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Special Relativity | Lecture 4

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I'm Steiger University today
with we studied particles less time the action of particles cause we compared it With nonrelativistic physics they with study fields fields armed for the data on Obama begin studying feel theory not quantum field theory but classical field theory From time to time I might look at an equation and say you know something about that reminds me about what happens in quantum mechanics and pointed out but that for the most part we will not be doing quantum mechanics just go classical field theory the classical field theory that you know most about probably electromagnetism electromagnetic waves and their interaction with electrically charged particles putting aside will easier than that it would be an examination at the end of the class from which field off particle physics to think I'm talking about but who will Ross this OK nevertheless I'm going to start that but start was based our space-time has always 1 kind important and some numbers they scored mathematicians or mathematical physicist abstract people could study spaces with any number of times in a number of spaces but the physical world even in those theories in which space time may have 11 dimensions 10 dimensions 26 dimensions however many like it's always 1 time and nobody knows how psychological sends out of more than 1 kind dimension so that I don't want to know why they chose factors my wife are only think about a storm of pity we think about 1 time dimension tea and some number of space coordinates let's call the X I find runs over whatever 11 and you are was a field of field is of a variable observable variable a measurable quantities which depends on space and time which varies from place to place varies across space varies with time could be electric field but because some examples of our fields are there are some dust from common garden variety physics with when you think about whether you could think about the temperature the temperature varies from place to place a loss of varies from time to time so the temperature is a function I love XNT it's a field of other fields be wind velocity the wind velocity varies also from place to place and may also vary in direction from place to place direction times for the with velocity is also a field the wind velocity is different than the temperature of the wind velocity is a vector feels it has a direction as well as a magnitude the temperature is a scalar field scalar as a word for a thing which has a magnitude doesn't have a direction appear as 1 component was vector field of ordinary vector has 3 components are relatively factors have components spoke on right so we imagine then some kind of field field is a function let's quality f 5 fires the Greek F and garnered the generic field is often just labeled with fine of X to card could will be thinking for tonight mostly about scalar fields now on the maker motors blackboard over here Paris will think about something else I wanna think just cairo from it and think about the motion of particles In nonrelativistic physics not the relativistic motion of particles I wanna think about the nonrelativistic motion of particles is always think of them we drive time axis now we used to joining the climaxes as vertical under the job for the moment the horizontal just indicate there because I think about functions of time and thinking about functions of time I want my independent faxes to be kind a horizontal and the same limit plot is the position of a particle position particle along let's say a 1 minute and access but let and Willie call axis what's a good name for Texas White why whenever X today but he added that the Close either 1 I'm with the but the stupid turned 5 while 5 fever 5 well that's the core warned the position of a particle so of course you know I have a perfect right to mathematical right call positional particle fight back after call X don't have recorded 1 and then what is motion of a particle on the history of the motion of a particle it's a curve is a curve which tells you what life is at every point of time no negative or positive areas and how we describe this we describe it by the principle of least action the principle of least action but them or the principally stature beat well we know that the act which 1st of all with expressed in terms of a branch the action it is a quarterly integral from some initial trying to some final time but safer beat from NATO me some liberals detail a similar nonrelativistic particles is a very simple Lagrangian we're right Darren it's a Connecticut Energy minus a potential so what is the Connecticut energy that's 1 half and v square 1 half the mass of the particle which put him over and what's the velocity let's start if I got right now that he . a fight that Pfizer Courtney so we will write off I got squares or IDP squared weaker writers defy square with us right away before IDP squares a mail on we I put some potential energy and was put potential energy and now a B minus the potential energy as a function of what OK 5 E cheat the
Oilers Lagrange equations led you work aren't of course this object and he would be called or grungy Granzien em all over to fight back squared my history of fighting and the equations of given by d by DT of the derivative of a Lagrangian with respect to fight not equals derivative 0 grungy with respect to 5 this is the 4th arrived and signed and this will be may at times acceleration acceleration double 2nd Prime derivative of we worked out you just get Newton's equations here OK tho only 2 or write it down and double docked equals minus DVD fight it's no use equations and that's the net upshot albeit just 1 more . 1 more point that's worth pointing out armed action principle or the order Lagrange equations of the solution to the problem of finding the trajectory with the minimum action on so I'm going to use the term minimum but you know very well that I mean stationery could be maximum as code the minimum action on a trajectory from year to year holding these . 6 of a point a N . be fixed find the trajectory of least action that connects them that's a course the order Lagrangian equations and the way that I like to think about it but we worked this out last time is divided climaxes while pieces and a set of thinking of the Lagrangian or instead of thinking of the action as an integral just think of it as a somewhat there a those terms depend on they depend on fly at each time another toddler action is simply a function of a lot of values of 5 if some functions of all whole bunch of values of 5 How do you minimize a functional fight you do you differentiate with respect the fact that the order Lagrange equations are no other way to say it is the solution of the problem of molding these points around until you find the trajectory which which minimize action think so that's nonrelativistic misses 902 because of the relativistic particles theory the nonrelativistic description of particles In a force field now let's come back over a year he advised the a field which is a function of time and base space Kamoke Sunday when our world there are 3 but we can consider world and dimensions of space 9 dimensions is based its even interesting to think a world tour dimensions of space that would be used stable things move around the tabletop at a world of 2 dimensions you could even think of a world With 1 dimension of space and time of course one-dimensional space could be be moving up and down on a wire particles I'd be like particles moving up and down our a wire but you could think of Zero dimensional space is 0 dimensional space means a point but Zero but what will the corresponding space be nor have time and no space a magical world now with time and Nos base camp at a field in each case the field was a function of time coordinator and what every space coordinates that happened the beat if we don't they don't do this limit this strange limit with space has no dimensions of all but only time there were talking about a theory of a function love the photoconductor This is a fury of a function but only depends on current Suzie you could think of you could think of ordinarily particle mechanics as a field theory in a world which has no nobody mentions of space only a dimension of time instead of thinking of the feel instead of thinking of fine now mathematics but a different interpretation instead of thinking of fly as the position of a particle think of it as a field which depend only on time calling field just the name you could call it the coordinator something you could cause a field in quote whatever you want but it's just a function of time around the call on the to think of it now as we feel that in a world with 0 space dimension so it is essential that it cases or not it could still depend not that would be of turned me energy which would depend on the value of fine you could imagine a field and you couldn't engine that the potential energy in the field depends on fire the cost energy to change life and that remind you could remember that flies stood for the will be positioned the particle of light but the sale is a potential energy just means that if you move this curve up and down the energy changes absorb With where of course are not going to think of funny as the position of the particle we are going to think of it as a field then will go any other direction with generalized from this pattern here this was a field theory In a world with no spaces and one-time River generalized field theory time direction and a collection of space caught the field that was not just a function of time it's a function of space and time 5 x unauthorized again areas conceptually editor of pungent state all are regional add overstates the you could it wouldn't matter it's very hard to visualize a world without coordinates so think about as a world without coordinates in which nothing depends on position only on part far right of quite so now let's try to guess what the principles Of mechanics the principles of mechanics but the principles of field theory are knowing what the principles of field theory are In a world with only 1 type a best guess my best guess somebody's best guess is that the world should be governed by a principle of least action least action was a very powerful principle and all of ordinary mechanics not just of 1 particle could step 2 particles and particles of lead particles and gas whatever it was a very powerful principle which encoded in Sunrise huge numbers of laws of physics and we're going to take it has a mouse and man of giving so that feel theories that the whole world is described by an action principle I know what is action principle look like now the world world is
not just a line the vertical axis now we don't have enough dimensions on the blackboard try to draw you a picture but on this side of the blackboard I am going to plot trying vertically but get confused time still carry of horizontally and not going to plot finally going to plot this next could stand for 3 or 4 to 11 many axes there are problems in a simple problem might be 1 time and 1 x that would be the problems of the world on a line and finally I don't have enough taxis to draw at both fire would just be a field which has a value at every point of space or like to draw a few withdraw this way he goes this way expos this way is intended to be a drawing of a horizontal plane and the value of the field whatever it is is on the vertical axis that however drop a driver as occurred not it's not occurred at surface to surface above the every point for every point the field by the way fire has a value and so if I want to draw the whole history just as it drove the history of particle here the the field the one-dimensional field of drug Ahold History in space time of the field I would drive as surface here but if so that from buttons Of course when I say Obama could also be below a few negative the reason why can't be made negative but broader reporting it would be a value of the field at this point 5 CDs at this point Pfizer Leahy a move back a little bit of fires p.m and that would describe a 2 dimensional world was real world 1 space at 1 time no legal incidentally the dimensionality of space time our space times call 3 plus 1 3 plus 1 3 dimensions of space one-dimensional time this would be 0 1 plus 1 world 1 spaced direction 1 directions and if we aided in the field Friday the history of the world would look something like that of a history of the field with some back now of course and not going to try to draw what this would look like and 4 space-time dimensions I can't even draw 4 space-time dimensions let alone put in another dimension to describe the field but you get the idea OK now what about the action principle In this case over here we found with time axis between 2 points and asked for a curve which minimizes some action would be analogous to conceal theory is the battle of the space time example by a rectangle rectangle bounding the both regional space and turned on here might be a rectangle like this and talk about the field within the rectangle for abrupt field the rectangle just as we talked about the field on the interval between 80 and be none is remind you I should have mentioned it while we were talking about the action principle over here where I said you minimize the action this something that you hold fixed in a more Maurizio hold fixed will minimize the engine the point you freeze the endpoints don't let them move and you say What's the minimum action subject to the condition that the endpoints of 6 this is very much like asking for the status don't the shortest distance between 2 points is a straight line and you hold points fixed and you look for all the ways you could wiggle the curve to minimize the distance between them In the same way for a similar way the action principle says hold fixed the value of the field on the boundary here the back of the doesn't have to be a rectangle it could be any spacetime region figures spacetime region freeze the field on the boundary OK so that means it's over frozen Holly Kia is boundary from diagrams getting too cluttered but there are we hold the field fixed on the boundary in them little wiggle up and down or let the value of the field Aleksandr wiggle until we find a configuration that minimizes the action and that would be the history of of the field space and time that will be that will provide for us a set of equations of motion you get the idea if not the details the details will be provided by telling you what action so the Roger formulation is still works as a put-downs absolutely analog would be something depending on space and time here now now grandiose with what doesn't work what doesn't work will grant you depends explicitly unkind division had to go around him formulation those work so does Hamiltonian formulation adjusted images served but Oh sent Korea yup is part of the whole electoral Baghdad the whole band river candle the stock all 4 sides right just as in this case we than the tool in the event of a door and here we bounded overhaul fix the value of the field here and let wiggle around salary in practice to its 1st of all we would need an expression for the action of the expression for the actions will be just as it is here it was in will over time another words a sort of terms for each 1 of these little segments here to assume that the action is a sum of turning 1 each failed if we divided this they have been through a trainee sells another word that's it integral over space and time so we take our region that were interested in break it up into a lot of cells but if this is god of course is the trekked implement going in integral integral over ex is just the sum of all the cells in each still we specify what action within that and then a them all out as the same sort of thing that we do he and refer the action was a somber over infinitesimal cells so are the only other question is what show we take for the Lagrangian through set of rules rules are quite similar and generalized mechanics of particles so most of direct generalization 1st of all to be an integral to the action is an integral but in integral not only all the time but of space and current DX team comedy DX is however many caudatus space there are so we can write this is D experts say DAY Romania 61 vis-a-vis but I
really kiddie we've learned a sort of think of the 4 coordinates are common plane has been all parts a space time and just give them a name or for coordinates call them X knew where you runs over 4 values kind 3 components space so it's frequently but frequently standard right integral all all 4 quarter myths of space is just will be 4 x Of course if we were dealing with a fury in Firebrick tool 1 they mentioned we would replace this body to XT for XT Rex while XP it was only 1 dimension would be done right so before Forex includes DT but also includes the suspects you could just think of it as DXT Desi and then some look around here of something as a shared which that is just to be integrated over time be integrated office space any time so often called the Lagrange density but stressed the analog of Lagrangian he and what does it depend on well of course exposed to build a grand jury in the field funny to tell you how far be so 1 of the things it certainly do depend on the side that's a catalog of saying Lagrangian wouldn't say over a year but that's the order on a lot of saying ground geology of depends on depends on position it may mean that the composition of course but it also depends on the derivatives of flying in this case with respect to time if time and space on the same footing in relativity we might expect that the Lagrangian depends not only on the kind derivative of fighting which put in never be fired by duty a lot of it's going to depend on defy by D. C and with dealing in the a theory of relativity with time and space are pretty much the same or very closely related footing we might expect it also depends on the derivatives of 5 with respect 2 X with respect to Why for harsh but Tuesday and however many more coordinates the Art Basel transient programs is a function of the fields 580 maybe more than 1 field incidentally just as In this situation over here a particle may have more than 1 coordinate locating its position the XYZ so for God and maybe more than 1 field but for the moment let's just concentrate our problem with 1 field called 5 programs in depends on the field and its various derivatives the effects are not just as the Lagrangian Mike depend on T over here it might now our throw closed system ordinarily system which has an energy conservation it means that the Lagrangian doesn't depend explicitly on time and the same will be true here would not depend explicitly on time or space but it got a voted out here 5 or a general at the moment at the moment that have so far I haven't said anything that makes a scalar field other than the fact that I only wrote 1 component vector feel there would have to have some more components but it's a scalar field and would only have 1 component a yes I certainly have the back of my mind here a scalar field but up till now it does not mean it could be anything I'd so it's so let's some let's condensed notation let's just write this as a function refined and the collection of derivatives D 5 IDX mural of which there are several time derivatives and space we could even convince the notation a little bit more time derivative we would use 5 . 0 used the space derivatives well remains convinced locations not to use this the ice condenser patient Mr. Right 5 he Pfizer X Y Y Y Z and these things mean by definition derivative of fire with respect to the river of fire service would be 55 IDT should be defined by the X and so forth as moral standard notation party but will tear an extra orders each year we could afford light right find some new meaning derivative of fighting with respect text but I will leave the waiters 5 IPX mail but such a problem problem is the to solve the problem of fixing the field on the boundary and wiggling around your we're going around until we find the configuration and minimizes this action if we replace based in time by agreed like there's obsessing about how we do it practice we would replace the space time by the a grid of points we would replace the Lagrangian it would depend on fire fired each point but what will really replaced 5 by IDX with good different those differences and the process we would actually write this action as a function on the values of society at each grid point who just iPhone to be a function of a lot of variables and many of them attend this way in this way BAT variables there are some who have a lot of variables finally make the lattice moreover would just be the action would just be a function of some collection of variables and how we minimize it we would minimize it by differentiating the function with respect to the variables and setting it equal to 0 exactly the same operational we do know here doing that operations and its senior notes in your classical mechanical in the classical mechanics notes we went from that principle the Maori grunge equation he or I will grant equations written in this form year minus theirs equal 0 is basically the equation which says differentiate the action with respect finally Ed each point considered equal to 0 this was going took a few steps work it out the question is How would that change every have multi-dimensional analog of exactly the same thing multi-dimensional iron ore on the same thing in a answer is it would change tho change in a rather recognizable way and I'm not going to work it out With right down you what the equations are which are the equivalent of saying that the the river the action with the valve with respect to fly in each good point is equal to 0 so we would do the corresponding or Lagrange equations all right the largest show you with the correspondence is well let's let's let's look outlets what's rewrite the ordinary Iran's equations over here and no changes originally was deemed IDT off partial of El with respect 2 0 0 before I buy DT with different Schuett the French Lagrangian with respect if I D T minus the director Grand and with respect
far-right equal 0 that was naive used but the simple version which there was only 1 cordoned off time has a changed the changes in a very simple way instead of having a single turned here with a time derivative you have a sum of terms 1 for each direction of space time and that turned looks like each term looks like a derivative it a derivative not was respected TD but with respect don't X New now worn out of the terms will correspond to cheat X new runs from T ex-wife Missouri some over a new 1 for each direction they started the day original Grand With respect to what but would you guessed back 1 possibility is that it's just 55 IDT now master derivative of with respect to x meals another words you go you look for the particular turned his a particular terms for example Debye DTD if which is 1 term which is new equals the time component it would read DT of parcel Lovell with respect the fight back but then there would be another term plus Deeb ID acts of the derivative low grungy and with respect to be fired by IDX than another 1 for the 1 fuzzy that's happens this Turner's at all the other turn is even easier it's exactly what it was before the derivative of grungy would respect the fight and that's equal to 0 that's it most of the oil Lagrange equations a solution to the problem of finding it because it doesn't solve the problem of finding the history of field but it does Is it reduces the action principle to a differential equation or collection of work well the differential equation let's do it before we discuss relativity and the constraints of relativity let's do it for a simple example which is patterned on missing structure over here but at it is evidently what we're doing here calling it field theory it is another name is called continuum mechanics continuum mechanics it could apply to a lot of different things Kofler the flow of fluids could apply to Alaska city of was a last this city then could be degrees of freedom Mike b the displacement for points in the elastic material it was floated the degrees of freedom might be the fluid velocities of accord and so forth so what we're doing here is called continuum mechanics ripping more what it he said 0 0 yes but it get no when this however a car Chen OK so that's exactly right that those are the equations it you right Baron if you are looking for a solution the field theory in which nothing depend on time which everything didn't change another words of fuel looking for what's called a static solution of the equations of theory if you have a feeling that there was a solution of youth theory prisoner book kind a tying independent electric field magnetic field magnetic field of a video a doughnut-shaped solenoid or something and you were looking for a solution which was not time-dependent but which was static you would then say you would look at the equations of West throw away all the time derivatives here since we're looking for a solution which is static and then we would only have the same equations but we're just space compiled that's good point on but let's try the build us an example study the example little bit see what kind of equations we get before we enter into a discussion of the implications of a of relativity OK Hillyard along with just take this thing over here and generalize a little bit instead of staying instead instead of just saying the law grungy and hasn't in over 2 0 times fight not squared and this is the number incidentally don't think of it in his anybody's Mets here just think of it as the number of it you could put until a grand jury and is just an arbitrary number Miller said equal 1 just 1 have fired up square but will the generalization be the generalization me you might be a guess adored Nanjing contains defying by d he squared it would put the goal will put 1 but we don't expect it the only depend on time derivatives will also expected depend on the space derivatives sold a simple possibility would be defined by IDT squared frosty 5 ID X squared plus defied IDY squared frosty 5 IDC cornered ominous stopped writing out all ex-wife is disease a little while and that's something we could write down on that's our right White what's up F. but not only having gotten the relativity we don't care too much about wrote experts parts not worried you right glycerite left now there would be about details relic this year you right but just make it more primitive than that 2 minutes From but pussy someplace In New York for traffic that was about when and if this move around and it would be good completely disdain for space and time don't billion be no difference between space endemic doesn't sound right don't sound right about the behavior of waves can't tell the difference between the kind scored world a away something different about turned space we recognize it but what's a red answer but a threat will Contel why closer white does have do exactly what you it has but exactly the same kind of a grungy incidentally Abreu doing relatively would be some speeds of light here anybody figure with the speed of light collected never remember whether it's C squared multiplying this 1 squared multiplying this war those units right see goes retained cell we microbe 1 overseas square this 1 here where word but this might not be later might be salary at From elastic waves of elastic waves and a solid they moldy what we're replaces West Soon lasting elastic wave Austria whatever
please these are Rosa yes yes you can think about this wedding you could think of this as part of the potential energy officially the definition of kinetic energy has do with car independent and can add up a tick kinetic energy and potential energy has to do with everything that doesn't depend on time derivatives sources we typically subtract potential energy that's yes that is that is part of the connection arms but ultimately the real reason His relativity now we could enter if we like minus some via 5 potential energy which is there just because the deal has a certain value potential energy In each little square where potential energy of potential terminal around June which only depends on the 5 so that's an interesting case to explore but that explores back and see what kind of equations emotionally get from take this didn't work out it's equations of motion apart as part of a Lagrangian here the whole of all of that but 1st step calculated the removal of with respect to ruled that start with the tying component of the kind parcel of our with respect fight not wares at read it is justified by right derivative of the slogans and with respect a 5 . just 5 . 5 . be 5 IDT that's a derivative organs in which respect the fi . and then was opposed to differentiate that with respect the time so that becomes a more outside the 2nd derivative of fighting with respect to Times Square it's analogous to the acceleration effect it is the acceleration of the field 2nd fight IDT squared now we do the same thing with acts was supposed to some this all a T x y and z and the only difference when we do it with respect acts will be that sign that might have put it could put the speed of the reasons we could put a 1 overseas scored he works but but that velocity and I said that velocity could be the speed of sound that could be the speed waterways could be the speed of whatever kind way talking about where the kind of rum they were talking about a 2nd fiber squared and then what happens for the space components the space components of exactly the same except for minor side 2nd 5 IDX squared and I won't bother writing y Square Square store for my that's the letter that that this term and then might steal by defied my the riddle of grungy another's low-budget independent signed sure about his you fight if there is a V of fight then we would have marinas minors my plus TV but 5 from minus 1 9 assigned becoming because the law Grosjean miners V the assigned coming from here equals 0 but pork but I don't think you try that a job at left back less thorough cases with Whoa with 0 well I think you're seeing that kind of equations before this is a wave equation is an equation for a wave moving along the axis don't think more solid tonight it's not our abuser I think most of you know this corresponds the waves which could more weapons barometz axis with the speed of light over the speech and girl we see how wavelike emotions come out of the same kind of Lagrangian formulation let's suppose that would put back the the view flight the backup particularly special kind fight a move put back in video 5 which is equal promote 1 1 half a quantity which I'll call squared it doesn't mean anything now other than the coefficient here times 5 squared what would this In the case of ordinary classical mechanics where fine really stood for for the position article they said they swing answered right this would be this would be no new Square times the displacement of the Serbia harmonic oscillator with a spring constant New squared and they are quite erratic potential energy like 1 he FKX squared this would be sold form right think of this equation Excuses plus plus and where where'd you knew fires quite right plus New New York DVD by 5 TV by defying his new squared plant you might think of this and a higher dimensional version Wal-Mart if this China Morin here it really would be just a harmonic oscillator acceleration plus of coefficient times displacement equals 0 the ordinary harmonic oscillator now makes a more complicated some sort wavelike motion and of course wrote 2nd 5 IDX squared here you might imagine but it was other terms to 2nd 5 IDY square and solve for over here because the derivatives public gives a factor 2 back period that's why the tools of 20 minutes of big disappear at a trip that is after it is the same for the harmonic oscillator be potential images always defined as 1 half kayaks squared just forest His acts right but well but that's
field theory nutshell but now girls continuum mechanics there are a lot of discuss its connection with special relativity when we built bustles a particles who basically 2 principles In addition to least action principle least action principle and 2 other ingredients the 1st is that the action was always an integral and for the case of a particle considered go along a trajectory not really that's that there with taking care of Europe's an integral space the other principal it is a thick action should be the same in every reference it should be built Out of quantities in such a way that has the same exact form In every reference had earlier manager had manager for the case or particle personal we built up however Wavell segments but we chose the action of the segment a quantity which was an invariant Illinois was the proper the proper distance of a proper time along the trajectory and its everybody agrees about the value of the proper time invariant proper time for little segment when you edit everybody will agree about the value of the action all expressed by the same set of rules and so the equations of motion when derive them in 1 coordinate system will have exactly the same form as they have in any of the court system and the words the the laws of mechanics particle mechanics will be invariant statement every reference when they wanna do exactly the same thing off in field theory we want to Eddie ingredient which we've actually really done it but the book nor a spell out now that let them implies that the laws of field theory in specific field theory here Lorentz invariant for that connect in we have the know-how of warm invariant quantities quantities like this which are in very good do that we have to know how things transform we could not have done we could not have done this case here if we didn't have a transformational off for the coordinates of we food from a dx dx along he added the Exs transform under Lorentz transformations that transforms the component for that once we knew that we but the quantities Dutrow squares which is equal to DXT he squared minus DX cornered we know that this wasn't there but we had to start with the quantity whose transformation laws we knew in that case the transformation laws low rents transformations applied the coordinates they also apply to differences of coordinates and they applied to differential little bits of coordinates here that was a guiding principle construct things which variants of the same in every coordinate system so have to start with a collection of objects which on are very high that brings us the nature of the transformation properties of field fields are functions of position what kind of field would be like the temperature just a thing which doesn't transformer all all observers no matter how the moving no matter how their around where they rotate coordinates whatever you do the temperature was the temperature now actually temperatures a little bit funny if you actually took a moving thermometer moving through the atmosphere it probably would not register the year be invariant temperature so it's not really a good example but a scalar quantity is 1 for which no matter what frame of reference book that from or what uh what took when system look from it's always the call the scale but scalar doesn't change when you rotate Lorentz transformation transform court you could say you said this way and say that the value of 580 a point at ex-mayor Austrians extant is exactly the same as the transformed field plan evaluated at the transformed court X could stand for are positioned in space time in your X X prime anti-crime who could stand a position in Michael display signed position in my Fraley fight is what Joe call the four-year-old Justin a you get to the field and I qualified time just because I'm the prime person it is a point in space time you call the value of the field phi of X X Aziz said excuse ex ante I call it by Prime of X prime but just 1 point sustained feel the same value which just invariant quantity of the transformation properties scalar fields are just equal toward next in importance of vector fields is mother 3 vectors these are 4 sectors we have 4 coordinates we already see where 4 compartments we've already seen some examples just displacement XVO displacement from the origin is of differential displacements between neighboring points DX meal is a if you multiply for sectors by scalar call by invariants you get back for vectors because invariants they're completely passive when you transform we already know and invariant deep house so if you take DX New by Carol which recalled before velocity that's also for vector Was it mean that's a 4 vector it's a statement about how it transformed under Lorentz transformations let's write it down right down the transformation properties are transformation if we consider talk warden systems moving past each other along the x-axis then keep Carla struck out out deprived feet and is able to feed miners VX all square 1 might be squared ex prime is equal to X by his
over squared of one-liners squared Prime because y pride Jose things which transformed in this way complexes of 4 quantities of time component of X component a component is component things which transformed this way of called for vectors so this it is also true for for the differential displacement With the differential displacement might be a differential displacement along a trajectory also the transformation property for the Australia you divide by deep How it tells you how the 4 velocity components transform anything which transforms this way is a format that I didn't mean a generic name for records on I will use vie for vector but these also already velocity used a but a stands for erode the vector potential used but stands for the magnetic field seized speed of light these derivatives he is energy have force I've run medical letters some will stick with age 80 80 a Arab how good Arab and later on they will have a dubious back so a floor has 4 components call at the time component but used to calling it a naught a Texas a wide easy and the transformation properties of these things are in my coordinate system I call on Prime that equal to the values in your coordinate system X my nest vii 18 naught divided by squares of 1 might be squared notice a not the same thing as the kind components a X minus V 8 wherever this 1 but that correctly not be did not a not minus X and this 1 is a X mines vii on my squared this 1 is right variable primed primed uh but but but an example of field which obviously transforms we can give you an example 1 that that that makes clear why the transforms gap imagine a fluid filling space time emerge a fluid filling space times every point in the fluid there was a need for velocity ordinary velocity for the fluid points but also for velocity and the 4 velocity could be called you knew Willoughby be a position of a position dependent thing if the flow was flowing the velocity might be different in different places before a lot this is you let before velocity could be thought of as a field before velocity of the fluid in space time could be thought of as a field instances of poor velocity it's a fought back there it would transform exactly this way except we recall you wins over and we got you would have a value of you'll in your frame would be different than the value of you in my framed the components and they would be related in this manner here but there are lots of other things which could be for vectors I want to tell you and your right now broke of 4 vectors is a possible kind of fields of 4 that there's are aII new also of and take a transformer in this manner here Twitter you could make galas out of full vector if you take the form of components of a force that you could make a scalar but if you take any tool for vectors to think In tool for vectors you can make a scalar out of them but let's just stick with 1 for you could make a scalar rather at the same exact way that we made a scalar of DX mu because now is equal to cow squared it is equal to D square above or below minus the exclamations 2 white women square in the same way every 4 vector could be used to make a scalar and it's just a naught squared minus 80 x squared say why square soulful that's a scalar exactly the same Rees wants to know that the eighties transformed saying ways X X and take them by the same little bit of algebra you could see that the difference of the squares kind component of space component will not change under Lorentz transformation St. little bit of algebra the see squared he squared minus X square my national minus square half wants to know that little bit of algebra the to riders How squares squared minus 6 square miles west square Manzi square for the same reason that this doesn't change in the Lorentz transformation neither so we have now constructed a scalar difference of the squares of recurrent components in space components last point was point is that given a scalar you could construct a 4th sector and not by just looking at it but by differentiating by differentiating it with respect for the 4 components of space and time goes for derivatives form of for I saw special case over 4 vector given the scale of collection of components the derivative of flying With respect X mule the review of a fire with respect to the river 5 with respect X through the fires but for wine and so forth these form of 4 vector not surprisingly not surprisingly these forms a before vector and so now we have a set of tools for creating Lagrangian awoke yes we know how things transformed and we know how to build a scalar and vectors our other objects What's the rules for making a Lagrangian on action invariant is very simple we required the goal of grungy itself is the same in every quarter but
Groningen the thing that we cannot always little cells is the same in every quarter that means a scalar but as means the scale of the Lagrangian should be a scalar so that's oral the Lagrangian should be a scalar under Lorentz transformations of squalor gala and that's all there is to it you take your field which happens to be fine in this case you consider old possible Steelers but you couldn't make out of it and they are possible candidates to put into a grungy and so let's what start with somebody running out a blackboard space Igor only discerning will work to make well fire itself as a scalar but now his fire scalar but any function of fires a scalar Everybody agrees on the value of 5 they will also agree on the value of squint fight fueled the hyperbolic co-sign the no functions of flight will also be scale so any function functional of is also scalar and therefore it's a candidate for something that you could around Jim well we've already done that here that next Wells could you put it well we certainly want to put in derivatives of the field if we don't put rivers of fewer fuel theories to be very very trivial it would be very interesting where to put the rivers and but we have to put the derivatives Ian in a way which constitutes a scale so what can we do what's easy we just thought with vectors build a 5 X X meal that's a erect and we simply construct the invariant scalar Out of the components of the vector sewers at me in practice it means 5 IDT squared but my AST 5 IDX squared but dynasty fiber you don't get too tired of writing a column I wish I had us shorthand notation for now an example passing law passed in 2nd derivative this is the 1st to apply RCN is like an acceleration of passing and uses a different thing this is the sum of the squares on the difference of the squares so there's something Yukon grew from around and you can be sure that when you do the equations that you get out of the same in every reference frame just because quantity is is the same in every reference in other words it equal to corresponding things in another Frank served years other other things you go right there we were messes none of the church
message the parameter it's not a dynamical thing you could say we put the Nets in their birth their mess of what would mess it will put numbers yarn numbers a perfectly good things political of we did that ready reserve booklets they via fight to be new squared over 2 5 square because the number so yes to corporate numbers and just numerical constants but something else you could do this would stop nickel complicated story but became this which is a scalar and you could multiply by another scalar multiplying Scalia's together gives galas multiply 2 things each of which is the same in every reference frame multiply still scale so it can multiply this by some function if we like a finite as a legal thing to do it starts the make rather complicated Lagrangian and when a considerate of the moment but certainly a reasonable thing Lorenzen variance thing to do we could do something even uglier this would this we give very ugly we could take this scalar and squared and have high you do really have a higher powers of derivatives that begins very ugly in fact it's not the
things which appears and burn off Chris higher were driven decision-makers allegedly was still but but you will but not within the can't find Our rules of classical mechanics rules of classical mechanics you could have functions of Oakwood nets and kind derivatives so if you're staying within the boundaries Our standard classical physics on you want to stay we just 1st derivatives and higher powers of 1st reviews but only for France or once we do that a citywide this was water while I chose it the way I did the restore the units output oversee squared here and purely for notational reasons meanings purely on 4 conventions with poor 1 half from the whole thing is same 1 half goes from 1 here FMV squared you don't have to do it warned he FMV squares you could have taken the liberty in the B and B and squared Of course the man's would differed from masked by a factor of tool that's OK you could say it's a convention to conventions that f equals and not articles twice and that's a convention so this 1 half as purely a conventions no content in it will want to fix it you fixed it so our examples example that that will focus on it will be the story and a over again over here From a believer I would see a Leivasy there for the moment minors but let's take New Square or 0 why squared that's simplest field theory fact even simpler if we don't have Mr. Freeling it's a very simple feel theory that a wave equation gives rise to wave equation written down with right the equation we directly equation but lost the are it's right down for the wave equation derives from this is it is 1 overseas squared d 2nd IDT Square the site acceleration and then as analogous terms to each 1 of the the X Y and Z with a minus sign he said confined by X squared plus Y squared Posey squared In the from here we get plus new squared 580 equals 0 analysis said as a wave equation it's a particularly simple wave equation because linear ruin the means it means that the field only if the field
as derivatives only appeared for the 1st power no turned Cineplex squared off fight the rivers of fire all things 1st top Florida voters said they waited as well as a where were how their use in this is that that said GATT says that this is a trip Cecil yes cast a S S E vote director of the metric space that is where you're out Easter that's butter Bermudez it is the connected with the method we haven't you lose the terminology metric Burma our for space time we will only get to general relativity but yes that's worth about Tibet because special REPORT young metric of space time was Deepak he squared minus square murders BY square Greg absolutely any other questions of this point is far more quite Lyons a component Ahrent just equal while it be a former record bomb I can't think of anything like that which we just have 4 components normal anything like that now of course would you can have a B C Taylor is the thing with only 1 component going components by Mexican Palmisano components Ozzie components component of the scalar vector has 4 components label by Swazi intake of course there are more complicated things cancers which have multiple components were X X components XY components TX components and so forth but fortunately we don't have to deal with them know from most of classical field theory not until we get Jarrow and so on now they are less Geller while you're still just about all that what is at stake I'm getting getting involved for now army in Russia Ukraine her that's right you 1st of all you could have a low rents transformation not along the x-axis along the Y axis in that case the transfer the transformation laws would think about a Lorentz transformation along the y-axis this would be a transformation in which to coordinate system moving along the the Y axis then they transformation properties would be exactly the same except that where select the quite have um day Prime is a photo he might be wiII a square root 1 mines squares X Prize equals X Y prime people's why minus V all square root see Prime physical fuzzy as it was actually and because you're much more complicated things you could have rents transformations with the frames of moving along some funny axis which isn't along the ex-wife of axis but these are all simply rotations of the same thing where you 1st take your axes and then you do what you would have done had he been doing a transformation along the x-axis so that in general the general Lorentz transformation general Lorentz transformation is a combination of rotations of space and trends and transformations along the x-axis every Lynch transformation Comey built up by thinking by airlines transformation 19 and arbitrary direction any 2 frames moving and 2 frames which are related by an angle in space and which are moving relative to each other and annealed direction that transformation and be built up at work combination of rotations space 1st and then the transformation along with no X-axis and their mother rotation the point is that as long as you make sure that your theory is invariant with respect to the simple Lorentz transformation X wearers of the X Lorentz transformation as long as you make sure that variant with respect to this end with respect the rotation vineyard business you're business it will be invariant with respect at any hour Marin's transformational roll up Watson interesting question out come back wears a Lagrangian here it is is is very with respect to rotation but to get ranchers nations about rotation yes because this is the sum of the squares of components of a space compiled a space that square plus Y squared Posay squared that's rotationally invariant so yes this is invariant not only with respectable Iran's transformations along with TX access but also with respect to rotations of space and that's enough that all you have to keep track if you know that your expressions are invariant with respect the spatial rotations and also guarantee transformations along 1 given access you're finished Romania should say you're beginning by so other we have the basic rules for quite quite classical field theory to take it feels wherever they happen to be you figure all they scalar xxx that you could make Out of taking the fields and they derivatives the fields don't have to be just Taylor's will take examples where the field themselves and not just Taylor's also vectors can't visit on point and you build however them and their derivatives all of the scalar circuit create want to have all the scale as you can create the new bill the grungy in terms of those functions of those galas level sums of terror and then you apply the or the Lagrangian equations and those give you equations of motion of the field equations describing the year of the propagation of waves or it is that few serious described not at some point will come back and study the wave equation keep losing it will study the wave equation solve it barely understand that does give rise to wave motions but before we do that I wanted the 1 last thing tonight we are we exhausted goes to repay quick using his blood you role and ends us labor the pair had all those of you visit was a lot better give you an not quite there are some rules of our have expressed fear the rules of our of the rules come from classical mechanics which we haven't talked about it but
summer some special rules a pair of there a rehab discussed gap they'll give you be classical mechanical systems almost all of them don't survived the transition the quandary can fix bats with the fund happen when you try to make the the transition the quantum mechanics you find it hold most or all but kind a set of measures 0 just are inconsistent give instead of answers skip over nonsense of 1 form or another and there are those are of course the warns or don't give nonsense whichever warns you focus on that level of C Ds at the level of classical physics yes this just enormous ambiguity but but Fields fields feel should be continued not continuous they have infinite derivatives of the derivatives and Ireland action we have spoken about energy yet but the Harvard energy will next time will come back to to the energy stored in the field but I want to spend another 15 or so minutes talking about the the relations between particles and fields are not talking about a quantum mechanical relations between particles and just talking about the relations between ordinary particles and how they interact with fields on if we if I had worked out the rules for electrodynamics instead of through a simple scalar field theory I might tell you how charged particles interact with electromagnetic field we haven't done that yet will come back to that but come on let's talk about how wave-particle might interact with AD scalar field with this scalar field 5 comrade and interact with it and how president of the scalar field effect motion oracle from we need a Lagrangian Ireland but there are many many possible around right end for the same reason classical mechanics simply doesn't have enough constraints and tell you in detail units to be rules let's think about below grungy and for a particle moving in the present Our up 3 established field somebody has solved the equations of motion for the field going all the field 5 exon take some specific functions of space and time now particle was moving in that field that particle might be coupled to the field it might be like a charged particle electromagnetic field how a particle moved well the answer that question we ask what kind of Le grungy genes which discovered go away forward was West toddlers but added that he had to be reduced the gentle person who was here these mosquito at the low-end skipped is too big to be mosquito but they all skaters Sunday flight and only flies you're well right so let's go back the particle mechanics the particle mechanics In the presence a field of so 1st of all we've already written down a Lagrangian particle is about the only thing available Our that was invariant and it was the integral Monday Talal Florida Reza's of conventional while not just for reasons of convention but there was a parameter here which was called mass which turned out to be the Mets we discovered that behaved so how it did discover they behaved like the nonrelativistic mess from a particle was moving slowly but it was also minus sign surprisingly was a minus signed him to a wee surprise me when I was a younger person winningest whitey put a minus sign but you put the minus sign there so that when you work it out and look at slow-motion a dire you get the right number of Mr. get to was on and D Tal was the square root of DT squared minus X squared all right I keep telling you over and over again I will right BY squared and easy Scribner do it anyway well I'm not gonna do it OK and then we divided by D D squared put the DT out here maybe this 1 minus DX Square T squared the explained he squared and that was all about it's a function or velocity which also call was velocity squared also terms for lions ate this was all Monday in which we see expanded out for slaughter lost cities we found out that it was the girl learned Tony and Lagrangian Grove conventional Lagrangian but this was it offer relativity now what can we do to theirs to warm to allow the particle coupled to field while the particle interact with field be affected by the field well in order for the particle to be affected by the field we have to put into Lagrange in the field somewhere we have to put it in a way that was Lorenzen variant and other words we wanna put it in a way which is the same in every reference frame the only thing we have around involving fields which is the same in every reference frame the scalar there's a lot of things we could do with a scale is but 1 that very simple thing we could do would be most with this was my best D travel the simplest thing you can do is you could say as the particles of the field you can and another turned him which has the feel of a planned deep how below many other things is not unique by any means 1st of all you could add the square of the field times town of the Cuba the few times towel or a visibly quite legitimate a function of the field times D town but as far as the particle motion goes since we want to use invariants quantities returned and we want to use detail see only thing around involving little steps along the trajectory which is very so what possible Lagrangian would be to dead miners class some functions of the field 5 let's just call plus 5 there would be a possible Lagrangian for a particle moving in a pre- established field for the moment the field has been given to us we're not going to ask how it got to be the way it was we're asking how was the particle moved in the field is it can like get help particle Mozer magnetic field a electric field you're right down a Lagrangian for the particle in the electrical magnetic field about have electric and magnetic field got that and this would be by Aleksandr T. why I'm doing this 1st of all to illustrate a point about derived about connection between particles and fields but 2nd of all just as a young In order to do a little exercise and to work out the equations of motion of particles let's work at the equations of motion particles uses Rogers and a workout equations of motion we
just do your of branch equation motion with just 2 they're ugly and pretty yes I'm sorry but should be in need absolutely was at a message functional reacts that's correct muscle have to be just a function of X this represents the position of the particle at yes it should be it should be an integral best we need to go because even know fight Mike did not depend on T TV X depends on of fire x depends on of next to him to decide the integral that the 1st all notice something interesting America's Sapulpa for some reason the field like Tom migrate some value for some reason the field like toast seek some staple a stable value which wasn't 0 I like to get frozen for some reason away from 0 constant completely constant firewood just be a constant approximately a constant the motion of a particle would be absolutely identical to a particle whose and they asked I was In the minus 5 who would be exactly the same as if the particle man was in the minors flight when miners worked miners and past 5 of romance would be and minors why this is the simplest example of how a scalar field can give rise to a shift of the article what I said those meetings and began what fielders most likely to nature Higgs field is a thing which is most likely scalar field nature and you could see how what possible that if the Higgs field did shifted by some mechanism its value shifted in turn it will shift the value of the massive particles in fact if the mess of some particle naturally began as 0 but it was coupled this field and this feel fine naturally sought some particular value beat particle would effectively have amassed which would be related to the so this is not exactly the Higgs mechanism but its closely connected with it and it's what the roughly what made when people say the Higgs field gives the particle maps Higgs field enters into the equations as if it were part of the a man but that starts but just the equation of motion would be and then maybe will try to take the nonrelativistic limit in order get done this for years and now but still it but strife OK so here is the Lagrangian and find that was just thought of as a function of X X function of the position had a pot of tea but so the 1st thing to calculate is a derivative Lola grungy partial the river Legrand June with respect to X . 4 X . right here was that that is In my mind this Friday Rexon tee times X Dr. although the square root of 1 my X . squared I'm doing this as if there was only 1 direction space will make it simple will make it a problem movie in which a particle moves of only 1 direction then really X direction which give I get this from well if you remember the last time we calculated the momentum by differentiating Lagrangian with respect next what we get we got we got the Mass cancel velocity over square of what might be squared nonetheless that was a momentum was equal to the derivative around June with respect of velocity differentiate the Lagrangian with respect the velocity will get exactly the same kind of thing except every place she saw and it will not be in this fight we did do it explicitly to differentiate this with respect to a component of the velocity we differentiate the square root we we multiply by this this is what you get is not in times of velocity but in minus 5 sets a 1st step was the next step we differentiate this with respect to of next EPA's differentiated suspected time is deemed IDT Of and minus 5 8 X . over square root of 1 minus X . squared but still it's 1 then what's the other terminal grand equations Our Lagrange equations he is a son of your after differentiated Lewin right what see general my dynasty OBX those X appear in the yes it does it's in fiery fright depends on Exon take sorrow at equal to 0 direct minus derivative of 580 With respect to X the Red Dress poses a partial live my partial derivative of firewood acts it find of Suez Lagrangian yacht would differentiating Lagrangian with respect acts he is the only place that X appears but this thing multiplies Square Square restore Hey square root 1 minus X . square but such that's the equation of motion of differential equation the left-hand side has something which is more complicated than ordinary acceleration the Debye IDT has all kinds of terms are a nuisance to work out it's not traders but this is it but there our hopes worker arts that's worked through a little bit of it OK so 1st of all we get I wonder if we figure out where the speed of light go now From didn't do that can do a lot of thought only a small us that Bush that were a little Granzien have finally Kansas square root of minus textile square that was in around right was with my assigned there was the plus sign was minus any costs and where differentiated fire with respect to x still contain the factor at that's their arms but this is not pretty this is not a it's nice the wind-up electoral lecture with a nice beautiful elegant result this has not happened in this case I know that word are but let's just she works here 1st of all the Debye he
hit this thing over here now this thing over here is what would have been in the canonical momentum had this turned well away it is what it is His work is term which is Debye dt on X . all of the square root of 1 mind up squared itself was not very pretty somebody by DT hits this 3rd times and minus 5 a derivative acting on this factor fire and then another terms which is plus we have to take a derivative of T of minors fight and is constant and as a constant derivative of the specter of minus 5 batters minors but 1st right was 5 . times X . over square of 1 mind 6 . glared from differentiating Miers and that's still equal to arrive he inside 5 IDX would get a minus sign square 1 minus 6 . square I bought off I got what I do if I got exercise because of space is also the 1st partial success as a simple case were finally depends on X just be simple finally depends on justice simplified if then it was 5 . equal to 0 now is now 0 because it depends on X an X is not constant price August 5 . fi . and physical to derivative fire with respect to x times X . bird so this is going going to be from here defined IDX and the annexed up square here among make it any simple vendors so says that river monstrosity on I'll leave you the exercise of putting the speed of light into it was a point but to speed of light into it and then study it when the velocity is much slower than the speed of light and see if you can recognize India many familiar anything familiar about she recognize anything familiar about are it will be it should be familiar in that case is to put the speed of light and and then take the limit in which the velocity is much lower than the speed of light the authority of the Arab world reaffirmed for more please visit us at Stanford .
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Metadaten

Formale Metadaten

Titel Special Relativity | Lecture 4
Serientitel Lecture Collection | Special Relativity
Teil 4
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/15003
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (April 30, 2012) Leonard Susskind moves into the topic of fields and field theory. For the most part he will focus on classical field theory, but occasionally will relate it to some of the concepts from quantum mechanics. In 1905, while only twenty-six years old, Albert Einstein published "On the Electrodynamics of Moving Bodies" and effectively extended classical laws of relativity to all laws of physics, even electrodynamics. In this course, Professor Susskind takes a close look at the special theory of relativity and also at classical field theory. Concepts addressed here include space-time and four-dimensional space-time, electromagnetic fields and their application to Maxwell's equations.

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