Special Relativity | Lecture 1


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Special Relativity | Lecture 1
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Susskind, Leonard
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(April 9, 2012) In the first lecture of the series Leonard Susskind discusses the concepts that will be covered throughout the course. 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 includes space-time and four-dimensional space-time, electromagnetic fields and their application to Maxwell's equations.
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subject this quarter dealers a collection of things but mostly special relativity and field theory we're coming back no not quantum field theory were not prepared for quantum field theory at Ridgewood learned classical mechanics those who've heard participate up classical mechanics the general structure of classical mechanics what is our work action principle a Hamiltonian foreground unions or what kind of stuff and I assume you know it we also learned some elements of quantum mechanics I can't believe I took a hole quarter canceling the the harmonic oscillator about the recover people say What did you do Salomon still that's good but but we learned the principles of quantum mechanics but we didn't spend much time examples in order to go forward with quantum mechanics and quantum field theory we're going to have to do some work on mechanics assured but not the scorer no Oklahoma characters caught back the classical physics but no back the relativistic end of things we studied nonrelativistic mechanics motion of particles thoughtfully motion of particles our ethical and made the Tony in physics we are not study anything about light and that's because light basically relativistic number from number that has to do with special theory of relativity and that's will be goalless quarter the special theory of relativity classical field theory classical field theory means electromagnetic theory basically electromagnetic waves forces and charged particles or what sort of thing but in the context very definitely in the context of the special theory relativity so as Hogan a began special relativity tonight bomb I I I suspect that most of you have it own pointing your lives learning the elements of special relativity and lay people would not fall into that category so I'm right and just about everybody has learned special theory of relativity how many people feel that I could completely dispensed with it without causing any Hill about the visit of the field that you would like to go through special relativity so I'm getting I'm getting the rough answer that answers yes and that's why I prepared to do tonight but had also been prepared the everybody said we really understand special relativity completely could you move on to the next thing I was yet but I'm glad didn't because I like the idea of teaching from the beginning OK so that's so that's come to Special Relativity let's talk about relativity bullets go back a step before relativity the idea of a reference frame idea reference frame you know what I I'm not going to spend a lot of time earned them not to be a basic philosophy of everything in order reference frame is we talk about reference frames in classical mechanics it's a set of spatial coordinates x y usually based on Cartesian coordinate Allah to specified coordinates you will want to specify where the origin of coordinated ambiguity in the sector could translated from 1 place to another and you will also want to to specify the orientation of the coordinates the most orientation freed up summer patient nor the pick x-axis that way the axis that we z-axis that weight or change it but specified To begin with and that's part of picking reference frame with spatial set of coordinates In addition to a set of coordinates which could be measured by meter sticks to like if you want to think physically or removed Kurt creakily about work order system means think of spaces filled up with the lattice needed so that every point in space be specified certain number meters the last certain Member meters option number of readers in amount and that's a accord with system for space it ought also to specify when something happens that just where it happens you have to specify a time coordinator so had a full coordinate system or a full reference frame consists of an X or Y and Z and words of time axis and that's a reference frame that you could have different referenced pipe from the ability to rotate from 1 thing to remember you could also considerate moving reference frame of reference frames that moving relatives some specific reference frame we could pick up our reference frame by our I mean yours and mine film in that city yours and mine you reference for you're sitting still I will assume that for the most part he Wilkins continued to sit still tonight but I will move around from time to time and march past you like this armor March pursue a bad day and we will see if I can think of my coordinates as a core frame centimeter sticks that I carry with me and move with me as I moved so that every instance I personally in the center of my own coordinates the my coordinates I'm different your accord lets you will specify any event by an X Y Z I'll specified by a different set of x y z keys to account for the fact that I may be moving past due in particular we won't agree about be axes if I'm moving along the x-axis relative to you our my nose I will always say relative to me is a 6 inches in front of my face because we have on made known as X equals 6 you'll say my nose is not lexical 6 will say my no is moving and it's some positions that changes with time to give a different coordinate erect arm in ordinary pre relativistic physics will also specify kind 3 special relativistic physics We will also specified times and is in pre relativistic physics we would assume that we all have watches all watches a synchronized we go through some summer operation to synchronize them will talk about operations to to synchronise watches will assume the synchronized the given instant of time all of your watches agree with each other they agree with my watch my watch agrees with your watch that was an assumption that was made in all of free relativistic physics kind time is time this time and there's no ambiguity associated with moving reference frames about what the time is a given instant off what is relativity relativity means that for all coordinate systems which are related to each other by uniform velocity all reference frames related to each other by uniform motion the laws of physics of the same the laws of physics of the same in every reference frame with the word inertial reference frame you know the chairman of commercial reference frame if 1 frame than any frame moving with uniform velocity
relative to it is what it is and inertial reference frame was the 1st semester if referenced family you start with let's just make a simple and say it's a frame of reference in which Newton's laws are correct and in particular where particles which have no forces on them move with uniform velocity that AT reference frame every frame moving uniformly relative to its not rotating relative to it not accelerating wrote to his mother inertial frame that's a feature of the Italian mechanics that the laws of physics at equals MAD back together with the Newton's Laws of Attraction cool Ohms Law of Attraction Seaman every reference frame so he'll feel like the way I like to describe this is we got to the proverbial railroad train moving down the X-axis with uniform velocity and think about you with him or me I'm in the train you're standing still you'll have a set of laws in the way I like think about those laws of juggling juggling is a good thing makes a lot of use of of classical mechanics and inertia forces all the good stuff and United juggle at rest you juggle exactly the same way in the movie reference rate you can tell a little later say is if you're removing reference frame of mind removing reference frame everything sealed kitschy outside I cannot tell them moving I tried a find out by doing some juggling and I find out that might standard laws of juggling work and I assume therefore that I arrest but that's not right although tells me is that I'm in uniform motion so the principle of relativity is that the laws of physics of the same in every reference that principle existed before Einstein was not invented by end of attributed to Galileo will not Newton would have recognized it and and it is called the principle of relativity what that Einstein stunning hated it won or lost physics the law of physics is at the speed of light is the speed of light sees the speed of light is a 186 thousand miles per 2nd or the speed of light is 3 times tend be meters per 2nd battle 6 now that you combined the 2 things together with the laws of physics of the same in every reference frame and that's a lot of physics of light was the velocity you come to the conclusion that light must move with the same velocity every reference frame white because the principle of relativity says the laws of physics of the same in every reference frame and Einstein announced that a law lot of physics of light with certain velocity that the puzzling and outlets let's quantifying and make precise consulate let's try to be very very almost Van Dyck about it let's start with reference frame of reference frame is a coordinate axis in space now drop purposes now spaces just one-dimensional I don't care about y z I care only about X so let's go x-axis his ex end also Timex used let's call list your reference frame and Eli reference frame light moves with the speed of light with the standard speed of light and so if a light gray is sent down from the origins kind XT and the position x it will move with a trajectory given by a X equals C T that's the most of light what does that look like on the map Iran's map of space and time it looks like a straight line lexical straight line White now Professor like moves the closer the horizontal that line will be another words in a given amount of time the further away more of salt some number if I actually put in the 186 thousand miles per 2nd meant that line would be practically so horizontal that you couldn't see the difference from horizontal it would mean that the slight Ray goes a 186 thousand miles in only 1 2nd that would be a very horizontal line I will use some of the units for the speed of light units in which I can see explicitly that the library is moving it moves a certain distance in the car and at his velocity talking I want to add a 2nd reference frames per 2nd reference frame moving relative to the 1st 2nd reference frame has an ex prime access its axis is called and the center of it just like the center of is moving relative to you so the cost of moving it means me at the center of my Courtnage system is moving also the straight line over depend as treasurer 1 also move straight line and that showed line is X equals V T we're really is my velocity relative to you so I'm moving with X equals VT my position relative to you keeps increasing as a function of time and that's the trajectory x equals VT now I don't call this x equals VT I just call it Mexico's Iran at me but Oracle x equals 0 at club X prying equals 0 ext time I will agree that the call my coordinates crying that I call my coordinates X Prize and key Prime next what about the art works was the the relationship between York audits and Mike warning that was pretty easy X prime is equal to X miners VT sincere Michael wouldn't it His Caltex priming your is called X my coordinate is your coordinate lest the distance between us which is so what X prime equals miners What about Clark's well if we make the assumption that no made made the assumption that all clerks can be synchronized and that kind clock registers has nothing whatever to do with how it moves then we can before I even started to move we could synchronize our clocks and now I could start to move a the rule will be my time is the the same as your this would be the transformation of coordinates between your Mike Ward final if you know what an event happens you could tell me in my court that's what happened OK now let's take this like light gray that woes of X equals or frame and as count was in my frame so I wanna find out how the library moves and my frame it's going to be X Prize is going to be acts along that light gray X is equal to see all along this like all along was like X is equal to see so that means along line X Prize is equal to CT miners which is the same errors C minus 3 times so my ear after all all which is also the same as
C minus V times he kind of support that's important to require my watches and my clocks and by measuring sticks that light gray which was emitted from here does not over the speed of light most of the speed of light C minus V as bad some wrong release of Einstein was right something's wrong because Einstein announced little ruled that all libraries move the same speed what about library going the other way incidentally light really going to the left would be ex equals Mike a if you plant that and 2 x prime musical text murders we pray you get X prime is equal C plus and he another words a lighter going Frank moving to your right a light gray going that way looks to me like it's going a little bit slower but this amount politely going in the other direction appears to me to be going the fact that was what Newton would have said that his work Galileo Water said everybody would stay until the end of the 19th century when people started measuring the speed of light extremely carefully and found out that no matter how things were moving the speed of light always appeared to be the same as the only way to reconcile let us say something is wrong with the transformation loss between your coordinator Mike warned that something is wrong with this simple-minded coordinate transformation a 1st step special relativity Mr. figure out how this has the chain your coordinates Rice's Mike Ward that in order the speed of light be the same in every reference they were murdered take it to that 1st 2 steps even before that even before that we should go back and Escobar perhaps assumptions that we made 12 unwarranted well this is William assumptions we made obviously a lot of them but the most important 1 and 1 which is most at risk and in fact the warm which is wrong is that Symington in every reference frame is the same that if we begin with ah clocks synchronized and now I start to move will my clocks remains synchronize with your clocks in other words is Mr. Right transformation between moving Clarkson stationary clerks and that costs in hindsight is the 1 that we know was wrong that the correct relationship is different and in fact that the whole idea of Siam is framed the pending his reference frame depending on it he is wanted a magic each 1 of you has a clock your clock your clock your clock you synchronize them by whatever means you synchronize your clocks I have an equivalent collection of friends was spread out exactly the same way relative to me the front row wrote this gentleman here each 1 of my friends has a clock I'll make sure that our clerks a synchronized and then we're moving removing relative to to you as we go past we check each other's clerks and we check whether there's still with the are synchronized and if so why if not excuse me if not how much out of whack his each clock depending on how far down the line you or equivalently we're asking for what Is the coalition is the same thing incidentally about a meter sticks because as I pay as you is my x equals 1 1 unit of your position relative to you if you don't forget point then of course this it is where Einstein made a great leap said we have to be more careful we have to be much more careful and defying very carefully what we mean by synchronous and what we mean by links what we mean by time I up and look let me let me a think about experimentally how I would synchronized clocks given his past whose postulate was the speed of light is the same in every reference frame how do you go about synchronizing your Clarkson how should I go about synchronizing right clocks miss a lot if like those that direction was B C in that direction with BT in every direction we will do track will trick it is if we have to clerks what's this gentleman at the end of here and that in the world and it was synchronizer clocks billowed up at seats delays they only get out of seat or stay with a sitting then they find somebody right middle now having all the persons in the middle they measure would sticks of afraid you'll have to get seats of your do that but the new measure would sticks and you find that that's 5 meters of 4 meters of the center you find that's 4 meters to the center for Bakhtiar seeds and now objector clerks checkered clocks by because also somebody sitting in the middle of error-free like we Clark's body when you all Clark reads 12 noon and when you are Clark reached 12 noon each send out a flash of light a flesh flashbulb if the tow flashes arrived at the center at the same time then your clocks was synchronized you reach said you understand that a light beam when your Clarkridge 12 o'clock if the light at the center arrives at the same time a clerk to synchronize Aceh where Einstein decided to define synchronicity now what about me you could look me who's moving I'm moving I get to this point over here when I get to the point of the year let's say that just happens to be the point that you would emit a simultaneous libraries at 12 noon according to your clocks you each a light that doesn't quite get the center at 12 Milner gets the center slightly later according to your class by which time I'm over here since I'm only here you off like race will arrive at me a little bit late relatives feel like so I will say you guys around a synchronous you made a mistake Putin synchronized clocks be costs the delight arrived at me at 2 different times I cannot say that likens say that the meaning of synchronicity is different in the 2 frames of reference OK so let's go back to war took systems and be skeptical about exactly what synchronous means in the moving reference frame and the stationary reference frame will assume the synchronous means 2 points on the same horizontal the same horizontal level here at the scene not the moving frame of of reference the Moline frame of reference will find it a minute that this point is not synchronous would disappoint those synchronous was another point and the fact that the whole surface here the removing reference frame calls synchronous not he had the wrong place that's like someplace else we're Matt our what the moving observer of calls synchronous and how we could do it we could do it by synchronizing the clocks calling Einstein's role OK so we need now is a better drawing whenever a big character relativity problem draw pictures of 1st thing to do 1st stroke pictures always the same Ericsson Thaksin it can be a reference
frame that you could think of as stationary in your frame of reference the other frame of reference think of the railroad train moving down the exodus but the next thing to draw drawing before you draw anything else drawing in the libraries how likely moved so drawing X equals CT and Mexico's CT particular X Accorsi keyword articles my see later but I said if we were to use Siegel's a 186 thousand miles a 2nd we withdraw practically horizontal line or do that so we choose the speed of light In some units were easier to draw a picture that particularly the natural choice of units is to make the speed of light water added make the speed of light 1 will use laid-off units which are defined relative to your time units appropriate way if you kind little a 2nd units to be like 2nd house because light 2nd Way 186 thousand miles but never mind 1 lights 2nd is a unit of weight and health ministers like gold in units of seconds of white sector because 1 white 2nd per 1st 2nd that means that on a diagram like this where x measures like seconds and key measures seconds the foot like moves at 45 degrees and show using this before I'm just going out HMOs backwards at the same 45 degree of adds that the trajectories of libraries next let's another observer another reference frame removing reference frame begin the reference frame is going to be moving X prime eagles X miners VT exactly like it is here St. picture superimposed on a wide array Mrs. X equals Vt it was a prime wrong here 0 was also ex prime equals 0 sewage put X prime equals 0 as fears problems that might be take effects is able to VKX Prime physical 2 0 so at removing reference frame and let's try to figure out which Prime is where the key Prime Texas's far-right I would begin by putting in free people the end the war in the middle and the war appearance here price or their initial time right away he we're gonna put in a moving person a modem a stationary 1 I should say about few people my friends my friends who were separated by equal distance is me a warrant from me and those who want to get in front of him but put it in the first one he always with the same velocity His most parallel let's suppose that he's 1 unit ahead of me meat but measured with your meter suspects that means that this distance right over here is 1 unit what is at say about the equation of the 2nd here says X equals VT plus 1 as 1 unit ahead in the stationary coordinates What about the next person who's units ahead and his equation is text Eagles VT plus your Prussia won over here has a watch and as it happens his watch said 0 0 was 10 0 part of the plan my watch I'll take 0 to mean 12 million time 0 cars right over here along his trajectory that's time equal 0 In other words the more observer and stationery observer a greedy by assumptions on what kind equals 0 the stationery observer people 0 all along the horizontal axis that essentially definition of a horizontal axis all times From the stationary observer of a single here but let's see if we can figure out where time 0 it is all along this 2nd axis along with skewed people named Fred Mary and Dessie Mora for those for Seymour him Fred answer whether the way is work now is at this point freighters and Allied-Signal signal to see Mary and at some point I don't know what point yet Seymour is also going to send them light signal for Mary but they get ascended in such a way that they riot Mary at at the same instant so friends since out was like sing-along here where does seem have to send out his light signal from in order that arrived at exactly the same time will start in this day and as such should light raise up the 45 likely to miss the point so I'll be smarter award backward 45 degrees stay here each life signal In the stationery reference frame Moses 45 degrees you could see because the 2nd reference frame is kill Biddeford relative to the 1st reference frame that the point that occurs above what the stationary observer calls equals 0 this I don't hear moving observer we'll call keep prime equals 0 why because in the moving frame of reference these 2 points send light raised to the central observer over here that arrived at the same instant of time so Mario you guys sent me the light signal at exactly the same instant of time because our armed me at the same instant China happened and all year equally spaced now becomes a little exercise the figure out where this point there's some take it that exercise due in detail for easy but give these points name and this is and this is being we won a figure out is what the coordinates of this point B R because nobody is synchronous with the origin In the moving reference frame that will give us some information words be must be point keep trying 1 points for which he Prime is equal to 0 OK What About . 80 I would find . 80 we find point they by recognizing that ended the intersection of two-line What is this line over here the Green Line that X equals C P this line X equal CT because it's a light rain moving to the right we decided set the speed of light equal to 1 I decided so it's really just X equals t emotional light just equals tea and his library that's this line and other line along there at this point a lies on is x equals VP plus war however would find the intersection we stops at the 1 equation into the other
senator says she is equal to V T plus 1 from the 1st equation I just said lexical or he 1 minus V is equal to 1 or even better he is Eagle to 101 miners very price their own or with a time of this point 8 years misses . 8 where about the X Of this point 80 hair would like a full set of coordinates of this point swords box around what she exit point while Yukon either people's 1 and blogger but it could be smarter and you could say Look this line is X equals he over here the Green Line as x equals so along the Green Line X is equal to take show the lurid acts is equal to 1 or more for now let's go toward buying a B next thing to do is to figure out why a B E which will ultimately figure out where it intersects Mexico's we people's totes a few steps a few steps from the fund to do so over from and I don't awestruck cut from their said on show we do this where the line may be every line which is at 45 degrees but pointing downward to the right of 45 degrees but they're going toward upwards of a left is a line that has the property that X plus he is constant along that line every line moving upward to the right with 45 degrees is X minus t is a constant 1 which are 45 degrees but upward to the left all lines X plus peoples constant sort its X plus is equal to something what thing well 1 easy way to find out what was put on the right side here is just to take 1 point on that line and plug in the value of X exports was to use the same all longs for a particular appoint an E X plus happens to be twice one-liners when I was just this blast this X plastered all along the line AB you'd like a B ex post please equal to what all over 1 minors so we have found that 1 ADB we want to find point you so 1 more step and that's step is defined the intersection of lying a B With X Eagles VP plus so we take ex custody was to over 1 minus V and we combine it with X minus V T equals To assist equation over here X minus V T equals 2 we solve the 2 simultaneous equations and that will tell us what the coordinates of Idaho we do that well the 1st step we just solved by subtracting the 2 equations I'll get rid of the X right sorts of track on a lot on the right on the left-hand side we have tea mind this miners VT which means c . times want was rigged that's equal to what 0 1 might misery minus 2 however Track Iowa subtract tool from a fraction so the best thing to do is to promote the same common denominator minus 2 times one-liners V over 1 like this week and multiplied in divided by war OK Hess you look at this carefully you'll see that too was canceled to minors 2 truck to remove here will become plus the whole thing becomes twice v over 1 miners that's equal 50 terms warm 1 more step to calculating it's just a divided by what was so what happens if you divide by 1 plus VAT while you divide their warm was vague on my industry Thames possibly thoroughly steps I hate doing our drummer blackboard both bit I did tell you do it but what 1 might be times when plus 1 Rosie square right so it is right this is warmer and squared 0 1 plus but he was not really I want warned ex ante is sporting so how or foreign to be acts of . be this is T of . be cause TB was happy X of foreign well all I need to do is substitute into this equation is equal TB so would see what it is X it is equal to the tool of either by 1 minus V minus TB right from subtracting key from this equation modest TB and TB 2 movies all over 1 my square looks although 8 it's not trips much these days EU Oliver the bad things happen right arm right off b is plus
or minus 1 that means that observer here is moving with the the speed of light and week were kind of generous situation on by just plug-in what does happen much less to offer will 4 over here with me is plus or minus 1 TB becomes infinite sulfur column so what's happening with gets ripple was warranted what's it plus 1 is blind 2 things over to become horizontal would like the horizontal parallel with it power with they never intersect so the intersection as part of a game here of course is to realize that we we will eventually disallowed things moving faster or as fast as pure white for the moment it's a fair question and something bad happens so begin deserve smell bad have somebody moving with to like weekend simplify this multiplied by 1 plus VAT and both the top and the bottom now is the fact that 1 might be times when closely is the same as 1 might be squared we get everything over the same denominator we must do all we have we have a Tovey we have another to movie with a minus sign a canceled Oh yes to the reef to sell assets of bad exhibition is equal 2 although 1 might be squared the only difference between them is it as an extra factor of V E N T know why did I got all that trouble I went to all that trouble because I wanted to figure out where there the surface of all lying to surface and higher than mentioned where the surface when the weather's also rather space dimensions likewise Missouri where the line is which corresponds the being simultaneous with the origin and it's a line which passes right through that point we can't have found the other points the line by using smaller on bigger intervals of the same argument saying argument more moreover bigger intervals found that old places which is simultaneous with the origin of of life on the line which passes through the origin and which passes through the point be where we know about that align we know that has a constant what is the slope of the slope of course is the radio of along that line and watch the race tax it's just TBA over B which is just V ratio of expertise is the ratio of this this is just plain V at all so the slope of this line here is Oh I Passaic this line is T equals a V X he equals acts all along this line the moving observers says Everything is simultaneous so however Clark however moving clocks work all the clocks along here if this synchronized if they been synchronized by allowing stunning procedure they will all read the same time they will read keep prying equals 0 all along white because this operation here was exactly the operation to synchronize these 2 clerks if this 1 reads T prime equal 0 sort of this 1 or I now look at least at diagram of a simplified diagram of media make it forget rid of a lot of stuff on a of 1st of all a light gray dueling at 45 degrees we have X equals VT that's the moving observer the center of a moving observer and then all here With equals V. X. notice the symmetry of these terrible Mexico's and vehicles VX the meaning of this symmetry is that this line over here it is just the reflection Of this line does related by that changing he index slipping about the green line here I'm sorry to say it is they're the same angle this angle here is the same as this angle here on this diagram we've discovered something interesting applying studying is right and if the speed of light is they Damon every reference framing you use light the synchronized clocks than what Cinco in 1 frame is not the same as synchronous the other frame of the 1st thing and the 2nd thing is we've actually found what synchronous means in the moving reference frame it corresponds to surfaces which on the horizontal but which have filtered a little bit tilted Barcelona reach back salsa the problem of work simultaneous meetings In different frames and then upper same OK next can we say more about the relationship between X T Annex prime anti prime Webster and we don't need this over our just look them up the Arab he is all we know we know that X prying equal 0 whenever whenever Mexico's VT Mexico's VT is the same as X prime equal 0 so let's begin writing X prime is equal X minus V T but we're not that that's correct only know Is that when X is equal to VT next prime is equal to 0 this could be incorrect by some factor call death that factor could depend on the velocity of the relative velocity of the 2 frames of reference so let's right f of the velocity between the 2 frames and I mean the magnitude how faces a growing officer across magnitude philosophy I use script they indicate that the magnitude of the velocity velocity veto be positive or negative the 2 reference frames could be moving moving frame could be moving backward could be moving forward so vii could be plus or minus but the function that would go here we might guess is just a function of the magnitude velocity let's leave the assume that for moment the and that's a general answer after what X primaries although we don't know yet anything about what the function from where about t prime well we know that prime is equal 0 whenever he is equal to XVT window that ex prime was 0 when equals VTU we know that prime equals 0 when tv equals X. they just convert the Texan so it must be then this is equal to T Miners VAX and some other possible function these 2 equations tell us that
X primer 0 when Everex equals VT tells us he Prime is equal to 0 equals VX another words it's just reflex this relationship that 50 primer 0 when this is show an X plan equal to 0 1 OK now we use some physics we use a physics arms Burns said that the speed of light is the phrase in both reference frames or the the speed of light as 1 In the stationary reference frame so must also be 1 in moving reference what that's theirs Is that would ever X equals CDU X equals he is along the Green Line it must also be true that X prime equals deep prime every have a light library and light rain woes with velocity won the silver right back to the 1st thing that we demonstrated that when we took a library that went like X equals C C C and then we transformed from 1 friend to another level we got the 1st time around we got C V C plus VAT at that do the same thing now say supposing the light satisfies x equals then it must also be true that extra Nichols to prime how could we arrange that about such easy let's suppose x equals t then curve we can write this as X friends what exports why would that to do make physical possess swarm Exs recalled 3 equaled the TV store will be the same Vista will be the same so what's in the bracket he'll be the same as Woods in the bracket he works as a quantity of everyone x timetable PrePrint is only 1 choice that's the Macbeth equal to g that's the only way that must be the same as
G if as is not the same as G most of the light in moving coordinates will not be be with velocity 1 cell we come to the conclusion Einstein's ruled tells us that whatever the connection between these coordinates is involves the same function here what what more do we know what more we want to know we want to know what dysfunctional visas or do they like what is once we know it we know completely havoc coordinates reference frames are related to each other so we use 1 more ingredient mentally I'm practically regurgitate in Einstein's 1st paper on the subject if I haven't read it for probably 50 years but it has left obviously of vary West Inc impression on me our wrist is basically what what he did in the 1st paper Robison a lot more OK so now said when their poster say which frame was moving poster said never mind frame is moving relative to you with velocity V or you frame is moving relative to me with velocity whatever the relationship between the 2 frames of reference are they should be symmetrical we could take the entire argument and instead of starting with the X T and drawing an explanatory prime we could do exactly the opposite the only difference would be that as far as I'm concerned your moving with velocity minus V as far as you are concerned I'm moving with velocity plus so we can immediately write down what the relationship between exit In terms of acts Parliament keeper here it already bound for you over here it must be X is equal 2 x prime now so I whatsoever I miners VT prime plus prime white because of all our city the relative velocity has the opposite side the function of the same magnitude of velocity magnitudes of velocity of fun movie wrote of you of velocity v still moving wrote me with velocity might be let's assume this function is just a function of the magnitude of the last of which is the correct answer the end and she is equal to is it prime plus VAT that followed the same function are these 2 equations of the 2 sets of equations compatible with each other after all I didn't really need to get the relations between now Exon T over here I could just a solve this equation for X these 2 equations for X and so I want to make sure that this equation In this equation are compatible with each other I'm going to do it in a fancy way and we know of course that will determine what efforts the compatibility his or you would do it is only he'll have X prime and Prime in terms of ex ante now supposing it take the ex- prime prime from here and plant them in over here what you get you get acts in terms of exon T and T in terms of I cannot make any sense to Kigali makes sense of what you get is Exs equaled X and musical that's going to involve choosing as carefully OK but I will show you there's more than 1 way to do that arms maker just do it by plugging in Yelets do by brute force was restored by brute force at a fancy way to do it OK but still X is equal to 0 X which is what so really after me x Missouri City but times SOV that's a sky here and plus VAT climb steep Prime which is T minus V X also plans effort very well times FMV right all kinds Acevedo so that was square here right now as I told you the only way this can make sense is if it just read X is equal to acts the only way to to make sense of I did the same think the T I would do the same thing for year plugging in a primer next time said did just like that being he is equal to teach but let's look at what is it they got Setzer combined things X column X Times Square over here X times of squirrelly here and that we have plus V squared X Times f squared up from here and there were about the T dependents minus but mobile portend is the key dependents the team dependence is mind is VT here minus V T Kansas squared and here we have V T kind of squares is minus Kansas square this is Class V good sign that these cancel automatically do we done something right research teams cancel automatically leaving only the axes and the exodus of X of squared minus V squared square X another words X kinds one-liners V squared times escalated and that but there has to be equal tax cut will arrange that well arrange that just by choosing where square squared to be 1 in other words it tells us work that has to be and it tells us that has to be I 1 divided by the square root of 1 one-liners square
cancel out IPX 1 physical toll one-liners these quick I'm just squared half his equal 1 divided by the square root that's it that's it will let us get but probably has summoned to change a term of plus velocity my philosophy was a bomb are so it would be like this some Isaacson version of course so something probably an inversion of coordinates Illinois never thought about it yard that's that probably corresponds to the possibility of reflecting the coordinates x Costa minus checks could question that much or be answers artists with it for the moment got to routes a much OK so now I have a hand ball minority answers square root of 1 might be squared the square root of 1 might be scored right now they would be answered is 1st of all of which route you pick plus or minus should not depend on the velocity argues shouldn't be some point in velocity which is suddenly jumped from 1 route to the other route that would be a very discontinuous thing to do was a function of the velocity I know well watered down from velocity V equals 0 so gleeful velocity V equals 0 in other words when the 2 frames are not moving relative to each other we wanted to friends to be the same that that X prime equals X. not ex prime equals minus sex to roots here we correspond 2 possibilities X prime equals X X prime equals minus 6 surely when the frames of reference arrest relative to each other we want a solution to just be X Prize Nichols ex ante prime equals t so that says pick positive plus scram warmers square OK let we right more completely there again the usual amassed only recognize this 3 prime is equal to T minus V acts all the square root of one-liners square notice they have the same form except XMT interchange next prime X Myers T P Brian this team is VX old abided by squirrel former industry squares these are of course the Lorentz transformation is a Lorentz transformations and they've been cooked up to make sure that in every frame of reference the speed of light is 1 when a the speed of light is not been chosen to be equal to 1 1 if it's just see with whatever you choose it however you choose you units then the figure out what's going on here all you have do say well if I'm going to use other units I must have these equations be dimensionally consistent I must have these equations now will velocity is a over a time wealth an X is allowing a time as a time this is a length over at times so the new rate here the 2 terms are consistent with each other that's fine no problem but what about the square root of 1 might be squared the units of 1 hours is 1 of the dimensionless the units velocity or leaned over it time so that that does not have units 1 so the velocity squared he must not really be velocity squared the only way to make it dimensionally consistent is to make his be squared overseas squared another words How we kept the speed of light in every wears squared really would have meant the velocity being units of the speed of light by setting the speed of light equal to 1 equivalently we said velocities a measured in units of the speed of light vehicles a half would mean half the the speed of light so the only way to restore the dimensions and get dimensions right in in this equation is review CV squared Privy square oversee square now this equation is not quite right what's wrong with this equation the left-hand side is kind the denominator was OK the right hand side starts for that time NYTimes exit time we need to the squared Be as a link per unit time over multiply that by violate goalie times a linked over prime that doesn't sound right so we must be V a simple away the savers among these although she square yeah the overseas square the oversee scored as the only way to restore the units and now this is the recognizable rents transformation notice that as long as the velocity is very small compared with the speed of light so that these squared overseas queries even smaller if these attend the speed of light be square overseas Queiroz 1 100 V overseas is a of miners for a 10th of a minus 5 square overseas quit is truly a small number and the square Otis becomes 1 where we get we get X primacy Quebec's mines VT that's the good old look version of things what about the 2nd equation here if the veal overseas is very small growers let's not plug-ins some numbers supposing these a 100 miles an hour but soon make it because Baker a hundred miles a 2nd CD was a 186 thousand miles a 2nd C squared is mostly big so veal overseas Square is a very very tiring number so this term here if the velocity on the reference frames was relatively slow this is negligible 1st also 1 might be square overseas square this is negligible therefore the rulers he prime equals t reference frames moving slowly relative to each other it's just the gold Newtonian type formula this goes away this goes away VOC
squared is negligible Moors reveal C in the denominator ignore that turns because it's just too small to count you still visit a guide as well as more slowly compared to the speed of light we get killed only get up near the speed of light is huge correction news Peter like big things have OK Bozo Lorentz transformations which soon is seen before you may be seen the derivation before incidentally what happens the other components of space now we've been very specific and talking about a frame of reference moving along the X axis the motion the relative motion of the Rangers along the x-axis but what happens to the Hawaii direction where there were renewed while physically wasn't mean if you have your arm was saying link is my home secure on a secure America and we went past each other the question is when we met at this point we're on match or would just be longer than mine planted by symmetry just symmetry of the problem last we can either refer to somebody sort of moving halfway between us it's clear that our arms of a match because no reason for 1 to be longer than the other day so there for the rest of this Lorentz transformation with involved why Prime Eagles y and z prime levels In other words things only happen In the plane XT plane where they X with emotions along the the x-axis excellently get mixed up with each other and funny way y and z it just passed a bill due anything they just stay the same your why am I why of the same the matter how fast growing Forex please get mixed up with each other in this way that's special relativity let's talk very briefly about some kind ovation the and space contraction while we're at it we might as well do it we have all the ingredients we need a hard core of this a a couple of illustrations again drop picture let's 1st talk about war would join I have a stick walking past your or do I want you to be the the warns of all the suspect you hold meterstick to music of over there right now you know that's a meter I walked past you as a world past I ask along with your meterstick measure relative to my meters 6 but 3 very careful what I mean by back in particular since I'm moving if I if not careful I will be measuring in points of view sticks had different kind in fact I want to be measuring the end points of view meters Stakes at Commons time which is synchronous me while I do that because that's what I mean by the length of your meter stick in my reference frame and no link stick relative to Miami to sticks where the tall and . 2 meters 6 are examined at the same time in my reference frame that's a definition of what I mean by going to your meterstick that I look at the end points of view on sticks at common instant of time and measure the distance between them with my measuring rods so let's go up picture to indicate what that means meterstick his arrest in your reference frame it goes from x equals 0 x equals 1 1 leader he is 1 end of the meterstick the other end of the meter stick of course they exist at all times the standing still and meters Dixon vertical tens of meters thinks vertical lines he now I want to know in my moving reference frame so again we draw and molding reference frame this see prime equals 0 pierced equals 0 x prime equals 0 misses X equals 0 alone in my moving reference frame 1 aim of the new as I pass arrive every year he Uzi other end of the meter stick act time T prime equal 0 that doesn't equal 0 but in mind woman reference frame I say that this is a the tool in the of the meter stick at time t prime equals 0 where all I want to know what X prime needed is at this point I want to know what text prime years at that point at the point keep he prime equals 0 it was not a figure out 1st of all let's see keep primate let's let's go back and said the speed of light equal to 1 side carry around the speed of light Piazzolla rents transformations let's look at this point right here and see if we could figure out what's coordinates are its X equals 1 and its steep climb equals 0 x Eagles won t prime from 0 sold trial equals 0 means that Gore's XV their service is T equals XVT all along he Sassafras paranoia Nova he is equal XVT I also know that X is Eagle toward so it seemed to find out what X primers X prime is equal to X which is wanted minors v but t E is equal to XVT that up the logical logic bit slippery here let me go back so it we're trying to figure out what the moving observer ascribes to this point at the other end of the Meda state an incident of this time cell which I figure out what the coordinate X Prize Mississippi and the meanest where we know we know that cheap prices equal
to 0 over here and we know that X equals 1 soul we use tolerance transformation no this is Ermina stick at rest meter his arrest your friend yes we're trying to monitor measured from the moving frame which means we're trying to find out what X Prize has over here so we have 2 pieces of information 1 is a T Prime is equal to 0 the others X equals 1 1st Deputy Prime equal 0 they equal 0 tells us that he is eagled the VX today is equal the VX which trying find next prime so it's ex prime is equal to an X which is 1 of my ears NYTimes veto XV squared X divided by square root of 1 mind a square peg acts that's just equal to 1 that desire and really need to say it's equal the 1 I could say text of us x equals 1 so refined we find it X Prize is eagled the one-liners be squared all the square root of one-liners squares which is just equaled the square root of 1 be square some of the more thing observer says that at the end the time meeting to say along this surface of synchronous times here the choice of the meanest are separated in his frame emboldened framed by a coordinated distance for a number of meters in the moving frame which is given by the square root of what might be squared another words meterstick looks in moving frame of reference to be a little shorter worker will start compared to what was arrest notice that they really talking about 2 different things meterstick arrest we're talking about the distance between this point in this point as measured by stationary meter sticks in the moon frame with talking about the distance between this point in this point measured by moving measured bribes and that really the same point of space and time he assumes no real contradiction that terror that 1 looks shorter than the other you could really up at the newly other calculations of the calculational leave it to you is the calculations opposite calculation think of the moving meterstick now he is molding meter it as 1 unit lawn where we know about this line here is 1 no it's a motor it's a meter stick 1 unit of late in removing reference frame that means it's X prime equals 1 In the moving reference frame is 1 meter my head on the other end the leaders of the tail and the stick resorts X Prize equals warned not X equals 1 and then the observer and arrests sees the meter the moving meterstick being this long so as to calculate what the point what X at that point you could do it and you find out that it's also shortened by square root of 1 might be squared the moving meter sticks looked short the stationary reference frame the stationary sticks looked short of the moving reference there's no contradiction that is talking about different things the stationary observers talking about links measured at an instant of his time the moving reference frames talking about Lane and of the other side of the talking about different kinds of things different different notions of what they mean by way of the history books well did look forget luck but be very precise about my home my meterstick removal of it and that some instant apparently you people however synchronized clocks and so exactly 12 noon somebody is going to see the pale and the meterstick past them and somebody else is going a front end meterstick pressed them so together watches synchronized and they have been instructed each other at exactly 12 noon see whole along here Is there contiguous are adjacent to the EU to ends of the state and the nifty done that compare notes and say how far apart you that's exactly would forget saying that looks shorter mean something very specific with synchronise watches and synchronized clocks just ask exactly where the ends of the meter stick war in your reference frame had an instant of your Pa I don't know know exactly the same thing I have on my body is lined up they also have clerks we've also done the synchronization my synchronization are the same as your synchronization but we've done our synchronization you hold who was holding stick before I was I was holding stick a 2nd ago now 1 of you people's holding meterstick a minority past and I instruct all my people at exactly Our 12 new look at the stationary meters thick and see how far party and in both cases to local little bit short from the frame of reference were in which the meterstick seems to be moving if the ever was but it would indeed but since we leave established in made sure that the relationship between X isn't keys and X primes and deep primes are essentially the same relations except with the Eagles to minors we we know that that can't happen the mathematics of each half it will look exactly the same as the previous 1 so you not so it check this out for last thing to it talk about before we move on to the motion of particles young as far as I was saying 2nd me this is not a meter we see indeed well of course that was battered the Fed's are Joseph time value could use a useful for time the time interval that it takes like to go 1 meter but they
nuclear equipment yes you're right you're right but I'm also right what about time dilation time dilation can the pretty much work the same way supposedly have a moving Clark surprisingly have moving Clark and for the moment I'm going to assume my clock is moving with uniform velocity and particularly have a clock that's moving with the moving observer my Clark by Clark moving the clock is right sitting on top of me it's not 1 of my friends clerks of my Clark and is along a trajectory like that it is a question or arrest at the instant when my clock reads To Meet me personally when it reads T equals 1 works said they would be a more general note people's 1 when it reads to meet people's won the question at that point was a time in your reference sell more I say Siegel's 1 anointing Eagles 1 adorning T prime equals 1 Michael prime so I had my standard wristwatch been made in the Rolex company It's a good watch it works well Our the guy so good to me for a 25 boxes has V got the art from a hasty right they fill it works well so it's prime equals 1 1 2nd 1 unit prime equals 1 I want to know whacked cars what the corresponding value of is all along the horizontal surface horizontal surface is a surface the EU called synchronous that you call an instant of time I know well so we know we need 2 things in order to pin down there so we also know that X prime is equal to 0 x prime is equal to 0 all along here and a prime is equal to 1 let's see if we can find out what series throughout over its transformations X it is a group X minus V T all the square root of 1 might be squared the that 1 and the other 1 is 0 I'm sorry is not the only 1 who won the other 1 X is equal X prime plus VT prime the square root of 1 squared and TV is equal to keep prime plus V and X Prize office where a woman's face court I simply you lose the other forms of the Lorentz transformation what I write Exon tea in terms of a permit prime and to do it replaced V might be by my services the other Lorentz transformation going the other way telling me the Exon in terms of prime innkeeper OK subtlety Prime is equal to 1 x Prime is equal to 0 and what I want to find out he that's right over here let's see what it is keep Prime is equal 1 X prime is able to 0 divided by square root of 1 might be squared bombs right so that time over here is a bigger or smaller than 1 a bigger and bigger time interval Ali here is longer than the time interval measured by the moving observer by a factor of square root of 1 might be squares is the origin of the twin paradox the twin paradox involves reversing the motion Of the twin over here the Twins 1 arrest 1 moving him back together again and they ask How much time has evolved in the reference frame while 1 of these reference frames is not inertial frames 1 which ends in goes around but we've already calculated how much time there was between here and here it's 1 or squared 1 might be squared another words we found that the bad I was a little bit less time along here that along he same thing is true for the 2nd leg of the journey the implication is that trying measured In the moving frame is a little bit less than the time measured by the fact that In the frame of reference of twins stays at home and so the twin that stays at home is a little bit older Anwar who are of reusing here a
basic idea that every clock participates in this thing exactly the same way and in particular the clock that has to do with biological aging also slows down but this is just a statement that firm whatever we said he replies to every kind of Clark not to of Rolex the Air Force said Wednesday at a mail last season offers guests is now than interest and we're yes that's right that's exactly right no it's not it's not right but I mean it's exactly something that you might be puzzled by a hug so now let's talk about exactly that with stock remember exactly the issue of the hypotenuse shorter or war longer than me the sides of year the way it goes away back less I passed for but wrote this we will not remember these 2 frames and not symmetrical with respect to each other 1 of them underwent an acceleration really not symmetrical rivers the twins will get send that sent back that he was the 1 who got a move went on curved trajectory because experience an acceleration was not that twin was not in the inertial frame that cod which you will think of it that way you could just calculated or you can think of it as the cause of the acceleration feel like it's a legitimate way to think about but it In mathematics is pretty similar a similar to the statement that if you take 2 points in the plane when there just 2 points in the plane did you connect them by a straight line you connect them by biker don't be surprised at along the straight lines not the same as the link along the curve by the kernel is in fact the lake along the curve is of course longer than the along the line in the relativity case we actually found that the time along the curve was a little bit less than time along he something going on that's different than ordinary links bath that's a similar phenomenon that that the lake along the curve is not the same as going along the straight line straight lines law shortest distance between 2 points well the 2nd relativity erased diet the IRA's Leon as was mentioned the time along the hypotenuse it is less than the time longer the vertical axis so something funny going on and there was talk about that funny thing that's going on in the let's say let's just think about Euclidean geometry for minute thing about a plane let's think about different coordinate systems on the plane coordinate system just related to each other by rotation that was the origin of coordinates the a scoreless Exon Y this is now ex prime why Prime This is not a moving reference frame now this is just the rotation of coordinates and I'm using it to make a point but on let's take up point somewhere in the space of a year as the point x the 2 or referenced but the McCollum reference frames a really mean court systems the took systems do not ascribed the same values X and Y to this point obviously the X Hawaii Of this point here are not the same as the ex- prime and the white pride in their elated to each other if you know X and Y and you know the angle between the coordinates you could figure out X private why Prime but did not the same but it is some something things but is the the same no matter how you calculated it with EU calculated in the prime coordinates only on prime coordinates and you visits what the distance from the origin everybody will agree on the distance from the origin and factor also agree about the square of the distance from the origin so what is a square of the distance from the origin from the point of view of the aren't on prime coordinates while it's just Pythagoras is theorem it's just X squared plus Y squared as the square of the distance where about from the point of view of the friends another words the quantity x squared for plus Y squared for a point here is in invariant invariant means that it doesn't depend on which coordinate system you work out its invariant under
changes of coordinates which in this case it is rotations of accord solely say then that X Web plus Y squared is invariant quantity and to saying the matter what frame around measured in question is a analogous quantity associated with the Lawrence transformations here which is an area in particular he is something we my guess what's try it out try out and see if something similar going on and he expects his please use 1st the point is characterized by TO next but also characterized by deep prime and X Prize and as a team next authority Prime and next prime has coordinates neon moving frame and the moving frame what is the relationship between them middle rents transformation but we might guess we might get is is something similar that's invariant let's make a guess that its key prime squared plus X prime squared equals T-square corn that's try out see flow we know in terms of ex ante we know privately prime would work out with David Strom OK so people of Saudi prime squared just take this equation over here a prime square Harris there's t squared it has plus 3 square X cornered and has minus 2 0 v TX the lighted by by 1 might be squared right Estee prime squared What about X prime squared that has X squared plus squared he squared minus tool v FedEx right question does this sequel fee square parsec squared L it was not that a particular 1 thing you could see and immediately is that X Teeter year Eds still EXT term here they don't canceled there's no XT here they can't be the same in general and they are not the same as simply not the same but now you look carefully at it you might notice that if we subtract them this V T X terms do canceled OK let's see what else happened so let's try subtracting Mrs. keep prime squared mightiest X Prize square and what's not prejudice but what's on the right-hand side he would find out what's on the right he prime squared minus X prime squared users different right it contains the movie cross via disappears they cancel because of the minus sign and how that X squared with 2 x squared has x squared hairs squared minus 1 squared minus 1 and then this past T squared and he squared has one-liners V squad 1 now my years squared although abided by 1 minus be square or might be squares canceled and you just get he squared minus X squared Rice's minors Miners plus OK so it discovered something we discovered that an invariant quantity under any Lawrence transformation the combination keep prime squared minus 6 prime squared is the same as he squared minus X squared it's almost like Pythagorean theorem that there a kind of notion of distance from the origin here which is composed Out of the base of the triangle namely X and the height of the triangle that stays the same if you do Arens transformation Ruiz a few other coordinates other coordinates can calculate X prime anti-crime you find out that the combination he squared minus X squares invariant because they the BX women's T-square incidentally women's consequently square quit picky picky choice and this is an important thing tho not very important to know what the invariants are weak components of the vector of a component of the displacement are not invariant they depend on which coordinate system using things which are invariant things which all observers agree upon are in this case we are given a name in a moment of quality on the space-time distant space time distance which is defying with this funny minus sign is invariant all observers must be important quantity OK let's Celia figure out what it what it really means in particular let's suppose that between here and here that we happen to be talking about the motion Nova was that's assume works with the stick a special case let's take the 2 points To be along the the motions of removing observer OK then normally observer what is steep prime squared minus X prime square well X at this point is just plain 0 this is point DEC's prime equal 0 along he so it is just keep prime square factory can write immediately that keep prime squared must equal squared minus X squared why would say that this point over here is point Prime and X Prize but such practical Zero so it's distance From the origin keep squamous expert just Twinkie squatted on the other hand be classed as quantity is invariant it must also be square where sex caught on it was quantitative Brian it's the reading of a moving Clark Clark will along here which starts at a price equal 0 over here will read prime over here that's what is the more it's reading of a clock that has moved on a straight line between 1 . 2 of a point that's how many units of time have occurred In the frame in moving well in the frame the connects these 2 points is called proper time it's called a proper time in everybody will agree on what the proper foreigners along trajectory they were calculated the
same way they will take the took points and calculate the squalor of the difference of the time interval between the 2 point and the space in the wall between the 2 points in any frame would have a picture frame and you calculate he squared minus X squared for the sporting and that tells you how much time involved between the clock reading over here and the clock reading over here called the proper time of that clock but you could also just think of it as a measure of distance along a line connecting 2 points it's not a measure of ordinary distance to measure of space-time called the proper times it's his mind is assigned here which gives the triangles the peculiar property that the hypotenuse is shorter than the height of the triangle itself why is that the minus sign makes smaller next more I put it In many other ways this function as the notion of a distant space time between 2 . 0 3 questions about young of us there Oh yes both of us federations we use stepped up up up up through the angles slowed down and that for that but this is a major dailies that lot on our rare but we never used any notion of links in the sweetest used court we you never really get that's right you can't you can't use ordinary notions of Euclid geometry space but still we were very careful and simply defined coordinates in terms of how clerks would be held and we do rollover the invariant the wrong invariant and that's it that's the American right so relativistic space time is current like would you clean space time but with a funny difference that you take the difference between them he squares elect squares and that's the thing which everybody will agree on a new could think of was cut right now there's 1 funny property of it what about the pouring let's take foreign goods go back this here let's take 1 appointed the origin critters and let's take some other pouring just use ex ante let's just use text team X until another point that ex ante is possible that the distance between these 2 points in these at different points as swans of the origin this will not be origin is it possible that this notion of distance just gives Zero distance between them even tho they are not the same point in a 1 0 Euclidean geometry if the distance between 2 points 0 0 the points are the same place that was not and this crazy kind of geometry with a minus sign supposing the distance from the point of origin 0 doesn't mean that means that X squared minus T squares equals right x squared is quick can be equals 0 without being 0 that X squared plus Y squared equals 0 0 implies that both X Y 0 0 thank you x squared is positive why squared a positive 0 0 why squares positive 0 0 if some of these 0 it means either 1 of them both of them have to be 0 that's where my squares equals 0 to saying such things would say this says E. The X is equal to T or X is equal to minus 1 or the other physical or was sexy host team means or like ready it means that all like could go from this point that point they connected by a possible like trajectory saying for the other 1 of Mexico's my industry would be over here again lyrate angle from 1 to the other if so 1 of the fact fears that if the distance between 2 points is 0 it doesn't mean the 2 points from standpoint it means that white break connect them still nevertheless it doesn't mean that the proper time between the 2 of them 0 another other words if a light beam could carry a clerk with it would be Zero ticks over Clark between here and he Clark said more with the speed of light simply don't take stake they don't take young will walk a young military pressure on that that depends on whether 2 points right here at 2 points which could not be the end of a moving object of a clock y because the velocity and where to go from here he would have to be greater than speed of light who is still nothing vessel was speed light then this can correct these 2 ports it that's the conditions Our basically that X is bigger than here Exs lessons effects is less than light breaking from 1 point to another sex vigor if we're talking about 2 points In this type of configuration called in award for it call the relations between those 2 points space like space like means that the space component is bigger than the time compartment or better yet that X squared plus t-scope school nights when minus squared is greater than 0 places up here so that a trajectory can connect those 2 things he squared minus X squared is greater than 0 this is called space like this distance and this was called Prime right arm if the to appoint a space like connected then you would call this quantity X square my squared you would call the proper distance between him if I'm like separated the new recalls he squared minus excluded the proper time so you could think of it also has the finding the proper distance but only they have 4 the key component is larger right were closed 30 and there's only 1 4 to bury um I was going to talk about some relativity paradoxes but Perpich we're tires 8 from Glausier thing award mention I forgot where questions yes there is a lot like the disease or yes absolutely so so where
Terwilliger privacy in this equation we could put it here see squared T squares or we could put it in the denominator here a distance divided by a velocity is a kind of this would be in this would normally be call proper time as units of time we got decks where miners he squared now we put the C squared here with Nicolas proper distance but that's a pure convention doesn't matter our where you put the C squared would be put here here just make a maker of things consistent the definition of proper time is usually taken to be he squared by the SEC's square overseas square please where were square Rwanda we were both Sorocaba time squared up from firm square thank you that would be proper Square got square it suffered Pythagorean theorem God knows distance is to take the square root of this thing must at times don't take a square reduced to be but there yes the proper Times square root of p squared minus X squared thank you support is a while I see on the trouble it's a rotation by the brand imaginary angles imaginary Mr. imaginary young it is literally a rotation by imaginary angle so yes it is sometimes useful to think that we're head the list of his was set to match their layers of from I I know that I know the history a little bit of of course is to have a history is opposed by Udo by a participant and know the participant caught on fire and I think there's every reason to believe him on I think Einstein was motivated by Maxwell's equations he took Maxwell's equations to be a law of physics values only 16 years old at the time we started thinking about it according to himself in 16 years on Wignall about Maxwell's equations server to give rise to these wavelike solutions and a solution was removed in a certain way and he was puzzled because he tried to figure out what would happen if you moved along with the light rain then you would see a static electric magnetic field away like structures move in he knew somehow that that was not a solution of Maxwell's equations sold according to him that was a motivation that he was puzzled by the idea of moving along with reproduced spilled like but I think beyond that he knew Maxwell's equations released at some point he learned Maxwell's equations and took Maxwell's equations to be laws of physics and Maxwell's equations say like moves with the speed of light so I'm inclined to believe that that was the correct history that he didn't know it time leader Michael experiment between rows know but city according ozone the historical work testimony about he did not know what I think is every reason to believe you he was truly small had our very well the logic makes sense that of Maxwell's equations Our and in on my own terms we would save differently you'd say here with his Maxwell equations and symmetry of the Maxwell equations some set of coordinate transformations for which if you do them Maxwell's equations have the same form every reference take Maxwell's equation which contain X isn't these K and you plug-in you know the old Lula and role for the Galileo X privacy X miners V T T Primus equal to teach you find out that the new prime frame of reference Maxwell's equations change therefore form they don't have the same form they had originally you plug in the Mac's video of the Lorentz transformations you find out that the Maxwell equations and no quorum are identical what they were revealed court so I I think In modern language I think what Einstein did was to recognize the symmetry structure of Maxwell's equations was not these transformations but these trips but was all encapsulated in 1 principally in that really know Maxwell's equations all they had know was that Max was a quiz were law of physics the law of physics said that like move of the velocity from very good his work with emotional white libraries young is frogs the Reds Fitzgerald transfers of that there is no he thought about that for the same purpose he didn't know about terror Michael but he envisioned them differently on the same equations but he envisioned them differently he envisioned as effects on moving objects caused by the emotions so he envisions that object moving through the so-called ether because of various kinds of the pressures would be squeezed and therefore shorter now worth you Ronk I suppose you could say in some way or another referee he was wrong but he certainly didn't have the vision that Einstein had of symmetry structure of what was the symmetry required of space and kind in order that agree with the principle of relativity motion of the speed of light arms nobody I think would say that the including Loretta would that Reds did Einstein he said further will instantly goes exact he thought was a 1st approximation of piggy moving through a fluid of some kind we get short that the 1st approximation would be the rents contraction and some high orders you would see some other things happening so he fully expected that Michaelson morally experiment was not exact he thought it would be higher corrections higher powers of velocity overseas where you would see discrepancies 90 cents looked as really offers expresses the principal so he a principal out of to likely to go away if by next I will talk about the motion of particles and how relativity optional particles would for more
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