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# Quantum Entanglements, Part 3 | Lecture 7

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song factory those here is mine 2 so the compulsively vectors can always Digby displayed as an entire symmetric concern over the entire symmetric answered can always be displayed as a vector now what's going on with geometry of the nearby symmetric cancer has 2 indices every component of it is associated with tool accident an x-axis a Y axis well Be X why axis X Y component of the entire symmetric cancer is simply identified with the Z component of the sector Our it's it's clear that it's ambiguous described a sector by describing the Filene's perpendicular to it which pays you describing it by a pair of vectors mainly via a pair vectors of finds a plane would you described by the Arab sticking out of a plane with 2 ways of describing the vector won by Justin arrows sticking out the describing it by vector components the other way of describing it is in terms of the planes but perpendicular Kovells vector components and that's the description in terms of me and symmetric cancer now they difficult thing it's very difficult rather easy it is to prove that the components of the Dyson metric tensor transform on the rotation according to the same rules as the transformation properties of a vector and they do they do so these Julie different aspects but 2 different ways of thinking about the 3 components of the vector 3 in 3 dimensions across product and the mini reasons for many purposes it's smarter to think about the cross product as a cancer rather than the vector but then again and visualizing it been trying to get a picture of figurehead you might you might wanna flip back and forth between vector supersymmetric OK that's the idea of a cross product now let's see but we do 1 more little exercise for you young Yamamoto exercise yet what's come magnetic field magnetic field is a letter but is also an nearby symmetric cancer but 0 0 0 be 1 still will be 1 ready but 1 which is mine is we want to of course 0 I won't I won't bother writing things below the bag just as confused 0 and then be 2 3 over here the other ones as simply minus the year for reasons that are historical reasons that a historical the components of the intestine magic cancers are related to the vector components by an extra minus assigned is across product but it is just a definition this is equal to 0 0 0 instead B 1 to study to be 3 it's might be then plus B 2 over here and my be 1 over here it's just that still assembly historical glitches actually summer some better reason for it but of course if a thing of that there's always the of it and so the connection between me and a concern and the vector that describes it as a plus ambiguous tho I describe this planes here as that vector format director is an ambiguity once a fix rectal usually fix the right here ruled by a cross they say by making right-hander coordinates but there was always an ambiguity whether we identified cancer with would be better components all with miners back the components and for the magnetic field as I said just as a matter of our history they were identified with with minus the components Of the cancer of the words of the opposite rule so that's the magnetic field can be represented either as a vector Olympus symmetric cancer where I want to show you was an interesting theorem actors in the forest I'm a charged particle is cross be now we can rightly be Crosby with the components of the cross from Crosby either by thinking of B as a vector or a cancer we show you what happens in the 2 cases let's work out the the components of what's worked out the components of me I'm only to work out the x component what's the x component vii cross that said busy component let's take the 3rd component of deep crossed beam a 3rd component of across it is by definition of across product literacy the definition of across product the 3rd component is you want to beat 2 0 minus V 2 B 1 X OK that's the 3rd component we could write down the others just by cycling through and cycle but now let's you is that identification up at the top of the blackboard to rewrite its Norris would be too is it's also will be 1 3 services also 81 BAT won the race what about now is my guests each I was BYE BYE is my V 2 3 might be authorities of change signed here multiplied by B 2 3 so the component of V Corps must be in the 3rd direction is V 1 B 1 3 was deemed to be 2 3 and then a and another Turner 3 D 3 3 1 mile out there that is they have been 3 3 0 B 3 3 0 0 0 it's a bad move from here the measure we have another way of expressing regret pretty nice way of expressing the component the components of Crosby's can cycle through as I said but we can also write the following way crossed and component of it and has won 2 of 3 Norris the three-year goes with a 2nd component of beach in each case here With structure of it In each turn you'll have the 3rd component of a vector being identified with the last component of B I saw that suggests that we should write down here something beasts something and we should velocity but not us let that the velocity components match with the 1st component of beef velocity component matches with the 1st component of B so that suggests that the answer is the M V M but well this meetings Sun over here this means want to be 1 and closely To be to end the Class B 3 B 3 M repeated indices get some over scandal I say easy way to ride across that's Ruiz that's another way of writing Clarance fossil idea of I ride it out in terms of components then at least let's concentrate on the magnetic piece of meat and component of the forest evidently there and is our electric charge the end b and end OK team that he better Marty little another will mathematical fact about particles moving in magnetic fields let's
forget electric field not electric field for more than just pure magnetic field and as a fact about particles moving a magnetic field that this be doesn't change speed is the magnitude of velocity magnitude of velocity doesn't change In a magnetic field that cause the the move is enough the forest it is perpendicular velocity but that looks that civic approve back but civic approved that a blasting that they lost speed velocity means direction and magnitude but that the magnitude of the loss it doesn't change In fact as a consequence of that you although the field there isn't time yet static static magnetic field that correct also said that you can do is get with the every is three-dimensional does no electric component in here now but I electric field equal to 0
the I set the electric field 0 if there's a time-dependent magnetic field the reason that they velocity the speed changes because of time-dependent polar magnetic field makes the electric field our but let's just concentrate on a independent simple situation time independent magnetic field will see how legal long will find out of the was important there was time independent I'm so among other things that means that the kinetic energy of the particle doesn't change only use static magnetic field that means that the square of the velocity we can write that as we can see him the square velocity this is 1 squared me to square the B 3 square Abedi BX square BY squared off easy squared this is a square of the magnitude of velocity I'm interested in finding out whether that changes with time so I wanted to I want differentiator with respect to time remember last summer repeated index I well what is this just twice the time derivative of me and Tanzania as PM squared the time derivatives twice beat us twice times the partner of of the Fort which is twice the acceleration twice the acceleration dotted where the velocity now let's write the fact that these acceleration we know what the acceleration is the acceleration is just the 4th divided by the mass but see where we wonder why yup substitute for the acceleration and I want to substitute the fullest abided by the masses but particularly I don't care about the numerical coefficients this is going to be proportional to to twice now velocity cross beat me and component times via the acceleration and a magnetic field proportional to the velocity across the magnetic field BN component of it farms but now that's use the formula that we have where it Oliver formula that we have for we crossed beam that race that started I erased it asked twice be lawyers and be Internet but but member because BV and component Was it is there that we can beat they have to be a M right that's crossed beat and direction but now let's multiplied by IBM they are so well really have we have the components of the tends to be bracketed by the components of the vector V on both sides now the trick is to realize but any expression like this is always 0 if B is an isometric for example is gonna be eternally which will be let's let's see what terms are there could be a term won over here and be won over here that will have to multiply 1 more at 0 but try another Soviet turned 81 over here and veto over here that will multiply B 1 2 but then they will be up partner twin termed the goes with it it which this is show which end is 2 and it is 1 that will be plus each 0 2 1 2 3 1 and now he is a fact B 2 1 and B 1 2 opposite in sign Be choir opposite and this is also 0 Joel if we take Eddie and by symmetric cancer now we multiply from both sides on both of its indices by the state sector we always gives Iran the consequences that withdrawal that from magnetic field the Debye by D. C of the square velocity is equal 0 another words that the square velocity is constant in time and the order that again Our Okada stomped on just go over very briefly again and we differentiated we got proportional to the product of the acceleration and velocity that I used for the acceleration Bill Iran's force along the magnetic piece of it because this would not be true incidentally of those electric field only because if there's only magnetic field then the acceleration is proportional to be Crosby and then use the across was and the end and the and then immediately from that follows immediately the symmetry of the N and is combines together with the enticed symmetry of BEN and to give 0 no 1 0 I do it that's where I could use elementary feelings about cross product so what I'd do it this way because we wanted generalize 4 dimensions were jammed with good with a simple theorems but 3rd 3rd generalized OK he so now we know something incidentally well over the course of an electric field violate if a particle is rest starts arrest accelerate beyond the electric fuel stop to accelerated and of course the velocity will not remain speed what remained constant it's only costs the velocity of peers In here in a particular special wary rhetoric has property doesn't change the magnitude velocity I don't think any place here that I use the fact that the magnetic field was constant in time I don't think I did our British rule is a time-dependent magnetic field will create an electric field and electric fuel boost work now it's more or less obvious that electric field F I and a magnetic field will get mixed up with each other when you go from frame the frame which are moving with relative velocity it's a well it's still easy to see this is under the assumption that the laws of physics of the same every reference the smaller less obvious if you have a pure magnetic field in 1 frame then other frames will also have an electric field words it a pure magnetic field want some access and you're removing some direction our depending on the direction you moving in you will see also with electric field away the that let's take a case let's opposes a magnetic field the up direction let I have a particle but say Let's I which is moving with
velocity v call this the y direction the X direction and this is the 3rd directions you direction 1 and 2 and 3 we have a magnetic field and the y direction we have a particle moving along the ex directory and of course in that case we will have our forests along the z-axis deep Crosby will be along the z-axis and we of course particle along the z-axis now let's imagine Shlomo or cities our Newtonian physics we're not interested relativity at everything is moving very slowly by comparison with the speed of light acceleration in the timely and kinematics is invariant so 1 observer season acceleration the 2nd observer will also see an acceleration and the acceleration will be identical as long as we're talking about observers from moving slowly wrote richer right now I have introduced a 2nd observer yet I'm only introduced the fact that particle is moving along with velocity but now I'm going to introduce 2nd observer the 2nd of the rock also happens to be moving along with velocity therefore a particle arrest from an observer the particle has no velocity but because acceleration is an invariant on their usual Tommy physics that observer must also say an acceleration of the particle is no Molossidae therefore the be field cannot create any acceleration momentarily momentarily at that instant of the acceleration 0 1 1 frame 0 every Friday so long as I'm go along with the particle I have the the acceleration on something else it can cannot be blamed on the cross product of the velocity and the magnetic field fact we know there's a force along the z-axis there's only 1 place that that could be coming from With only that tolerance for slaughter is the same in every reference frame and it's not coming from across beat because my frame EU 0 it must be coming from an electric field and so the must be an electric field sticking it Out of the blackboard in the frame of reference moving with the the particles in this particular case so any result frame of reference starter with no electric field but particle moving velocity the 2nd frame of reference a particle arrests I must have an electric fields sticking of a blackboard so the consequences are that electric and magnetic field must up with each other on the velocity transformation but in order to get the equations really relativistically correct we going you work with 4 factors so let's remind ourselves about 4 vectors and how the electric and magnetic fields it until 4 vector notation and then see if we can derive a set of equations of motion which will be the same in every reference frame and less said we will have to mix up electric and magnetic fields they'll have to transform I will work out some of the transformation properties and here we do a little bit of it but will work it out mathematically and see for example that this really does happen OK we have to field electric and magnetic 6 components altogether a 2nd object which has 6 components in fact the only object which has I think this will be the only object in four-dimensional space which has 6 components is made by symmetric cancer notably faults you could have 6 Taylor's Ukiah vector and scalar think that's all of the vector galas our 6 galas but neither of those are interesting or other object which has 6 independent from former Cosby and estimate cancer so it's natural and that's natural since electric and magnetic fields transformed into each other the guests that they fill out an isometric cancer import Minchin works our he is a matrix associated with that let's just guide my on provided for but for now my notation my notation seems to differ from allowed Burke's my notation is always at 0 0 1 2 0 3 No . 2 3 0 that's because I'm T In my youth I used a 12 0 4 now what's called 0 well I'm not that old but 1 2 3 0 . 1 2 3 0 1 2 3 0 West filet and electric and magnetic fields and some of this of course is convention some of this it is that arbitrary drink what invariants factors visitors in an isometric cancer describes all of us exactly how we are still in the components is a little bit arbitrary not very but we fill in a 1 2 3 3 by 3 matrix exactly as we did with the magnetic field before 0 0 0 my USA zeal beat 3 defenders as Class B 2 0 0 minus be worried so this part of the Soviet bloc about 3 by 3 part of a 1 2 3 is exactly just the magnetic field and on off diagonal places over here we have the electric field room for the components of an electric field I forgive you want each 0 3 the 3 components of electric field and of course in the lawless spots here we simply reversed sign plus Be for example down here will have my he 3 etc. And that's a magic cancers 6 independent components constituting b electromagnetic field cancer here who bid all as well as I'm only in interest in an hour and components 1 2 and 3 as long as I'm labeling things of 1 2 and 3 it doesn't matter if they're opera law here I simply means you're right down her I switch the patient up-and-down they know attention to it let's see what would be consistent I guess it would be consistent the lead them his upper components but it doesn't matter 3 components of electric field and the 3 components of the magnetic field exactly as Maxwell and parity would define them that would go into those places they sold weed paid much attention to wear the way the indices are this it is the object made up at exactly the same fields that Maxwell and early will perform and this object is called asks you know it's nearby symmetric format for cancer will you run from 0 the 4th 3 0
3 2 0 end of is former fought cancer called Marshall White called Africa maybe for clarity but I'm not sure every was called Efram until maybe Minkowski amateur Einstein did not have a picture yet so they didn't think about what I fought cancers when he simply worked out in detail was the transformation properties of electric and magnetic field I think it was Minkowski 1st put them together cancer T or come now that's stepped-up object Kretzer electric and magnetic cancers thing that has to be responsible 1 way or another but of force charged particles can now only wanted to study possible ahead possible laws that would be appropriate in the special theory of road salt let's begin with the idea of acceleration we already defined for velocity for velocity you knew is equal to X new by not by D. but by deed to as for components or hours for components the other 4 components of never velocity plans Gamma Gamma equals 1 divided by the square root of 1 might be squared XVY in easy times gamma and the Norwest component is as gamma another later right then is the expo ID TDP by D. Powell that's where the gamma comes from the is just a by above the 4 components of 4 velocity b momentum is and plans you yield wraps Peel In the 1st 3 components of it are we ordinary momentum their momentum but notice that they're made up of mass kind proper velocity that's what goes into generalization of Newton's equations goes in not then the former DPP by D. but P P Peabody tell white because we want to make a former if X is a former rector and is a scalar Powell is a scalar foreign invariant X by was also P is a form of that NDP by D. Powell was a for that's the left side modes it's entirely analogous to 0 would place In the notions equation cheer except that it be buddy and Dubai Duty we can write that if we like what puts lead the way for the moment for free because where will rule right slightly different life if the lefty inside is a former rector the right hand side must also be a former rector it's got electric charges from it is galling that have electric and magnetic fields in it crew obscure can thing we can better there's something not let that works for the moment just call the right hand side scrip death left New OK now let me move here on the fear his that s will Year's is the statement F New you knew is equal to 0 no this is an analogous but not identical analogous to a statement that the force due to a magnetic field is perpendicular to the velocity but the force due to the magnetic field is perpendicular to a velocity is analogous to its of four-dimensional version of it because now our works out early projects would probably not by using any special form but by using the back to return probably mentioned less time if I haven't followed our review it or I will review it will do it you knew you knew it is equal but warned could we do that but it would be a firm is briefly remind you it just says that the extra by D. Powell the Exon you'll tidy is equal to 1 man Paula really says is that what was formerly should be lower new Fuller really says is it d that the sum of the squares of these differentials is just cow square and then we did that I could remain twice already you'll you'll some new plans New super is equal to 1 Valerie differentiate that with respect apparel the receivable while means doesn't change along the trajectory of a particle is very much a very analogous to the statement that the speed Berlin change in a magnetic field at the same thinkers his foot components here but um birds mathematically analogous but to differentiate it with respect to power I would get 2 terms when I differentiate a product I get 1 terms From the derivative of the 1st factor times a 2nd factor in Southport survey will be for example new kinds New Super Mario and also plus you some new menus and use of the same day Her wanted to use some current you super new by but these are the same thing it doesn't matter if you have when you put 2 vectors together like this summer will be industries it doesn't matter which 1 has the upper index in which 1 has a lower decks rats no content and that doesn't really just twice but where's is objects this is twice be proper acceleration times the proper velocity this is the proper acceleration here members the proper role last the proper acceleration Of course is proportional apart from factor message equal to BPT by so this is just aimed times proper acceleration the aren't summer left here far from a factor and is deemed to be you but towel then a fog multiplied by you you want underwrite inside it must be equal to 0 so followers it follows that F new plans used some you must equal 0 I'll we finished 5th pressure says this is equal
to 0 it's equal to 0 because the right hand side is just 1 and when you differentiate 1 you just give 0 the right-hand side is a constant when you differentiate a constant you get 0 and what is the expression here it's just proportional to the force plans the velocity of here so you could see that F 2 times you you must equal 0 this is a kinematic factors doesn't depend on anything it just depends on the definition of 4 velocity for momentum sorry for acceleration for the acceleration sample and so tells us that a four-dimensional stands before on on object must be perpendicular orthogonal Tula before velocity in this of equal 0 that means we can't just stick in any old thing for for got satisfied this relationship here really satisfied well it's as I said quite similar to the fact that the magnetic forests is proportional is perpendicular toward ordinary we satisfied Mac by a trick made the trek of making a magnetic field in cancer Lehmann and multiplying it by the end Chris is we across be there or not in the end it was automatically exactly the same trip exactly the same trip we take f you want to be death Nuuanu would be and isometric cancer times the 4 velocity the talks Everybody know what I mean when I put a lawyer index for velocity it just means lowering the index by changing the sign of the space components sell AFI in and the end isometric cancer to put here and I multiplied by you know guess what if I now multiply it by EU yield the answer will be 0 years after a America at all as right explicitly if force proportional Daphne you know terms you know and I got an NTU meal the answer will be 0 because ever-present by symmetric so this is 1 way 1 way of satisfying the criteria that of course is prepared to the forests when multiplied by the velocity is equal to 0 is to build the force of investment and tactics the simplest way while we have by symmetric tensor factories in quality you know and has built up out of electric and magnetic fields so let's see let's take the surprise here is our equation let's get rid of the intermediate step here misses Q. Forms FCU times you know let's look at this was slow velocities 1st let's 1st look at a small velocities weekend we can study of larger velocities later but let's check that the lower velocities this really is beyond the Lorentz force but see related to Iran's 1st of all Q that just goes just electric charge as you know is going to involve 2 terms electric and magnetic In Newell has 2 kinds of turns the time component and the space components of you so let's work out some example of worked out deep P which when that worked out let's work out D P X by detailed now the 1st thing is as well as removing slow to come back a while and discuss were moving faster as long as we're moving slowly I could see the towers slow-motion deeply indeed cowardice saying so in the nonrelativistic limit we can just take this to be DP by said we'll chat rule will do better In a moment I just wanna make sure that slow-motion we have all the right things that's equal to kill will obtain a let's see what we have our new and match here and here so this has to be F X and then could be any 1 of Florida's anyone of fora for entries the 1st of all and half X X X U X that's not fair because ever an x x 0 gunmen as ever X Hawaii you white plus F X zee Uzi plus F X short time Illinois OK with apart F X 0 0 path X 0 0 it is electric field if 1 of these In the sees is 0 and the other 1 it is space then this is just electric field so this year is the 1st of all returning which is Q times the electric field they're both X components so this is the x component of the electric field but what these X Y and the X Z components and magnetic fields so let's see what we have X Y that's 1 to what F 1 to F 1 through his mind is B 3 of us will be Q you why 3 With the minus sign and then from here Waters X Z that's beat to know where they are said to be too so this will be plus you you 3
B to Yasuo getting forgetting the cross product of the velocity with a magnetic field crew to the extent that we can say Red Yowie is is the same as we reproduce the boss loss Of Loretta's um recourse B plus times we we see where they come from let's be a little more careful let's right but try the more carefully and not do the number of the court which the more carefully 1st of all they have deep by deep that is d PDU by BT kinds Gamma France DP but cow duty by County strike is gamma square root of 1 mind square will be is a velocity of the particles ordinary ordinary loss so this is by BT funds gamma but see what's on the right hand side on the right side we have things like status journey Q O E F x y you want but you why is the same as BY fans gamma likewise our likewise for all the terms you while you Zee News 0 plus a armed have Q FXY another words this is gonna be cross B times gamma and Journal is just electric field plans you're not you not just Gamma From the UCI viewers on these times down and the last 1 here it is just but OK so was cancel and that's what we get exactly what sure of what Lawrence Tech except the something slightly different and that's that P it is not aimed at times V but it's an times you so there is a hidden gamma in here the relativity is hidden in the definition of gamma with really write this as being by duty not of any movie about love and NYTimes Gamma the gamma changes along trajectory and sell gamma is not constant but it is a by DT still looks exactly like low rents with the exception that we have to calculate key heat in the right way as proper over and terms a proper role Ossipee sub we dweeb purses reproduced the offers I have got worse he is not sure OK backseat bad is killer rents for slaughter as derived by an unwanted was nice 1st proved that the acceleration and velocity are of Cardinal four-dimensional cents that told us that the force is orthogonal velocity the only the only simple way to do that Is the bill before 7 AM by symmetric cancers so that when we got back in the velocity we give 0 and then we there's work out which is worked out making the identifications With all the fashion kind of notation is just filling up taste definitions here this as being this is a when it's all put together it simply has a simple form of a before vector equation because it's an equation between 4 sectors on both sides it will be the same in every reference for our variation man is part of blast at the bottom the numbers but this is because the heavyweight physics heavyweight physics is the laws of nature of of the same everywhere an Dell that when and the other thing that went in before doing it really really derived from nothing other force laws of the kinds of forced laws all we certainly have a right to have the back of my mind that we were talking about things of electric and magnetic fields but our own a nine-month viewers we use the idea of through equations of 4 vector equations every reference that was the most important our the pieces are the things we used we use for example the fact that it's a 2nd order differential equation This is the X by deep as these 2nd that but deep have squared D 3rd X by have square before text physics goes into that farmers 24 because throughout the equations of motion 2nd order which is equivalent to the statement by physical statements on mathematical statement that in order to predict the future you meld the position and velocity particle you don't need to know the acceleration equation gives you the acceleration the equation could then but the derivative the acceleration is equal to reporters remained you would also need not be acceleration the beginning of physics tells you that opera Experimental Physics tells you that that's up the correct so to bail out there in the parlor that is what they got as that caught you want Bermudez burgers it is regarded let's talk about the transformation properties of
electromagnetic field cancer Our transforms off again as a homework assignment I can assign you to work out transformation properties all or a broken marriage of electric and magnetic fields In particular the transformation properties when you do a low-rent transformation along the x-axis when your 2 observers will be relative to each other along the x-axis Enrico octagonal Y axis but there would be no different except X will become White of the way think about it is the 1st right down the the transformation the product of 2 vectors the simplest cancer we want to work out the transformation laws right cancer all cancers transform the same way Joe we work out the transformation will offer a simple Cancer composed of a product of 2 vectors you'll see as a show that that tells us everything we did so for example suppose we want to know in the trying reference frame the prime reference frame is 1 movie for the sake of our right with velocities I'm trying your prime I moved to the right parliament just right down for you to remind you need transformation laws I'll be coordinates and let's call them X 1 X 1 which is were usually call X primaries next prime is equal to 0 X which is X 1 my is which is minus V X the wider by squares of 1 might be squared X nor what climb to Mrs. deep prior ropes 1st it is equal to 0 that is equal to 1 which is equal to X's naught is V X 1 divided by squirt former squared about X tour next 3 x 2 a X 2 prime is equal to X 2 is a wine This is the transformation property full vectors all that this transformed the same way so now let's worked out the transformation property of Cancer let's for example take 2 vectors looks like a cancer out 2 vectors AT&T another words 18 you beat No let's concentrate on the 1st component of the electric field sorry 2nd component of the electric field which were once now the 1st component of electric field but In the prime reference not was EX Eric Israeli asked for naught 1 crime I'm trying to find ethanol at 1 time assuming that I know things in neon primed frame now and what Prime is like right not low but is sort of like the product of 2 vectors aid not be 1 this would also be a component of the cancer factor would be the lucky 1 component of cancer so 1 does not mean prime let's calculate b warned Prime would direct well we know ignored prime areas Israel the not component of a vector changes we know what the 1 component of a vector how it changes so all we have to do a substitute but sapsucker was in a lot a not time is built out of a lot and they want the same way that x North Prime is built that of the components X Notimex 1 but Ringus a lot there's going to be a non minus V 81 sorry a member perfumes season Place a V a warning divided by
square root of 1 might be squared that's a not prime about B 1 prime B 1 prime we have to see how the 1st components of vectors change that's gonna be B 1 minors being not divided by square of one-liners square the see what I've done Our shared but E X is a component of the cancer of the mower 1 component of cancer just as an exercise on a calculate how a similar namely if you will be New transforms undersea what happens to the north 1 component of it In the prime frame to do that all I have to do a substitute in the know component of trying not component that I get from this equation over here the 1 component of the prime beat that I get from the scope of equation over here that's these 2 factors that clearly enough but right now it's more of a plan together and see what we get 1st of all where they get 1 divided by 1 one-liners squared that's easy thing as multiple are used to together now mags we get 18 or what we want that's a component of it tell the corresponding cancer in neon Prime reference for will use that now we get mine is V 81 B 1 minus V a no being on that count was From a not be not turned and a 1 B warranter and finally plus least-squares 81 be naught now the only unusual thing about that is that it's an isometric we could get so so we could get now what the transformation property Of f not 1 areas where we could make begets have not won prime is equal to ask not wanted earlier From here miners V F 1 1 0 4 more miners V F naught naught plus squared warning on all divided by 1 might be squared said 8 times beat transforms it cancer it is its transformation properties and all straightforwardly from the transformation properties of the 2 vectors and now I identify the components of a B With the components Of the cancer this is how it cancer transforms but the impact symmetric consumers especially simple F 1 1 and if not not armed they'll so we can erase them what about F. nor 1 and therefore not they're the of each other so this becomes ever know 1 times 1 of my squares and all of is just f North Warren Oh my God it looks like have not 1 Prime is the same as if not 1 that's correct was at say that says that if 2 observers are moving down axes and is an electric field along the axis would doing a Lorentz transformation along 1 axis and we're looking at the component of the field along the 1 axis it says that be component of the field along the direction a relative motion is unchanged is same a true back that be component of electric field all and it's also true of the magnetic field to do it ah but also a magnetic field the effect electric and magnetic field transformers similar to whichever but now wants do a case which is trivial where is an interesting transformation long let's work out the transformation property of a component of the electric field perpendicular the direction emotions well what's the
transformation of a we had with them over here the transformation of trying to start a transformation of 18 knots is on trying day North miners V 81 again the blighted by squared 1 squared that's 80 knots prime and then this is incidentally is being here is not the magnetic field a bit confused is just 2 vectors any 2 vectors What is a transformation property of the 2nd component the white component of this case where transform along the x-axis nothing changes visitors be but still we have we have a not be to Nevada by square and one-liners squares not be 2 0 0 square 1 might be squared well that suggests that we replace this fight if you want 2 0 divided by squared of one-liners square says that's the 1st jury and then minus V France 81 beat to that F-1 Until F 1 tool by square 1 miners squared OK let's see what this says this was applied component troop each own time is equal this is 8 0 why doesn't read no at 42 foreign to is closer Mike B 3 which is it is 3 might be Korea for that miners betraying must be possibly B 3 survivors square Warner's risk word so we see Our misuse to Ito Pioneers does change and a sequel to eat we'd be free firms Gamma if velocity is slow is close the and adjust now now that the electric field mixes in a little bit the 2nd component of the electric field mixes with a little bit of the 3rd component of the magnetic field Christopher transformation along the x-axis along one-act Yukon worked out the same way details of all of the transformation properties of all of the components of electric and magnetic field and find out if you know the 1 frame stationary frame what they are the other for our particularly you could check that if you have only a magnetic field in 1 frame if move relative to it you see an electric and magnetic field Our Chanukah workout directions of electric and magnetic field point these transformation properties for the electric and magnetic field will always be such that the equations of motion any frame of reference for the charged particle aren't saying a for getting questions about the chance for him the as his heir apparent lawyers said they would set lost CDs are white gold would be with that charge yes there quiet get she it's a be wired with loss CDs widows you wanna keep up with you electrons From Is it that's what when I signed relativistic velocity mu that's it's Media Milosevic yet but the move to work it out Work it out of there it very slow not now that's To get rid the magnetic field for the velocity of light from the wire is slow there were lot worse Musa site right last much because of Old Man well well were weary of careful that yup yup yup puzzler charges that 2 armed nor does trims agree I think we just have to work at the magnetic field and do it the severe driver oriented toward him I suspect intermediate slow fast its Cyrus he herded the nite . years now classes only when it was over but that's as the rocket this is an electric field on a large electric field you commit might be able Europe magnetic field nor transformation property of the electric field will produce a little bit of magnetic field which could cancel the original magnetic field for all you have is a magnetic field begin work Mel you kept but for example I mean that's what I was thinking which was inappropriate for the problem but if I had moving collection of charges at the moment collection charges makes a magnetic field it also makes an electric field images what the charges moving the Nixon electric field makes magnetic field than I can answer very quickly what velocity delectable to get rid of the magnetic field I just at would object no magnetic field that the current to the wider stricter and the electric field cancels out from the protons and electrons and is no way to get the snow in Europe How OK as Aziz however that there is a lot of work to get guess we also you also study Maxwell's equations but before we do with our earlier war the little bit about waves scalar where you scalar wave equation is part the main point being but we have we haven't talked about derivatives of fields if but chaotic BX which might be a little displacement along a world line now can to an entirely with the apparently is an entirely different subject Ophelia fields with forget now particles of concentrate on field fields of things urged time so they depend in general all 4 components of position let's call firefighters field field
scalar field which means it has only 1 compartment the simplest possible thing we can make and now we can differentiate it we could differentiated To create D fight by BX meal this consists of 4 terms 4 components 5 ID X 5 IDY B 5 by busy refinery Friday by BT 45 IBX 1 IBX to be primary extremely fiber IDX nor the former Florek but they formed be called variant components or for that it transforms slightly differently than the components DX meal however they transformed they transformed the same way not as VX but as the DX me with a lot component let me remind you would be lower component the law component simply means you change the sign of the kind compiled not by the side of the basic trend of super X Muir was a set of 4 numbers which former Contra variant component of that in this subject here is just minus the X 1 that that the X 3 my yes also the X and DX not change the space components and those things defined the called variant component of record sustained geometric object just being described differently is being described a different notation it's useful because we can make scale errors by contracting Lauren upper indices contracting means Utica Laura index and upper index to identify them and so on and that makes so it's a useful device it takes into account the sector signed in things like he squared minus 6 square miles 1 square mile square I'd OK if you watch a workout the transformation properties of the derivatives how you do that you normally Exs themselves transform you know what the coordinate transformations are 5 x 1 trying is equal to or whatever it is all Ourense transformations you know how they transform you know what combinations of the old coordinates no quarter you could work how you could work out the derivatives Of the same field flight with respect to x prime I don't that blackboards a little bit tedious we can work it out and see what the connection between the derivatives with respect to the prime variables the moving coordinates and be coordinates of the year on prime coordinate frame and the answer is that they transformed the same way not as DAX news here but the DX news with Laura index that's fairly easy to Pearl and so this object is a former actor but if forms the cold variant composed for vector it are called early on the Commerce coal variance components of for vector physically or that means that just transformed way forms a complex of 4 things can we make an object with upper components sure we can make an object with upper components we just change the signing of the space components and we decided the time component along right so we have 2 sets of objects we have defined by the X what's quoted 1 tool and 3 together with 35 IDX nor what and that's what we call defied IDX meal how we could also consider this is a definition now if I buy X with an upper decks several Laura index now has an upper decks and that's just the same thing here now with a minus sign here another words the object of the lawyer index and the upper index are related the same way that all 4 vectors of upper and lower components are related the effect every time you differentiate among object if you have any cancer already vector scalar a new differentiated let's suppose you take a vector field here some vector field depends on X let's oppose it as an opera index simplicity and you'd differentiated With respect X new that makes it cancer but makes a tent with 1
opera index and 1 lowering index every time the derivative operator acts it simply put another index on the object source b is a vector renewed differentiated to get cancer and of cancer and you differentiated to get an object of yet another index in store for the differentiation always put the index downstairs so differentiation is another rule that allows it perform vectors vectors cancers are and so 4 wanna discuss for the remaining kind today His wave equation and be variants of wave equation invariant of wave equations follows wave equation has the form that some scalar is equal to 0 scalar as are the same in every reference frame so sofa we can write he equation for some weight in the former East Gaylord is equal to 0 then we can be sure it's sustained in every reference we coordinate transformed let's I try to do that wave equations by their very nature incident we always have to derivatives not 1 derivative that correspond to 2nd derivatives acting on fields got cut Nagin wave equations with 6 derivatives of the simplest wave equations Maxwell's equations of a wave equation and 2nd derivatives that means they were written Ahab a derivative another derivatives acting on some field equals 0 this is a typical example of a wave equation there are wave equations perplexed so look for 1 like that we want to be the same in every reference frame so that means we want to be beat the left side abuse left inside these scalar would be a U a 1 way of going to make a way course single wave equation is only 1 way to do it was really only 1 way to do do it and that's as we got the begin by differentiating so put new the only way to make a scalar out of this is to differentiate again and to rule put the index you're upstairs was amenity next is upstairs it means that the space components of the derivative are treated with the negative side Miss it is the simplest wave equation its invariant because what ever you contract or lower index with the upper index automatically gives you something which His invariant under wrench so this is simplest example of a wave equation and let's right that would actually means but tried out of actually means in detail Our take 1st time components the time component is just being knocked 5 and now the time components don't change sign when you raise the index services just the 2nd derivative the 2nd derivative with respect to the 1st derivatives don't change signed when you raise an index and services might be 2nd sorry be 2nd by the X squared same thing for y squared think busy square this equation Russia's debate geeky squared this equation Harris b the virtue of being the same in every reference had a check you're writing in a form in which it manifestly just a scalar drop the 1 index upstairs when they expect that always gives you a scalar bats the simplest wave equation the reason I'm writing it down now is because we wanted it to Maxwell's equations this is an example of a simple wave equation Maxwell's wave equations of more and more complicated will I get the Maxwell's equations we want to right there In a form which is equally manifestly Lorentz that's or record that might good quit now next time will come back and work out Maxwell's equations in a coal co-vary notation their equations which makes up electric and magnetic fields of somewhat a intricate way but they have the virtue of being written for or the way will write them has the virtue of being X blessedly Lorentz invariant I'm getting tired of people quit our while it fully now could be a constant in fact it could be any number of things it could be any scalar right inside this is 1 particular example of a wave equation another 1 which is interesting is a writer constantly at times body I'm what's on Ironside constant times by squared you could have all kinds of things on the simplest equation is just 0 . 0 discussed those equations with work among worker study this solutions understand how they behave but this is just the simplest example of a wave equation and next time I will I will study them in some detail a long time dinner rolls the preceding
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Drehen
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