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Special Relativity  Lecture 5
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Erkannte Entitäten
Sprachtranskript
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rule on Steiger University of Romario
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questions I can answer 5 minutes a whatif about as look Who's depends on which weighs gone our before going that way and your moving with it and the wavelength looks long refused stand there were few moving the other way it looks about what was soft and hard remit careerhigh you worker you could work it out this geometry from geometry Lorentz transformations are desert you do 1st of all still the usual drawing of took orders frames 1 of them is arrest but your frame and my frame is moving from right From fraying it looks like there's carts draw more more afraid withdrawn the speed of light a draft final imagine that they woes a
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wait and think about braves and the wave is moving to the right as it was right recurring follow the peaks of the way of bright points of the way of our withdrawal Amin red there also grew more with the speed of light so little quite says Norris as you know pop as as time goes forward as time goes forward the way we moved to the right answer right moving way of laying the In new or frame of reference is this distance over here the distance between successor Our Maxim work 3 week wavelength In my frame the wavelength and my frame is this Mr. bug this this parents as measured using she wrote using Lorentz
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transformation of our words this distance here is the distance between the 2 lack at what you call an instant of foreign terror wave waved to me is the distance between the 2 maxima what I call them instant the time saw if you work out the Lorentz transformations and you look along here and ask What's the distance between these 2 points in your own mind reference Barry and compare with the distance along here you'll find the different that would give you the publisher you have the tools to do that you could do at you know Lorentz transformations you can figure out what the quarter if let's say the waiver wealthiest some way some particular wavelength you could figure out what the point is you could figure out what the point is and you could ask What's c X crying distance between X Prime is my coordinate axes your the X distance between news Islam and you could figure out what the X Prize distances strode forward to calculate busy publisher so I recommend as an exercise and arm and working with Lawrence has relations young off last yes courses next fall yet level is not sure of the
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plant gave us a lot quicker broke assure Jr for drinks was possibly Jarreau committee possibly what quantum mechanics normally College of quantum mechanics was reasonably deep but not very Braun having not gotten through a harmonic oscillator that's a bad thing so I think before we study quantum field theory we do have a study of more or a karaoke bar there are natural which way to go next quarter ahead all us now it is dead Of back in red all Wall who met with the general refers to gravitational issue we studied gravity that's an issue combination of quantum mechanics and gravity you may know enough about corner character was poor amount of program here the sale the last where do you mean the goes on August and normal just over the bar for your half to do Lieurance French version right column right Prairie do it let's make this wavelength warned just a simplified so this 0 Miss 1 was to store for right now are like OK with disappointed could be important for me to locate where that point pointers this point is X Prize equal 0 that's same Prada's Richards repair feet equals V. X still firm that the surface a prime cause auroral is a same he equaled the extra wife our lines transformation halls keep crime is equal to T minus V X over square 1 I might as was with feet prime equals you're always a prime equals 0 that's what I call on the incident happened to be the instantly practical 0 and that's clearly teen equals V. X. This is equally X 1 over here for 1st or was a story up I XA JET moose white Siegel's warned that checks equals was a former broker Mark T.J I actually most people's right a or at the intersection of X equals people swarmed and vehicles VX Forbes work at our 5 pursue what that point this X equals teachers which is the X plus warned X 101 miners equals warned or X Is won all of 1 mind 3 firstly XO pouring nor about the sea of this scoring well we know the acts of this spawned or say free of EXO this point we can read off saying is equal tour V although 1 Marguez right now we would like to find out what about the simplest thing to do now is the figure out we know TV and we know X what I'd really like to know all I really like the point here I'd like to know X prime our would like to know the distance between successive Nunavut in my friend prime frame so we gotta Brian frames X Prize was is equal to X Miners VTE or square worn a square now I just plugin for X and they were our fear guru over this year's ex prime equals X 104 Missouri minus V crimes feed sorrows NYTimes these squared or want miners very all divided by by square root of 1 mind as a square right we do it right example OK sought to now where 1 might be square water minors square provided by 1 minus V and also in the denominator Syria's square root of 1 square 1 might be watch our oneliners square is was 1 might firms warning possibly right it's 1 minus squared as 1 might be warned caused 3 of these canceled Our wrote what else can I do with well I can put this in the square root just by
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squaring it and then putting it in the square root of all be calm square root of 1 Plus NYTimes 1 plus 3 toward because we squared provided by 1 minus square which is 1 plus fires warm either city Saleh the answer is square root of 1 class we over warned minus that's what X prime is over year is a bigger more Small World War I think 1 is warm Saul Meyer reckoning of this this gives me something bigger them warm said the wavelength was 1 I say the wavelength the square root of 1 plus 301 minors OK so wraps the arrests a calculation the high cost of giant you question warrant but OK so it's all there In these pictures just draw a picture of a thing you trying to figure out and then converted into some simple or owned some simple equation wrecked so for which purpose are watch from there as of course is the relativistic publisher formula square warned real woman's we end the really be correct about it misses the overseas you are not collector was correct but put units which is the same errors square root of steep losses overseeing minors where they happened to be going the other direction if I was going in the other direction you adjust team assigned velocity from plus 3 combined the U.S. just going actually my Missouri and you would get square root of seeing might booked C Minor overseas Class B and now this is less than 1 so a few shooting against the direction of use you you see a shorter wavelength if you're moving along a light ready to walk away and what happens if your velocity gets up near the speed of light or this 1 over here at your lawsuits up near the speed of light than the wavelength get bigger and bigger and bigger and eventually if you reward or library ifyou caught the wave wavelength would look in the with young shots fired was we got like Our bread prices are no further questions we will start tonight's lecture threw its weight feel the particles now require mechanics fields of particles of the same thing but would not go Inquirer Kerik's ferocity a field of fields from particles of particles I want saying that the plane will meet because they will meet it will meet tonight marked are Rivera not the same thing in particular were Why has questioned if a field effects upon particle a certain way for example by creating forces on it accelerated riddled must not also be the case let particle effects of field or while I say that while they have to be the case why if a effect be Busby necessarily have to effect a and that's built into the idea of action built into the action principle of the Lagrangian principle we see it deliveries in the simple context but it's also called action and reaction our basic idea of a effects B B will affect soda a simple example it but it is suppose that you had to quarter any number of corporate less just suppose for a moment told court an action principle so let's call according Texan white nor grungy is going to depend on X and Y annexed white XMY do not have the stand firmly orthogonal coordinates Our Cartesian coordinates of anything we're just took not Our Engel grungy and depends on more trees now 1 possibility is that the Lagrangian is simply of some of the terms will grungy and for XNX not plus a Lagrangian for YMY got a different around you can't just continue court payroll might be the same might be different if this is the case and you look at the equation of motion for acts of fraud there are very bad equation emotion will involve by IDT of partial about respect backstop vehicle departure will respect X and Y won't get until altogether likewise if you write the equation of motion for Y X doesn't give a direct answer on this case X doesn't affect wine 1 was reflects what do you say if you know that why effect X that tells you go a grungy and must be more complicated and there must be be in it which somehow affected both X NYT for example it all around are X . square plus Y dark square over tools completely independent could have a potential energy for X miners maybe another potential energy primed Hawaii so far as say some of 2 terms an extra a white at 1 item affect each other what you need to do if you want to see the effect each other put something in additional into a grungy which which involves both X and Y in a way that you can't travel for example just expletive if Exxon's y the equation of motion Fed Ex were white equation emotion for wine will involve acts if you'll have them both in Lagrangian Norway which is coupled together in this way there's no way that it's possible that 1 can affect the other without the effect of a affecting awards such as simple as act that is why if a effects be be effect OK now wants company issue which talk about particles and fields we talk a little bit less turned abound How Warfield might affect the motion of a particle that was just a simple model but we will go back to it but go back With talk about how the field effects of particle motion and then come back in his now the particle the presence of a particle effects of field this is very much the analog of electric and magnetic fields affecting the motion of charged particles and charged particles creating and modifying the electromagnetic field for example just by having a charged particle critical Longfield so for and those are not too would dependent things they from the scene will grant here so let's all work out this example right so let's begin with what's supposed 1st of all the some feel fog by depends on Exon for the moment we forget about How will it got the however have a value of had somebody also equations for 5 and found out that fire and he has a or measured Pyrex and he had some specific born
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Violet my way he might not be waived Harlow said effect emotional particle I would not get a test of the dynamics of fine now we're going to ask for those grungy and for the particle misses particle Lagrangian sorts remember From where is begin or if begin whether contains minors and by the square root of warn Martinez's X stock squared what was that this was a terminal a grand June correspondent joint action which just ended up old proper times all we moved segments of pack multiply by my an army in the role of us will grungy our integral to seal off the action integral to this and this is the same as mine an transient the will of the proper times 1 and the trajectory toward backtoback RFC or Nehru around you for particle but that doesn't have anything in it in which the field affects the other particles let's put something for the hero a simple thing the simplest and think of symbols and I can't think of all this a little bit and broke plus foggy not fires as a function of position and time in general a function of X and bar allegedly for the moment normal functioning of the coordinates of the park broke find the same thing again 1 minus square that's a log on June 4 particle I could work out or the Lagrange equations and find out how far effects emotional particle I could do it but I'm gonna do it In this simplified approximation with the particle moves very slowly just as illustration I don't relativistic square root of 1 might be square and so forth this of course is square and we could be thinking about particle we don't need widens EU those just Exs enough to restrict the point article moving in onedimensional when they put back the speed of light it's useful to put back the speed of light in order to see what the nonrelativistic limit is the nonrelativistic limit is basically the limit in which the speed of light goes to infinity or speed of light is bigger than any other velocity the problems of the nonrelativistic limit are the Newtonian limit Granzien has units of energy does not have units of energy on lesser speed 1 but the speed of light is not 1 we put my yeah M C squared now units of energy find going to leave that worry field is defined all let's talk constantly at astrology in here this constant is a constant called or coupling constant measures the strength by which they field affects the motion of the particles forwards Fujian here he could me anything for the moment we don't know what it is and might have receded my privacy and important at the moment which is G plans farright fight plans are excellent now let's fix our and there are of course the square root of 1 minus X . squared overseas square vessel a graduate of all the speed of light enquiries and then let's go the nonrelativistic limit the nonrelativistic limit when gets very large we expand the square root will expand the square root using the binomial expansion 1 2 minus a small number of mightiest onehalf power were factors becomes 1st thing becomes warned minors X squared Over twice she squares twice columns 1 half columns because this is something of a 1 have power using that 1 plus Solana warned minus steps or this case however 1 half is just 1 minus 1 half epsilon plus things which are really very much smaller because they will involve higher powers of the race TOO velocity velocity right so this is what we get plus tiny tiny things such racists system right warned that stuff square over to see square no best how we tried flying the biggest pieces the pieces which are almost important when the speed of light gets a la the Army's 1st pieces and square MC Square Warner but and she squared was the number it doesn't do anything when you start different it's a Lagrangian with talking about the or differentiating it will respect the position of Australia doesn't do anything just number so that is of no interest bidding constant grungy and doesn't have any effect on the equation of motion so they see square times 1 is of no interest the next germs has the speed of light canceling altogether so it's part of the limit in which the speed of light goes to infinity and survive survive that limit so the next turn is the product of NEC squared punch extort square to see familiar that X stop squared and over to the squares of canceled and that's just good old 1 here FMV square the nonrelativistic kinetic energy I want about Mr. G. flight times 1 that has no speeds of white so it clearly survived so we're minuses a minus sign here miners GE fly are X and that's all is when the park was moving slowly emerges compared wrapped With you know the oldfashioned nonrelativistic Labonte years no fashion 1 motorist circle around June was Connecticut Energy minus potential energy this is clearly kinetic energy and that identifies forest G. Kimes far as the potential energy of the particle potential energy of a particle in a field is a constant butt In for example electromagnetism and the strength of the coupling on a field of particles just electric charge the bigger electric charge the bigger before summer particle in a given field so it's natural that maybe some constant multiplying something about the field will come backers but July In this among wrote only reason I do the nonrelativistic limit is so we after Mark around always square roots of a former squared just illustrate a point this is effect this year represents the effect of the field on particle OK now about the effect of a particle airfield important thing to understand is doctors only 1 fracture total actually altogether and Republic actions and close action from the field and the actions of particle physics problem with trying to do could be if we wish we should drop picture was trying to do where a regional space in
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car and prime space and we have a field in it right of a few euros a bunch of red Wash there's the field 1st of all what scientists are now have a field day that involves knowing although grungy and for the field and minimizing liberal Lagrangian for a field and in addition the mechanical system of the system or studying also happens particle moving freely store were trying to do a study of combined system of field particle moving through the field were trying to to minimize the action not just with respect to the motion of a particle with respect to the motion of a particle and promotion of a feel for how a few behaves in for that very simple we simply explore around changing the field letting it will in different ways change the particle trajectory until we minimize the action and that will give us trajectory field correspond to a principal with faction OK so let's go and write a whole actions including the field and particle will 1st of all we need an action for field we discussed his last time is going to write down what we wrote where starring are a simple case of it we're grows in the field we rode Elsa better since the Lagrangian from a field that would erupt down Western or are well the action the action itself before action integral but D 4 X before acts means and the X the also Foreign Region X Y and Z and he said we wrote our Western was the integral of onehalf partial afire with respect to teach squared Martinez 1 have partial fired with respect X squared fish was freed white for the moment don't care about them anymore I just want to keep track of them in this little exercise over here but we want to keep track of the speed of light think it would be a oversee squared here this is a you feel that it doesn't bother particle but now I have all reaction was reaction Ferraro from the euro for the field mallets also interact reaction particle theory is right over here I'm going to take a very special simple case you could work out a more complicated case but I'm going to work out a case where the particle is at rest but put the particle arrests were starlets assume for a moment that there is a solution to the equations with the particles just rest the situations when electron might be addressed after and asked what the additional termed the actions we could make electron move it doesn't do we will but there are about 1st simplicity let's just take electron arrest not moving faster trajectory are more the action and that case it right over here X 5 0 which is 1 square by itself was no interest and so little of bungee and for a particle or contributions programs from this piece over here is just the integral of foggy fire exit goal or where what kind of integral persimmon dual right With particles our fields and will and integral along the world line article but it involves 5 balls the reason the particles responded to the field was because the Lagrangian for the particle involves finding the reason the field response of a particle was because that's a Lagrangian also has fine and the positions of the particles so while how how cold air this turn here I wanna it in a way that corresponds to an integral over space and part How could I turned this the willing to roll over space and time role sorry for bags that go back respect West put particle for convenience and simplicity just put it at x equals 0 it's particle position x equals 0 tomorrow it put in here what we put in the position of the particle effects which is 0 5 0 and in the integral now this term here is just an integral over the are in this town here is an integral all space time and factor where I wrote it spaces only onedimensional so this will really be only a twodimensional integral but OK can I write there's so that it looks like an integral overall space time just put in the 1 terms for the is simple the answer is you used the Dirac delta function the use of Dirac delta function all you do it so I wanna get cake 5 0 the time was not port here and they right In the following wary an integral call space of fire prices a Dirac delta function the proindependence Islam port here just girls along for a ride to recognize this When you integrate the Dirac filter function Dirac delta function is a very high spike of a function at a high spike of a function of 0 everywhere except at x equals 0 so when you integrate the field with the delta function you just get the field at the origin and you feel that the origin or the field at the place where X will follow in the delta function has expired a Cinderella volume if this was a onedimensional axis who just right DX works take a case of our workers are a review our guest I really need to tell you what built a fortune really means what I've written as just of X feel more precise score was dealt a freer vexed 2 3 X Delta free of X means the product of 3 delta function of X novo Delta where is functioning nonzero it's not uh 0 when the ball all 3 of these delta functions on on 0 the only 1 place for all 3 of them 0 x equals 0 1 0 0 equals so what is it is a function of free coordinate stop exwife Missouri a function of a freak space but it only has a value at 1 point of space and at that point the space it's enormously big where you integrate the field was
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right about refresh our way detail integral fire Our exwife Missouri I won't bother writing your car independence Delta of X over a white grows it XTY busy bus robbery here at a meeting of the symbol here tribal belt function and integrating all wall space I had either the role was not very hot integral to do remember when any integrate a function with the delta function it gives you the value of a function at the origin or a place with Oprah function is big but has spiked with just using the equation and Demirel F other X belt of X eagles at 4 0 0 0 defining property of the dope function picks out for you the value of a function at the place will don't perform has a spike Our acts of I do these integral over extra 1st this 1 tells me cassette X equal to 0 visited meters that White equals 0 listened to local news of 0 so the whole thing is a fires 0 so I've written here right but Of course not less I also in order the calculate the action I also have to the integrated Over time we are actually was integral on safari over time so the actions we have the integral all coordinates due for X means BXQ argues EDT until felt killed the wrecks was Dr. killed the X is doing nothing but picking out the value of fiery at the spatial position of the particle board doing truck just a right this thing here it's a trick full writing the value of a field of a point as an integral 1 0 do it well I just wanna do it cause I would like it here plus the list something Mr. Judy very likely because of this yacht Ma yes GE Delta killed other ex wife of X and now the whole thing is just 1 big integral or all space and all current areas and this thing beside the big square bracket is is thing inside the big square this the action equals the action now an integral or space and time were getting good field Lagrangian good although field actually and this turn we hear represents the effect of a particle back on the field is just firm over here for the special situation where the particle happens to be arrest we could do a jazz laughter of particle moves as fire we could do that by making a delta function move around with time possible to do but that's not important right now OK now let's go take a on June and work out the equations of motion for inside the bracket is low Grosjean work out the equations of motion 1st of all we have to do it 0 Grosjean with respect through partial afire with respect to tea IDT that pressure state or by defied go ID to justify right so was what really give us a 2nd derivative of fiery with respect he squares some like an acceleration Ave a kinetic energy fight about squared we work out the equation emotion we have something that looks like an acceleration of party now are other terms coming from defined X squared exactly the same sort of racist just go for it and might these 2nd fiery by BX square there if that's although I was a research that equaled and this would be a good oldfashioned wave equation and warned them into case of rural they had more dimensions we where miners 2nd followed by a BY squared minus 2nd fire you nor was set that equal to 0 that would be the wave equation describing waves Of this far but now we have more fine dependency below Grosjean depends on in another way what happens if we differentiate the Lagrangian will respect the fiery was also put Iran right here are right side is now 0 was supposed to put the derivative of Lagrangian with respect the fight that's the other side of ranchers equations of the side of orange equations currency from here with the Renault the L. respect the fight exit is just a GE times the delta function which just G and Dokic you Chad Our equation is just this is equal to g Prime's dealt Q other books you recognize this equation you may have seen before a minus sign was a minus sign yes was a modest this is not equation awaiting equation except on the righthand side at 1 point in only 1 point something different something going on at 1 point which is different from the rest of space what's going on is a foreign court it's affecting the equation of motion only 1 port let's just to try and make you little bit familiar what's supposed there were looking for solutions of the equation after a particle standing still I wish example of the particle standing still there might be a chance that as a solution of the equations with the field itself doesn't change with time just be effective standing still particle affecting the field is likely to make a field which also doesn't vary with tar and so we might look for a solution equation in which 5 High Noon depending news not here then again chase started of everything costs plus plus remote if you recognizes equation passed on accord was passed on equation for example for the electrostatic potential energy electrostatic potential was sometimes rewritten as Dell squared fine using corporal source on the right hand side charge density and this case a charge density is just the delta function of rural high a sharp spike so you see no not really doing electrodynamics at the moment we're doing no sort of
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simplification involving the a scalar field Mr. electromagnetic fields but to see how works the same turned or grungy which tells the particle To move as if there was a potential energy G flight which exert forces on a particle also exchange terms where you was for the equation emotional find fueled cars or I feel has a source these are not independent thing the fact that the field affects the particle Carlos particle effects of field and with a specific simple example it even tells us exactly how the same parameter GE the parameter gene determines how strongly effect particle on the field is also tells us how strong the effect of the field on a particle so we have a nice story their fields of particles affect each other commenter Lagrangian or era How might we are right so let's just explore how how we might change is a bit less tobacco a onedimensional case just arrest 1 space dimensions make it 1 space and this just becomes the 2nd a river of fire reversible equations 2nd the river fire with respect to x squares Okojie times of sex simply suppose it I want to put the particle someplace else I wanna put at the I want to put a X equals or 18 what I do fear however changes X minus a doctor of X minus a note X minus it has it's spike when Exs equal pay now supposing the particle moving positions work AT of tea will you would prove his right don't X minors a year for a year what this would tell you it it is a source of the field the source of fuel was moving at any given time it's a position of Sicily can accommodate a moving particle we can accommodate more particle the only thing is if we did have the a particle moving we wouldn't expect the field to be kind depends if particle moving around obviously the field will depend on at 4 Source the particles moving around so we will have to put back we'll turn he minus 2nd fiber IDQ squares there's no way we could get a solution with kind independent fly ball with the right hand side does depend on time the only way to make it consistent is to put back the timedependent so Moly particle accelerator particle or migrating particle will car effective field give a time dependence horrible will radiate waves but were at the moment we're not going to solve wave equations woes somebody else would promote kind of just talking about the location of relativity rotation mean the mathematical rotation we used to make equations look simple and pretty and not have the right to every day and time equations 2nd derivatives part X squared sectors y square LeBlanc but have a need protection notation was partly in housekeeper partly but I young visitors classical and in that way by them it is what it 5 None of this crew convinced that is quite a while a bomb is what we really want to do Real monolog Mr. Blue Water keep of the overseas nor do we would not want to broker a number of Mr. Quayle but even nonrelativistic limit the right hand side this time so it will create few create ways by mushers accurate arrest if where road where if for tea now the tomb of formal request to their side of the equation to square yard form for a he yacht Japan's fires to be called G F well let's say emitting figure out the units how we knew you for your appears Pulitzer fire happened the units of energy and that means a unit of G R dimensionless numbers to depends on the dimensions face that's not always true Burke nurtured from all this knuckle port for 5 hours cars just reviewing for some of you will be reviewing for some we're learning Standard Notation for Roth of vectors for America's Taylor's and you make we talk about last time our back Amex galas ever had a mixed day was elected as directors of Skelos and how you condemned nor patients of the things we meet short of Austrian province but remember where next CEO of course stands for the 4 corners of space time and space time and which TXY owns 8 usually this is called 1 2 1 3 innings was called naught effects on portal mule runs over 4 notice now I'm going to pay pensions outerwear repetition next I've put the index up on top above but who need something for Strawberry the penumbra because think but this quantity of thought as a displacement from the origin is a former rector of 4 vectors a statement about the way it transformed transforms our transforms on orange transformation anything that transforms the same way on tourist cancelations ex ante is called for vector I want to write it down for a moment format there has 4 components same back 3 of the components of equal 0 for example that might be true ballads that would mean that it was a displacement that Our only involve free of accord fourthquarter but it's not an invariant statement observer 1 you only Barbara was displacement from object which was purely time white rulers move in space I will say it did move varies so the statement let sector has 3 components equal to 0 may be true in some frame of reference but not invariant statement that everybody will agree upon other stated that all 4 components of 0 that everybody will agree on a solid says it is about yeah it is 0 no displacement that all nor agreed last year over there was a 0 fact 0 transforms itself armor Lieurance transformation of all 4 corners 0 before a transformation the Zero after Seoul on that's motion of a format vector
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Germany for vectors differences 4 sectors of 4 sectors sold differential displacements of 4 sectors and so for our format that we can make a scalar of a scalar meaning thing which doesn't transform which is to say that friend I know we've done this before going over all material the West but Allah Ave displacement we can construct a proper time to Bacau squared million or about it is that he squared minus the X squared square feet square and that the property that it is the same in every reference frame it doesn't have components only has 1 component 1 value and it's a or course for vector scalar now the Belo is equal to 0 in 1 frame of reference it's 0 on every frame of reference y because the scalar was the same as a thing which is the same in every frame of reference for some frame of reference a scalar whatever practically definition of scalar which is every frame of reference 0 1 frame 0 Anthony framed sought variants statement to say scalar Our is equal to 0 in here we have an example of how you make a scalar for that's very general if you'll have a former rector of any kind scored a new say let of 4 vector means it transformed the same way under Lorentz transformations then the difference of the square of a kind components of space components are scalar Barber writing at the difference 8 times square mile space squared are a scalar is a scalar it's the same in every reference price 1 my sweet right that I don't wanna have a right AT Times Square Marseille square Mercer why square murders scored for Matt Williams may matrix the matrix name called the Matrix In special relativity is usually denoted a dash Peters Greek letter are with the Greek letter a tea a hillside bat here is a matrix is a Greek letter which varies from critics and the Matrix has tho indices new and but it's a very simple matrix it's practically the identity matrix but not quite it has 3 a elements which are equal the 1 on the diagonal society unit matrix for the 4th element was minus 1 of off element corresponds to Prime minus 1 0 0 0 0 0 plus 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 hour to make this plus warned it it would just be a good old unit matrix but is not now let's take this matrix we're going to represent vectors by columns column vector for example a new will consist of 4 entries time component and free space component X warrior Missouri must the Matrix and multiplied by the wreck created look by matrix multiplication all we can say with who are of course a new with Susan Newell he knew runs also from 1 before of multiply the matrix private sector we can represent back a them you know a rule some you that's 1 way to write it just this warning are just matrix multiplication but it can be represented by summing 801 warned Monday 1 8 0 1 tool a tool and so forth and this represents a nor object which will call 80 With a lower index here let's see what to a figure out what is what a indexes it's not Buergel if this really were the unit matrix and you multiply vector you would get back exactly the same thing but not quite reminder sworn here means that when you multiply aided by a what happens it changes the signing of a 1st plan Everything else so I can write down immediately without me the work last 82 for mule you would know it doesn't matter who represents the column vector of vector Martinez AT T A X K Y a hazy another words when I perform this operation I simply change the sign of the times component parts about 1 piece on dislocation this is killed a patient of ice is no real content the 2nd occasional the rice which no doubt you're also somewhat familiar with the Einstein summation convey the Einstein summation conventions when every you see a downstairs index and upstairs index was seen in this case no budget automatically the rule that you samples were you don't need to write summation every time Aurelius repeated the index as long as 1 of them is upstairs and 1 of them is downstairs don't Barber riding summation signed its imports and plus and the conventions of the notation Blake why of this operation here just has the effect of changing the sign of the times know now I should warn you depending on who was riding the equations this can sometimes be plus 1 minus 1 minus 1 minus 1 for minus 1 plus 1 plus 1 plus were different convention I happened like this 1 almost everybody else I know what the other 1 from West there wrote to study general Pavle like the convention are used our sold it's 1 of these conventions and never quite settled down and so for Republicans and Democrats Jews and Muslims they hate each other like all hell there are those who use 1 conventional also use of convention where were born a group of people or use 1 when I would be of a group of people or use other so far have been killed about over progress will use is 1 null 1st gaffes are duty his arrest location just gives you a year more blankets multiple crises multiplication really means multiply this by this process by this course the artists forces progress on this case it's just trivial or more generally Ada was not the Unity matrix was somebody else Guatemala matrix John Major at will be this time this close this time there's plus this this force turned back the other would be a summation we're but I suddenly got fed up riding summations mold realized every in every equation was writing Baron
1:05:49
whenever those an index 1 upstairs and 1 downstairs you've always something over cleverly said I would the Einstein summation ventures but but and so badly and next like no here doesn't actually have value dummy index thing you some this expression depends on you both on depend on no nor was a summation Texas summation something expression of depend on Mr. some you know going all right now but go 1 step further Morga but quantity a new AD your Assembly missing 17 the vector would be operating next year is called a kind of area next over operator Rex's called Contra next end after the removal or index is called a cold areas factory operates Contra variant Florida corners our use at terminology because there was fear which is which score upper and lower fees you remember a with upper index and a with or backs new goal between them with his matrix aid you could also go back by but we've got to figure out how to act the same way OK until stake 18 you'll this is wiser some Nova be caused by the artist convention repeated index 1 operand son go so let's see what this means our this is equal to 8 T 8 he plus 8 X a plus the same thing for the Kauai component Boris V. compartment now the space components of a coal variant and contrary and vectors are the same so this is just a X squared officers just a X squares and doesn't matter review put upstairs likewise the swimmers a wife squared no no not yet why doesn't have a minus sign acquires when you plant and the fact that a lower index to give you minus 80 this columns Martina AT T sweater mail this notation here says some literally some ominous by my assigned comes in for the simple reason that when you will always be index of called Loriinae index the operational Loriinae the index changes the sign of the time component a key opera times NET lower his minus 8 square mail so you don't have fact you you must put any extra minus signs in here all you have to do is remember what the meaning of a opera and a lower well this is exactly the quantity that we think is a scalar tapes different the square of the prime component of space components in other words there's quite a year except for minus sign starts however might decide you're an a plus sign here or whatever it is it's clearly a scalar clearly a scalar so we see an example of buildings they lose our vectors and how we do it we do it by soaking up all the index of called contracting industry contracting the industry when you see 2 indices which the same nocturnal your summer weather this quantity with Labrador index armed user a scalar also utterly is he grown so I'll be right Will 6 cases a quick word but our yard we're probably wonder doesn't he you're right I think of lovely purists we might but there are a we would bite this is where there's this is what components are misses what compartment OK missing is a scale it was set How do but short the yet until his her OK OK I'm like about who we are we get your site we could also write this another way we could write this since a law New West let's so it's changed summation next year of collecting knew I haven't changed expression just changed the dummy index but now works look of all Norton right March change want Ramallah where I want to do is use fraying fears Nuno 0 a new is equal Ada New no so I have no way to write some or all of your window with a matrix a there is the same thing while it stands for is AT crying squared minus a space square sorts stands for boats and waiter right it will we don't have to keep me writing X square y squares you square so so that the 1st notational device it easy does there is little the old guard when every Greek index Nuuanu refers goal for a comport with your refer just over 3 components usually black arms I injuries in Indian depending on was writing young right OK so this is a scalar now who now of what's what's considered summary of stick to 4 vectors take to for electors AT&T the now that looks like and might be a scalar it's got knowing this easily indices Osama question is a a scalar yesterday is a scalar must prove it all will really want things to prove it to promote 2 scalar or we really need are used the statement that the sum of the scalar different to scare was was also scalar you have 2 things you when I agree upon what they are because there were different frames what we agree we can and track will still agree about the values of the some different so some differences of scalar scares let's prove that this is a scary that's easy all you know is you write a plus they b mule primes a class we do not charge annual growth this is a scare a course 3 it is a former actor you take any for vector and contracted with itself it's a scalar so this is a scalar now subtract from theirs but with with but will buy back they already have returned a new we have reduced the AAA terms and BB firms on the same card all different as they may be sold you left although we do this with an aide to being we'll always
1:14:54
something prove this is the same thing a new being killed it doesn't Larry occurred but the ups downs and the bounds up you get the same thing the difference between these 2 things is just 4 times a new New song here's a scalar fears among us gala the difference between them is basically just 8 harms being a newbie so and I take any who wore a former actors and contracted in this way I get among scale slight generalization of just saying that 8 a scalar 84 vector with other for records also scale fertile source me he wanted more from all that would let Baja more like a thought can be back products would not products except for the 20 minutes are is still low rents the Minkowski version of the dark may be either facilitates the year of the change of sign of sorts for OK now let's take a scalar a scalar field striker scalar field phi of X X now since the accident a writer thing like this don't say otherwise it really means is a function of war for court and let's considers the difference I'll Fly to neighboring pouring so if you wanna ever find a scalar consider it different at 2 neighboring points and we agree on the values of fired each of those points will agree on the difference between them and so the allwhite if Europe Radio Theron but a which are won't prove I will tell you the statement if something you could easily approved by yourself is supposing he'll have supposing you have no aid for vector off vector means that transforms and set is not just as they prepare for compartments means when transformer Lorentz transformations it transforms in a certain way supposing you have a fortnight record No. 1 and No. 2 0 0 and another quantity being mural such that there will you multiply a new being you win some over new you get something which is a scalar another words you get something which doesn't depend on the frame of reference you know all this is a former rector and you don't know about that as possible to prove that a you funeral With this product is independent of reference frame the B is also for also transformed for vector will use that a moment it's obvious that is a former vector and 8 times B is a scalar there be is a 4 right now work OK now let's take the difference 5 at 2 neighboring point so if we agree about the values of 5 rule agree about the values of the different soil the different something which we will agree about will call for the moment the scalar I want is the difference between fine at 2 with neighboring pouring in from a test was separated the answer is you differentiate of 5 With respect to each of the coordinates multiplier at this means the river with respect to keep forced Rural Fire with respect X firms VX plaster afire with respect Hawaii and so forth this is a small change in in going from 1 pouring all scalar and so we will agree about the value of a difference of flying DEX itself was clearly of 4 vector it was very basic for which started with bomb small differences of small displacements this will format and the whole thing is a scalar was a value about this it tells you what a a derivative of 580 with respect the complex of 4 quantities stands for a fire with respect seat deliver fire with respect X and store for us be covered the issue is not where the vote now that the yes but far that that's that's what I was drawn right Rock In order for Mr. makes cents as promise as a year Kerry Proctor was tight this had better road defined by the X had better correspond port call variant that there is a derivative corresponds to a Contra variant so we find our is the reviews of fiery with respect extra complex of members them former fee sector but they form the cold variant components or for call various components like emu knowing or something else we know that derivatives of scalar With respect extramural Oracle variant vector and that sometimes just written as sub you'll find next meeting Lauren next year to indicate protocol variant very will now we could use an artificial device Cohen event being new fiery was being you fire with the upper indexes talk it's actually just exactly the same things here except the time component having the opposite sorry case or works right about the meaning of these symbols but they 205 status for complex of things he flying 5 IDT 75 X Y Missouri what about Newsom will be new fire With be index upstairs is the same set of things except my mystified by duty and idea Lesotho budget allocation votes keeping track of years squares of piping components minus squares of space comport for squares a space components where squares with particle for fraud doing 5 but need finally we can construct a new scalar phi was a scalar the derivatives of 5 former former vector but now we can form 1 scalar moose gala was composed Of these 2 for Brown or a before vector or is it we just take a new fine theme you fight From Lemieux was a stand for stairs something we wrote back earlier 5 IDP squared Miners plus defy by the X squared sorry before I buy the X squared plus a same thing for wines remote or it's a thing we wrote down or with minus signs from
1:23:59
it's the thing that we wrote own from around you for scalar field was telling us installing us the scalar field Lagrange is a scale the grungy itself not scale field skill feels a scalar because we settles a scalar music and transform when your warrants transformed the new exercise here Americanowned notation tells us this quantity here is also a scale why then God why it good verbal grungy and is a scalar and the reason they flee as here but reasonable past this is grind you pass is composed of 2nd derivatives is of squares of 1st around European this language Grosjean for the scalar field minus 1 have been your fire new fight that is just kind of those squared 1 half time squared minus space square we get service for Grosjean how do it now but we could other things those grand with the rule is will grant it should be be a scalar was the value of single a scale is says actions is the same every reference price if you have a scalar quantity which doesn't depend on frame of reference and new integrated or space hard and the result doesn't depend on which frame of reference the evaluated Sofia principle of physics is the principle of least action and the action doesn't depend on which frame it's written in there and be seen action principle will give you equations of motion and each frame which have the same form every reference number or equations which are are going transformations much of studying feel fury is really the construction of invariant Lagrangian construction of various kinds of fields skate lose vectors are kind of fields which we have discussed here cancers spinners constructing out of these objects will grant genes which escape White because you want to build equations which are saying every reference frame you want equations Bureau rent that's use that a user over all over again more follows the tricks of love index manipulation but then only actually come the right down real equations that we have to solve river course after you know what this means is that the means prime square mile OK that's so that's rotation for you yard the run of for Woodgate on buried if Hugh talk much of our lives Say F right we do there are far are we didn't talk about wrote to much but that a nonrelativistic version of wrote purity yes there is half the special theory of relativity entails 2 things it entails will rule over these laws of physics are the same in every reference and 2nd a loss was accessible white wars with the boss together White with same austere different are duly that statement about life and ordering the Tony and all Galileo physics articles and make physics does have a relatively associated with it's not the relativity of Lorentz transformations it's the relativity on the limits of raunch transformations when the speed of light goes to infinity let's talk about that we talk about but right there were arrests transformations and take X Prize musical acts GTE provided by square root of 1 mind square oversee square feet prime physical team minus XVE or square of 1 square overseas squared but not quite ready the virus but she square fear units right Mrs. Lee Of course we have to orders wife prior musical twice you primes they our winning we didn't speak much about us but we could take the limit of these transformational rules as the speed of light it's very very what happened well the square root of nominating you just becomes warning so is just X prime physical X miners Vt but from area that's just see a changeover quarantine burned before anybody ever heard of the special theory of relativity this is what they were written there were about this 1 over here again thinking Szeto infinity this becomes worn by now we have X crimes overseas squares when she goes to infinity was Costa 0 has just keep chronicles Solheim yes there is a variants of nonrelativistic Yukon mechanics it's called Galileo and variants and it requires the equations could be invariant under these replacement as replacement is trivial it just says that all observers all frames of reference share the same time simultaneous saying for everybody so that's easy and this is just the usual change of reference frame from 1 inertial frames for an hour or so when people say that the equations of physics are the same in every inertial reference frame it's a statement of wrote is a stigma or and fact will write down if equals and they are typically goes equations Oregon variant of Sweden discussed the payoff are Austin's the transformation about for over an half a world on Monday nite and are quite but you're your progress but it almost almost all talk about a primed to jails that 80 miners being a X square warmer and a square so 4th Baron a crime X she is of course a track coach really a your jaw fared now it is right that there are gay pride is the same as minus 18 a team here as a same as of my ears a here are a would operate next she hears a errors a with next year where
1:32:53
about 80 superscript X Vassar saying Tuesday at Loyola superscript text or kind components change so just go downstairs so of starting to write a think there's just a change to just jails Assad now I close Caesar what also said that CAA ladders drive off some that that were so why is there my I was last Tuesday for what it is a work of that have also would lose imaging and of our energy density energy and energy density of a from a gas theology of again and a gas airframe reference which gas is a box boxes arrest the energy density if you're arrest when you look at the energy density of certain images if your moving fast growth that about boxing energy density in boxes a lot larger so energy density is something which we won't agree on but said would be a better sailors on on physical 1 dead Harare is it is world then a draft that Rice say we agree about value I mean if we measure we read about value now we agree because when I still whichever we agree because remained measurement we compare right on so energy density is a good example Our Owosso them on her the euro components of a 4 velocity of radio system kind component of a formal lottery is not the same indifference referenced new questions osha OK if not sure who move on with this this is all very easy stuff it's very easy it's just not patient bike the until you learn to rubrum from area but phone waves 1st look at the wave equation was for the wave equation solver but saw the wave equation go let's imagine we have some wave field where it's satisfy our roads begin by writing down or grand Jim Wright family were ready written down Lagrangian but why a slight extension number here it is 5 IDT squared minus defied by India Fernanda arise about detail Pierce our also widened Zia might right down the wife of a common woes China's this we wish for a war half from the whole thing and when a won water from 1 more turn is also scalar it does not involve derivatives it's a function of fire and the simplest function of fire that I can't think of why we could just I that would be a simple functional price but slightly more complicated by square including 1 half arrived with minors a parameter which recall New squared fire square over to Charlotte North a ready go to work this is the field theory analog of the harmonic oscillator for was a field theory among the harmonic oscillator if we're talking about a harmonic oscillator and we called the coordinator of the oscillator hcfa he called Cohen the farright we will write down I got squared or tool for the kinetic energy and then for the potential energy minus a spring constant which I'm calling you squared minus New square by square of this would be the gold harmonic oscillator annealing differences in now has some space and hire us will granted let's take this work out the equations and Saleh find the solutions well as 1st wall derivative of hello with respect time derivative just 5 IDT early differentiate with the time we get 2nd a river of fire with respect square this of course is the acceleration terms RV acceleration in articles are very very young minus a 2nd Friday by DX square the same thing with white my same thing with and that has to be equal to the elbow by defined so radical true mightiest New squared fight when you differentiate New Square 5 square With a factor of 2 you just give new Square fight off we want everything on a website for everything on the left side plus new squared equals 0 doesn't simple equation has a name anybody the name of his equations Joe yes claimed were decorated was Brookline gaudery greater Cloyne gone eh classy Anglican I don't know if they were connected each remarks but have feeling Klein was a German Gordon was probably of great so it probably will not a collaborating case some combination another wrote this equation calm this was there and attend before strode go the right the equation for over a year of Quantum Chemical particle are very made the mistake of trying to be relativistic have not tried to be relativistic they were written down the Schroedinger equation and very very famous they wrote down the relativistic version of it and became much less famous famous among physicist from our figures outside of the world only 1 person you knew that it was acquiring Gordon equations are referring requires a somewhat similar but only has a 1st remove whose time has customized but the score the Kleinwood equation its connection with quite characters port right now or the most solid we hope there are lots of solutions zillions of solutions but were old Bill topped out of playing wave solution to this kind of equations or some of playing way so much right back playing where In exponential form it's useful when thinking about oscillating systems to pretend that the coordinators complex and just take the real part of it we do after a harmonic oscillator we also oscillating systems look for
1:41:50
solutions which we allow to be complex complex functions and later Picarille parts or your example your our cut solutions look for your look for solutions which 1st oscillate can remember warm off from writers here but this was it raises he reminded I major TV that's oscillation with time with frequency omega and no room for solutions which are so even space which means there needs to be IKX of some pride he really RIA idea KX Pires X plus KY times y cases he can't be what is entering numbers kayaks just 3 numbers recalled wave numbers for the wave vector archaic sphere warring easy and on was a former year the solution I will look for maybe there is no solution workers were Jack 5 our 1st of all we can rewrite this McCartney way rewrite it as he earlier on May opens you with X at come from well if I take if I think of Katie as a former rector with components my guess Omega my assuming thinks KXK wine is but Everything cover for components are very complex of things barring Omega KXK winding Kinsey grouped together was formed here call figure them for a moment that may not really be they are actually a year before think of them as the a former them what company X here is just King KUX prime component Omega primes T X component of Katie projects Sosa nicer location but will likely use of marriages Carita were discovered using form he West see what happens if we try solve this equation what is the 2nd derivative of fire with respect he squared well what happens when you differentiate eagerly RIO with respect to carve scored twice over multiply what you have here by minus Omega square so where's scoreless services misses are guests at 5 what happens if we differentiate twice restricted we William minus Omega squared terms flight what happens this if we differentiate with respect acts we get kayak square with respect the white twice KY squared so altogether would you mind this Omega square plus KX quoted gross KY square plus squared or furnished we still have cost and Square here stock rose plus and square but folly Fuqua 0 well it's very easy now defined the solution you simply say homemaker must equal the square root of kayak square plus KY square close Casey scored a and square and our words you just at work in a bracket here equal to 0 and you have a solution this is just tell you what the frequency years In terms of a wave number frequency square root of Kerik square pose KY square puts Casey squared blossoms square that's a solution of acquiring or equation there are 2 possible solutions a square work like this commit plus or minus little the equation where you put plus some may go minus on radio and cost for every KXK wine case there are 2 solutions with fossil my little girl for summer route of the square root are Bozo solutions of a corn Gordon liquid their of we're back with me many cartoons quantum a or misses a classical field version of an equation which describes a quantum mechanical particle with mayors an energy Omega momentum Katie Mr. each equal square of key square for some squatters Clark's constant Mary argue about the Solheim on use of these equations aha a simpler linear equations you solution at the end of the the day after you've constructed solution In this way you could take a real part of it and the real part of the deal was supposed to be real there was a real number you just real part folk any questions getting guys along narrow with talking about for a rooftop we are aware of the plan were but he is but he's on the the glass as a little larger I may merger if you look for a local this is pretty much the unique will run which would lead the linear equation a call will come back so why we're looking for linear equations for the moment this is just an example Martha at an example or new a set of bras by order of Minnesota next to the jury is at the you're out the Granzien with respect to a thief IDT best justified new different shape that with respect the card your same thing he said Porsche with respect now they remember remember what the rules are OK what's right back the auto arrives equation for a field Debye theory Ericsson where now right and reform IDX mu of partial of L with respect to defile a body DAX mule that means some Lemuel equals or minor um miners quash or row with respect of 5 musical 2 0 talk theme IDX Newell This is a kind of cold areas index of downstairs index is denominated here you have 5 IDX knew that makes upstairs acts but progress it is of the form of a lawyer index plans opera index Our there you could see that this equation has some advice relativistic structure operator seasonal or indices contracted Narrandera Vera everything is a scalar so this could
1:50:46
scalar particles Higgs most on the Big Board would go back out but a summary of what go back Lula are the your actions particle we wrote I mean the sort of course I made the but I know I was using some rules the rules that should be relativistically invariant and fairly simple so we generalize this thing Times Square Root over 1 might be squared with generalized put my and plus 5 I'll just have to get summing up something you Yukon right down to your figure anything else want you compare what but this is 1 thing you could write down the notice that this has an interesting property if for some reason or another the fire was not equal to 0 for some reason or another you had a region of space in which fine was not equal to 0 but was equal to sum fixed value for some somebody for whatever reason then sector this particle moved With a mast and plus twice for all practical purposes of fire is a constant it just shifts the of particle well it's even possible that you might have a massless particles With no turn at all and the entire match is due to war is for some field not being equal to 0 in some regions space for some reason was a environment which fires radical the 0 acknowledges shift it can create email particle and have want to begin work but the nature of the Higgs bullshot create maps that 18 roughly was on and because it could masses particles such as quarks and particles because it creates is the presence of those particles will where Ali well react back from the field and grief field particles create field fueled effect particles pigs balls on as a particular case Is it a lot so that it is very hard still got USA just means you're right Paris's from a real partner and America are what's right Steger solution so where was our solution I lost it well from before he I Omega T minus K . King KDK dialects of scholar might escape directs a REIT multipliers everything on OK sorrow this is Eagle Cole signed are all major team plus REA signed will make it T might scare both the the real and imaginary parts if something is equal to 0 we went through an exercise that all of this equal to 0 if all of this is equal to 0 it means both real and imaginary pottery your support the real and imaginary parts are solutions of the equations of construction like this gives you 2 solutions to real solution cosigner McGerty and sine cosine and sine could and and together you could get them together with real coefficients still solutions by Our no exponential Zoran trek for facilitating picking derivatives as the reason we use exponential not be course the field is imaginary or complex but because it just helps us to be algebra picking derivatives on but once we have a solution like the real and imaginary parts of both separately solutions that both plane waves of more waves and among the 90 degrees and if this is a way of them was the exercise such that at times equals 0 Sierra are the waivers because located here Of course signed the assigned is just come our way of displaced right back you go there together Yukos track them and this'll social I no questions here he usually a a war who once worked who we were we're aware who Irwin that come from us from give particles vast articles acquired vast where are your guests were how can I Camex when I can explain superconductor work a lot yet we have no electricity and magnetism of superconducting cases a situation where the electromagnetic field get some rest and get a mess From a similar thing for us but about this point normal 51 yet Soviet connected now but we have got tour for its would the answer but however it is In effect where a scalar field where the president of scalable started present so scalar field effects of folk art and gives it a book tour firmer over oil is a bad idea voters is bad and that is where possible that visual artists have less need kiss him is here is and as a whole where do we follow on Tuesday as said that as the both of which of were fibers stewardship shift nor the mayors of Eros but this does is thought those flat but track radii prodigy here day so look at right so different coupling constants could lead to different classes and do the strength the strength of the strength with which a particle couples the Higgs field will be turned death new choose former manager Bob Master but never respected functions as we is a shield reserve all both the past not all particles No no quirks would be Jimbo's arms but we woes arms would be best are expose on wouldn't be wearing a proponent of neutrons would be was for partisan neutron do not get removed from age electron neutrinos with electrons between protons and neutrons not quirks in the songs sent shares of our arms from further Higgs field away
1:59:41
1 day electrons become worse that would be a disaster for Adams because size of Adam was governed by the inverse of mass of the electron and the massive electoral were grow 0 the Adam would become infinitely begin that would not be good for you say Da protons and neutrons would be virtually unaffected very very weakly effect on March the art well I mean it is it is true that a world without a hint of would be a better world figures many many things will go wrong as a set up a waste of which would be the electron would become vessels would sold alone Adams mergers Her have Noah nor callers nor were they like Chara become completely on localized for more please
2:00:55
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Metadaten
Formale Metadaten
Titel  Special Relativity  Lecture 5 
Serientitel  Lecture Collection  Special Relativity 
Teil  5 
Anzahl der Teile  10 
Autor 
Susskind, Leonard

Lizenz 
CCNamensnennung 3.0 Deutschland: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. 
DOI  10.5446/15006 
Herausgeber  Stanford University 
Erscheinungsjahr  2012 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Physik 
Abstract  (May 7, 2012) Leonard Susskind answers a question regarding material covered in the previous lecture. Following his explanation he continues into the concepts of fields and particles as they exist in special relativity. In 1905, while only twentysix years old, Albert Einstein published "On the Electrodynamics of Moving Bodies" and effectively extended classical laws of relativity to all laws of physics, even electrodynamics. In this course, Professor Susskind takes a close look at the special theory of relativity and also at classical field theory. Concepts addressed here include spacetime and fourdimensional spacetime, electromagnetic fields and their application to Maxwell's equations. 