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Special Relativity  Lecture 7
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Erkannte Entitäten
Sprachtranskript
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Lance Steiger University 1st
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announcement I will not be here next week I have to Belgium's biggest some lecture I will be back and we will continue this course the folk set of 10 lectures were given I have no no idea where we are now because get lost there are some farmers firewall chisel you you will have a full set of lectures but I will be gone that's what I think we missed at least 1 of the very beginning I can remember was 1 or 2 1 I will go this 1 next week they will be made up I have many reasons for wanting to make a mark 1st of all I don't watch is all but 2nd of all I wanna get the full set of lectures at Paris earned store we will continue until the part let's talk about 1st on principles and I want to I I realized Kashmir I did that was really up to my ears of all kinds of stuff I have notes from tonight but I simply wrote them opera with Norwest 3 hours 3 hours and I will get them out if somebody would like to take them and scan them and put them on the Web site I would really like that much scanner takes 4 hours and hours it's the slowest in the world but there was somebody direct form me case let's a berths for just a moment reformer person it like replaced page on I appreciate that the last time I did some pretty heavy intricate mathematics on blackboard and I you would guess that some fraction of you'll follow it fractured already I pretty bad I understand that little again I will bet on we'll be might teaching policy is to teach until you cry a flat fee always been but if we are going to take seriously the the goal of this course which is teach the theoretical minimum necessary mysteriously go the same for the next level there's no alternative if I think I've said this before a fight you will wait to to do it without the mathematics would it but I don't Leon it seems the sort Beale law of nature that nature is expressed through mathematics I don't know why that's true will actually do think I do know why it's true because the extent of the laws of nature makes sense on exactly have to be quantified have you quantify things were mathematics and so forth are but it's true that the mathematics gets more and more are came as you move as you move forward and this is not because physicist want to persuade its not because his sister a bunch of high priests who were trying the height with doing in expressing it fancy language and nobody else can understand if it just seems that it is that way maybe someday somebody will crack the code some young person will see us simplicity and the laws of physics that we don't yet see and be able to express them are in a very very simple equation x equals 0 and that'll be it a final point that I can always do that you could pick old the equations of physics square them some them up instead of equal to 0 because of the sums of squares is equal to 0 when it returns 0 but that's not helpful songs to top off his head but I armed with you I got idea that I got list of Europe right on absorb we write we are we say somebody young in spirit Peru weary warms to add up to death on yet will find lot St last last week but you're out of it we while words all are the official role is a coordinate this is arbitrary which is up which is back Our its arbitrary as defined in your left hand to be this 1 in your right hand the this way back when somebody Court of called this right Ms. weft started to apologize but you get but which is open which is down of course arbitrary is nothing there and it's not that some indices a lighter than air and some indices alone heavierthanair bottom just arbitrary but so basically the distinction is the court themselves somewhat arbitrarily you label with them upstairs and that makes me differences of coordinates for example along a trajectory you label as X knew would be index upstairs William upstairs derivatives with respect X for example the derivative of a scalar with respect to x you think of having a downstairs index figure housing having
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downstairs and backs and there are many many other things but also have downstairs and to use things transformed under Lorentz transformations as coordinates Kerry upstairs industries things which transforms derivatives carried down there is not much difference between you go from a thing with upstairs and that the downstairs index just changing the sign of a party component that bulletin changing the size of the current component but it saves you writing out a lot of work to right drive D Best Buy X knew what X you downstairs index of upstairs and this is for the stand for it stands for the essay by a VAX current component times BX naught plus B S VX warned the X 1 plus blah blah blah blah on 3 Carroll and the whole thing is just a change in a test in going from 1 . 2 with neighboring porn Delta so here's an example which its much more efficient right yes by DX UVX that the right out the whole thing but we need some conventions we need our we need some rules about when it is were allowed a somewhat over the industry's without writing summation needs we need to keep track of things carefully and so the official will always whenever you see all lawyer next next door an upper index ended the same letter it means on that's that's all that's the whole rule there is a deeper geometric mean to it but I don't wanna go into it now will go into wouldbe general relativity theory is a geometric mean toured are about but for us it now the only difference every vectors every sector was caught EU has both cold at Contra variant life and core variant life and the only difference is the component of the vector with downstairs index is the same as with upstairs and except for the time component will you flip side so it really is just always just a way of keeping track and on the right forward over things like he squared X squared minus D square squared things like X squared plus Y squared Posey squared minus D squared times square is just equal tax new acts as space compose go following the index does nothing store really stands tracks where postwar script Aziz square and as far as the prime component here the current component has signed changed relative to the opera 1 so because my guess is that the whole point of it is just helpful in writing simpler expression Black Bart if you follow the rules and you follow the rules are of upstairs and downstairs indices rigorously you'll find out that is very very easy to write things which are In a variant the warrants transformation X squared plus Y squared Bob Bob opposition aware minus D squared it is in area that doesn't change is the same for 1 observer and another observer so it means to be a variant cyclic X next meal upstairs index bounced is index when you could track them which means some of em like this that makes a scalar Chilo some nice rules to keep track our of our ended the notation not show the vices but very efficient or Wash more border yacht any time you take a partial derivative the always carries a lorry next born yes yes yes recall areas for the call thing stands for me by the X naught which means D. C they by the X you by this is viewed by VAX new its equal to mean the components of the subject here August set up operations if I want what's court let's call it east of here these you any time I have a fake women index so I can construct a related thing which is the same collection of objects with proper and that's all it stands for is my Debye DX mark might X see by DYB by DC so the upstairs thing is the same as downstairs thing except the sign of the times component was changed at it's not a deep and mysterious there and its purpose is just to be able to write things like he squared minus X square may efficient way arms things like he's glad Hasek squared of things which are less invariant it's a trick for easily recognizing and writing warrants in various things quite let's let's discuss I wanna go back a little bit drill thing we did last time and just sort of sketch
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out again talks since it was probably kind of tough there may be seeing it twice but not before they might help but before I do at spoke about principles or spoke about the basic principles Norris said principles I the principles in the sense of a theoretical physicist at 8 discovers a no phenomena are Zhu experimental physicist Scotland from our but then a theorist wants to make a fury of the usual rules are some labels might break down but so far they haven't usual rules for whatever reason are backed the physics should be described a number should be described by an action principle there is nothing that we all in nature that is not describable by an action approachable it is an accident probably not use is something that's a weary of physicist certainly not we don't normally anything form the service pattern is it important that added has formed yes it is for example the derivation of the concept of energy conservation momentum conservation all conservation laws of physics especially the relation between conservation laws and symmetries is entirely through the action principle ridges right equations equations may make a perfectly good science but if they're not based on action principle they will not have energy conservation momentum conservation even charge conservation was also an aspect of this bar there are but a particular focus on energy conservation that is a consequence of the action principle together with the assumption that things are invariant under changes Of The year of the time axis changes which we call translations of the car and variable so yacht writing down physical equations that don't derived from an action principle we would be giving up the a guarantee of energy conservation momentum conservation so far as we can tell a deep principal plus principal world number 1 we look for it actions such that the equations of motion which follow from our actions described the thermometer of discovery laboratory we look for an action description that guarantees that it will fall into the category of things of energy conservation so for the action principal means of expression for the action which is minimized by minimized by the true solution of the equations of motion and the equations of motion fall by you or I will grind equations stationary equations for the action number till lowcalorie well calories basically the principle that if you poker system at some point of time and space it has no immediate effect except on things in the immediate vicinity of ripped off its direct effect on things in the immediate vicinity of red Of course here Friday Have a string violence train and I popped over here it would only affect the neighbor instantaneously bustle calories but then the label would affect the Labor damage chains off for time poking your he would effective everywhere but the instantaneous effect or the short time effect his local your body and that is guaranteed by requiring a principle of locality is the action mainly that action a correction is in the world if it's a particle trajectory that would talking about send them along a trajectory becomes an integral a field theory that we're talking about a field contain the on volume of space and time the space time region green to represent the existence of field and that action action the action is an integral not only over time but also wore space and from right that before X the principle of lowcalorie high locality says that the thing we integrate he vote grungy and form the Of course grungy here this'll Granzien in here depends are only are
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the coordinates of the system and that a particle it depends on the positions of the particle positions components of a particle and only the time
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derivatives textile and other words runs at this point he depend only on the position at this point neighboring points points during your body as principal aboard things only affect new barring neighbors near my neighbors he never in space or in end for a field theory says the program in depends on the fields on just called generic field then the various derivatives partial with respect the X meal which just quite I knew that means the derivative of fire with respect x this is enough to guarantee that things are only affected nearest neighbors directly your neighbor our it's a principle of locality that With use depend only on fields in their 1st derivatives not on fields at 1 point Times fields on
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a very distant point for example you could Majid is no question you can Ledger a world in which poking something over here as an instantaneous effect over here In that case you would not write the book Grosjean was an integral or something only involving nearest neighbor derivatives but more complicated kinds of things but all yard last at Wells last I don't talk about Quantum normal cowardly he never use the word you'll guys keep using might do not we talk about entanglements triangle is entanglement is not on what powers you cannot affect the thing over here distantly by doing something over here I try to make them extremely clear tractor make it clear by showing you for example have density matrix of a degree of freedom or he will not change if you try to do something over here now you guys have got to here kids something some Scientific American men however you some kind of priority that the something mysterious normal cat nonlocal about entangled and this it is not true you cannot send signals you cannot affect things at a distance by doing something over here Korean that's not what entanglement entanglement is basically the quantum version of the discussed that 1 woman quantum version of me having to coins in my pocket a nickel and dime closing my eyes shuffling a mop giving Allison house 1 of them don't tell which she does no I don't know because someone I give our Bob the other 1 and then send them off our heads there is no way that Alice has any direct effect on bomb by looking at her calling but instantly she knows or Bob has a few observers riots correlation that is not a fact good a right now book locality is fundamental a 3rd 3rd principle you require the physics Toby Lorentz invariant another word that rules the rules of the game the equations of motion and should be the same in every reference right that is easy to do the rulers or variety and should be the same in every reference frame and that means it should be a scalar how he makes galas makes Gaillard used by following the rules about indices so arrives it should be a scalar I that's Lorentz invariant what's right without warrants a variants I'm aware school which will be discussed tonight this is a mysterious world this is 1 that you won't get the 1st time for you to get your are hopeful you get on but you won't understand the depth of it and what it really means on West we go through it from various directions which will go forward Is it prints are not principle of invariants this is a principle of variants that physics doesn't change in going from 1 frame of reference store it badly this includes rotation variants and so forth and the falloff rule is a rule called Dejan variants and think people have heard of Dejan of the majority of you have heard of gage dangerous Kigali where there's lots of them I have thought what's Ieft it you can get away with wearing stuff from grand jury certain kinds of stuff Cajon CA but now None of us there is a U.S. market with yet you're right you're right you're right this is this is special rocket let's suppose that with the experiment there has discovered in a laboratory has nothing to do with gravity is quite sure it has nothing to do with gravity and these will be the rules a word yes there very general but not completely Jarreau you're right and yes yes you're right I heart was actually the ovary general they Avery general aren't making grungy and be a scalar has to be slightly modified it has to be a scalar don't worry about that Her that's a concept which comes up when we think about our curved spacetime so for but their path remember a minor change in the Member in words is of the same rules that use them got the Eastern I that's right that's right bigger invariants called the variants and not arbitrary Ward that's correct what density your Joel Bob you're removed or can a resolution that at a lot about at biodiesel it turns out yards and so is it where as required this this requirement but don't get this'll graduate have units entergy Mel this will graduate Prost here would have units of an energy density and the reason what's right for us where tried deep team time was XVY YDJ OK now the XVY busy has units volume so as a factor of volume here that is not here In order to compensate for a grant of field theory has to be a energy per unit volume so that were integrated with the volume it becomes an some as a that you're a writer bus full but you really have to worry about it because tender work units where every has 1 of the a ah right OK so let's go back through the example we did last turned her I want the work were to act quickly on a blackboard again because of this all centralist allimportant if we really get to gage
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invariance song Burford former go back and just remind you what we learned last time China do it really quickly and back and do it quickly I think if there are we call the questions and until we have a not in request for P and of their demonstration but solely said let's just think about the the emotional particle in an electromagnetic field the 1st thing we did was introduced off for back that which is before that the vector potential components on where that it comes from what we will see where it comes from but it is necessary it is necessary In the water who has an action principle which leads to the equations of motion of the charged particle or electromagnetic field while those equations lose equations at sequels and maybe but with the force given by the Lorentz force loss electric and magnetic forces you cannot arrive that you could you come right back now it we can write down was equation our times acceleration is equal to electric charge times electric field plus velocity crossed magnetic field many view of seen that our before little better potential and it that's correct you could write the equations without the vector potential but you cannot express these laws of physics and terms Inc action principle where about Oh we're are carrying are a vector potential Norway right away about action principle and the actions charged particle contains good terms reactions I want diffuse action that the potential of right afterward action the action is an interval now this is a particle it's not a field theory which is the particle is a given electromagnetic fields Justin and they will be to feet higher than the usual term which is this and now that's that's what's not right that what's right just grow action is integral minus and plans deep was How along the trajectory of the particle we break the trajectory of the infinite test will segments and calculate the proper time along each in the most us segment multiply by miners and this is guaranteed to be a scale so Sorensen variant and any other time shutdown was minus EU times it's still integral under the intervals are you off ATU DX saw just as specify exactly what that means for every little segment here along segment can the position of the said right at the position of the stars of the vector potential of the potential a field it depends on space and Todd so it's just call it a day of X X stands for XYC YCT 80 at the position of the segment pardons DX and this is a scalar This is a scalar phase of 4 vector DX meals of 4 vector this combination now we have a potential candidate for an action and from it weakened the rival a grungy arrives in like this as integral on and how each EU member arrives you always has form of an aide to go over time forgery cars still how is the same as DeKalb ID DTE if you remember what how might be he you Yukon work it out is the square root of 1 Servoss square for years square off square root of Warren minus the sum of the squares of the components of the velocity we worked out a prime misses were granted for the particle without electromagnetic field and that in the presence of electromagnetic field another turn minus and now we can write this trick is always the same right ABU DD X TOO D T DT just a trivial will operation here were a multiply the divided by D TD but now what is DX we'll buy DT was nobody has 4 components were for Cabarrus well this time comport with Thai component is a silly thing by Serrato 1 and this year are the components of a velocity component ordinary components of velocity so what this become what this big arms is a sum 4 terms tried hour the title Parliament guards DTD teacher paused Of course ATstyle M X and . 5 didn't get back I'll show you overhear VX middle overwrite VX the X Newell by GTE GTE now what's go through various terms here 1st all the time component transit time component that just a notch times deed TDT would after right back down and then the next is a sub x times DX by duty and a of why BY by duty and so forth dresses over here the index can it doesn't matter from space was based components you could read you don't care right so I knew right we could put this upstairs but it doesn't matter was based Kabul space components our same upstairs and downstairs 5 so this is the vector potential dot the velocity and that deterrence DT so now I can read out what will Lagrangian Grosjean the part it is my visit and square root of oneliners X . square next ask live music stars Quebecois got square Posay squared around here and then mightiest electric charge kind a Of Accident plus a area of ex ante times X . him at school graduate ropes With door Her practice here From the points from the starting point where we get grungy we were very mechanically mechanical rules all straightforward sometimes a little bit tedious
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sometimes a little bit tedious and some deep trivial just mechanically falling out of the about oil a Lagrangian equations for a given Lagrange and this is what gets tedious and physics facility comes in the sometimes is huge amounts of tedium were granted can be very complicated over could be simple but the steps necessary to work out and solve the equations of motion can be tedious so basic principle lots of fun part working out what book is that may be far and then did do see the consequences of its battered due to graduate students preparing let's right through it I remember what be consequences of this grungy we begin With detailed by the X . let's say we pick sunken parlors and component and we want authority we compute the Elbe IDX stopped we get something from here Russia Quebec area and if you work it out X and . velocity the mediums directions those abided by the square root of warning Miners X . square that by Dom Rucker work Iraq you could do it also happens to be for further for future reference and times D X and by D. how they also happens to be it is kind of 4 velocity are sorry X M M P OEMs component before velocity and cars or we called you would have yes as chairman yet book employed alert right right that's
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correct I'm sorry I that's correct this is not a this is an equation which is nothin is being right written yet wise in the ways be ready because as another terms which depends on extort here saw it should be on another PC which is just my miners electric charge times AT & Partners a N A MOX Tati and let's now but that's it this years of stores right correctly was right the equations over really is an equation this might miss this happens to be equal area times you would end before vector of Austria Miners Ace Obama Major jump up and shout and scream like Hello my equation is not consistent I have upstairs and this season's downstairs and this season's inquiries but it's OK because it's purely the space component space components don't care whether indices erupt as a barrister but trust me it's OK this is the Elbe IDX founder do would be held by the X Will Rogers equations we differentiate it with that to time right Dubai did on he held by DX stock Hey by Didier there's soul where we get go right down to it all here the left side of the 0 Lagrange equations the left side of the or will grant equation is deemed IDT May IDT on Myers speed d by D. C Of the end component of the vector potential What is the kind derivative of a given component Of the vector potential now where we need 1st of all by the time derivative 1st of all they may have explicit dependence on trying but also depends on X we're talking about value waving the vector potential at the position of a particle along the trajectory sought Aitken for 2 reasons it can vary because explicitly depends on time or it can vary because the position of the particles varying with time so it took contributions to trying derivative we have to take the time derivative Our this In all the years miners he the to terms of writing down derivative a and with respect to see which our right to river respect tax not saying they that's when terms and another which is my I wore compute the change in a sub member cost axes changing because he is changing because X is changing with time and that's the derivative of and with respect for the position In New Times X In . 2 some both in terms of derivative of AM with respect to x times sexpot to remove a with respect for times white dot derivative of a M was busy times E . services Osama here it's a sum end that's the lefthand side of your Lagrange equation it's not so hard to understand it comes to terms 1 coming from the 1st terminal a grungy and and the other coming from the 2nd term will gradually losing each year were black starring right yes Travis right but electric charge multiplies all minus electric charge from the horse that's the left side your about the right side what's the right side of the over grunge equations which is right or wrong Runge equations right side crib if it is a derivative runs with respect XM attic OK so let's go in compute wearers Rogers his lawyer Roger what the derivative will grant June with respect to the exit out of the pen on the extras this doesn't depend on exit along depends on stopped only depends on the velocity around you depend on positions and velocities this journey only depends on what these fouryear misses of Osceola here Notimex board a naught in a depend on X remember a orderly we're just fix scenes which somebody has told us OK so it's our right the right him starting mind this derivative of 8 North with respect acts
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that might have a right I think I have a right to say times the derivative of naught with respect to x yet yes that's correct derivative of naught with respect to and from here by her warriors he times derivatives all 8 was caught and with respect to rule X but see the real armed scored 3 derivatives are a M she thanks energy mix begin with respect to x end a X a 2 . 2 words came from it came from here Our yacht Ursula bronze this is a summation in next year I can change it again changes name cortex and expand the matter on the left hand side and evaluating the equation of motion for the elite component of position was an M. here it's not a summation index that's why I didn't use a summation index and more over there James variables submission X act so here I differentiated 8 and with respect to X and then just multiplied by the excellent but which is already he was already there but just differentiating this whole thing with respect X and and that's what you get bite or stuff that's a lot of stuff for a for a for a simple equation emotions or we can combine it together now combined together doesn't look so complicated once with combined together 1st of all the left hand side the left hand side it looks like an acceleration you was a velocity is a former velocity but still velocity was slow particles that looks like particles are moving too fast as just outside the left hand side acceleration a derivative of Austin so it looks pretty much like acceleration what's leave that on the lefthand side and looks like him maybe that's a little bit different because that's the former velocity not the 3 varsity slow particles at approximately the same the lefthand side looks like and let's leave it that way for the moment we're gonna back to it change it will come back to it manipulate a little bit let's put everything else on the right side what's on the right side will evidently an obviously be before UAE his acquittal before so it put everything on the righthand side of our temporarily just call this mass frenzy and component of acceleration off with a little bit of cheat here about what I mean by acceleration little bit of our looseness in what I mean by acceleration or whatever it is it's the left side of the equation and that is equal to now on the right hand side Everything over the right side Till kinds of Turner's 1 contains a velocity this want in this war it contained velocities and this 1 and this work not contain lost the use of his X nor do not contain velocities let's 1st take the 2 terms yacht as if they were grabbed and very low or you are it is not good I want to sort what I wanna right so Eagles 301 right people something else were an order right I don't wanna write something which on the left side saying is on the side the OK I'm much are missing of let's take the pieces which don't depend on the velocity 1st of press is former trends the right hand side is going to have a plus sign a G 8 by IDX nor but I get that right by the X nor that's 1 turn comes from putting this summer righthand side and that might just be a Norwalk by DXM Japan need little expression DAM by DX Norton XOR of course means time services DAM time minus D a card by BXA Meisel symmetry to it the 2nd terms wearers act that's the 1 that involves a velocity a in Basra and plots did they can buy VAX that'll multiplied extend not but as another term
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which also involves XM part of mind this be a hidden by the EX there's a electric charge even then Missouri X end . so I have I have mixed on best times acceleration is equal to a 1st term he error which doesn't involve any velocity it was not now velocity dependent forest a force which only depends on position because position and time in general but not velocity and this object has a name is called capital E Emily against to space component it doesn't matter if it's upstairs and downstairs this is called electric field Barkley Pepsi
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electric field electric field a combination of high noon derivatives of the space component of vector potential and space derivative of kind component of the vector potential that's the notion of an electric field as described in terms of for the better the potential were not be overturned when I did the term it's quite apparent that multiplies of velocity but that is exactly where his mouth is class Yeager lost crossed magnetic field state component of V Crosby and component of a chorus be let's check to reject and see it really is brawl last remaining velocity X 5 crew I think there will march jacket of a Mormon I will leave it for will but 1st since I haven't yet told you the detailed connection between this object was it with it the other money recognized what mathematical structure be AM by DXN might just be a by VXM theirs the it's that curled Mr. Kroll Abu vector potential buyer right down the components of the promises curled back the potential and the curled vector potential is the magnetic field I so it's not obvious the moment that this is a cross product on a velocity and magnetic field but if we recognize that this is the Curragh will come back the Pacific curled or something and that the curl of the vector potentials of magnetic field this is some kind of expression involving the product of components of the magnetic field Bob with components of the velocity but slight wait a little bit and later on prove that this is our but this Crosby we have to make some identifications but what they would be used so for but that's what's going on at force who are derived from a vector potential OK services they just tell you how works art do you Our for strokes Way much weight on the identification in detail but the occasion with a magnetic field prickly edits looks something like a hawk 8 1 0 0 last step this was it to truly be acceleration was not truly acceleration it was neither the 2nd derivative with respect to kind of X it was not that that would be a derivative with respect the kind of me but said there were no effect kind of EU what you'll get itself it is a derivative of respect the proper time so I have this funny mix thing he deemed by Neil ordinary time times EU which is a derivative with respect proper time I would like the right everything in terms of proper Todd proper time was nice relativistic description of time or longer trajectory I saw 0 let's place What's rewrite the left side of it as if you must DUO sub by D. but that's not right it's not that we have the multiply this by pal by the derivative of you with respect she is a derivative view with respect to power times a derivative of power did the derivatives are really ratios ratios of differentials of functions with respect differentials of year of the independent quarter could this is lefthand side has an extra deep power YD t pilots for throw Booker exactly what we have always here at battery right at times eggs and I was X X thought it is a derivative of X with respect the teen saw what fourthyear DXM by decaying and here we have a deep our my duty what we're were well under way there's a bit of a suggestion heard is I'm really multiply both sides of the equations by DTD power to get rid of them aside so we multiplied by DTD pal both sides TV County about put a T by Powell here at the end of October Vijay body towel here was a saying that's DXM by D. Powell which is why you would ever not the end for you wear so we have over here for Class E times a member In my and him times before a vector velocity on the lefthand side this is we have the derivative of the former vector of velocity and rightly inside we have before us itself nor about by tally was I was like a belongs then that's right a M body X marks norm DX and now website team by Powell as D X by pal that's where it is that he trial and the X nor by it just the 4th component of a full velocity this is the 4th component bow before or after its you withdraw I'm sure that 0 our lost charge he was charge would have something is shaping up interesting way may have plunged proper acceleration times proper acceleration withdraws and these 2nd by Square you will is DX by your body Carroll 2nd derivative this is off formal acceleration but acceleration where we're accelerating relative to however 2nd derivative of recorded that with respect that call the proper preparatory should a high and it's a for the important thing is this is a former on the right hand side we have something which involves an electromagnetic field is elected feel this is a magnetic field and a 4th comport and they will for vector of of Reuter right this reform elected charged Times f but stiffer can get us right arm yes their war you'll Millwall Ontario are not quite right here are quite right here and we see a subject before I hope I think pointer I think we talked about the subject before but 1 West Bank 1 West Bank Mr. an index years of upstairs index or downstairs index it doesn't really matter it doesn't really matter of we want to phrases as a former vector equation we
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would like to put our mule here and put mule Hill after now Roger properly constructed for vector equations and I will tell you what these things off this year it is the acceleration the music component of acceleration is a former allwhite by the way this highly equations idea 3 equations are here for what was I laugh too are removed from were rejected while yes equations are Lorentz invariant and the 1st 3 components of or for vector are equaled the Samara 1st 3 components of the 1st 3 components were former rector of 0 then it automatically true that the 4th component tours of 3 components I'm ready for vector equations are correct it is automatic with a 4th component is also clearly that's the consequences of Lorentz invariant and since we assured Lorentz invariant by requiring that the Lagrangian was Lorenzen area and we are all allowed free to put in the 4th compiled yard How is it it that the constant factor know what it was it was a constant but may be best for us you might not always want to win 3 4 numerical values 1 2 3 4 death or your know them more what's is back 3 vectors for a moment of just over 3 characters but supposing I have but still ended 2 vectors took upon directors is 3 vector whose components are 1 and 2 what does that mean but that's a vector whose X as warm and was white component is tool but if look that from another frame of reference effect works pick a frame of reference a particular frame of reference but pick a particular frame of reference that happens for why along this exercise In the rotated frame of reference the x component of the of the selectors Europol and the Y component as what square root of 3 were forced to a 1 our square fire square so now are the they want to what could be the components of a vector in some particular free but not it's not a they Inc which is 0 4 vector romance it has the same components of every Friday no thing which has numerical values are as a former components of a vector transformed the component so that the transform sole Nolte numerical set of coefficients defines a rector To see which transforms yacht column in rural rural that he 1st half rather sad to see a U.S. moon or carry that's it In here it's in here put cycle look at let's take a look at the art arms and take this term he at now structure that appears here is a sampling Summerlin new war but Estrada bounced very exact take me old but said the case would yield is equal to 0 and new was a space index and let's see what is really at stake the case Newell eagles and that NewWave is a space index on our part Aja high and mass on these 2nd and by Powell squared is equal electric charge now was M services F amd limit no no could be any 1 of 4 industries however it can't be and why can't be His because as his and by symmetric will come back it can't be the diagonal components of deaf 0 0 0 0 Stop the next thing I can is 0 no could be 0 and that this would be you 0 and that is the other possible reduce costs that's Daewoo or scored you will there was a solid here f and 0 0 is exactly what's written over here this is Ephrem 0 Pondview 0 so this is already here In this expression not a lot of success a North right at 1st I thought he know wound not not expression like this over here are no we weren't trying to get rid of all of the city's all always trying to rewrite look this DTC at this is Dedie by Kyle we're trying to write the things in a form which is manifestly of 4 vector equations are but left inside as a 4 vector it's a derivative right X Mueller was a former DXU by Powers of 4 vector biggest hours a scalar and the 2nd derivative is a former as a result for vector I just manipulated around until everything is written in terms of 4 vector of rotation this is not the important thing here important thing here is that big cost both of the principles again I lost the principles of the principal 1st of all was the principal of the principle of lowcalorie calms down the statement in terms of equations of motion that the equations of motion of differential equations it says that force only affect things nearby said at the same time almost the same time it says that that Lesotho differential equation is is a star things are only affected by a force of a given card in the neighborhood of that time that's different request a 1st locality tells us that the equation emotions of differential equation what what what could you have that would be a differential equation you could have the acceleration at 1 time is related the force a completely different that would be bizarre Paducah habit acceleration a onetime related to be a force of different or even worse could not be acceleration but the difference of velocity at tool completely different car which could be equal to a force of some 3rd time that's exactly the sort of things that locality says the acceleration time is equal to the force at that time I differential equation 2nd Lorenzen invariant the equations can be written as for victory equations that's the important thing here for vector equations that they will arrest How do you at Miller was a rank to venture as a right to aren't you are put off on the blackboard here it's components in terms of electric and magnetic fields and then Ucon go feed them a year and check was really use Lawrence force was Maxwelton the field test runs this way you would
1:08:37
runs in this way Newark 0 0 1 2 3 0 0 0 1 2 3 when a right of all 16 components really are a 16 independent components bar all 16 components in a form by 4 Ray here as a matrix budget is the rewrite it ahead which is the not normal this is actually F for our you know the only difference now remember when you when you raise index if it said time index you might sign of space in which to raise your was 1st causes a stir it may be a sign Libya signed change in our some of the components gap and basically it's just a question of if you raise the new index series of newest kind component was minus sign space component of this a good exercise through our sort out the relationship between the things of the upper and lower industries but this is which is roses that Eurydice this way new runs this way solo Our was due in New for example F nor what we that would be Wilson walked with the a happier and knowing Sosa subdural and 18 so see world her was a tough time change will cost more got out I cannot use a sign on a convention that I don't see how this is probably the standard convention but it's just a way of displaying this is the way of displaying them and I'm using the convention new runs horizontally new runs our but you just freedom of no equals 3 new equals 1 would be over here write our strongest displaying them for you that the bag overall 0 Cedar City that the diagonal elements of 0 to entice the cancer 5 and a half years minus of X minus East on why this is busy is of the components of the electric field and stand by symmetrical into class he's of Beck's plus he's a wiry plus EU subsidies 0 or but 0 0 0 now magnetic fields a slightly surprising way and not too surprising this 1 was busy minus BY and B X and this 1 was mine is busy because by some that the plot BY and miners BX This is the electromagnetic field cancers and identify which could portions of this complex over here after you know a quarter which components ordered a magnetic field so what you can do it this is you cannot take and plug it iand this equation and say Hey let's look just were serves this it a space following mixed component the space behind makes components we are parole kind base everything a parole has 1 index which is time but then these 3 things a space component solidly is electric field is component are mixed space time is an electric field over boat and this is a hut misses a spacebased parliament the in or both spaces and the spacebased components of his pants for over here so Rosa components of the magnetic field now will come back that but young Eva with 2 upper or lower them both impressed America 1 Oberon 1 not quite there is our arms or maybe I would like to get back I think by accident the and symmetric farmers I think all the changes if you're 1 opera 1 for his electric fields change signed wrote the magnetic fields and ridicule stay the same the electric fields change foreign so I think act all selector among what there's for me I'm wrong but does write to me but there however were about those of fish or other things a toll or indices ought to operate this usable that there were a total of yeah I think that that sounds right that's right work think about where are you get me distracted on right OK so that's that's the basic set up but we've never gotten yet to gage invariant so I went back through what we did last time because I thought it was sufficiently important end Saul illustrative of the basic principles who led the gone beyond that of our understanding it would be a big loss OK now let's go back Let's go back to action for the particle along with trajectory electromagnetic part of it and discover the concept of gage invariant 1st always invariants in invariants is a change in a system that doesn't affect anything or doesn't go their a firm that doesn't affect the action strictly speaking it to change a system that doesn't affect the equation of motion for example or give you some examples is an equation of motion could it has exactly the same form if you translate the Xaxis translate center of the Xaxis from 1 quarter from 1 . 1 not as translation Yukon wrote to a quarter of the give you another example let's let's take offfield were granted years now of field theory the simplest field theory that we wrote down couple lectures ago it had an equation of motion which was these 2nd this was a real scalar field funds deepsea square a hot skillet my was decertified by ID x squared allocable why Square Square and we will and everyone is equal to 0 Our were put there could this as a log share which is given by defying DE New 5 New 5 figures from a minus sign baby minus 1 half of a for right now what kind of invariant this equation of motion Havel has all kinds of variances including warrants but has additional invariants which is supposed just take a field 5 and a half constant it growers opposing I take a field which is solution of the equation emotion and just had a constant everywhere is the same does the result satisfy the equation of motion was sure divers why because the derivatives of a constant those euros or fight take fight but take 5 and I know I satisfies the equation emotion and then a constant is still satisfy the equation motion wiII because the derivatives or a constant does euro like this it is in the area Of the equation of a scalar field of reversible scalar field you could get a constant defined doesn't affect the equation emotion you could see it in action the action supposing here a
1:18:01
constant defy what happens the reaction nothing the the Riverdale a constant doesn't contribute Saleh 0 if the action is such a or if you have a particular action and you have a field configuration which minimizes the action if you get a constant of the field it will still minimize the action or constant field is a cemetery the Tonto which is more than I used to work to symmetry what happens if I had for the right here side of the equation for use with the costs of typical of vast new squared fire that was the equation with an extra terminal Grand June which was minus New Square over 2 0 5 square number that we talk about the equation of field equations where we aided by square this is also scalar perfectly legitimate thing to do now what happens if we ate a constant the equation it is it a solution that war is not is no change in this tournament is no change in this terms but we a constant defied the changes were good at the action this term doesn't change if we had a constant the fire but this 1 by Square not the same as as 5 plus a constant square jaw with the extra charters This is not symmetry this is an example of a cemetery of an equation of motion which depends on what Grosjean buzzer buzzing contained this factory the book both a perfectly legitimate sometimes dosimetry sometimes symmetry yes I really yeah there should do the work John for a while back not bad an access for you set Asher B. just 1 area next to him right is now field is just a number of years just a ability reactions Pfizer from those a case I 0 0 0 why she would do or do it would be fine your acidified egos last acts were glad that Our he but see on that does just that does of to those doesn't satisfy this equation young right at that it does it does that's right 2nd this is yes that's correct so let's see what happens so what's with getting rid of its right now this listing listing stands for flight got squared minus D X 5 square salt if we ever sure are or your happens on your right subtree where comfortable soon enough on if we able to what happens we have D X 5 plus acts squared OK but do 1 those days is terms because X the removal of his but few 0 Prieto changes the action by an amount which contains DE at our the expo fiery squared OK survey gives us starters D acts Friday like plus 1 square heard now there issued an order Texas ER independent variables now I'm a reserve of functionally depends on acts doesn't depend on so to 5 due back against bronze 2 terms and we warned 1 square Friday at constant will grungy is completely insignificant does not so we don't worry about that and the other thing is that a Eds D 5 IDX actually twice defied by extort Roger gave a total derivative of around you doesn't change anything we haven't done that we haven't done that and we do that in in a moment with exactly why that's true in 1 hand white ending a total derivative or derivative tour Grosjean was changes we can't do it that's exactly where I was going and you sort of storm I found by body during this example pocket let's go back to the integral part of a new VX we need to get the need to keep going in order to get the concept of gage invariance which was appointed as set high might need to be exact for right now consider the following the operation on we were going to get that we were wanted to Woods Norm gradient of a scalar a fourdimensional gradient of a scale a physical wary vector and yes by BX was also Cauvery vector and don't asked whether that changes the equations of motion was a change bits of the particle follow make any change what in the dynamics of the particle let's get this AU which aging was considering different views and aII without the BS by the X muon anywhere and this could be effect on the motion of a particle will be or 1st of all it changes the action but see what does the change reaction changes the action from the the original actions no which is minus times the derivative transient the grow up AU UVX middle plus they esp IDX BX meal it changes the action of the particle here's a fuse the audit the dollars from year to year from . 1 2 . 2 0 enter it introduces a new term until the action which is integral on by X me now what is the role of DS but gets new BX each 1 of these little things here represents incremental change of as long as small little piece of the trajectory user the trajectory broken up a small little pieces yes by IDX fans DXU is change in Essen going from 1 . 1 of us could if we end the more it gives us the change going from the initial profile point so this expression over here the grow at the enterovirus slots for begin grow and but they they give forget order charge but forget where interested in this quantity over here Pierce forget Shmuel integrated back it was equal to 0 St evaluated at the end point to what might best s at the end point warned it's just different of this function S as Fonda Charlie function probably arbitrary functions of Indian it just gives me a pass at the point where the IRS at this point might just as bad for now does 40 in such a as to the vector potential change the dynamics of the particles from our maintain Norway were following resumed I've changed the action by something which all we depend depends on points of the trajectory on the other hand if I go back to the basic action principle reaction principle states that we search for the minimum action we with all the trajectory and search for the middle of a trajectory of action
1:27:29
not subject to of the constraints that we don't move around the end point starts the principle of waste action only in pouring a wiggle trajectory in between if you want the solution of the equations which goes from . 1 2 . 2 will hold a . 1 point to fixed and we with trajectory round between what will you trajectory around this totally action will not change so if you find a stationary point a minimum of reaction it will continue to be a member reaction when you put in this change of the vector potential changeover vector potential of this the big particular kind of change of a potential mother any change of the potential of a crime which is heading for the vector potential gradient of something before the ritual Grady that has no more effect on the motion of a particle white because of only affects the actions rule be In point and the robbers the harbor the trajectory keeping me and points as communities but this is exactly the operation of putting a derivative getting a derivative of the action adding a derivative actions typically doesn't change anything be cost you could do the integral of putting derivative until a Lagrangian doesn't change anything because we knew integrated just gives you some important contribution but here it is Ochoa we discovered we discovered a principle of invariants promotional particle and electromagnetic field was not changed if you take a new a new Class B S by DX new Fed NED S doesn't matter where this is called Dejan area was the concept of gage invariant but the vector potential itself can be changed without any effect on the behavior of a particle of a charged particle sudden changes not averaging certain changes those which obvious Buicks rights for any of us but BX neutral vector potentials called gage transformation Wu will award gage came from a historical artifact that had nothing to to do with our just became the terminology for some confused reason and the beginning of the 20th century this was called a gage transformation with stock but don't really have anything to do with bird Geagea's gages meaning things that measure away from her do you fear was course refused to say why transformation a swarm Amin are critical of fish they look killer of large the fault that or reserve really each sure that video that the equations of motion don't change if you aired so that the potential such a gradient defined Alex we just go back to the equations of motion here they are noticed that the equations of motion do not directly involve the vector potential for they involved is a field and new so change that we make on the vector potential that doesn't change that you know which means doesn't change electric and magnetic field may change it we make on the vector potential it doesn't change these quantities here will not affect the the motion of a particle so let's thing we what do tonight is show that that electromagnetic fields cancer is Dejan variant that doesn't change when you Eds yes by DX knew vector potential that will establish we know that it's gotta betrayal there's gotta be choked by this argument but what Probert directly ooh whatever owe wares and Chigwell derivative Of the youth component of the vector potential with respect for the quarter ended minus the Ala order whatever backwards I might have a backwards bomb Ji we forget is a new new DAT YBX new war is being away the extra popped or starring Ms. was correct could right now supposing we Ed aII derivative of X so what will it do there's world Ed delay Deeb IDX of the shift in Dementieva was the shift and that be asked IDX and then we subtract or here my best by X mule R D by BX no what I've done His shift a E by the gradient of a scalar and and the resulting contribution and have me and now was is over here this is a derivative other derivatives Mr. 2nd partial derivative this is deemed by DEA X new VAX of S this I think it doesn't matter which order you do derivatives a fact that partial derivatives Camille doesn't matter which order you do the partial derivatives This is derivative backpacks nor the removal of respect X mule and this is just feel derivatives and Pappas quarter but they're the same thing on importance of order of differentiation so this is 0 another words as you know itself does not change when you make a gage transformation so this sort of closes a circle here it was really the main thing I want to illustrate tonight is where both the actions of C actually principle as well as in the equations of motion was for this war cemeteries Norwood variants in variance with respect a gradient I tonight last natural thing they ask is why the hell all we were trying to get was equations involving motion of particles an electromagnetic field wanted we bother in this vector potential particularly if the vector potential isn't unique if you could change it and it doesn't affect the answers given change it that doesn't affect electric and magnetic field what popular why we bother the answer is that there was no way to to write an action principle for the motion of a particle that does not involve the vector potential there is no way that we could Brittany action for the particle to go from here to hear directly in terms of electric and magnetic fields all the possible to write it by the auxiliary dove isovector potential and yet the value of the vector potential is not a physically meaningful thing because of you change it by a gradient the physics of change is just easily they it's uses of yes we sort utility how high sometimes you can get something of a vector potential to make the problem simpler for example 1 of things you can do is you can always choosing cassette any 1 of the components of the vector potential equal to 0 that can be very useful particularly set the time component of a potential equal 0 that samples or from very useful are exactly the way it usually goes is backed by choosing different asses the make vector potential have this property back property you illustrate different properties of theory by choosing gage call choosing engage choosing an essay that something further simplify the vector potential may illustrate want property of Yuri at the expense of obscuring another property are barred by looking at the fury
1:36:58
from the point of view of old possible choices of this gage older properties or with young a little over there is a lot is she says there's no rest areas for mayor drive was a no not really not really but that's good for on some variances have obvious physical meaning they are In particular it has physical meaning because you cannot imagine tall reference frames in the same problem and translating between our 2 physical reference frames yours and mine New Era of observer on toast 2 meters 6 nor gage invariant is really not In variants in the center freely a redundancy of the description many descriptions all which are basically equivalent to whichever barked From a mathematical point of view of it there what's know about what's know about it is involves a function of position if you were talking about rotation of course you would not want to rotate the cordon warned plays differently than you wrote picric quartan that's another place that would be in variants over ordinary physics to rotate over here this way rotate over here this way you rotate by some angle once and only was no functional position involved here as a function of position arbitrary function position whole function a position which is involved in the transformation that's what's called a gage so I don't know why we lost our fundamental principles black for Duke a mail it the locality Lorenzen variants actually principle you can and Jaeger variants gage invariant it is feature of every known fundamental physics car bomb theory of physics electrodynamics standard model YangMills theory of gravity or how big agent usually a a what killed great gray are head of the house bad said the general said in a while they can get a little more complicated depending on the system farm now that can be a little more complicated the YangMills theory is a little more complicated than this but Spirit descend on we are find different ways that define gage invariant away that Her has some more interesting geometry and physical picture associated with it and we understand that will understand the better way of thinking about the more general at a car that I will be back and a yacht while yes yes yes the match part of overall maybe transformation meal a useful device for solving a problem for illustrating a feature of a physics but should not change fine where record erupted different the loser the early are rare and that there was any other CEOs or other your theories about difference kind of 80 it was another matter is sorely and former cookies let's wait on which way was whether typically they are vectors and may have they usually call case there almost always called a eat they haven't index Mueller to represent the barest of us back he's from and say Look restrained but they have some additional structure of associated with other symmetries and Williams were not prepared to discuss how we may get it next week a little bit a little bit but not yet ratified a half a half but they are not yet that the shares that a lot of the field yes that's right is not all very of our opera as normal and easy to prove furor warm Pruett that you can knock expressed so we have actually of charged particles in the Lorentz invariant way in terms of the electric and magnetic field position and velocities of particles security so if you want to do it Norris invariant local way and so forth and preserve these principles the only choice is to introduce back the potential and then you find out that invariants that you could change the vector potential work changing the physics of what's our that's a very odd and peculiar situation where you'll have to introduce some auxiliary things in the equations but the auxiliary things are things which can be changed without affecting physics very remarkable pervades physics all over the place but we know why saw if at the classical level there's no reason at the quantum mechanical revelers a deeper reasons go but a would say we've learned bad series of this type have a great deal of beauty symmetries simplicity that kinds of fears don't have but not we don't understand why you nature always chooses please gage theories but others very perverse you can measure that would attract can measure a because you could measure a you would know what value is on the other hand changing value doesn't change in physics and part of making a measurement physics you could only measure things by by physical process use the physical processes are insensitive to be additional tests of the SPX you can't measure it right through the correct what can you measure about well you can measure these things and maybe some other things but you can only measure rose features which I did OK so observer on a lot further efforts polka Li Na obvious symmetry of electromagnetism at this point it should be mysterious will try eliminated maybe next time a little bit To bad the fad that it also said that any change that is not of this form or not any change which is not that will be detectable in particular there are vector potentials which have absolutely no effect on motion of particles if we just take the vector potential itself to be the gradient of a scalar nothing but the gradient of scalar that will have no effect on emotional particles that should be clear to your FmHA think about it if the vector potential had see of full but the potential is just a derivative of with respect acts will have no effect from motion of particles particle loom over straight lines with no forces on for more please visit us at stanford . EDU
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Titel  Special Relativity  Lecture 7 
Serientitel  Lecture Collection  Special Relativity 
Teil  7 
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  10.5446/15008 
Herausgeber  Stanford University 
Erscheinungsjahr  2012 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Physik 
Abstract  (May 21, 2012) Leonard Susskind reviews some of the heavy mathematics from the previous lecture and discusses how at times complicated mathematics is the only way to explain high level physics, briefly sharing his thoughts on how and where math and nature collide. 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. 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. 