Special Relativity | Lecture 3


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Special Relativity | Lecture 3
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Susskind, Leonard
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(April 23, 2012) Leonard Susskind begins to discuss particle mechanics and the role that they play in the special theory of relativity. This includes how particles move, the idea of momentum, and some topics on energy. 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.
Theory of relativity Hot working Speed of light Evening Photography Day Vehicle Amplitude Klemmverbindung Mitsubishi A6M Zero Group delay and phase delay Roots-type supercharger Ship class Limiter Spare part Orbital period Negation Tool Hose coupling Water vapor Star Striking clock home office Mechanic Power (physics) FACTS (newspaper) Classical mechanics Series and parallel circuits Standard cell Cartridge (firearms) Fahrgeschwindigkeit Frequenzumrichter Electronic media Measurement Miner
Animal trapping April Effects unit Bestrahlungsstärke Hypothetisches Teilchen Cosmic distance ladder Bird vocalization Separation process Canadair CL-44 Limiter Book cover Transformer Water vapor Mechanic Device driver Power (physics) FACTS (newspaper) Focus (optics) Excited state Belt (mechanical) Cartridge (firearms) Fahrgeschwindigkeit Signal (electrical engineering) Mass Food storage Plane (tool) Book design Measurement Shock absorber Ruler Trajectory Hot working Speed of light Transmission line Schubvektorsteuerung Newton's laws of motion Year Crystal structure Cash register Interval (mathematics) Roots-type supercharger Automobile Halo (optical phenomenon) Alcohol proof Low-pressure area Spare part Steckverbinder Watercraft rowing Hose coupling Cylinder head Doping (semiconductor) Key (engineering) Electronic component Car seat Two-way radio White Classical mechanics Kopfstütze Focal length Spant Moving walkway Reciprocal lattice Week Last
Analog signal Bestrahlungsstärke Quality (business) Cosmic distance ladder Bird vocalization Turning Limiter Sunrise Transformer Tape recorder Magnetic moment Line segment Programmable Array Logic Reaction (physics) Combined cycle Display device Single (music) Power (physics) FACTS (newspaper) Bügeleisen Excited state Belt (mechanical) Fahrgeschwindigkeit Cartridge (firearms) Wear Mint-made errors Frequenzumrichter Food storage Rotation Book design Level staff Ruler Hot working Trajectory Speed of light Schubvektorsteuerung Year Emissionsvermögen Force Compound engine Field-effect transistor Roots-type supercharger Brickyard Automobile Steckverbinder Spare part Watercraft rowing Leistungsanpassung Propeller (marine) Transmission tower Fax Electronic component White Tuesday Ice Classical mechanics Elektronenschale Spant Game Hour Cab (locomotive) Powell, Kleinschmidt Inc. Remotely operated underwater vehicle Miner
Girl (band) Effects unit Submarine Bird vocalization Roll forming Limiter Solar energy Water vapor Magnetic moment Reaction (physics) Mechanic Ground station Power (physics) Cartridge (firearms) Fahrgeschwindigkeit Mass Food storage Book design Gas turbine Morning Holzfaserplatte Ruler Trajectory Hot working Speed of light Newton's laws of motion Differential (mechanical device) Year Summer (George Winston album) Force Crystal structure Interval (mathematics) Pen Weather front Roots-type supercharger Potentiometer Brickyard Caliper Steckverbinder Musical ensemble Spare part Pressure vessel Transmission tower Holabird & Root Inc. Kilogram Electronic component Stationery White Classical mechanics Kopfstütze Spant Metre Sizing Game Universe Miner Convertible Week Ground (electricity) Last
Girl (band) Weapon Saw Effects unit Watercraft Monorail Group delay and phase delay Bird vocalization Packaging and labeling Cardinal direction Tape recorder Magnetic moment Felt Initiator <Steuerungstechnik> Mechanic Ground station Electronics Vertical integration FACTS (newspaper) Specific weight Excited state Buckelschweißen Belt (mechanical) Wood Fahrgeschwindigkeit Cartridge (firearms) Mass Mint-made errors Measurement Noise figure Hot working Speed of light Bomb Schubvektorsteuerung Year Force Roots-type supercharger Ship class Spare part Leistungsanpassung Mail (armour) Doping (semiconductor) Particle displacement Electronic component Kilogram Stationery Data conversion White Kopfstütze Spant Fulling Finger protocol Game Bicycle Miner Week Ground (electricity) June
Weapon Hot working Speed of light Day Year Force Separation process Elektronenbeugung Meeting/Interview Fog Wednesday Spare part Wind farm Magnetic moment Water vapor Collision Mechanic Minute Data conversion Ground station Composite material Power (physics) AC power plugs and sockets FACTS (newspaper) Kopfstütze Yacht Spant Direct current Weight Fahrgeschwindigkeit Cartridge (firearms) Hour Mass Food storage Book design Measurement Ground (electricity)
Prozessleittechnik Scale (map) Aerodynamics Bird vocalization Turning Wavelength Roll forming Continuous track Cardinal direction Limiter Tool Rail profile Magnetic moment Energy conservation Mechanic Electronics FACTS (newspaper) Weight Direct current Fahrgeschwindigkeit Wallet Cartridge (firearms) Mass Frequenzumrichter Electric current Measurement Book design Orbit Speed of light Schubvektorsteuerung Spring scale Year Lunar phase Force Gemstone Apparent magnitude Roots-type supercharger Brickyard Automobile Steckverbinder Spare part Leistungsanpassung Wind farm Cylinder head Bus Stellar atmosphere Electronic component Neutrino Kilogram White Chemical substance Positron Kopfstütze Classical mechanics Fulling Spant Sizing Mechanic Potenzialausgleich Photonics Strangeness Hour Radioactive decay Radiant energy Miner Amplitude-shift keying
Weapon Girl (band) Saw Effects unit Rolling-element bearing Watercraft Sommerreifen Rope Bird vocalization Turning Continuous track Limiter Magnetic moment Striking clock Glättung <Elektrotechnik> Mechanic Electric power distribution Combined cycle Refractive index Hall effect Photon Power (physics) Belt (mechanical) Direct current Fahrgeschwindigkeit Cartridge (firearms) Narrow gauge railway TARGET2 Measurement Throttle Volkswagen Golf Boeing B-52 Stratofortress Trajectory Hot working Speed of light Spaceflight Day Year Measurement Force Apparent magnitude Brickyard Low-pressure area Spare part Musical ensemble Electric charge Hose coupling Mast (sailing) Eraser Two-way radio White Noah's Ark Positron Kopfstütze Fulling Spin (physics) Potenzialausgleich Photonics Magnetism Electric current Radioactive decay
Lalas Stecher University part
of this evening I want to get started on talking about particle mechanics and a special period relativity help articles Wolf notions of momentum energy group vehicles M C squared there are a Repsol affairs for next time or maybe even tonight will be In 1988 there move past free fish with stop talk about classical field theory are just do 1 thing on the blackboard that we will we currently in all and over half for various purposes mostly 1st purposes the showing that relativistic formulas that we write down have the limit of these nonrelativistic formulas that we may have written down that we did right down to quarters ago in classical mechanics we are going to want to show their work objects move slowly compared to the speed of light that the no Feingold relativistic theory are particle motion area particle mechanics reduces the old Triangle from Orchid throughout lured me that To make approximations which become better and better for slower and slower velocities and particularly they have built at approximately 9 0 U Aung most prominent do it just for completeness on blackboard we have this quantity which recurs Over want miners square square root of 1 square they really means square 1 might be squared overseas square if we measure the velocity in units of the speed of light 1 might be square all remember that he represents a fraction of the speed of light of the object which is more removal possibly be typically is very small strike down how we approximate for small velocities in particular how approximated told water velocity squared after this is just an example of the binomial theorem all right down once and for all we use a couple of times tonight varies from when I was at the binomial films says that 1 plus a small number Epsilon as a matter of what's more but our 0 1 plus more told he where he can be anything as a matter of fact be positive negative imaginary complex was doesn't matter that this could be expanded In powers of epsilon Taylor's series if you like Taylor series we just expanded powers of epsilon for Epsilon equals you're always just give my 1 and then the next time he kind epsilon Mecca work much progress this is something is a standard by binomial Ferrum Annex standard peak I can't be true he times P minus 1 of 2 factorial factorial isn't too and we just 2 times epsilon squared and the elite teams going like that all right the next warn people right all warns after homework assignments right I will be included in the program AP and it was about Yao party the show all 3 factorial killed himself what are you epsilon issues not so big day in this series will converge in the media saying it emerges it that each turned was significantly smaller than the previous is such a way that our that it's a good approximation you have to terminate series after an appropriate number of terms were appropriate depends on just how much research precision you want but the secular after VfD epsilon really very small is usually the case at the 1st hurdle was the most important of course it is your Turner which of course it is the most important of his Europe 10 was awfully boring our defined for various applications the Zero photo the 1 it is just so boring to be relevant to us the 1st order term here which is important for what's Terminator series of that point of course does break down the steps or was was small but that's so supposedly want to expand when I was a quit or be square where epsilon is now would we are using Epsilon equals more might best be squared roads 1 minor steps or class epsilon is the same as 1 might be square and the power he has cases equal one-half square root of work or quantities quantity to warn him power and service becomes 1 plus Epsilon which has worn star 1 plus P Epsilon which is 1 plus one-half of P and then epsilon might be squared be squared which is the same as warning miners square over Saul that's a pretty damn good approximation for small velocities in particular from nonrelativistic velocities Mosquero 1 might be squared is just 1 of my misery squared over before a lot of tools in classical physics come from if thought about was the limit of relativistic physics most of the Toulouse of some examples calm from here are ushers examples of regal
along with our the other example but we will leave will be 1 provided by the same thing in this case is the same as 1 of minors these quarter power minus ahead the only difference relative to we just did about is that he is my half that has the effect of changing the idea here so it just becomes 1 plus square over there through signs have conspired to cancel each other sign from half open the number meets Parliament and the might be square he give us 1 paucity square over parts that is just by way but are setting up rupees calculation in mathematics that were used various situation I know we're really interested in particles now not pineapple can be anything that holds itself together it doesn't have to be an elementary particle it could be a sign it could be a box of chocolates there could be a cookie it could be anything and when you think about the location of our radio whatever happens to be committed by what's it a case of No problems will we mean by the position of the object is generally center Matt Egyptian or we talk about the mechanics of motion of particles was thinking really about mechanics of the center of mass of an object in the case of a doughnut the center messes with there is nothing but that's OK we know where the center masses and motion according lose some of last year's governed for example by Newtonian mechanics of articles are made our momentum as mass times velocity doesn't matter whether it's of elementary particle aura a ago a cookie not its momentum as the focal mass times velocity it's comedic imagery there's 1 half empty square removed from his accepts the energy of motion doesn't have any other energy of the Obama it was as Obama have many other energy besides 1 FMV squared yet calories exactly as all the internal chemical energy and that sort of stuff that's inside the doughnut that the energy was not the case in our midst overall motion state emotions cast it's a constant would respect how fast the object was moving so you could say the energy of a moving down I was born here and be squared plus the internal energy and the internal energy doesn't depend on state of most OK so those are those are just some reminders about classical mechanics another reminder about classical mechanics used to go back and read the classical Mechanics book where you will find things like the principle of ways action which will use tonight grungy and canonical momenta and all that stuff and we're going to use them so very quickly review in your head narrow the previous quarter of classical mechanics be important idea about the principal allegedly stature was the the idea and a trajectory in space not just the trajectory space trajectory based on where if you plot it was a shock Our system it could stand X could stand for a one-dimensional coordinator could stand for a three-dimensional quarter and factor could even stand for warned that a large number of particles Rangers Coach based or coordinate space vertically Park kind and the trajectory of the system between 2 0 instincts closed because from 1 . courage call a worldwide if it's a particle of single-particle we call a World I caught the world of the particles this concept is just as good a concept the nonrelativistic physics as it is in relativistic physics but they that's certain primacy united in relativity be cause of the death of a connection between space and kind of space and time to morph into each of the Lorentz transformation we tend the focus more on the idea of space time these trajectory of a particle is time right there will you mean by car and let's define the notion of our time might begin with just 2 points do with any pair off point in space time and think of the separation of the coordinates the difference between the coordinates of the talk points difference of admits norms speaking about space end time coordinates we could call them dealt and Delta Air I discussed the cover for a couple of times already know some of the invariant proper kind between 2 points wrote back Delta squared minus felt X squared is equal to opera Powell square where town where Delta Tau itself the square root of this quantity is the proper time connecting rose points now this quantity can be positive was quantity which written Delta Tau square conclude be positive or negative is positive if separation between pouring is such that the In the row is larger and the spatial distance between them and our words interval was likes
frog if a special distance His last time and in that case Tau squared is positive and the interval between these 2 points it's called pine white is got more kind of space when this donor list kind right you could always find a low rents frame of reference Yukon always transformed to to frame of reference within 2 points are the position where hasn't moved in fact you simply gotta be a frame of reference where this line is at rest you're always exist structure of such a frame of reference sort of till points space time time separated relative to whichever you could always move in such a way that they warned of being at the same point in space but they have moved road which are OK as what case the other a case is where Delta X is bigger than open seat in that case be separation has more space and the violent crime all of us at him it said less than 45 degrees relative however grow viable incidentally should or I should say right now when I'm thinking about four-dimensional space-time recourse space one-quarter of part of what Delta X stands for is 3 components all vector dealt X and Delta X squared means the songs of squares rose 3 components square postal was corporate store was I suppose we could wear white won't be a fancy who will add role here indicate the the river water however the free vector Dr. X is bigger than Dr. T interval becomes space like space like in the rules ah resist ever more space than there are trying in this case you cannot find a reference frame which took points the position and space what you can do is you could find a reference frame in which they are the same time you can always find so that means that is not only in very insignificant for 1 in being later than or you could always find a frame of reference which 2 points are exactly the same time an even worse drawn so that point B could be an 80 . be appears to be later than . 80 but there exists a reference frame as you could work out trees you work are you provide a reference frame which point B earlier pouring so no very significance 4 points which separated by a space like in April was no invariant significance the which 1 of her 1st log 1 being later than the other for what being simultaneous misses the relativity of Siam opened OK particles move on Primark trajectories everywhere a long emotional particle if we break the trajectory of the little in doubles every move rupees survive trajectories crime as just a statement of particles can nothing could move fast speed of light rural seen how frustrating it can be to try to get something to move restrooms pure white you driver compiled up a whole bunch of velocities each of which is nine-tenths a speed of light on nonstationary observers shoots out something with mind and speed of light the movie thing out something else will velocity 9 cancer speed of light something else shoots out of 3rd mind Kansas like you never get anywhere Xenos paradox you just get closer and closer to the speed of light without ever getting there so it seems like a reasonable hypothesis but nothing ever goes pure white and that's a good thing because it would give very confusing if we said a particle goes from could be but we can make sense out of which 1 was earlier the starting point of a particle the end point of particle would get us were very confused with you also questions were refer kind travel and so forth but the physical factors nothing including a folk move fast foods right then and I would generalized David information information was carried by physical systems cars signals signals a signal information from 1 place warmer but also not property that was pure white we take that a posture with if a war or art wise we do some very paradoxical inconsistent Our situation right most of time in space like the notion of the world wine every system and worldwide the system is made up of small cars white intervals us focus on almost time like intervals the 3 draw over here here's what firm argued the imagine you had small because eventually who regard work very it's and more graphic we talked about we talk about relativistic for velocity where do we didn't do it we will do it very I'm sure the trajectory of particle moves from 1 point to another point for the separation of X but don't the X is a four-dimensional go exit consist of a belt and a doctor of the 3 Oliver coordinates which 1st called broker was extort barracks sometimes I will call these spatial Dr. sometimes call and when you see I just runs over space when you a you runs over all court this is a this is it a young elite of all along the trajectory here was former from component story then it defined a doctor How war between and points beautiful trap equal square root of war of property square back scored a set just I want twice every so often with the speed of light just word goes he has units of time excess units a space to make this thing has no end on surfaces called proper time Arts assumed has units of time this was OK as a senior officer left inside the this was was not OK it as units of Lane as well as units of time with the speed of light vibrancy square right elbow amount nearly every so often who once more over here are and then suffering because return oriented now have a concept of velocity your basic ordinary velocity is the river of X with respect to see whether notion of four-dimensional velocity is of 4 vector it transforms as before but I felt that Mueller transforms for Delta Howell is an invariant dozen transform every observer how the moving prescribes the same Delta power with different dealt the X mused to reciprocal being velocity ordinary loss ordinary last job score we were very vector here who ordinarily velocity it is its components are simply be X are by C for the limits no limit the dope Exs gets more it just becomes the limit on dope XI provided by the see that's a three-dimensional emotional velocity and of course this also has you as a condition for V R a musical to be ex vii by as 1 or several last week a more is valued relativistic notional velocity is instead of using a key down here refer the separation of
coordinates of a proper parliament cordon and servers beyond our framed dependent ordinary cars at gives rise to something called for loss it doesn't have freed components it has 4 components has you are Peter newsstands components and directly runs from 0 3 0 is the part called this is a quarter or the X mule by be Taiwan not tabled a foreign and they can't through was called before loss of four-dimensional analog of velocity 1st question where's the connection between the former last city and the ordinary lost so let's work that out not a relativistically of course there only 3 components of velocity that must mean something funny happened the 1 of the components will see where I would start with BX abide by EU at least a sequel to the exit of VXI right table what's right of errors these acts on by Howell our provide provided by reality a judge for a provided a multiply by Carroll but it's just for sorry our yet is correct fight RBX by D. Powell where that for the use of ice use of where I think you're concentrating on the space component of you find by cab I will come power a moment vector from Rafah right now barred works Croats conserve the are young before your hair let's consider exactly this quantity EU nor was you nor you is the 4th component of this relativistic velocity and while it Is it is team power exactly what rivers the EPA by half component of Xerox comport versions of X is just she he knew nor is just a key by detailed how be he is just the immersive salt were interested in figuring out what did how right he does figure out what they cover fears Carlos square 0 2 both warring the square root of p squared minus the export that but just rewriting restrained here with differential locations to square by her via Twitter this means all 3 components the sums of squares of all 3 components let's take out of the square root take it outside the square root of factor off staff and then we get in here 1 that might be x squared by a squad Ford war just DT firms a square root 1 might be square so where we know we know that How by D. C is equal the square root of 1 mind square as soon 1 squared it is for the square root of warmer industry squares a rate of change of prime relative ordinary part of these closer is small by comparison with the speed of light of its costs Azura this is close the 1 who was in the nonrelativistic limit these key by big powers just 1 no difference between Cairo and see if the object is moving with close to the speed of light and be a larger bucks you were war we are you off 5 or more years this equation over here of credit to Monee equations was working on next year's where was working 1st I'm little bit of a mess terrorists blackboard case 1st of all this is equal to be by detail was by pal become IDT Izquierdo might be square was baby tower over right so you're gone whatever it is is 1 divided by the square root of 1 mind as its FC which nonrelativistic we just goes war in the nonrelativistic limit will be is much smaller than the speed of light this is just 1 SOE never think about it we never think about it he by power because how is in tabloid uses 1 the other hand wrote eristical you worry about it for last year these it use of Iive probably but they have is again just which where Frick off probably few square former square provider by research has got resupply BXT Carroll County multiplied what is it surprised not proposes abroad that's correct that's correct Malloy can some will together was the meaning of a former Wassily turns of normal Ross 1st of all EU the kind component of a former loss 1 or the square root of 1 might be square what about the space component the use of are you right we off from here that's cyclical these Somali provided by the same square a former and square services a basic computation you would do defined for components of 4 forget if we're talking about the nonrelativistic limit will be is very close toward 0 the square was trivial and use of iron body between notions of Australia was saying as call so close to the speed of light books argues buyer is much bigger than these I got you survive much bigger than survive much more new supplies OK this is the notion of for Milosevic was an emotional for archery and everywhere as along the trajectory of a particle of particles characterized by a position you could it the time at which a characterizing it so let's ask for vector of positions and the also a former rector of velocity now a former actor of velocity does it really have for independent components BOI has 3 independent components Let's go through war let's take due note OK hearts binary quickly doesn't have for independent components there is a relationship between the 3 components of spatial velocities or from use of hearty and use of nor and just a relationship that use of Of course you square which happens to be the least squares all over square of war these square Over 1 miners if space compound squared these squared provided by former we scored a F I a the issue 0 yes We Can we care about halfway through its auto emission of
State Duma squared minus the thing here I've sort from minus signed here plus 1 all square for miners square I'm making is more complicated areas the answer is right here marks were just former injuries were boy for Syria's I want for my has square or America's risk word White store all 4 loss to use them and how fast the particles moving moving the square of the we square the difference of the squares off components cotton component of space component always equal 1 always equaled a warm and that's why they were not for independent components of the euro for Ross only 3 independent components of before related by a single relationship there's otherwise their independence on that part of was set on elsewhere 11 and yes the that it we are sure that there is an easier way idea I should badly but served but let's go through a very simple belt that have square is equal to doubt he squared minus dealt back square for their Mccomis over there let's just provide both sides of this by Delta house Qualia 1 is equal to Delta see squared all Dr. Carlos Queiroz mimes don't X square or Delta Powell square broke Halbur or propel that's you not 1st areas as you nor square and or is you were vector square so that's all around him complicated we say it's not true throwback America the of the world is a worldwide is characterized by a 4 Australia for elected only 1 redundancy and that's before components of your mouth completely and we want to know what law governs the motions website a free particle 1 with no forces are what the law governs motions Milano abstract From a quarter's worth of classical mechanics the principal factions are the principal waste actually don't think you'll see action is going ahead we want our laws of physics be the same in every reference phrase that means we should try to to cast them in terms of quantities which oversees every reference rate of return or principle that shows that and going from here to here the particle goes along a trajectory which minimizes or maximize and minimize the quantity quality action we are doing good business if we make sure that that action doesn't depend on which frame of reference evaluated here like the action to be something which is the same in every reference frame we not only once all that's 1 for reaction should be invariant and the other thing that we've learned from experience is the actions should be built up incrementally as a psalm although the trajectory of little infinite possible pieces for each hour segment of the trajectory that we could extract from classical mechanics actually being song or integral better yet and Interpol trajectory and we think of which is an integral should be a quantity which is the same in every reference for this really up only 1 thing voters in variants and roaring from a small from opposition Kuwaiti displays position and it is a proper Parham along with trajectory for-profit Torre from here here is a variant meaning to say that all observers around or velocity they're moving where it will not agree on the Delta Exs but they will agree on the bill House saw a good guess right yeah right guests for the accident is the somber will provide better when integral moments of all who don't the powers From what I trajectory toward although the powers back is the action not quite quite match and has to being minimize holding me 80 . 6 2 a classical mechanics reaction principle your whole initial and the final configuration fixed a new search for the trajectory that minimizes or makes a stationary reaction between 2 and points of the trajectory as you would go around the path until you find the path of least action the ax is composed of invariants everybody will agree our drive a given path minimizes the action so that's our row that's our principal and a mop sauce each we have a freedom and eventually where freedom we could multiply the actions by a number any number will make any difference after why not because the 1st thing stationary with respect to changes of the trajectory of our multiplied by pain or 7 0 minus 15 it will still be stationary grocer multiplied by a negative number I will have turn upon a maximum for minimum but I want change the fact the prep trajectory makes the action stationary yard that while Norway's beginning in the game which saw natural energy we deride will go that rule all before they will use although we will discuss why we didn't start with energy the answers it's not variant are it's not even very an ordinary younger from asserts the energy of a particle is not independent of the state motion so if you see a particles standing still and I see the particle moving described the kinetic energy to it so Stalin invariant concert to go idea to start our discussion of particle motion by talking about variants everybody agrees on that we make sure that our laws of physics are reversing every reference got up Tuesday it White well as ignorant 1st yes I good guess it's a good gesture simply because it's the same for a review of moving that's what proper time was proper kind was particular combination of faxes nobodies between 2 points that everybody will agree on the value of it is already thing which is invariant Lorentz transformation if we want our laws of physics to be invariant there would be a good idea based on quantities which all observers sale Our do have the same value for suffers amplify just say a really Watson how his following this is could entirely in analogous the following year supposedly I'm interested the shortest distance between 2 points on a blackboard I'll wear white for white do from interested the shortest distance between 2 points obviously a search for the shortest distance search for the curve which is the shortest distance between 2 points now my only find that we get different answers In different coordinate systems record just more backcourt system more not of classroom and the reason is this this and you could be in place is invariant with respect the rotations of quarter and particularly everything got this From 1 . 0 along the curve the be us some of a lot of little incremental distance and everybody agrees on incremental distances curve that we get will not depend on the quarter system that we used to evaluate soon concept the shortest distance between 2 points or we do then is we ate up you little you clearly in distances here northerly exactly the same kind of thing
is not all proper crimes along with trajectory sack the formulas are almost identical apart from sign right now we have a freedom we could multiply this action by a number says we arrive at half time a word this ls would further regrets will there is the reason why there should army all we will always choose the path square yard Our yup proper time is by very definition of positive quantities of prime rib along with Clark From the past the future growth Our we could use the negative wrote we could if we use the negative route everywhere it was simply changed sign of the action that's OK what would happen if you change the size of the and it would not change the equations of motion all instead of saying we were looking for the middle of the action we might to looking for a maximum of the actions remember to minimize the function or the maximize exactly the same equation you just say stationary so it doesn't matter whether you take the action and multiplied by the number including the possibility of a negative number it won't change EUR we trajectories if the rule is the trajectories should make stationary reaction stationery our our means it could be a minimum or maximum a also 0 that this is war chest change off for a while now as saying yeah yes searched for fire and I'm going do 1 more little sign 1 little changed from minus sign here Garza said multiplying by miners and changes nothing it's a convention complete convention but the convention was established not by Einstein was due in relativistic mechanics but long before I'm starring by people had no idea what the special theory of relativity waters but who had some prejudices about the way classical Waconia mechanics were sold their combat why this morning and there will combine it's for the purpose of matching on to nonrelativistic form it would not change anything if we didn't put it there but In order to match would nonrelativistic formulas will put my and there for a time OK so what's I think that the reaction and and now all we do is we will be in the form of trajectories were already been imagined be broken up at intervals and now we move that point man in space and time and look for new trajectory which minimizes or mixed minimize which minimizes the action and that will be the true trajectory as a principal Louis actually but before we do that lets convert destroy integral but converters by taking the limit in which they differential in which these fine they differences replaced differentials solve this will become the miners and Pansy integral these how along the trajectory in are explain exactly what this means more some of little infinitesimal has more power from 1 place where there's a reason why the integral it is written by this curly symbol here your grows symbol means Osama Delta tower just becomes D Karel from 1 end of the trajectory of the other from the initial and the other and now we can write down a Formula B. towel musical per square root of 1 of my best GTE squared might VX square and yet square meters VX where post square prestigious square that this is a funny thing in the role the deep however underneath us figure square root beer that never seemed to go like that we you have Werribee differential things around the square root but see effects it's easy to make it look more standard factor out a DTE factor that DTE but the and a outside that means under the square root we have to divide IDT square this just becomes 1 more about by BT that's go fashion velocity squared so this becomes 1 miners velocity squared or 1 X I got square XI not square means X squared plus Y squared was scored normal writing this summer father writing ex-wife just X about and not now means derivative with respect for ordinary kind of respect ordinary so by factoring out EDT I converted this something only wore recognizable there might have been before said if the growth of something that depends on the velocity remember about the ordinary principle of least factions your principal list actions for an number particle for example the action is an integral Over quantity called Iran's nor runs in the pens what positions and velocities What about the case where there are no forces but always depends on the velocity of the positions and he'll like that we have something like that here we have the action is minus and frenzy the integral of 1 minus a square velocity from AT this'll starting point scored 8 could be so how right now and now I can just take off and do all the things that we would do with Lagrangian right with foreground years while Grant was my best among which I say it was a convention where similar here times Square Root of 1 minus now all right expects where plus White got squared Lizzie dark square a it's not simple as usual Grosjean there were grounds for a we're Granzien that growing grunge right it is wrote down from Lagrangian the kinetic energy you know we made our often was energy he wrote down Lagrangian just 1 half and next up squared that's simple around here right all also wrote that was just an man X fat square bullet 2 1 here and be square where excess square does mean extra square was square proceed this is all about kilograms of written down this is grungy and that a relativistic version of a branch would have written down what's the connection between while the 1st connection between them most important for our purposes right now all before I say that was the the value of the sovereignty and the I have this Lagrangian is that encapsulates the motion of a free particle in a way but independent or Rosa seemed for every reference was that he be quashed quantity action independent of reference frame may keep stationery you get the same answer in every Friday so this is a translation of that through grungy language OK 1st let's ask what our work approximately White not approximately works like in the limit with velocities a much smaller warned 1 being the speed of light services 1 miners and square root of 1 might be square velocity squared that we go back to liberals he quipped binomial fear binomial fearing tells us the square of what might be square was approximately call was approximately equal minors yeah you can't use 1 in course these square so firstly wall there is a contribution
which is just minors Tim it doesn't depend on anything and position doesn't depend on velocity it just remember what does it number of Lagrangian does nothing and the number of water Lagrangian does nothing because the trajectory the action of the trajectory road comparing different trajectories constant PC doesn't depend on the trajectory was the Expos the velocity plays no role at all so sunporch the important piece collapsed around sorry Minor Threat 1 Murray square right so pot from the constant peace the peace with the Jewish center at the priesthood juice it is plus square or miners and plus can be square all over 2 priority is just grungy in or around this will weaken research that the laws of motion would arrive in this way will in no limit of small velocity be the same Goldberg last Lagrangian 4 so far OEM put in anything that resembles a potential energy force of or was simply working with a free particle so that's a good thing now next concept that don't work out the equations of motion I'll tell you right now the equations of motion of the brilliance of ability to simply make this particle move on straight line with constant velocity will see that brought the equations of motion not interesting things the structure the interesting thing is to identify the notions of momentum and energy FOR girl who are understand where the relativistic formulas From momentum and energy come from they're not quite the same as the nonrelativistic formulas but got momentum and energy Our appointing because 2nd serve the recount smaller particles in the universe orally solar particles of closed system they may be interacting with forces in this matter but the net momentum is conserved what was at a consequence of the board that consequence of translation of translational medicine translation variants of Korea on we're were actually even a special case of Lagrangian mechanics everything we set about Lagrangian mechanics will around jeans and calmly and conservation laws it all applies here it's just below grungy was a little bit strange has a square root of 1 mind of cripples white but other than that the rules of the game of the rules a Lagrangian Hamiltonian whatever we were we derive previously Wikinews but the 1st question them it is what is the meaning of momentum it will be considered it will conserved because this will grungy translation invariant only depends on the velocity was not depend on position its translation invariant moment will be considered as a consequence of everything we were in the 1st quarter OK are like identified for example P sub X X component of momentum so we go back we open up Bob book on classical mechanics only go back to the definition of momentum not the definition which began by saying only that it is equal and but the definition which came from the Lagrangian point of view and that was it have a better remember definition a momentum firms or on Riviera fingertips I was partially or should part of the Rio Lagrangian with respect to the component of velocity around you With respect to X . from Barbary doing classical mechanics in front of your arrives here it's over Kuwait arrived at this point thing to do between now and next week is words to review procedures for westward B sub x as partial road backstop same thing for a piece of white jeans or for so let's apply that were grant and from the square root of this thing here right so this is going to be present the mightiest and then there's a derivative up on this square room with respect x parte where you get when you take the derivative of square root beer wine or a square of 1 square wrote everything inside the square which happens to be 1 miners we square yes the calf thank you as a hair fair now we have to take the derivative of olive the baby inside the square root with respect X . some ideas from here was among later sorry lost and then the derivative on this thing with respect XI which is twice X . that would eat up Guillaume the tour of the Mahmoud there will put annexed Oshea or v sub x provided by the square root of 1 minus the the cargo velocity squares well we seen this before we knocked are here these Hawaii in this case piece of X provider by square one-liners squared is just use of X store get we get a piece of X piece of acts is nothing about the man's not timetable ordinary velocity but by the ordinary velocity divided by squared away when squared which is the use of the relativistic for velocity times of mass is equal to to work the x component of momentum likewise so why component and so forth and so on a white component piece of why a physical and use use wine and likewise was it and his arrest and the Mets OK let's talk about earlier convention our views use of Turner may have stressed national Mom was all convention and a new convention new convention was older than I am cloned is more arrests nobody I know those physics uses the term rest mass more rest mass is an anachronism Soren anachronism words it's a macron the all the place would be used as and arm undergraduate textbooks on the graduate textbooks cannot seem to get their around the idea that the idea about mess of a particle is a kid goes with particle which characterizes the particle and not emotion article made us on an electronic you look up an electronic you won't get something that is different the
electrons moving your stationary will simply get whatever it happens to be the rest of electrons kilograms of small number of ah so away the mess is used in the Modern Language is number that goes with but it is exactly what used to be called the rest were used to all the rest is Meryl just call us what used to be called Harris is what energy energy divided by the speed of light square or something we use the terms energy what the things which characterizes were moving object and energy at rest we just call not resist BOY factor just a conversion factor will come for quite a while when when when we talk about the mess article we mean or use the recall arrest man from now on I will never use the term rosmarus I will always use their mass meaning the energy arrest for I was a observed portend the stationery are normally don't think cromorne would arrest Maris Torre mentioned butter was good the white or yellow Warriner abroad about our so we're a bit confused OK the let's talk for a moment about 4 directors for vectors of things like dope built the but but but but but but but how so far they have a force in this case it was simply represent a little extra for vector of displacement guess you would call for rector of displacement of Isabel property and tracks Nossa Pollack told you don't that acts as opposed to go dealt the X was equal to 0 could that be In invariant character rosaceous this simple while others daughter acts were you will range from the changes right in particular they will pick up a little bit of the city so it cannot be a variant description of his record of vector would not dare chicle 0 would be a vertical Red Bone some other frame of reference a war movie past their past a stationary taxes the same vector would be tilted another through moving past at the beginning of the end point would be separated by some backs so the thing the belt that physical your all that might have some significance to a particular frame of reference but it can't possibly be the variant of distinction what about the statement said no acts end the Treaty of both equal to 0 all 4 of them not just matches a Delta Erickson Bovidae but Delta widened and opposition can that be is that in invariant state all components of the vector are equal to 0 for us for the original rhetoric with 0 well you think about the way things transformed adult X where transforms will pick up a little bit adult could be a little property will pick up a little bit of doubt the X but delta T in belt that's 0 and 1 reference frame then it will be 0 in every reference frame so all same let a vector of 4 record particular Izzy auroral the whole thing all components of it that's invariant state say some specific component or even a specific subset of components less than all of em is equal to 0 that is not invariant state the same is true of the 4 velocity before or after a year of components is it invariant statement say they use I it is equal to 0 I mean being forced Justice space components now it isn't because if you means the city use bicycle 0 is just that you're in a frame of reference particle happens to to be stationary Jean frame of reference use supply will not be 0 thank bomb so in general if you want to if you want to express invariants statements you want those statements to be about all components of 4 vector before wrecked means of thing white X or you you want to be a state which is true for all the components you want to be a full-fledged vector state full-fledged for Retrospect OK let's come to the concept of momentum conservation of momentum conservation that yet we so far identified through reef components of moment from only 3 components those we components of momentum happened to be proportional for further components Our for velocity it happened become a proportion of the components of loss but that's incidental from moment the important thing is that momentum is conserved if I have a collection of particles the initial momentum must be quarterfinal moment Melessa Victor statement that's a vector statement about verify read components Our momentum remarks label in this way in this show must equal the total proposal initial momentum must with problem among over from war Rico subtract just write a lot of momentum conservation says of this is 0 initial momentum Miners final moment musical 0 this itself is a vector equation Is it in invariant vector equations carriage drawn Frank isn't necessarily true and another not on West weakened make an equivalent assertion about some 4th component yes there was off for what component that can the momentum in the war for effect and then we say all 4 components are conserved then it becomes an invariant state and it becomes a very strongly and various statement if we can identify of 4th component and save 4th component it also could serve a 4th components such as 3 components together with a 4th component airport for vector well it's pretty darn clear what the 4th component of this thing has debate on what was the 4th component be there must be an times use era we have for 3 components if we didn't have a mail will have 3 components of the former last week and the 4th component of the 4th velocity would just be the thing which fills out the full force quest through Is there a significant taught and use of not wherever it is can we can court piece of nor are we even name 0 significance to it and do it this is something that we already recognized something with daughter To be conserved and of course the answer is yes there is a 4th thing which is conserved particle McCarrick energy so it's natural there's the question is this energy users 4th
component of 4 momentum nothing but we merger particle why we decide that I will decide what villages nobody got a clue about how the site would be will Hamiltonian right must only worry was too only systematic systematic procedure look at the Hamiltonian sold the next step is to figure out a Hamiltonian room erased some of the blackboard here and wrote a memo from me the court did must From the point of view of relativity all the work that we do in the same quarter was just a set sales for this kind of problem so it's gotta Hamiltonian I know all that not all of us remember what the Hamiltonian is Soros remind you the Hamiltonian is given terms of ground you again I don't expect you to remember it but do expect to be able to drop and their remind yourself where it came from worry this was an important quantity to conserve quantity of important part of our ah I will write down to its song look ordnance saw more quarter starting off Wassily associated with record a sometimes is upstairs from conflict with the downstairs pain From the moment and about what multiplied the are Ramon associated with the same direction proper that 1 upstairs just for variety remember that's finished well so might martyrs full around here but with but some of if you want a systematic way to think about mechanics wouldn't it make up stuff as as we go along we fall back R B basic principles of the carrots all around which Hamilton so for OK so now we can arrive at their America Lagrangian over here he was peace of mind piece of Im X so why Dr. provided by squares of one-liners square that was the same errors and use of Mardi Gras use that war discourages forward piece of ours that an order on June was my and square root of 1 mind is a square where that means VX square costly war square was square for graduate of everything we need now to calculate I'm or what was essentially energy OK so we have some additional vii X plans P survivor in acts of God square over I think square 1 might be square the 1st term here the XMI about square of war abide by from he ordered from he amassed over here now minus the Lagrand years from earlier saw grungy and his class and squalor of 1 mind looks like mess brews not visible 1st of all X abide Bob square as just total velocity squared square was why group of Edith Greswick or be square for the 1st time here is just right or some anymore it's just to the where all over scorer warmer and a square where it what area times Square Root Noriega Hull they had thing of the square of the numerator with the square of the army Robert we'll put everything the monorail so we have the multiply provide for square 1 might emerge as a square that gives numerator one-liners squares and square root of one-liners square where we are now a dozen or so fearsome anymore because we notice immediately and these squared over here cancels a square over here and offering him upon square root of 1 we square I at the Hamiltonian sort it the momentum a 4th component here which started P Norwalk which is and be recognized what they're orchestrated you a we sorry but parcel path from a possible yet but Moor not a normal girl force no this formula a Hamiltonian is extremely general all systems that we've ever studied this is the formula there was a general general formula for it any will grant during any action principle we derived the conservation of H and very very general grounds pending what you put the our depending on what you put In from gradually might be relativistic physics could be any number of things but I know a business a conservation of momentum b Collins conservation also for momentum conservation X moment more momentum zoo momentum and the conservation of energy before your wine which is figure out what this no concept of energy has to do with the old concert how different visit the energy that we already established woods but the wanna finger we have to be careful about the fact a speed of light arms they are convention was largely conventions if the energy defined in 1 way is concerned that won't make any difference its conservation if we in in order to make a match with units if we are multiplied by the speed of light or something conservation was conservation and we will probably have to do that OK let's look at immune off 1st game all over the square root of 1 mind square vessel calling the energy of the particle of course if we want to restore some what we fully restored young before we restore you let's use the armed but on your fear but use binomial theorem who approximate Asturias here's a binomial clearance for 1 provided by squared away my great because was a square Our Sosebee carnage area posture and be squared over To this so causes completely recognizable again it is just a ordinary Tony kinetic energy and no nor about energy this would've been went to a renowned for kinetic energy use with kinetic wrote down I have no idea who 1st berth for Rice's
energy broker summary 1 us this term here for 1st of all this equation is not dimensionally consistent not dimensionally consistent and annually and times of velocity squared From make it dimensionally consistent always have to do with you know with number ways make a dimensionally consistent but they give EU audits of energy with Vicki let's see energy give it units of energy this already has units of energy just the gold Connecticut Energy and forgive this term units of energy we have multiplied by some power were of the speed of light what power of the speed of light oil ready this thing is missing 2 power was of velocity compared this 1 with the big B and square Rudd of are both right this is the energy Georgia wishy comment Out of the basic principles of mechanics conservation quantities which were evaluated for a system that rest is just square for a system that rests its energy is related to its inertial Marist with may have led to square power we know that this is a national matter what what's inertial nets because it enters into the energy the same old way that entered into the old energy it enters into momentum at least 4 slow velocities in the same way that dinner from our centered in here so this must be the inertial mass Times Square used that's the origin of farm Einstein have another hour motorway think about he's EDS himself in a coalition of objects of various kinds given the EU would go but different he didn't know about Lewis's theorem he had work from the ground up service he talked about collisions of objects object from together break up scattered or flying off work I know that the moment was what so I have to think of this energy in order that the energy and momentum together would formal complex that would be conserved in any frame of reference for this short-circuited and found a formula for the energy Matisse Standard Methods OK so that's the origin of equals M C squared it doesn't apply only to a particle it applies to any object or are to be identified as a close objects that its energy when it's up of the object is arrest nor was the object being arrest me inside here there are lots of molecules moving around like crazy so in that sense the subject is not arrest on the other hand center of mass is arrests of several arrest will we mean what we mean by young by velocity views the velocity of the center of mass I was left over when used at the center of mass velocity 0 His arrest energy the rest energy is related to the mayor's by factors will say that is the speed of light is just a conversion factor from 1 way over the foggy and were not as just a commercial above all also that is what I it is above all that sort of all I know formal has reduced this Wednesday is 1 right good point good for tried on purely dimensional grounds we never could deduce that this would be 1 here but on the grounds that has reduced miss Wednesday's eagled 1 only 1 thing we could do it Le want old absolutely and there are errors but right yes there are higher returns and they have simply very very small velocity well I leave for some purposes room I care about their brought Berger Leedy peace force which we can evaluate them see how small a offer a baseball moving at 90 miles an hour and believe me it would be really really small as that Russia a corrected their corrections everything everything gets corrections those corrections Harold quite significant when they object is moving closer was pure white the fact that we may become he'll just think about 4 minutes arm if I worked a plug in the speed of light heroes forget pretty darned big but would merely be given there's further refine right on the other hand plug it into this formula ghost awarded we Connecticut Energy of object moving with close to the speed of light becomes interfere with another way to see that you can never get up because would take In the ensuing energy to get the speed of light also incident momentum you was also are Nevada by square in 1 nite squared store has you push the particle faster and faster its momentum and its energy increased explode February become infinite when the velocity becomes pure white that means it would take in force to accelerate a particle speed of worry and use of force was part of nature I was on water for it said show for preservation right conservation for to Dracula book a yacht is more this is a war if he a war How for a while yes welcome back are less us a statement that momentum and energy is determined in terms of momentum that uh after all a lot of time he said Of the total idiots yes of course on if you wanted the mass of the combined system by that I mean the energy of a combined system when the combined system arrest was a mean the combined system arrest exactly what you said objects might be moving in opposite directions and the energy is definitely want to remove some of the best day of death is rest right we we have to be careful about definition something's a purely definitions some people might react the point well I mean by the rest of is simply the sums of arrest of 2 particles that that is not very good and very concept or another person I know I need what I mean is Think of the things we unit and discussed its energy
in the frame of reference for total momentum is equal to 0 Northbrook call the center mass frame or the rest frame of the combined system now since all ordinary systems are made up of particles of removing hidden the role the greatest are a invariant concept is the energy of the system in a reference frame where its momentum 0 if we call that the Mets then the mess of a composite system in Europe case in your particular case would be greater than the sum of the year of the 2 mayors if your forces particularly
attractive forces might be less than a summary of what was as nuclear face there is a need to Italian squares grants square from so health warnings rest but expression was slowly there is a lot supple abstract quarters but have worked been leaders list where Paris of steps is a time of maybe it's not the thing conserve is a problem not the individual ministers apart so let's go let's talk about example of leader before a terrible example followed me 1st pick a little bit aggression in the massless particles massless particles are a little bit strange use the energy of a museum here and riven by 1 might be squared now what's the velocity of a mess with all say that they were dropped well they had troubles not too bad the numerator of 0 and the denominator 0 it doesn't tell us what the answer is but at least it tells us at worst 0 0 that's better than 1 over 0 maybe lot votes a little over 1 0 0 it doesn't seem like such a good idea to try the think of massless particles in other energy of massless particles in terms of their last since all massless particles with exactly the same velocity can they have different energy of the or more of the same velocity pull yes they can the reason is because you're always euros not thinking about Nestle's particles it is not good to think about distinguish them by their velocity although velocities of the same that think about it differently waiter think about massless particles and fact reconfigure by many particles worry but think about messes particles in particular is to think of energy as a function of the momentum instead of the velocity I think we could even do we even know that quite often ordinary mechanics when we write the energy of a particle kinetic energy but it is one-half envy square up writing for another later right there is momentum squared over twice and this is I'm talking now not about wrote Mr. mechanics just old-fashioned character we write the energy in terms of our start ready energy in terms of the moment of the way when alternative way to express the energy 1 happens if we do the same thing in relativity can we find the expression for the energy of a particle in terms of its Mormon well over a simple track a simple trick is just to use the fact that you know what squared my ears you what X squared off with more and SUY squared and my Susie square say using square physical born who worked out before and now works I'm gonna sexy for the momma promise of See you go toward Seahorse warned Friday if I multiplied with beyond the components of momentum all 4 of them are the same as the components of 4 except for a factor so aware those multiply this equation by him square and squares and square them squared m squared when this becomes this is square after he squared but a lot is just energy this becomes a square of the energy my ears each 1 of these is a square of your corresponding a component of moment he squared minus peace glad that multiply the size and squared minus P. squared equals and scored turns over for velocity it's you walks squared might you vector squared equals 1 turns the corresponding components Of 4 momentum and you momentum he squared might be squared equals and squared for can back as Hey he recalls the square root of p square Austin scored I just threw a move the piece Cuero tho a right side of the square root West put back the speed of light I'm going to leave this is an exercise for you or are you would be answered is square root our squared becomes P squared square menswear becomes and squared seeker before check that out just a exercise and giving your units straight show Her murders of energy in terms of momentum square a woman from the match from this formula we can immediately take the limit of mess goes 0 we had trouble thinking about a full time in terms of velocity but we have no trouble and thinking about 4 times in terms of this formula orders I now if war drive from yard that's energy becomes equal to the square root Of the square of the moment the speed of light which is just the speed of light and the magnitude of the what's called magnitude of momentum square appease Queiroz just going from the momentum vector energy from Nestle's particle is simply the magnitude of the momentum vector but in order to keep a our units correct you have multiplied by the speed of light is is for photons it's approximately trophy neutrinos neutrinos have pretty a little bit of amassed a true program bonds is not true for particles which moved significantly slower pure white our right now we know what our expressed 0 mess was particle we could solve the following equation rare following problem the pope let's take a particle this particle remain school positron happens to be an electron and positron in orbit around each other but some important recalled the example what this from what we call a book or positron it's an electrically neutral particle and has amassed its messes approximately the mass of 2 electrons but don't kill
a rebuke auto electrons or not to move it Is more than 2 Emerson tour on Western and wiser Western back yard right now the kinetic energies so told the more than 2 charms but has even more negative potential energy so well whatever the positron particle wears it doesn't matter what made our it's simply an electrically neutral particle with a Mass Our diagram promote market has to look up the rest of the electron yes funniest you look of the of I'm back gets fired I were meant to be that's something in jewels Our sovereign somebody would give jewels further ahead what it Rice is I have no idea I notice Michael Owen times and that this electron nurse I'm so 9 . 1 yards a 1 . 8 a mental or some number of Jules that's the positron 1st positron and an affiliate positron wall around for a while because from became a wallet decay into it will make two-photon through photons will go flying off positron in will disappear the 2 photons will just go and other words you've made our electromagnetic energy out of them made just plain old electromagnetic energy positron positron in disappears and becalmed place tho photons going off separated from each other can you calculate what a mess moment nightmares would be on energy and momentum of those to 4 pounds of sorts go through the exercise this is something which would not make any sense at all in nonrelativistic physics in nonrelativistic physics the songs of the masses of particles are always unchanged chemical reactions happens some chemicals and other chemicals and so forth but a few weighty the system you wait the massive research from the masses we ordinary man never change borrowed because he is a son of ordinary masses changing this has amassed or wherever workers in amassed or reducing kilograms in each of these is less selects photons as normal hours some of the numerical stones of the masses of particles is not considered of this process happens busses does happen but the rial role is not that the songs of semesters conserve energy and the momentum was refresh questionnaires what is momentum momentum conservation 1st let's assume that positron Liam Adams past strontium particles is at rest and offering of reference was not arrested your frame of reference just movie bottom given to a frame which it is arrest might get into frame in which it is at rest so it's rest from what momentum of its arrest 0 salt call the momentum of the system the begin with 0 not case to 2 photons firstly conclusion is a photons must go off back-to-back in opposite directions if they don't go off in opposite directions it's quite clear the courtroom momentum will not be 0 that war from opposite directions after by this yummy will go off in opposite directions and they must have equal exactly equal moment back-to-back equal and opposite from white judicial however 0 momentum the final 0 5 so that means that this particle goes off for momentum he and this part all those are for the moment the miners ready now let's euros energy conservation the energy of the initial mark your positron it it's may ask whether it's messages and speed of light squares in there that this is not the mass of the electron not exactly very very close to it but not exactly a little bit less than twice the rest of but is a mess of the positron so you get out Euro spring balance or reduce 0 scale for positron Mahmoud New way and about the best from Leonardo and when rest in the rest frame it has an energy eagled and square lastly equal the energy of the resulting 2 photons going off so this has equal the energy of the so far currency theology of the photons the same because they have the same magnitude from moment and just equal to twice the speed of light for the momentum terms the absolute value the momentum p so that tells us for the absolute value the momentum has be Anna although tool that's the momentum of reform of farm prices that steps simplest example of calculating a relativity decay and calculating what happened this is the moment the number for particle out we could convert this using a little bit of quantum reasoning we could convert this tool or wavelength the year photons corn now we use the connection between wavelength and momentum that we learned last quarter we could convert this to a statement about the year of the waiver for users more murderous more optical so those the mechanism by which may ask turns and energy of cost and this was old was always energy but it was in this frozen form of rest energy of the year of the positron mark your pass from Adam decays results in 2 photons going out to photons collide with the atmosphere with everything else in heated up but can be absorbed by electrons generate electrical currents and so forth so warm so comes the energy gets converted from the frozen form of rest mass through a dynamical useful form of electromagnetic energy houses but our city mammography have to begin with 0 angular momentum these guys are going off the back of the head of Office of the moment so the easier but must try went up together they have to take a always so no enough freedom your governor freedom and a folk harmed her angular momentum Rickett chief Europe Ricky she is and what does so well let's put it this way in order that have signed in order that have a finite world before immunity
if he goes to warn the nominator goes 0 the mast numerate ahead of them also abused say user us however say laughter some fraction of the time the euro positron Harold Ford who overcame a pair of rabbit now that percentage of the dime is 10 the the House of large numbers but there are some percentage of the time to be electrically neutral positron will dictate the careful problems Russia now by friend resisted don't look for in a laboratory young that has been His I His name is means war it was a mean that set up to more of it apart from our days is tired while you're not part of Europe are what is what experimental question ask you could try you could tell you could detect the photons earned her some sort of chamber of tracts and you could start Patricia to photons back and you will discover that they both common together exactly the place where positron were present from me more surgical there were decade from the decay was peculiar and the positron became 1st producing 1 photon aware here and there later creating yoga full time which only 1 out here both moving with the speed of light and you trace them back they would appear to come from some was over here that doesn't happen if you traced back trajectories of the photon America our was leave tracks chamber or something like that they will come back and intersect exactly where the will be initial positron more much reverse question asking to repair maker were there for surrender you end up what it is worth eastward that that a was I he said the last word for it last eastward what yet what said Raul Ruiz to Ovapa go through it again refer again for you Ray had 2 versions of 1 of them was garbled among grovel oranges cancels out of grumbled 1 extra a time so they remember this formula about the cows square musical belt he squared minus now backs backswept but off movers ran over him that right it is but but a mystery buff so you last year is deceased out of products with that of the not that doesn't it doesn't those strike deep chord in a year yeah no it's true 0 Brazil really ringing something powerful only in the stands well I was you want to d TV count nothing nothing really restored Of course it rest TV tower was warned most its not worn this telling you how sorry you Harold B. energy changes as a function of velocity but nothing more than a yes are but part of cities and says what that 0 yard to both awful let us see it is from rent Ms. Costa on well you in a laboratory situation in many laboratories situation of course there are contaminating effects on even if nothing more power of the presence of some kind radio field magnetic or electric field or something could corrupt the thing and and lead to a slightly different answers remember the rollers momentum as considerate for a truly isolated system truly isolated system means a of that you've accounted for everything in the system and nothing else is interactive and the moment of our systems could share their if you're laboratory is a sufficiently isolated and regional space is cold enough particles bombarded and so forth and so Warren positron together with the young and the photons constitute and almost closed system but not this it's really not completely closed as Grandma if nothing else is gravitational forces between the Earth and the particles are completely negligible within 0 so Yarborough are contaminating influences that these start however that doesn't affect momentum is concerned where is better than all the momentum and particularly the better you know the relationship between the moment may be there you have less information about the relative positions but that's a small effect and drove for our purposes and I want a separate quite mechanics questions but but some real questions on burnt we're at war will eradicate this part of what once was a proxy this was from while this is not a part of this virus where's our loss this way EA both what sort hasn't France mission I just E equals at while this is it that Bulgaria's America back over here 3 have we review with calls were in provided by square 1 squared so I look at the units and and then it well like right now or 1 and of these squared have different units the only way to make these squared have the same units is 1 is survived by square is no other way the 1 might be squared overseas Queiroz dimension however the whole formula was not the left inside his energy the right hand side now has units of a man message
energy don't quite have the same units a related by the square of velocity we know that the use of energy the unit's energy or quarterly units on mash primes the units after a velocity v square example would be Connecticut Energy is 1 half and squares that's consistent so it tells us that we have to provide the square of all velocity year and that tells us that square go but but set he 0 renewed just plain M C squared all he said said at each of us president came to photons Vermont says I of the Ark Young became what does this off-white think the Czech if you if you if you had positron and rest now remembered because of quantum mechanics is going to be some uncertainty in the position and reserve as a region in which you may have known of the positron was then the positron indicators and column if you trace it back world lineup of photon comes our two-photon there with the sector could drop to could withdraw space-time picture of this a space-time picture is a positron time goes up as worldwide of the positron rest and the positron indicators and photon knows our persuading ropes for tires are this way and from China was actress worry traced back or warmed the worldwide and we thought target you'll find out that they will intersect vessel hall at the spatial location where from him woes but they will intersect with exactly the same kind of barley seduced is kind of we ignore quantum uncertainties your vote legitimacy Estes lows due vote us and I lost rare from bond that way sometimes that way sometimes I the costs around the corner and or your girl taking spin orienting in this way in measuring other acts sometime this way sometimes comes out with some angular distribution but the but randomly and at random times through some that 1st of all so go around you are we did you the 1 have said the action principle being a statement about minimizing something a rubber about trajectory will be invariant if you you choose the action to be in there was more violence no room so they wouldn't it less enters that we just Yashin for that that as but that provide you're not quite as been saw sort see Beatles rated other ongoing reason we have put a concert before us now are what we could do was we could take this grungy and work out the equations of motion and see what the equations of motion but we really know what they say that the energy and momentum and therefore the 4 velocities constant always just as armed notions equations when there's no forces around just say well received and IRMS times acceleration of 0 a sustained saying the current rate of change of momentum a 0 or the moment was constant prices will save the momentum is constant nothing more or that before velocities constant well before Molossidae being constant is volley a magnitude but direction it says that the particle moves along the space a straight line even space time that we however we motion of a particle a straight line through space turn and that's is moves along a white space with constant velocity that's a whole continent Lagrange equations for this particular case now we could try putting we could try asking houses get modified if we try various kinds of forces and we will do that 1 of the things we will do what it is allow the particle to have AT and charge and the couple would tour electric and magnetic field but so far about that's a little more complicated than would have to narrow Hall content of your over orange equations just moment straight line motion or not just rely emotions loses Toulouse laws become the same law when those laws a particle moves erode further of long straight line and there once a prior of particle moves with constant velocity here becomes a particle moves a straight line through space time which entails both straight line motion and boss 1 more cautious I thought that there were velocity very love is was a lot of problems for laws of conservation measures they were derived from and it is this will back asked you want back all over about yes the law but 2 entirely different women but there was a loss of life the plus combinations of various things structure a mess of the object in the important thing is the mess of the objects and the smoothness of these forces backed by shouldn't be a brought some that's another regrets separate things this this got to do with the speed of light are quite mechanics has to do with plants constant and you can have a situation which is highly quantum mechanical by nonrelativistic you can't have a situation which is highly relativistic and quantum mechanical and you could have both Mr. can quite chemical 2 independent limits is as good Abdul I have always gonna do everywhere you thinking about gravity gravity our puzzles about prefer we haven't gotten there yet so far not doing gravity all from OK I think it's for more please visit us at Stamford


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