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Special Relativity | Lecture 2

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last time we worked out basically rents transformations relating to frames of reference on weather and I said it but missed it now will mainly dealing with problems in which all of motion and in particular the relative motion of different observers is a long 1 particular axis we didn't try the things go fully three-dimensional in the picture you go have head is we have a long railroad track one-dimensional sunset of observers is sitting still at the station because Yukos the station observers say 5 and other observers are in train and then moving relative to each other along the one-dimensional axis when worry much about the other taxis we we I 1 of 2 things about how we talked about how you relate it coordinates 1 observer relative to the other and particularly wrote down warrants transformations they on the basic hypothesis of Einstein's that all reference frames see the speed of light exactly the same factory said would be unit so we used all observers see the speed of light being 1 that course choice Of units Oregon years light years on a 2nd white 2nd whatever choice of units if we choose and correctly we can make speed of light 1 and that simplifies equation of course if we really want the plug even the real observational physics we might want to use the fact that the speed of light common units in the common units the experimental physicist would narrow use is not 1 its 3 times tend to be a and some units and so we would talk back the speed of light there's always a unique way to do that the unique way to do it is to make sure simply stated that you modify the equations by appropriate factors of the speed of light see so that the equations of dimension we consistent Our I will go back and forth mostly I will use the speed of light equal 1 but every now and then just to illustrate a point I was stick speeds of life and there you could go through the equations sells OK joy observers 1 moving down the axis with a velocity V relative to the stationary mowing down track of velocity v wrote the stationary observer stationery observer season observer of Warrington v units of space per unit time leaving velocity and of by symmetry just by the symmetry of a problem if we believe that all coordinates frames are equally valid after we say relationships the same kind of Lorentz transformations will relate the stationary observers coordinates Tony moving
court the two-way street stationary observer ascribe stationary coordinates moving observer ascribes link warned his coordinates and they can each be related to each other reciprocal I'm a reciprocal relations truly reciprocal exactly the same relations X set their whereas if I were moving to the right you're right a file of moving you alright you would say my velocity it is positive you would be moving to the left as far as I was concerned I would say your velocity was negative and soon relating the 2 frames of reference the only thing we have to remember is assigned change of velocity when you go back and forth OK so for example it X probably it is coordinates as scene In a train is a train or a yellow structuring trained cases of observer in the trained as an observer on the tracks or observer the station is observer in the station it observer on the train and observe the train is moving with velocity V 3 The Observer and the train as stocks and scandal on floor year performer great observe also has a pint beast Clark likewise be observer at rest arrest respectable rests with respect to the station the Observer arrest with to station also has played out and also has and make various comparisons in hours worked in the event which takes place an event which takes place in the event means a event happening at a point the space and a point of time and the words some point of space time I like to think of it as a flashbulb exploding someplace going off some someplace doesn't matter whether it's in the train or outside the train but in the train for committing flashed bomb off over here at a time that these stationary observer reckons the be punchy at position Exner opposition X means coordinate X in the stationary reference frame extra at at time t according to be kind piece of the stationary observer so the stationary observer describes to coordinates extant cheat and observer describes the same and coordinates X Prize enterprise work at the last time the relationship so necessary between XT an expansion team Friday such that everybody will always agreed the speed of light is we want learn down quickly X Prize is an old French transformations next time musical acts miners VT and no 1 would recognize that body when the recognized was a square root of 1 might be squared downstairs and if you want to Pope back the speed of light it goes right over here put in and take it out these squared overseas cornered and of course if the velocity is small by comparison with the speed of light is a terribly tiny correction was just 1 might be square and Prime is equal minus V. X. divided by the same score 1 might be squared if we want it and the other 2 0 directions and particularly the directions out of the board and vertical in other words a directions perpendicular tracks we could have been very simply perpendicular directions don't change under a change of velocity along the given access if the changeable at velocities Of the 2 frames wrote the velocities along the x-axis than the coordinates are unchanged by Prime very wide equals y and z chronicles but I won't write it better
if we needed will use it we can avert these relations misses simply a matter is nothing the sophisticated going on here this is simply a matter of solving for exit in terms of next time and keep trying but may remind you what the result would be a it would be x equals X Prize last VT Prime provided by that Assange square root and T equals prime plus V X prime divided by square of 1 mind squared the only difference only asymmetry is where you so of velocity over here the sign of the velocity just to account for the fact that the road velocities are opposite direction you could also really off if somebody view this form for of the relationship between the coordinates you could easily read off what the relative velocity between the 2 observers if you look at the X equation here and you say they only observers coordinate his ex time equals 0 right at the position of the origin of um coordinates inside the train at Prime equal to 0 ext time equals 0 corresponds to X equals Vt it just look at this and you say molding observer pride of their effect on moving the manager over Our is at rest lowly position next time arrest of equal to 0 excuse me X prime is equal to 0 the origin of the prime coordinates How response that X equals VT you not to know about the nominated area just look at x equals VTE that specifies that chronicles 0 that shows you X Eagles bt because of the relative velocity between the 2 of them is by now we want to do another exercise the donor exercise is to assume there's a 3rd observer of the fair observer is moving relative to the railroad car of the train is going Kitty car inside the train but his relatives thirdly passenger with score and the passenger is moving relative to the passenger with velocity you want a passengers it is a little kitty car will kid in the car peddling then are via a video I O of the train with velocity you would relative to himself a question What does the stationary observer ascribed to a car 1 velocity does the stationary observer unless all this is just to use that I know of no waited DSC answer the answer is some velocity widow given name we can call at velocity W I don't die you could use V-1 veto would be free but I hate subscripts and so I prefer to say the velocity of the train Is VU relative to the stationary observer the velocity of the Colorado the train is you end the velocity of the car relative to the stationary observer so-called W.
stationery observers car move with velocity W and the question is what is w in terms of you would think this is just the use of the logic that of logic is but the relationship because 1st of all we should give Fabio be also excuse there also coordinates in the car moving with the the car this could be a 7 sticks laid out on the floor of the car and also kind peace that the driver of the car has was really sets of coordinates in this case ex ante are the coordinates of the stationary observer uses prime anti-crime RV passengers coordinates bid for a 3rd set of coordinates which would give them the name X double prime anti double prime X double prime and he doubled and other coordinates that a kid in the candy car uses to describe things relative to the position relative to ozone no frame of reference right now just it's a little bit of logic say we know what the relationship is between the double prime coordinates and the single prime coordinates those are related by velocity you you is the velocity of the double primed growth the Frank would silica right down those relationships straightforwardly but we don't need this over here but them down here X double prime is equal saying that kind of relationship want that kind of except for put financier and set of v e we will use the relative velocity you serve as X Prize you keep price divided by square root of 1 mind this EU square services tolerance transformation between the double time frame and the single primed for a prize minus you X foreign over of warm Tigers you square writes about how we following the connection between the double prime coordinates in the car and they under unkind coordinates an arrest in the railroad station and that's simple always do is let's take this equation over here with focus on this 1 of kind equation also works out very nicely but must focus on the space equation he and we know what X primaries In terms of ex ante how we know what time in terms of ex ante so all we have to do is to plug a do some good Rice paid much O X Prize is X miners VT divided by the square root of 1 minus V squared that just X Prize here so far this research price might this you keep trying must couldn't prime prime it T minus V X and all thing again divided by the square root of 1 mind square now so far I've only written this in there so I have to put in the denominator to put on the Data I just put another factor of 1 mind you squared 1 minus you Square Square of 1 square so that is the relationship between X double Parnham X writers Will more simply let's focus on what the votes some big denominated dominated involves a product of the 2 square research fibers per square inch square makin a ride out what's inside the square root it's just a product reviews to square respond interested in the numerator really the normal writer has an X and it has a plus you will be X X here and plus UV acts was plus UVX as a minus sign you a minus sign X times 1 plus you 1 course you read x more about t t you will multiply my guess viii plus here all kinds minus V and minus you thirsty and this is the answer but Favre transparent what it means but all we need to do to figure out the relative velocity of the double time frame relative to the untried adjustable exactly what we did over here if we want to
find out what extra nickel 0 means just set X might equal to 0 and tells you that X is equal to VP tells you have a fellow at rest here who is an expert legal 0 how is moving Road prime frame we do it exactly the same thing here said X double Prime will be 0 X double time of course being 0 means the position of the kid inside kitty Garcia X double prime equals 0 is the same thing as the numerator here being 0 but leader who cares about the denominator it's not a but we can set the normal rate a quarter 0 here and that will tell us under what circumstances X double prior musical 0 so what does it say X double prime is equal to 0 when 1 plus you X is able to you'll plus she said this equal to this To make the numerator 0 war if I divide by 1 pursue me because me X is able to you'll plus v all over 1 class you really fit so what is the relative velocity so I guess what it is the velocity this tells us these excellent teacher Jack theory of the kid in the car here the trajectory of the shot over where it is in the kitty car is X is equal to you closely over 1 plus UV times that corresponds that X double prime equals 0 but another way to say it is just that the stationary observer the kiddie she's the cake are moving along with velocity you plus veal over 1 plus you so we now know what W is that's exactly what W is W worked out we can now we can write it in the form of the form X double is equal to T minus W X provided by square root of 1 minus W squared and he double price is right there right now X minus W C & C minus W that makes Over same square root from the square root and modifying W. lens W is just that's that's how fast Katie cars moving as seen from the stationary Frank Searle yes you have because we sets eagled 1 right but that the back admitted young will come back and get your era you're having a step ahead of me fokker 5 so X is accomplice of W speed W. physical to double EU plus VAT divided by 1 plus you need enough I want to restore the units would that this could answer question again if we set the speed of light equaled 1 Of course we're working units in which velocities of dimensionless but we want restore the dimensions we simply look at this equation we sailor W it was you pass me that's dimension with find its a getting 1 U V which Curia but we restore the images by putting in square 1 SUV oversee squared UV oversee squared is damaged so this is the year equation with see being restored was the Newtonian equation early corresponding 3 Einstein's equation wealth stationery person sees passenger moving with velocity you passenger is cake are moving sorry stationery observer sees passenger moving velocity v passengers cheesecake are moving velocity you the answer naively would just be you plus lead but let you positive view but divided by something which is a lot as you and significantly smaller than the speed of light you ve overseas squared will be very small world will put some numbers of Ameritrust protest about but as well as you and these are small compared with the speed of light that this will be negligible it's a product of told ordinary velocity 100 meters per 2nd or whatever it is you are putting divided by 3 times tend to be 8 meters per 2nd all squared this is a very very small number it's a very small correction on the other hand when you and we get up near the speed of light it can get very significant but to at 1st but still a case where you and we are small velocities compared to the speed
of light they're weaken me that will remember that you would mean velocities measured in units of the speed of light so viewers via measured in units of the speed of light and let's say for example supposing you is able appoint all 1 1 101 that's descent of the speed of light and let's views also . 0 1 The both 1 cent the speed of light this turns out to be . 0 2 without which known would recognize provided by 1 class and now you is pouring all who all wore old ones who do the right yup and all 1 is a very small correction now this is a very sizable velocity incidentally you being 1 per cent the the speed of light and that's us that 3 times and 38 3 times that the 6 meters per 2nd so it's pretty fast but the correction is a small 1 plant something notice is a little bit smaller then what would have estimated this number here is a little bit bigger than 1 and so amid the murmur due to the small but took to the other extreme but suppose you win VER 90 % of the speed of light we do 99 % muscle glamorous that you and we are . 9 V equals . 19 Newman said Ricky cars moving faster than the speed of light relative to the stationary observer the I probably would have said 1 . 8 times the speed of light you headed east numbers but Einstein would have put in the denominator here sorts you will get you get . 9 plus point 9 that's 1 . 8 by . 9 square . 9 squared it is 1 . 8 1 slightly bigger than 1 . 8 result is that the net velocity is slightly less than 1 another words we have not succeeded in making nicotine gum cargo at the speed of light even tho blah blah blah you you know the rest of the story this is the answer to the question What happens if an observer's moving faster than the speed of light well you could ask that question but I think we are pretty well protected against abuse of people moving faster than the speed of light is the way they are made to move is relative to some previous frame of reference moving Speedo slower than the speed of light sources Norway by cutting another observer inside the speed of Corolla cars here making illegal 90 % of the speed of light etc etc we're never getting a fast speed of light so it consisted throughout a consistent thing to say all observers moved slower than the speed of light even tho they can move arbitrarily close to the speed of light relative to each other any combination the net result will still be smaller than the speed of light I don't see what kind I suppose you will be all wore C in 1 of the same thing then you plus 2 and 1 plus you too so the net result is speed of light again serve to moving various each 1 is moving very very close the speed of light the net result In other words of the kiddie car was moving relative per passenger very close to to the speed of light passengers moving both of the station very close to the speed of light result will be the key cars moving out of the station and even closer to to the speed of light but not in excess of OK that's is using him last fall the train I assume trained at us a glass walls don't see how that mixed in we're not talking with talking about how appearances a lot with talking about her measurements of phenomena of by Peter sticks and by well-designed clocks correlate with each other what somebody sees is much more complicated for the simple reason that wouldn't event happens like has to come from the event and the could be much complicated would you visually city where wrapped up in a usually see we're talking about correlating these locations and times of these events In frames of reference which is defined by meter sticks an arrest relative to observers and tiny pieces which are also expressed relative and doesn't matter what kind of walls the car has B a B transformation woes are universal OK now next the next thing we talked about last time woes the notion of proper time of proper distance a property the I'm just like you quickly about bad very quickly therefore have a event taking place at point ex ante we found out last time that there's an invariant notion of separation space time difference or space time distance between the proper time and the proper term of a proper the purpose of Pieter here Eckstein the proper interval between them is called Powell and defined by How squared is equal to T squared minus X squared and the interesting thing about 1st important thing about it is the same in every reference frame of reference frame is moving them we would use kind coordinates purpose in quantity here keep prime squared minus X prime square it's also recalled the double prime squared minus X double prime square into to also observers agree on the value of this interval between here and here Tibet don't agree about the coordinates themselves but they agree about this notion of property at the proper time it is also the time written by Clark moving between these 2 points clock is suit equal Coetzee 12 noon at this point and moves along this trajectory than the kind that it reads at the end of the trajectory is the proper time we
worked out time relation last time so what I did wanna say about this I want now put Back the other 2 coordinates y and z y and z and let's put them back into the game for a moment for back errors there is another kind of transformation that we can do will not just a transformation between 2 moving coordinates moving along the X axis Villa can also consider rotations of coordinates for the moment not even think very much about relativity that's just talk about 2 different coordinate systems related by ordinary rotation with respect to whichever show the stationery observer might have 2 different coordinate systems 1 oriented along the x y axis of the 1 oriented along so prime why Prime not the matter saying Prime some others set a vaccine located foreign angle wrote To original ones what happens now we can forget part of the prime importance for the moment what about this x squared here this squared is really the distance From the place where the ever where the clock started the Place where it ended up in Leon Friday court which supposedly take into account the other direction scholars why for example then this becomes a plane here this point might not be located directly over the X-axis it might be a point in space time which it is not at the same value of acts as the origin here what then is the interval between here and here the invariant quantities Willis is actually fairly simple as long as the event is located on the X axis it's squared minus X course that it's not located on the x-axis only make retention than what was originally x squared becomes X squared plus Y squared Posay squared becomes the spatial distance between the origin and this point over here is really becomes minors X square minus 1 square mile 3 square but that's ever so if we're networking strictly along a one-dimensional access the invariant proper time in between a restart of a clock and the place where the clock gets is given by T. squared minus the squalor of the spatial distance which is with Pythagoras sphere of Pythagoras clearly apply to X Y and Z and that's the notion of proper interval Nassau or more if we make any combination of low rents transformations and rotations of coordinates will always find any coup inertial frames which agree at this point the way he will find out the squared is invariant is the same in all inertial reference frames and did a very similar the idea that Back in ordinary Ukrainians geometry different coordinate taxis will describe different coordinates told point in space but they will always agree about the distance of a point from the origin here it's a funny kind of distance with a relative different signed between the space components in the time and that is probably the most central factor by appears to be that this combination it is in area that's really what it's all about OK solve rights keep that idea in mind and introduce a little bit of occasionally get tired of writing X Y and Z and we try the convinced the notation so what's convinced the notation standard wrote no pitcher yacht just going back to what it used to be that point plan that very much an EU that the car that a flashlight might can ask a and worrying about these 2 measures in order to try pick 1 up there is 1 of what I see is the speed of light that means it's warm foes these 1 0 hour which when your 1 great you want archaic make fuel 1 that's 1 plus provided by you was 1 1 plus they advances 1 the speed of light speed of light that light gray woes of the speed of light you 2 of you going out the size that portion the blast walls trying I believe Dokic if you understand that I have let's talk about light rays how light rays moved like let's go back for the one-dimensional case and 1 them once based 1 upon go back light rays moved for example along 45 deg taxes like this that means they will move from the origin of the point X P E but only if X is equal to take just saying the light was velocity 1 in a certain time see the distance of moves is equal to that time so it moves to appoint our same coordinate that means that he squared minus X cornered squared minus X critical 0 4 that the space time interval or college space time intervals appetite and a number of different usages it is 0 that's different then ordinary you could be in distance you could be a distance of 2 points have genuinely 0 distance between them is sitting on top of each other in spacetime if the points have space-time distance of proper time equal to 0 between them that simply means they are related by the
possibility of a light Brady going from 1 to the other now we introduced additional coordinates y see squarish donor was originally x squared the distance the polite being traveled along the x-axis well up the square the distance obviously become X squared plus Y squared poseuse square square of the distance and that will be equal to 0 for a light ready so the emotional like crazy please squared minus X square 1 square mile squared equal to 0 again squared is equal to 0 how so that's 1 concept of How light removes it moves along trajectories such that the proper time along the trajectories equals 0 photons move that way drop picture favorites a library moving to the right as a 45 degree access to a right 45 deg lying right by removing left moves exactly the the same way except in the backward direction whereby a lightweight moving outward while I'm on the same way except that a 45 degree angle outward direction more generally we would drop her corn now I can't draw the full 3 dimensions plus time two-minute dimensions draw on the blackboard but if we had only dimension X squared plus Y squared instead of with get busy square he would find that the motion of light rays is such that in space time and singing along Colin created by 45 degree of light rays coming out of the origin that's COBOL icons like Cone is set of points of light can I arrived back if it starts at the origin of the motion of the light and this should be called the future like called the future like Cone is all of the places light can get too starting at the origin is also a thing called the past like all the past light cone is all the places that can say the light great toe the origin of the future like owners all the places that the origin concerned rate to and the past light cone all the places Dickinson the light weight the origin but it just the because turnover on not past like on the future and this is terminology terminology is often very helpful but that's all losers terminology young steady but the military there must be some areas defeat Everybody agrees about the time interval between 2 and the time interval is just right that's a good point are prior were getting closer to the start of the concept of for vector the most primitive count the most basic example of a vector in ordinary three-dimensional space I am not talking about the kind of vectors we talked about last quarter we're not talking about state vectors and quantum mechanics we're talking about vectors is based on the most the basic most of the example of a vector is arms Our interval between 2 points in space due in points as vector which corrects sector could measure would be could have to do with How far somebody walked in the the sector as a direction that has a magnitude and if we wanted to we could think of it as a vector beginning at the origin an ending up someplace else doesn't matter where it begins but the vector This is the vector original at around the same vector but we can think of it as being arm and excursions starting at the origin and ending at some point acts and has coordinates in this case X Y and Z the location of a final point always could call them X I I mean 1 2 or 3 representing X Y and Z 3 coordinates x I would really stand for x y and all are X 1 next to an X-ray now we have another edit component worry about not only do we want to know where an event but we want to know and what time that is supposed that measuring space and time relative to some origin that they have the edge in another another words the vector becomes a 4 dimensional object with Thailand component space components the normal occasion for it is the represented 2 different ways the 2 different ways we can represent X Y and Z by calling them X New Newell goes all the possibilities usually it's normally 1 arranges them as c x y and z X mule and what new run over what are the values of I went because everybody says 0 you got to do the the sobriquet right for whatever reason historically T was not considered the 1st coordinate X was considered the 1st court while was considered the 2nd Lindsay the 3rd you might have thought but time should be the 4th comport component for whatever historical reasons time was thought of as zeros component of the stands for x 0 which is tied X 1 which is X X 2 which is why Annex 3 which is what about be space time distance between these points the proper time that's T-square minus sex with Square Square tower squared is X squared sex with minors wife square mile disease squared but we can also write it as not squared minus X 1
squared minus 6 2 squared so 2 squared mob of our sorrow that's just notation is just imitation when NBC and knew that means the index runs over 4 possibilities of space and time when and that means the index runs over only states Hi status for space in newsstands for space and car exit just as X I can be thought of as a very primitive version of a vector primitive basic version of a vector space X mule with 4 components becomes a notional for vector just as vectors is transformed when you rotate coordinates for vectors transformed when you arrange transformed the space where you go from a moving coordinate system to another moving coordinate system and the annexes transformed exactly the way Lorentz transformation tells you they transform coterie right this as X 1 is X 1 9 is V. X. nor X Prize naught x North trying is equal to 0 X naught miners VX 1 store for this is not content this it's just a way of organizing the components for vector by calling them all by the same name and giving them an index New also will use that will use that just a 3rd make formulas mice a simple yes transform linear shapes it's I made absolutely absolutely you could read off from here on matrix the matrix would be 1 minus V 1 absolutely you could think of romp even figure of Lorentz transformations having associated with them matrices and you could write that X crime is equal to the current acts matrix Times column vector these components of the column vector would the ex isn't he accidently gesture certainly could use matrices and I advise you do so because it's a good thing to do right now let's talk about some other examples of 4 sectors in particular instead of talking about the components relative to an RJR and logistical a little interval but just take a little on intellect could be interval along a trajectory Kobe having we could have a trajectory we might wanna consider all along trajectory of small intervals valued his small think calculus eventually we're going to be talking about differential displacement along here for the moment let's just called Delta instead of be sorrow this differential element here corresponds to it already not quite different should get some discreet the distance corresponds to would be new Delta X Mu doubt that's new means that the change in the coordinates the change in the 4 coordinating going from the tale of a vector the beginning of the vector and composed hour Delta T and the X Delta don't does DEC's meals what I want to do now is to introduce a notion of 4 velocity four-dimensional velocity which is a little different than the normal notion of velocity velocity in this case this could be the trajectory of a particle expect this to be the trajectory of a particle from here to here and I'm interested in these notions of velocity at a particular instant over here well I do if I would don't ordinary velocity I would take a little built the X and divided by the and then take the limit and that would define from me ordinary velocity that velocity has 3 components the x component of velocity the Y component of losses E. component of velocity there is no 4th component of that ordinary velocity did not introduce now any notion of four-dimensional velocity do that by taking the dealt the X new and instead of dividing it by Delta T where are divided by the invariant distance between these 2 points but scored built the house Delta Tau is defined so that square is equal to 0 . 50 squared Miners belt acts I don't backs are you might amount the sums of squares of adult OPEC's go back square Boca y squared each quarters another words it's the invariant space time distance between this point and that pouring we take the square root of this is as Delta town and that is called before velocity it's labeled by a instead of and has an index knew was so it runs from 0 2 3 4 components now resisting related To the ordinary velocity we're getting there should probably go through legacy the will continue next time about Talal going will going toward a fury of the motion of particles that a theory of a motion of particles we have that notions such as velocity position of course momentum and energy Connecticut Energy whatever for moving toward a them 1st of all just a motion of particles and then told the dynamics of how particles a generalization if you like of vesicles while hired them notion of acceleration all the things that Newton had except the relativistic generalization of and they will be in terms of 4 vectors OK let's ceiling rare birds was too I have attempted leaders it was right but talks at Rivera because just of the street for more please visit us
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Metadaten

Formale Metadaten

Titel Special Relativity | Lecture 2
Serientitel Lecture Collection | Special Relativity
Teil 2
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/15002
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

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Fachgebiet Physik
Abstract (April 16, 2012) Leonard Susskind starts with a brief review of what was discussed in the first lecture -- specifically the use of vectors and spin in three dimensional space and in relation to special relativity. In 1905, while only twenty-six years old, Albert Einstein published "On the Electrodynamics of Moving Bodies" and effectively extended classical laws of relativity to all laws of physics, even electrodynamics. In this course, Professor Susskind takes a close look at the special theory of relativity and also at classical field theory. Concepts addressed here include space-time and four-dimensional space-time, electromagnetic fields and their application to Maxwell's equations.

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