fraud yet that for that's the reason and various community playoff there is more Brut are br after the more joint is best described when we go a general relativity for us right now it's just that need full of our manipulating indices from manipulating indices and minimizing the number of indices with at a thus if and a new yacht and a new being research team lower work guest contact did right that's not your context of relativity be cause there are 4 coordinates indices for he ICI but I'm not worth while keeping in mind that go with their work but that's correct and lack the analogous version of it for threedimensional space is essentially the same but slightly simpler in that you don't ever have to distinguish upper and lower industries that's appeared more control areas I certainly Karen Tanaka might has a geometric significant but you know the geometric significance as some something everybody figures out for themselves and then promptly forget because you just really you never really think about it you just learn to manipulate begin the season so quick and so efficient so
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fast that but there are yes to to both have maybe another hour it does have a geometric meaning the geometric meaning is well OK I don't want to get into it now are wanted to go with the lectures really about electrodynamics forget all right now let's so let's talk about the different kinds of cancer as I'm talking about cancers with towing the seeds of stock about a who cat it is in any case industries now this is not necessarily the same effect is not the same in general TO BE about seeing mail say order early in the season general for example 18 you'll be is not the same in general as a new being killed this could be a Norwalk B 1 and that's not the same as a warned me not it is not sale so until oral cancers are not invariant under a changing from 1 intent on changing the order of the industry but there is a special case called that attempts symmetric all or where bill have this property the team you know is equal to he New York struck 4 for a new being new class a new won't be mill if you interchange Mulino this doesn't change what happens becomes this this because this was the same quantity so you could construct cancer a symmetric tensor by having together we are well by adding together sometimes a symmetric others are some Kansas electric cars or in a symmetric tensor has a special place in general relativity not for in special relativity well we'll come out more reporter for us tonight is the end by symmetric can't West afternoon cancer which when newly interchange knowing you changes sign weekly construct such as cancer but putting my signed here and put was assigned we construct the cancer which would change signed when you interchange new and if you the change we this gets swap with this but there's a lot of my time so there a symmetric cancers and the entire symmetric if by symmetric cancers have you were components and some cancers and the reason is there diagonal components vanish for example equations says that f naught nor is equal dumb ideas naught not said you no equal to kind have not minus F not not the only solution to have 0 so if you think of are they as already a matrix of feel like if you think of it as an array already than if Tyson symmetric tensor Missouri be interested in is completely 0 on the diagonal and they also diagonal elements a given needs now will be given names work call this 1 might guess E won this 1 minus E sub tool and this 1 minus east of 3 end these names are of course chosen for future purposes but on bound half of is labels for now might B 3 tool miners be Warne put down here I haven't written all the elements of Earth but I'm assuming that this f terms were very by Sinatra a lesser laughter it means that is a matrix of sent by symmetric down here we put plus he wanted plus Ito plus he 3 but said reserve with Darice here which must be Class B 3 mightiest beat tool and possibly 1 the sent isometric which means you just flipped the signed when you reflect the back that's the notion of end isometric cancer and it plays a key role in electromagnetism where each stand of course for electric field and be spared from magnetic field electric and magnetic fields will say nothing about wholesale electric and magnetic field combined together before and aided by celebrity cancer in relativity electric and magnetic field are not tool independent things and particularly bad news is that the rule out when I will arrange transformation is before the electric field can become magnetic field just like X can get mixed with seat eking get
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mixed would be so what you'll see is a pure electric field I might see as having some magnetic component to it Apple not there yet so hours just by way of world notation Stricklin rotation reps now begins with a physics there are part of tonight's lecture was all notation abstract draw Foley although crazy if I had the right nor what 1 1 2 2 3 3 every car equation so it's good have OK now we could begin I either studying the dynamics Of the electric and magnetic field old or we could begin by studying the motion of particles in an electric and magnetic field the dynamics of electromagnetic field it would be the equations of motion that electromagnetic fields satisfies the Maxwell equations to put the briefly whereas these equations of a particle moving electromagnetic field are the Lawrence for swore they remind you were warrants for sources familiar the rider from something else will arrive it from a combination of relativistic ideas and the action principle while overall does remind you of Mass acceleration this is a loyal velocity version of it may times acceleration we don't have to worry about the relatively much mass times acceleration is equal right inside your have electric charge on charged particle times electric field they stars acceleration is electric field firms electric charge and the other terms is the magnetic charge of magnetic start up a magnetic charge the magnetic force which also involves electric charge plus I think it is velocity of the particle cross product with the magnetic field for crossed product I assume you know how to think about cross products will use them over and over again I actually have removed section here cost products but some them the tonight because enough crimes that there are some speeds of light in their promise of the more legal toward our this is usually taken the B overseas he divided by see that's the only thing that's speed of light and that formula will is just electric charge measured and appropriate Europe's measures the appropriate like that they have let's nonrelativistic version of Lorentz foresaw all of which would go into the arrived the relativistic version tonight and fullblown order find that these 2 patterns I really part on the same time there really part of the same thing written in a way where all reference frames will get the same was so to ensure that that are answers to respect the rule of the law of motion of a charged particle is the same in every reference frame and we want to write an action principle where the action is invariant under Lorentz transformation that's a key for action invariant under Lorentz transformations singleparticle we have a formula works back leader snow will come back to it for a particle without any field we just write down an action which was the integral along the trajectory from wanted to was to be the starting point and the end point of the trajectory minus the mass times the proper time interval profit time from 1 . next minus the best times totalled proper time from wanted to show you real wrote that but as the square root Of 1 might X . squared where x stop with more X stocks Querida's ext . and is a component of velocity here and is just a 3 vector of symbol X and back X Box squared means X 1 . squared plus to dark squared plus facts we got square and so forth lowers the total square of the velocity that's what I thought word means here summer want to and 3 lots of hours the hours the actions the rope firmer identifies minus and times a square root as Lagrangian of the particle we work that out we found our work he will mental water was the energy was there firstly that were early Ed an electromagnetic field while I haven't told you much about electromagnetic fields yet but the basic structure which enters an electromagnetic field is a 4th vector of 4 vector of electric and magnetic fields themselves are derived quantities the basic underlying quantity is 4 vector a meal that a function of position at the end time it's the field describing electromagnetic waves if you like or electromagnetic fields in general and forth Vector B from electric and magnetic fields are the right things which were going to the rival abroad trial drive tonight but the basic starting point of 4 before vectors core over the year of the vector potential of 4 component vector potential will learn what has to do with electric and magnetic fields of for you do it it said who construct and action for a particle iii electromagnetic field near was very simple it's in many respects simpler than the thing that we did with the scalar field a you'll field Lauren barracks the natural thing to do it along a trajectory Mr. take a little segment of the trajectory described by DD X knew will for vector from here here and what can you do with such a 4 vector if you have another 4 vector in particularly rough for vector with covariance indices to make a little scalar quantity associated with that little gap there you take the this new multiply
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together With 1 operating next 1 Lawler index idea that AU was a function of X which means accent and this summer lock you integrate from warnings the other From wanted to 0 and this is just another way of saying for each little gap to take the differential distance along the segment and multiplied by new performers Taylor TX New Times Diaz and a scalar everybody agrees about value on each 1 of these little segments and we had them together you could them as well as I can and then I will get the same answer for the actions of course what I wrote he was a scare them all up quantities that we agree about will continue to agree about them what more can I do with this not much the only thing I want do it as an multiplied by constant however call I constant it is electric charge and be class on a notation that was 1st put in place by Benjamin Franklin was a minus sign here as arbitrary it depends on definition of electric charge and have the proton and set forth well where Braun trucks yup OK so signs reviews the other terms in the action of Porto of particle particle boring electromagnetic field with Trevor appear Maria's he finds an integral from St. pouring from wanted to PEX a meal of X no 1 0 do we want a white family were bronzes equation we want our because equations for the most particle and really my goal is to write programs as equations and stole that they just Lawrence Waslaw but I would like to do in a way which is relativistically invariant in a mass times acceleration I'm very nonrelativistic per hour Formula One for example is is relativistic acceleration Watters acceleration mean that doesn't mean the 2nd time derivative Of the position with respect to time but doesn't mean the 2nd time derivative of any 1 of the 4 coordinates With respect to for those in need a 2nd time derivative with respect the proper time while I think we're probably all could always guess that these variant of the year the best definition of acceleration of relativity is differentiate the coordinates twice with respect the proper time along the the trajectory of suggests it is the right definition and ask for acceleration was a fourday we want to arrive for victory but we're going to begin with our thinking about velocity being derivative respect time and later we we calculate the equation will go back and make it Lawrence very now we all change will we write it in a way which will recognize as an equation among 4 OK so let's leave it this way for a moment are this quantity in here would be the logon jail cell this would be the Lagrangian for the free particle but as other contributions reaction over here so let's write this summer forum which would make it look like a Lagrangian this very easy we just don't violate by these team and multiply by DT 5 so let's see what we have here was the X naught IDT was VX naught IDT X noise is so VX not by is as worn in her 1st day we get here it is a not DT today not as a function of X city that's a 1st R D C by D. C History of our next terms is minus E In the role but cause for now X sought BX by which is X . starring a sub x bt likewise for wives saw the rest of this it is simply X a bottom a federal court X dock the end component of velocity dotted into product ex combatants component of velocity dotted with the M's component of the vector potential as better this is an ironic this way with here is to make it have a familiar look on 8 will grungy and action and action as an integral DT OK so let's tried at Paramount he has integral 80 Norwalk class X and . 80 and a DEC now I can identify Boulevard yeah calculate what's the roar of the satirical Granzien this year it's just everything inside the integral here or transient for this system firms particle in electromagnetic field of minors and square root of 1 minus X thought squared which means Exxon square got square Placido squared electric charge a Norwalk Aswan bias electric charge 5 and a M of Texans C B acres are just functions of X to their fields just like we did was still a partner with a scalar field work your Maginot somebody call about what a of extant somebody course it just unknown function each 1 was known function of X and B and now we're just exploring the motion of a particle In that normally field and he is action Florida where we want bill if we want to write your Uruguay granted equations or want to show that they look like the Lorentz force that's our goal take this will gradually and see if we can derive the warrants force Mass times acceleration is equal to it electric field plus may not but you can magnetic
1:06:16
our forces Willie began let's consider the equation motion for X and so we saw with partial of L with respect to X and got us a starting point it was obviously a contribution from the 1st term he already done whether back we ought batters it's incidentally this and is not the same and the services index this was a mass of socket confused love I apologize for that but it's too late to do about was mass this 1 is component M and tieins X M Dr. divided by square root of 1 mind X parte square that's the derivative of this this'll grungy in here with respect to be and component of velocity as another that we've done that before as likely this happens to be and my time the X MYD how it's the X by DTE but this extra factor here just makes it before velocity before vector of velocity this it is in arms of 4 vector of losses DXM rugby towel but will come back act for the moment cautiously with this way similarly that we take derivative of is that the time so all we get ended by D. trying X and got provided by square root of 1 minus X . squared vessel left him sorry that score 1 for some this is not right is not what right good now more stuff there's more staff there are other here was put back here what is it to my EDS he finds it a piece of em From he delivered although Grosjean with respect X . gives us on terms which is just the vector potential electric charge vector potential sorrow or we have to do is be careful 1st of war this is Emily times this minus V them With whats that's a left side as a full lefthand side of the order around equation comment the righthand side of the right hand side the right shy should just be D held by DXM right it is a cold up yes I did remember but this is we could put it back on EIA I did I did a remember that when you switch space components it doesn't matter for it it doesn't matter Fahey either right or threedimensional components it doesn't matter where you are while it only somewhat of repeated overseas and don't count the messes index will well few if I want to be where might have done that yes all year did here I did so right with Aladdin indices it doesn't matter whether they're upstairs and downstairs and if they repeated use some of them are your only some from 1 of the 3 nite from 0 to 3 of that's part of the rotational tracked by so Healy have the lefthand side of the order Lagrangian equation would have on the righthand side right hand side we have the will of the derivative of Granzien with respect excess whatsoever that the whereas it their wares at depend RXA a lot depends on X it's just club now has a fixed not only function of the positions of the particles songwriting inside rural hammering this he trial derivatives all 18 Norwalk with respect to x and now as I looked familiar with anyway on the left hand side something that's sort of like mass times acceleration and on the right side something which is money the derivative of something this is clearly he formed a nor is clearly electrostatic Aziz electrum as is the potential energy saying which mechanics who would ordinarily called V or you whatever we call potential energy so this year is just familiar Over years lefthand side here has something like mass times acceleration if we didn't have this thing downstairs here which is a or very close toward a slowmotion promotions for all this literally as mass times acceleration this Aomori about later not righthand side we just have my electric charge the derivative Of the electoral firm of electrostatic potential services potential energy over a hero of the gradient a potential energy which is essentially electric field will come back to that part of now was more are leader these terms also depend on X I have to differentiate them with respect to x these Our mixed with depend on both velocity position we've already take it or kiln a factor they depend on velocity that was on the left side of the equation on the righthand side of the equation we have arrived Marine this he called us here then him it's a summation of X X and the dot kinds of derivatives I love they did and with respect get X and my wife we're was come from it doesn't matter whether a call was an array and some of our bot From back here Lee M. index was not some were so I don't want to call summation index and are cold a switch that I why am I differentiating with respect X M because the righthand side of your Lagrange equation derivative law graduate with respect to x In job we have excellent back through the and with respect X so look at this look out carefully the lefthand side has an index which is ensemble over the right hand
1:14:11
side has an index and which is ensemble over not some progress but it also has an index in which it is some of this is me or I will grant equations for it each component of the position X 1 X 2 extremes of 4 it doesn't look like anything you are my recognize yet but there are fortunately there's no simple way to work the deal would a deal what so who ever OK move the site over here that's easy that's just to reward doctors that he by D. C R mn money outside or X thought and in the matter lowered the matter over the square root of 1 minus X . square at best this over here now how league differentiate this thing with respect the time the sudden may and made it may depend explicitly on part but even if it didn't expect depend explicitly on car and it still wouldn't be constant because the the position of the particles from the position of the particle is moving so even if only depended on position and not on hard a sober and as attracts a particle is as follows a particle would depend on time stopped there 2 terms when you differentiate a and see what they are minors EU and 1st Turner is just the explicit derivative other AM with respect to teach me Aleksandr but there's is another turn and the other turned issues warriors part of the change in aII when x changes howitzers X end a M sex did I thought laughter have gone to sleep gets us a COA Harris we have the time derivative of age over here a depends on time explicitly if the field is changing with time but it also depends on target implicitly from the fact that the position of the particle is varying with respect to Part II so we differentiate 80 with respect the position a multiply by velocity this dummy index at some the sworn is the index that appears on the left side of the equation story have this side of the equation right here I restorer finished restored the right hand side a bit of a mess our mind this he had fires D a North Sea but DX In a sister on us a potential energy and there was Estella here Maria S E X. editor Dr. D. ADD end by as a lotta stuff is more stuff that I usually like to Dormer blackboard bark no way around it so let's take this call list and a mass times acceleration that's not literally acceleration but let's just call it but think of it as best times extortion lever on the left side and buy X . Emil square root and put everything else on the righthand side in the group things together we are grouped together the kinds of 1 kind of terms is not proportional pool velocity this is no velocity multiplying it here is another term which has no velocity multiplying it the a 2nd term here on the left side has a velocity and the last term a righthand side has a velocity cylinder group things by whether they have a lot of people don't have a boss that just remember going we're going to electric field times electric charge which is does not have a velocity and v Crosby you which does have a velocity saw going forward trying to identify now goes 2 terms left inside Is let's just call it nest times acceleration but put some quotes around to indicate that it's not a literal 2nd Prime derivative of extra with respect to just Her left here side on the right hand side now we have the 2 terms which don't involve Ferrero velocities we have each time D 8 AM by D X naught AM by DX nor that now reverted to calling X not my list same D not by IDX Norris nice symmetry here it has D. a M by X nor what has been a nor by VAX and with a minor obviously this thing wants the B electric field has no velocity in this term over here so it any sense for us this must mean mass times acceleration is electric charge times electric field and indeed busy electric field the electric field us 2 terms 1 of them is the gradient McGrady and of the Ark of the kind component and the other term is the kind derivative of space compiled that's what we later see this fall's nicely and the place places electric field now or about the terms of velocity let's get them offered readers from our IC Insights or plus iii we have both of them have X In the doctor source for it X got our future 1 of them has sought see where is it could bracketed why has M by DXA and the other 1 has DE a by DXA Musser patter each term he or yacht Parra apart from the X Box is each of these
1:22:07
terms has a derivative of the Nikkei with respect will coordinate if symmetrise norms that symmetry that 8 M
1:22:20
buybacks not Matos a lot by XM a M by X and minus a and my ex interchange of the unit in an old here and interchange In and Norwalk over here let me ask you and they those something or we're here to make us to make this a little more parallel to the rights of the stupid where I T . she 150 . this by IDT its is warned young 5 or I could also write it as X naught Procter but the new anything just a just threw in for free R D by D. C which is just 1 now starts a kind of 1st symmetric with respect the space and time a little bit here we have a enticed symmetrise thing with index mn and here we have a nearby symmetrise thing with Nordic and NAM misses XM . sorry yellowstriped that right right OK this it is a form of the equation motion this is electric field this of course is the magnetic field is the magnetic field for identified detail however components of magnetic field was this operation over here what mathematical operation of our done on the year components of the vector potential the curled right of mathematical curled will come through I didn't want to do this it is these are are the components of the cruel AT & the curl of a use the magnetic field this year contains want terms which is just the gradient of these kind compulsory possum terms and this constitutes electric field but private everybody everybody happy with it's a bit March but I tell you go through this this is something here are a few really care about their so here Halaby June 8 where reserve the caesura stars stop their kilograms and Gore the Lagrange equations collect together the terms 1 side your call and they terms have terms with velocities and terms without loss of course didn't have a loss of was just 1 group them together and show that this is a worry group together you learn a lot most of our field theory about rubric calculus especially about dynamics Hooker Yale the 3rd inning for it actually reserve cost just react sugar palm on part for look of the particle Rauschenberg where he he will be a year or more aware that bristled a bull minus currency integral part a DX 1 dead us was put the user experiment but ot bark on what constraints I think the right it's not white yeah right exactly this that's partly experiment but what is it that this integral has what properties as a half which make it a good candidate 1st of all the action is in it integral along a trajectory that's a wrong up till now we respected that the action to be thought of as little incremental pieces thought much of incremental pieces and the other world is simply that it should be a scalar that should be composer scalar the X New Era basically do things you could do away to make a scalar 1 is for multiplier by itself that that's just D Tower squared that's already in kilogram here it is b towers just squarerigger that's in or around here already it doesn't involve any field was the other thing you could build you could not afford vector of water you could not be X or not adopt but you could could multiply DX fight AT & contracting industries if you can't find another way to make a scalar you welcome reply where there are other ways there are other ways are we not drive yet the concept of Dejan vary but I prefer to work with it at this level 1st and then the study its gage invariance after we are gone as far this is basically the simplest thing we could bound you go right analytical multiplied by eons may need new weighing new figure scalar multiplied by most scary combine analog job but this is very much the simplest selects explorers but stored in that the arms uh our friend of mine and exploration of a simple action what happens to start writing it out you get a set of equations of motion which are the Lorentz forceful Of the redirect what's that yeah you your neighbor right Bible having you lose we have Zola principles that we regret that we have identified principles that Oh yes we're going to do it but we will do exactly that will leave another trial in the action Germany actual will be field part of the action not the particle moving in the field but just part action that governs the field by itself before we did for the scalar last time OK let's keep going In the last great resource your user it was weird Her OK most summation here this is a component of the acceleration at sea and composite the acceleration and is not some additional here was just an M index and that's all there is In the index over here that is combined together with an index over here a repeated in X X a Debye DXN and and not and DAN projects are locked in his summation index literally this some all over and if we wanted to to provide some summation some more about the rollers repeated indices gets some of our non repeated in the cities explicit and there young on both the left hand side of the right hand side the sale let's see if I can write this formula in a way which is manifestly that means obviously Lieurance at the moment it is not the reason is that the U. lefthand side the right hand side are not themselves for vectors enough for vectors but we can convert them for vectors
1:30:59
source do that young Wednesday said about what they is but a far I well I will arrive with elected work everyone I rural right E but yes we want or I eat currency and component of electric field what's special about this is it doesn't involve velocity it just depends on a electric field which depends on space and time but this trend was not containing a velocity works on velocity dependent for she forced Osnabruck velocity only depends on where the particle is the sorrow here we have a which is like that Over here we have all terms which multiply velocity so it better be that bad corresponds a magnetic field but I think I think which is saying is right there if we started from this principle we started from the principle of least faction we wrote back that action we eventually got this equation we might look at it say are let's think this whole thing here it doesn't matter whether it's part this isn't part that let's just call that he knew were invented discovered electric field so I think that about that really is the natural water natural logic is to identify this call electric field and all like manner we will identify this with components of the magnetic field V kind sampling but not yet because we were hurt or do they are right let's see if we can write this and of former 1st of all which is obviously obviously invariant means but you should write it as the left hand side and right these cancers of the same kind while this obviously wants to be a vector type thing it doesn't want to be has come as a component of the component is a special compartment is solid is part of some kind of 4 complex which you my identifies a former vector but not quite and there are lots but see what we can do it OK so what was it really what really was and was was Emlyn Cairo's Debye city of DD X knew by did the X they are likely to be for XM got provided by square root of 1 minor such a score squared the parte squared before this quantity X stop over square root of 1 might be squared X thought means DX by DT offices expiry bt but this happens to be exactly DD X M by these cow remember what we call that variable but that quantity the BXM Carol we call it before velocity and there was a letter that identified with you will of course is also a false compartment was a 4th component the 4th component would be DAX not like towel that would be you for particles is very close to 1 but nevertheless there but at the moment what we have here he is literally Deeb IDT R DD X and by D. Tao for your ears the time derivative of before velocity and that's equal to a hand side the right hand side also manipulated awhile but I don't wanna have a Debye D I wanna have a there while I wanted towel because how was the invariant quantity I don't want to have a non I don't wanna have this kind of makes things where towers these might be how that's Lorentz invariant doesn't change from 1 frame to another but he does so or I do with this and multiply when a multiply this year or stuff here stuff that comes from right side of the equation murders where multiply both sides of the equation by D. T by deep how for Auriemma left side they buy DTD Didao this is a budget how canceled and this is just the rate of change with respect to Hal but after a multiplier but ability to use must pressure this is just they a mail storms or 2nd derivative of the component With respect the cow squared this it is Park Ave for vector a fight and the 1st 3 components x y and z if I ate that are 4th component which is the 2nd derivative of time with respect of how squared that is a 4 for vector and these 2nd ex New with respect the power square as a kind of acceleration but acceleration just as a Lorenzen variant concept velocity is to differentiate the 4 components with respect to the proper time they Lorenzen variant notation our Lorenzen concept of an acceleration is to take the 2nd derivative of or for components of position with respect to a proper time so you know that on the left side right that's gives us where is standard called proper acceleration of called proper acceleration on the left side the full only particles it's close to the ordinary acceleration let's look what's on the right side of scrotal right here inside here and narrow multiplier body DGD cow what's multiplier produce so we that parents X naught by d d towers the same thing as the X not cowl multipliers multiply With the X naught by deep towel but what's already I this is D X In my view she now are multiplying by body child so are that make it just makes it DXM by now now that they stood terms they have exactly the same form the same form for the different index here the index's 0 0 0 here the indexes and this can all be summarized into a single equation or single terms a single turned he
1:39:32
cars end by x mule minus a mule might X Denmark 4 times will be X by towel the lefthand side of our right however left inside its D 2nd by deep cow squarish and of X and equals this his ex an early nite New Wave big nite East Africa cheaters and told you this is an equation is 3 equations for the space components of these proper acceleration on the righthand side we still have space components here wouldn't it be nice if there was a 4th equation this Sastry quite 1 fridge and want to would be nice if it was a 4th equation which was just the prime component of theirs same equation except with and being the time component or where he and the current component over here if that for the equation was correct then we could summarise the entire set of equations by at basically 1 equations last proper acceleration in all 4 components BX New he prices a derivative of a mule what have you she mule crew With respect X New my sister of of a new war with respect to x mule the X New but because your BX nobody pal I think I should have next enemy air end game with what we like that equation down again your home workers to go home and arrive the following equations store in all the 4th equal let's look just we write purity this is a complex of 4 equations 1 for each new 1 2 3 and for the 1st 3 of them over the last 3 of them 1 2 and 3 are just the equations away written he when and was warned to Winfrey's an equation the equation I've edited is for the 2nd time the river vexed nor how I know the drill why should be true 1st the boss of Whoa the lefthand side is literally of 4 vector so as our right here side This is a cancer as with 7 index if you take a cancer when you combine it with a vector you go back there this equation is a cancer equation and the Senate's because it transforms stock be call last week took care of from the beginning to make sure that the equations of motion while Lorenzen area and how we did that by making sure that the action was a scalar is the larger Kizil logic prevail Everything and learned and feel theory of particle physics armada of physics make sure that grungy and respects the symmetries Of the problem of the symmetry of the problem is Lorenzo cemetery make sure grungy in Lorentz invariant once you do that you don't have to worry again about whether the theory is In the variant with respect tolerance transformations of you after worry about whether looks same every reference wants to know that's the laws of physics a Lorentz invariant end but the 1st 3 components of a certain for are equal to the 1st 3 components are some other force vectors thing you know automatically that a 4th components will also match the only way a system of equations Kabila and variant of the 1st 3 components 0 0 let's is for the 4th component also could be 0 automatically conclude from the fact that we built the Lorentz invariant theory a 4th equation must be true the other 3 equations of a complex equations which include all 4 components another like to say it is if it not the equations for the 1st 3 components and transformed them you would simply pick up above the 4th equation that way but you need to do that for you to do was say I know my theories Lorentz invariant and therefore the 1st 3 components Of of a vector equation our true so must be the 4th component yacht then a she has where are you yours but Torres but on your house just as world what that how we visit that they would visit to its energy conservation beyond well OK sovereignty concentrates the on the 1st 3 equations cannot and will not energy conservation they were the 1st 3 equations are a former D P PT physical before us they'll be PDT is a writer and fears and PPD and so there were roughly speaking Road did not roughly speaking a exactly the equations that the current derivative of a momentum is equal fervor force 1st 3 equations and the being everything it's on the right hand side of the equation before and they tell you will not momentum conservation but they tell you have a change of momentum with time is doomed is a response to the existence or force the flock equation tells U.S. how would the kinetic energy changes with time how the kinetic energy changes with time how does the kinetic energy changes our these kinetic energy by of Coach K IDT well as cold work by workers for starters velocity that's the 4th equation is the 4th equation is the equation that tells you that energy changes In a way which is consistent with the amount the work being done and that's not so surprising that I remember that we momentum and the kinetic energy form of 4 so it's not surprising that there were video component equations are related if not to energy and momentum conservation at least for the way in which energy and momentum change in response to the existence or forests are for the 4th equation it is the energy balance because you have a Connecticut Energy changes when you do work on the system and the work being done by the field that's what this is about armor
1:48:08
but we exciting thing is that you could represent the equations in this very neat Lorenzen variant form tens of form with the left hand side as it can certainly afford and the right he inside it is the product of a cancer with indices forms of 4 vector again we could track the indices and the contraction of the indices produces of 4 vector MOST inside matching for wrecked on the right side that's the exciting thing about that we you'll symmetry from the beginning that called us the kind of Legrand deals with the right down would manifesto symmetry and would plow through it and discovered in fact that the equations of motion really do reflect the symmetry of a problem the symmetry being weren't now there is another symmetry the other symmetry is called gage invariant we have got to it yet will get to it next time but laughter 30 yours is next time but say that he 3 really fundamental principles that calm time and time again which will see in this simple example have 3 principles the 1st principles coal locality 1 locality says is the way I think changes only depends on what's going on in your mind the things you're thinking about a field over here doesn't change brutal field or they're having some value it changes because he'll be a buyer has some very emotional particle only depends on the values the fields new buys it also doesn't depend on the values of the fields a later time earlier time the equations of motion are equations between things over a year by a small vicinity of the car being related things in the same small vicinity of time off and space how we represent has highly implement that we implement back that's called cowardly we implement backed by making the action the action should be integral all wall space and time of some kind of Lagrangian density which are only depends on the values I'll things at a particular point neighborhood the core eccentricity and well so it depends on feels an end derivatives Sadie fields and their derivatives With another words it depends on data which has to do with local nearby concepts in your head up the action is point by point by point in space what comes out is differential equations differential equations are equations which relate how things change from 1 point toward neighboring things you could imagine the not having this former you could imagine reaction has In things which involved products of feels over here products of feels far away but they equations when you worked out of orange equations they would have performed that the way things change over here depends on what's going on over here not that's not the way we actually principle works the action depends on fields and neighboring feels neighboring fields now means derivatives of away the field changes at 1 point is only determined by the value of fuel about point and neighboring . 4 yes for components for compartments parts of represents a 4 derivatives of fighting was ushered the theater's fields change in a way which depends only on what's going on new bodies in space and car as the principal locality and it's no more Norwest saying that the action is an integral at quarter of all by making derivatives by saying it depends armed fields and their 1st derivatives fields in the 1st derivatives that you could try putting in 2nd derivatives and her there's an interesting subject there but but the but were too much are actually principle together with the idea that the actions built incremental pieces each of which depends only on a field of neighborhood serves local neighbors the ITI what talking about particle yes when we're talking about the particle motion then we can substitute the statement that a grand jury in all the world line of a particle only depends on the positions of the particles at neighboring positions along worldwide but also it depends on the values of the field at the position of the particle acceleration of particle over here does not depend on the value of the fuel over here it depends on values of fields in the neighboring region that's the principle of lockout that the responsibly the fields for or particles depends only on what's going on is that incident test would cost 5 2nd Lorenson variables there are in the in both cases for both the particle and the field the rulers Bill will Grosjean out of scalar build the Lagrangian make sure the program itself is a scalar way of saying that we all agree about the action bar within a certain volume of space and time L equals scalar With postal principles are very pervasive was 1 more principle which we have our work back yet I have it in my notes and alright bound bus could be the subject of next wake gage invariant and that means we will find that nicely these 3 principles are extremely pervasive every fury that we know about where its general relativity quantum electrodynamics a standard model of particle physics YangMills theory are all conform with these 3 principles and basically I would say anything else on Copper vector you need a sort of incidental and incidental not in the sense that stance port coincidentally or a remote right word that varies from this example example in ways that don't really have to do what was was deep principles the fact for example the standard model has 3 species of quarks well let's not contained in here that's a sort of random new edition of the basic deep principles are these so next time were worn with gage invariant said I advise you to to growing go through these equations and figure out where the various things came from because they are expected to world our goal for more please visit us
1:56:43
at Stanford down EDU
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Summer
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Energielücke
Supernova
Walken <Textilveredelung>
44:22
Schaft <Waffe>
Fahrzeugsitz
Summer
Intervall
Lichtgeschwindigkeit
SEED
Computeranimation
Rotationszustand
Bildfrequenz
Kristallgitter
Passfeder
Fahrgeschwindigkeit
Nachmittag
Array
Eisenkern
Kaltumformen
Seil
Elektronisches Bauelement
Diesellokomotive Baureihe 219
Rootsgebläse
Übungsmunition
Magnetische Kraft
Werkzeug
Jahreszeit
Jahr
Elektromagnetische Welle
Matrize <Drucktechnik>
Lineal
Ringgeflecht
Luftstrom
Frequenzumrichter
Stückliste
Magnet
Drosselklappe
Rips
Masse <Physik>
Kraftfahrzeugexport
Newtonsche Axiome
Bergmann
Schnee
Elektrofahrzeug
Schnittmuster
Teilchen
Spiegelobjektiv
Pfadfinder <Flugzeug>
Brechzahl
Energielücke
Bahnelement
Klangeffekt
Stunde
Verpackung
Relativistische Mechanik
Entfernung
Windrose
Kombinationskraftwerk
Elektrizität
Elektrische Ladung
Trajektorie <Meteorologie>
Schiffsklassifikation
Nassdampfturbine
Großtransformator
Source <Elektronik>
Magnetspule
Ersatzteil
58:19
Summer
Magnetisches Dipolmoment
Mechanikerin
Hammer
Spannungsabhängigkeit
Nacht
Zelle <Mikroelektronik>
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Array
Elektronisches Bauelement
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Diesellokomotive Baureihe 219
Rauschsignal
Rootsgebläse
Knüppel <Halbzeug>
Jahr
Elektrostatik
Stückliste
Kosmische Strahlung
Drosselklappe
Masse <Physik>
Irrlicht
Ruhestrom
Reaktionsprinzip
Nanometerbereich
Teilchen
Franklin, Benjamin
Gasturbine
Kopfstütze
Brechzahl
Energielücke
Stunde
Entfernung
Waffentechnik
Eisenbahnbetrieb
Tag
Elektrische Ladung
Druckkraft
Trajektorie <Meteorologie>
Nivellierlatte
Band <Textilien>
Schiffsklassifikation
Amplitudenumtastung
Lastkraftwagen
Magnetspule
Ersatzteil
Schalter
1:14:11
Drehen
Target
Masse <Physik>
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Elektronisches Bauelement
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Arche
Elektrische Ladung
Jacht
Rootsgebläse
Band <Textilien>
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Gleitsichtglas
Teilchen
Bark
Gasturbine
Source <Elektronik>
Ersatzteil
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Hohlzylinder
Ausgleichsgetriebe
Drosselklappe
1:22:20
Greiffinger
Gesteinsabbau
Magnetisches Dipolmoment
Leisten
Sturm
Energieniveau
Satz <Drucktechnik>
Leistungssteuerung
Juni
Bark
Bildfrequenz
Fahrgeschwindigkeit
Vorlesung/Konferenz
Array
Kaltumformen
Explorer <Satellit>
Elektronisches Bauelement
FaradayEffekt
Urkilogramm
Rootsgebläse
Magnetische Kraft
Jahr
Luftstrom
Ringgeflecht
Kugellager
Analogsignal
Walken <Textilveredelung>
Drosselklappe
Direkte Messung
Erdefunkstelle
Feldeffekttransistor
Teilchen
Hochspannungsmast
Digitalschaltung
Gasturbine
Brechzahl
Mark <Maßeinheit>
Waffentechnik
Gruppenlaufzeit
Eisenbahnbetrieb
Maxwellsche Theorie
Rauschzahl
Schmalspurlokomotive
Druckkraft
Trajektorie <Meteorologie>
Nassdampfturbine
Source <Elektronik>
Ersatzteil
Mittwoch
Zwangsbedingung
1:39:31
Toleranzanalyse
Schaft <Waffe>
Leisten
Satz <Drucktechnik>
Nacht
Bohrmaschine
Schwache Lokalisation
Woche
HDTV
Fahrgeschwindigkeit
Vorlesung/Konferenz
Array
Eisenkern
Kaltumformen
Pulsationsveränderlicher
Elektronisches Bauelement
Unwucht
Teilchenbeschleunigung
Übungsmunition
Panzerung
Elementarteilchenphysik
Atmosphäre
Flüssiger Brennstoff
Jahr
GAL <Mikroelektronik>
Lineal
Starter <Kraftfahrzeug>
Drosselklappe
Kraftfahrzeugexport
Jacht
Reaktionsprinzip
Feldeffekttransistor
Teilchen
Digitalschaltung
Spiel <Technik>
Gasturbine
Brechzahl
Reisewagen
Kontraktion
Gasdichte
Behälter
Antiquark
Omnibus
Energieeinsparung
Rauschzahl
Druckkraft
Raumfahrtzentrum
Videotechnik
Großtransformator
Fernbedienung
Zylinderkopf
Mikrowelle
Ersatzteil
Feinkohle
1:56:42
Computeranimation
Metadaten
Formale Metadaten
Titel  Special Relativity  Lecture 6 
Serientitel  Lecture Collection  Special Relativity 
Teil  6 
Anzahl der Teile  10 
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  10.5446/15005 
Herausgeber  Stanford University 
Erscheinungsjahr  2012 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Physik 
Abstract  (May 14, 2012) Leonard Susskind dives into topics of electromagnetism and how it relates to quantum mechanics. In 1905, while only twentysix years old, Albert Einstein published "On the Electrodynamics of Moving Bodies" and effectively extended classical laws of relativity to all laws of physics, even electrodynamics. In this course, Professor Susskind takes a close look at the special theory of relativity and also at classical field theory. Concepts addressed here include spacetime and fourdimensional spacetime, electromagnetic fields and their application to Maxwell's equations. 