Special Relativity | Lecture 2

[Music] Stanford University all right we were last time we worked out basic Lorentz transformations relating to frames of reference whether or not I said it let me say it now we were mainly dealing with problems in which all of the motion and in particular the relative motion of different observers is along one particular axis we didn't try to think fully three-dimensional in the picture you could have in your head is that we have a long railroad track one dimensional some set of observers is sitting still at the station we call those the stationary observers and other observers are in the Train and they're moving relative to each other along the one dimensional axis didn't worry very much about the other axes we we I said one or two things about it and we talked about how you relate the coordinates of one observer relative to the other and in particular we wrote down Lorentz transformations based on the basic hypothesis of Einstein that all reference frames see the speed of light exactly the same in fact we said with the units that we used all observers see the speed of light being one that's of course a choice of units we work in years in lightyears or seconds and light seconds whatever choice of units if we choose them correctly we can make the speed of light one and that simplifies equations of course if we really want to plug in to the real observational physics we might want to use the fact that the speed of light in common units in the common units that an experimental physicist will ordinarily use is not one that's three times ten to the eighth in some units and so we would put 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 C so that the equations are dimensionally consistent I will go back and forth mostly I will use the speed of light set equal to one but every now and then just to illustrate a point I will stick the speeds of light in and you can go through the equations and do that yourselves okay so two observers one moving down the axis with a velocity V relative to the stationary moving down the tracks with velocity V relative to the stationary observer stationary observer sees the moving observer moving V units of space per unit time V being velocity and of course by symmetry just by the symmetry of the problem if we believe that all coordinate frames are equally valid the same relationships the same kind of Lorentz transformations will relate the stationary observers coordinates to the moving coordinates it's a two-way street the stationary observer ascribes stationary coordinates the moving observer ascribes moving his coordinates and they can each be related to each other reciprocally and the reciprocal relation is truly reciprocal exactly the same relations except that whereas if I were moving to the right you're right if I were moving to your right you would say my velocity is positive you would be moving to the left as far as I was concerned I would say your velocity was negative and so in relating the two frames of reference the only thing we have to remember is there's a sign change of velocity when you go back and forth okay so for example if X Prime is the coordinates as seen in the train is the Train over here let's draw a train that's the train okay there's an observer in the train and there's an observer in the tracks or observer in the station is the observer in the station here's the observer in the Train and the observer in the train is moving with velocity V V the observer in the train has meter sticks and the meter sticks can be laid out on the floor here the former grid the observer also has a timepiece a clock likewise the observer at rest what rests with respect to whom what rests with respect to the station the observer at rest with respect to the station also has his meter sticks laid out and also has his timepiece and they make various comparisons you know how all this works an event which takes place an event which takes place an event means a event happening at a point of space and a point of time in other words at some point of space-time I like to think of it as a flash bulb exploding someplace going off someplace it doesn't matter whether it's in the train or outside the train let's say in the train for convenience a flash bulb goes off over here at a time that the stationary observer reckons to be T time T at position X now position X means coordinate X in the stationary reference frame X right over here at a time T according to the kind piece of the stationary observer so the stationary observer ascribes to it coordinates X and T and the moving observer ascribes to the same event coordinates X Prime and T Prime and we worked out the last time the relationships that are necessary between XT and X Prime and T prime such that everybody will always agree that the speed of light is 1 let me write them down quickly X Prime these are the lunch transformations X prime is equal to X minus VT now Newton would recognize that but what he wouldn't have recognized was the square root of 1 minus V squared downstairs and if you want to put back the speed of light it goes right over here I'm gonna put it in and then take it out V squared over C squared and of course if the velocity is small by comparison with the speed of light this is a terribly tiny correction let's just call it 1 minus V squared and T prime is equal to t minus V X divided by that same square root of 1 minus V squared if we wanted to add the other two directions in particular the directions out of the board and vertical in other words the directions perpendicular the tracks we could have them in very simply perpendicular directions don't change under a change of velocity along a given axis if the change of all if the velocities of the two frames relative velocities are along the x axis then the y and z coordinates are unchanged y prime sorry y prime equals y and z prime equals e let's I won't write it if we need it we'll use it we can invert these relations this is simply a matter there's nothing sophisticated going on here this is simply a matter of solving for x and T in terms of X Prime and T Prime let me remind you what the result would be it would be x equals x prime plus VT prime / that same square root and T equals T prime plus V X prime divided by square root of 1 minus V squared the only difference the only asymmetry is where you saw a velocity over here you change the sign of the velocity just to account for the fact that the relative velocities are in opposite directions you can also read off if somebody gave you this form for the relationship between the coordinates you could easily read off what the relative velocity between the two observers is you look at the X equation here and you say the moving observers coordinate is X prime equals 0 right at the position of the origin of coordinates inside the train X prime is equal to 0 X prime equals 0 corresponds to x equals VT you just look at this and you say the moving observer the prime observer let's not call it moving the prime observer is at rest or they or the position X prime is at rest or is equal to 0 excuse me X prime is equal to 0 the origin of the prime coordinates corresponds to x equals VT you don't have to know about this denominator there you just look at x equals VT that specifies x prime equals 0 and that tells you X equals VT tells you that the relative velocity between the two of them is V all right now we want to do another exercise the other exercise is to assume there's a third observer the third observer is moving relative to the railroad car relative to the train we've got a little he's got a little kitty car inside the train and he's moving relative to the passenger that's calling the passenger he's moving relative to the passenger with velocity you the passenger sees the little kiddie car little kid in the kiddy car pedaling down the the aisle of the Train with velocity u relative to himself question what does the stationary observer ascribe to the car what velocity does the stationary observer see and the way to solve this is just to use I know of no way to guess the answer the answer is some velocity we can give it a name we can call that velocity W I don't I you could use v1 v2 and v3 but I hate subscripts and so I prefer to say the velocity of the train is V relative to the stationary observer the velocity of the car relative to the train is U and the velocity of the car relative to the stationary observer I will call W alright so the stationary observer sees the car move with velocity W and the question is what is W in terms of U and V the answer is just to use a bit of logic the bit of logic is that the relationship be tourists of all we should give the the cool are there are also coordinates excuse me there are also coordinates in the car moving with the car this could be a set of meter sticks laid out on the floor of the car and also a timepiece that the driver of the car has so there's three sets of coordinates in this case X and T are the coordinates that the stationary observer uses X Prime and T prime are the passengers coordinates and a third set of coordinates which let's give them the name X double Prime and T double prime X double Prime and T double Prime are the coordinates that the kid in the kiddie car uses to describe things give to the position relative to his own frame of reference all right so now it just takes a little bit of logic to say we know what the relationship is between the double prime coordinates and the single prime coordinates those are related by velocity u U is the velocity of the double primed relative to the prime coordinates so we can write down those relationships straightforwardly let's we don't need this over here let's write them down here X double prime is equal same exact kind of relationship same exact kind of thing except we're put in primes here and instead of V we will use the relative velocity U alright so this is gonna be X prime minus u T prime divided by square root of 1 minus u squared all right so this is the Lorentz transformation between the double prime frame and the single prime frame T prime minus u X prime over root of 1 minus u squared all right so now how do we find the connection between the double prime coordinates in the car and the unprimed coordinates at rest in the railroad station and that's simple all we do is let's take this equation over here let's focus on this one the the time equation also works out very nicely but let's focus on the space equation here we know what X prime is in terms of X and T and we know what T prime is in terms of X and T so all we have to do is to plug in and let's do so oh good nice good pen all right so X prime is X minus VT divided by square root of 1 minus V squared now that's just as X prime here so far that's just X prime minus u T Prime and let's put in T prime T prime is t minus V X and the whole thing again divided by another square root of 1 minus V squared now so far I have only written this in this I have to put in the denominator to put in the denominator I just put in another factor of 1 minus u squared 1 minus u squared square root of 1 minus u squared so that is the relationship between X double Prime and X we can write this a little more simply let's focus on what what's there's some big denominator the denominator involves the product of the two square roots all right let's put it in square root square root I'm not gonna write out what's inside the square root it's just a product of these two square roots but I'm interested in the numerator really the numerator has an X and it has a plus UV X X here and plus UV XY is a plus UV X because there's a minus sign here and a minus sign here x times 1 plus UV 1 plus UV X and what about T T will multiply minus v plus u times t minus V and minus u times T and this is the answer but it's not very transparent what it means but all we need to do to figure out the relative velocity of the double prime frame relative to the unprimed frame is just to do exactly what we did over here if we want to find out what x prime equals zero means just set X minus VT equal to zero and that tells you that X is equal to VT it tells you how to fellow at rest here who is at X prime equals zero how is moving relative to the unprimed frame so we do exactly the same thing we say X double prime will be zero X double prime of course being zero means the position of the kid inside the kiddie car here X double prime equals zero is the same thing as the numerator here being zero denominator who cares about the denominator it's not a it's there but we can set the numerator equal to zero here and that will tell us under what circumstances X double prime is equal to zero so what does it say it says X double prime is equal to zero when 1 plus UV X is equal to u plus V T [Music] I've set this equal to this to make the numerator zero or if I divide by 1 plus UV it tells me X is equal to u plus V over 1 plus UV T so what is the relative velocity I'm sorry yes what is the relative velocity this tells us the X and T trajectory of the kid in the kiddie car here the trajectory of the child or whoever it is in the kiddie car is X is equal to u plus V over 1 plus u V times T that corresponds to X double prime equals zero well I put another way to say it is just that the stationary observer sees the kiddie car sees the the key car moving along with velocity u plus V over 1 plus u V so we now know what W is that's exactly what W is W we've worked it out we can now we can write it in the form well let's write it in the form X double prime is equal to t minus W X divided by square root of 1 minus W squared and P double prime is did I write that right no I didn't X minus WT and t minus W X over the same square root that on a room for the square root there all right and what do we find W is w is just this that's how fast the kiddie car is moving as seen from the stationary frame so yes because we set C equal to 1 right we're gonna put them back we're gonna put them back in a minute yeah we're gonna put them back in a minute yeah you're you're a step ahead of me ok all right so X is equal to so W the speed W is equal to W plus V divided by 1 plus UV and now if I want to restore the units well just to answer your question again if we set the speed of light equal to 1 of course we're working in units in which velocities are dimensionless but if we want to restore the dimensions we simply look at this equation we said W equals u plus V that's dimensionally fine it's adding one to you V which is peculiar but we restore the dimensions by putting in C squared here 1 plus UV over C squared u V over C squared is dimensionless alright so this is the equation with C being restored what is the Newtonian equation the corresponding Priyan Stein Ian equation well stationery person sees passenger moving with velocity u passenger sees Kitty car moving sorry stationary observer sees passenger moving with velocity V passenger sees Kitty car moving velocity u the answer naively would just be u plus V that's not u plus V it's U plus V divided by something which as long as u and V are significantly smaller than the speed of light u V over C squared will be very small well we're putting some numbers in a minute just to test it out but as long as u and V are small compared with the speed of light this will be negligible it's the product of two ordinary velocities a hundred meters per second or whatever it is you want to put in divided by three times ten to the eighth meters per second all squared this is a very very small number and it's a very small correction on the other hand when u and V get up near the speed of light it can get very significant so let's see what happens first let's do the case where u and V are small velocities compared to the speed of light we can leave out we'll remember that u and v mean velocities measured in units of the speed of light so if u and v are measured in units of the speed of light and let's say for example supposing U is equal to 0.01 one one hundred one puts 1% of the speed of light and let's say V is also 0.01 they're both 1% the speed of light this time have to be 0.02 that was what Newton would recognize divided by 1 plus and now you V is 0.0001 do it did I do that right yeah and point ooo 1 is a very small correction now this is a very sizable velocity incidentally you being 1% of the speed of light that's damn fast it's not 3 times 10 to the 8th it's 3 times 10 to the 6 meters per second so this is pretty fast but the correction is small 1 plus something notice the answer is a little bit smaller than what Newton would have estimated this number here is a little bit bigger than 1 and so in the denominator it's a little bit smaller let's go to the other extreme let's suppose U and V are 90% of the speed of light we could do 99% but I'm not so good at arithmetic U and V are 0.9 V equals point 9 Newton would have said the kiddie car is moving faster than the speed of light relative to the stationary observer in fact he probably would have said 1 point 8 times the speed of light you would have added these two numbers but Einstein would have put in the denominator here so let's see what we get we get point 9 plus point 9 that's 1 point 8 divided by point 9 squared point 9 squared is 1 point 8 one slightly bigger than 1 point 8 the result is that the net velocity is slightly less than 1 in other words we have not succeeded in making the kiddie car car go faster than the speed of light even though bla bla bla you only know the rest of the story so this is the answer to the question what happens if an observer is moving faster than the speed of light well you could ask that question but I think we are pretty well protected against people moving faster than the speed of light if the way they are made to move is relative to some previous frame of reference moving speed slower than the speed of light in other words there's no way by putting another observer inside the speed of a car here making him go 90 percent of the speed of light etc etc we're never gonna get faster than the speed of light so it's consistent though it's a consistent thing to say all observers move slower than the speed of light even though they can move arbitrarily close to the speed of light relative to each other in any combination the net result will still be slower than the speed of light okay let's suppose U and V are 1c and 1 are the same thing then u plus V is 2 and 1 plus UV is 2 so the net result is the speed of light again so if you're moving there if each one is moving very very close to the speed of light then that result in other words if the kiddie car is moving relative to the passenger very close to the speed of light and the passenger is moving relative to the station very close to the speed of light the result will be that the kiddie car was moving relative to the station and even closer to the speed of light but not in excess of it okay that's the what if the train had glass walls if the trainer I assume the train did have sir glass walls I don't see how that makes any we're not talking we're not talking about how appearances look we're talking about how measurements of phenomena by meter sticks and by well designed clocks correlate with each other what somebody sees is much more complicated for the simple reason that when an event happens light has to come from the event and it can be much more complicated what you visually see we're not talking about what you visually see we're talking about correlating the locations and times of events in frames of reference which are defined by meter sticks at rest relative to observers and timepieces which are also at rest relative to them and doesn't matter what kind of walls the car has the the transformation laws are universal okay now the next the the next thing we talked about last time was the notion of proper time or proper distance or proper interval let me just remind you quickly about that very quickly if we have a event taking place at point X and T we found out last time that there's an invariant notion of separation space time difference or space-time distance between them proper time and the proper time or the proper let's let's put this up higher over here XT the proper interval between them is called tau and it's defined by tau squared is equal to T squared minus x squared and the interesting thing about this the important thing about it is it's the same in every reference frame if the reference frame is moving then we would use prime coordinates but the same quantity here is T prime squared minus X prime squared it's also equal to T double prime squared minus X double prime squared it's an invariant all observers agree on the value of this interval between here and here they don't depend I don't agree about the coordinates themselves but they agree about this notion of proper time the proper time is also the time read by a clock moving between these two points if the clock is set equal to let's say 12:00 noon at this point and it moves along this trajectory then the time that it reads at the end of the trajectory is the proper x and we worked out the time-dilation last time but what I did want to say about this I want to now put back the other two coordinates y&z y&z and let's put them back into the game for a moment or put them back in there is another kind of transformation that we can do not just a transformation between two moving coordinates moving along the x axis but we can also consider rotations of coordinates let's for the moment not even think very much about relativity let's just talk about two different coordinate systems related by an ordinary rotation with respect to each other so the stationary observer might have two different coordinate systems one orient that along the x y axis and another one oriented along some x prime y prime axis not those not the same primes some other set of axes rotated or at an angle relative to the original ones what happens now we can forget a prompt the prime coordinates for the moment what about this x squared here this x squared is really the distance from the place where the of where the clock started to the place where it ended up in the unburned coordinates now supposing we take into account the other directions let's call this Y 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 is not at the same value of x as the origin here what then is the interval between here and here what is the invariant quantity well this is actually fairly simple as long as the event is located on the x axis its T squared minus x squared if it's not located on the x axis and we make a rotation then what was originally x squared becomes x squared plus y squared plus Z squared becomes the spatial distance between the origin and this point over here this really becomes minus x square minus y squared minus Z squared etc so if we're not working strictly along a one-dimensional axis the invariant proper time between a start of a clock and a place where the clock gets to is given by T squared minus the square of the spatial distance which is X squares is Pythagoras theorem it's just Pythagoras theorem applied to XY and Z and that's the notion of proper interval and that's the one we'll work with if we make any combination of Lorentz transformations and rotations of coordinates we will always find to any two inertial frames which agree at this point over here we will find out that tau squared is invariant is the same in all inertial reference frames and it's very very similar to the idea that that in ordinary Euclidean geometry different coordinate axes will ascribe different coordinates to a point in space but they will always agree about the distance of a point from the origin here it's this funny kind of distance with a relative different sign between the space components and the time components and that is probably the most central fact about relativity is that this combination is invariant that's really what it's all about okay so let's keep that idea in mind and introduce a little bit of notation we get tired of writing XY and Z and we try to condense the notation so let's convince the notation to standard relativistic notation if you assume that the the speed V let's say it was point nine when you said it was very close to the speed of light and speed u let's say that is setup a car that guy has a flashlight in his hand so he's actually sending a light beam and that's what at that time and I'm kind of wondering like how do these two people measure the speed of light whether they get the same number why don't you try it take one yep here it is what if what if C is the speed of light that means it's one suppose V is one oh I don't know which one do you want to make one make you one okay make you one that's 1 plus V divided by U is 1 1 plus V so the answer is 1 the speed of light is the speed of light that light ray moves with the speed of light about something and you pointed out that generally it comes down recycling too near to definitions of stop to me if it's not the question I asked a minute ago I realized with the glass walls to try and look at both of them is trying to reintroduce simultaneity ok if you understand it good let's talk about light rays how light rays move light rays let's go back to the one-dimensional case then one down one space in one time dimension let's go back light rays move for example along 45-degree axes like this that means they will move from the origin to the point X T but only if X is equal to T that's just saying the light moves with velocity one in a certain time T the distance it moves is equal to that time so it moves to the point t T or the same coordinate that means that T squared minus x squared sorry T squared minus x squared is equal to zero or that the space-time interval are called space-time interval proper time any number of different not deaf usages is zero that's different than ordinary Euclidean distance Euclidean distance if two points have genuinely zero distance between them they're sitting on top of each other in space-time if two points have space-time distance or a proper time equal to zero between them that simply means they are related by the possibility of a light ray going from one to the other now if we introduce the additional coordinates Y squared and Z squared then what was originally x squared the distance that the light beam traveled along the x axis will after the square of the distance will obviously become x squared plus y squared plus Z squared square of the distance and that will be equal to zero for a light ray so the motion of the light ray is T squared minus X square minus y squared minus Z squared equal to zero or again tau squared is equal to zero tau square so that's one concept of how a light ray moves it moves along trajectories such that the proper time along the trajectory is equal to zero photons move that way we could draw a picture for this a light ray moving to the right there's a 45 degree axis to the right 45 degree lying to the right a light ray moving to the left moves exactly the same way except in the backward direction what about a light ray moving outward well that moves the same way except at a 45 degree angle in the outward direction more generally we would draw a kind of cone now I can't draw the full three dimensions plus time there's too many dimensions for me to draw on the blackboard but if we had only two dimensions X square plus y squared instead of let's forget the Z squared here we will find that the motion of light rays is such that in space-time they move along the cone created by 45-degree light rays coming out of the origin there that's called a light cone a light cone is the set of points that a light ray can arrive at if it starts at the origin that's the notion of the light cone and this would be called the future light cone the future light cone is all of the places that light can get to starting at the origin there's also a thing called the past light cone a past light cone is all of the places that can send the light ray to the origin so the future light cone is all the places that the origin can send the light ray to and the past light cone is all of the places that can send the light ray to the origin but they just the cone turned over on its nose the past light cone in the future light cone and this is terminology terminology is often very helpful but that's all it is is terminology yeah the the fundamental invariance for these systems equals T squared minus XY there must be something that equivalent for eternity panic Cimmerian go with that just keep just the everybody agrees about the time interval between two events the time interval is just T right that's a good point all right we're getting close to let's talk about the concept of four-vector the most primitive kind of the most basic example of a vector in ordinary three-dimensional space and I am NOT talking about the kind of vectors we talked about last quarter we're not talking about state vectors in quantum mechanics we're talking about vectors in space all right the most basic notion very example of a vector is the interval between two points in space given two points as a vector which connects them that vector could measure the could could have to do with how far somebody walked in the or whatever that's a vector has a direction it has a magnitude and if we wanted to we could think of it as a vector beginning at the origin and ending up someplace else it doesn't matter where it begins but the vector this is the vector we can move it around that's the same vector but we can think of it as being a an excursion starting at the origin and ending at some point X and it has coordinates in this case x y&z the location of the final point or we could call them X I I being one two or three representing x y&z three coordinates X I would really stand for XY and Z or x1 x2 and x3 right now we have another added component to worry about not only do we want to know where an event is what we want to know at what time that event is let's suppose that were measuring space and time relative to some origin that means we have to add in another coordinate T in other words the vector becomes a four dimensional object with a time component and space components the normal notation for it is to represent that two different ways I'll tell you the two different ways we can represent XY and Z by calling them X u mu goes over the four possibilities usually it's normally one arranges them as T XY and Z X mu and what does mu run over what are the values of mu well why do you why did you say zero because everybody says zero yeah did you hit me saying I didn't hear I didn't know I said okay right for whatever reason historically T was not considered the first coordinate X was considered the first coordinate Y was considered the second and Z the third you might have thought that time should be the fourth component component for whatever historical reason time was thought of as the zeroth component so this stands for X is zero which is time x one which is X X 2 which is y and X 3 which is Z what about the space-time distance between these points the proper time that's T square minus X square the square of it the square root tau squared is X T squared minus X square minus y squared minus Z squared but we can also write it as X naught squared - x1 squared minus x2 squared and so forth x2 squared blah blah blah so that's just notation it's just a notation whenever you see a mu that means the index runs over the four possibilities of space and time whenever you see an eye that means the index runs over only space okay i stands for space mu stands for space and times X just as X I can be thought of as a very primitive version of a vector not primitive a very basic version of a vector in space X mu with four components becomes the notion of a four vector just as vectors transform when you rotate coordinates four vectors transform when you Lorentz transform the space when you go from a moving coordinate system to another moving coordinate system and the X's transform exactly the way the Lorentz transformation tells you they transform we could rewrite this as x1 is x1 minus V X naught X prime naught X naught prime is equal to X naught minus V x1 and so forth and this is there's no content to this it's just a way of organizing the components of a four vector by calling them all by the same name and giving them an index mu so we'll use that we'll use that just to make formulas nice and simple yes they transform linear rate such they could be they could be it set them up and operate them with the matrix absolutely absolutely you can read off from here a matrix the matrix would be 1 minus V minus V 1 absolutely you can think of you can think of Lorentz transformations having associated with them matrices and you could write that X prime is equal to this times X matrix times column vector the components of the column vector would be X's and T's X isn't it yes you certainly can use matrices and I advise you to do so because it's a good thing to do all right now let's talk about some other examples of 4 vectors in particular instead of talking about the components relative to an origin let's just take a little interval let's just take a little interval it could be an interval along a trajectory it could be hovering we could have a trajectory and we might want to consider along that trajectory a small interval now when you hear small think calculus eventually we're going to be talking about a little differential displacement along here all right for the moment let's just call it Delta instead of B so this differential element here corresponds to or not quite differential yet some discreet distance corresponds to a DX mu or Delta X mu Delta X mu means that the change in the coordinates the change in the four coordinates in going from the tail of a vector to the beginning of the vector and it's composed out of delta T and Delta X Delta Y and Delta Z Delta X Mew what I want to do now is to introduce a notion of four 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 let's take this to be the trajectory of a particle from here to here and I'm interested in the notion of velocity at a particular instant over here what do I do if I were doing ordinary velocity I would take a little Delta X and divide it by a delta T and then take the limit and that would define for me ordinary velocity that velocity has three components the X component of velocity the Y component of velocity in the Z component of velocity there is no fourth component of that ordinary velocity okay I'm going to introduce now a notion of four-dimensional velocity and we're going to do that by taking the Delta X mu and instead of dividing it by delta T we're going to divide it by the invariant distance between these two points let's call it Delta Tau Delta Tau is defined so that it's square is equal to delta T squared minus Delta X I Delta X I - you know the sums of the squares of the Delta X is Delta x squared Delta Y squared and Delta Z squared in other words it's the invariant space-time distance between this point and that point we take the square root of this and it gives us Delta Tau and that is called the four velocity it's labeled by a u instead of a V and it has an index mu so it runs from zero to three four components now how does this thing relate to the ordinary velocity we're getting we should probably go if we want to get seats but we'll continue next time I'll tell you it will going well going toward a theory of the motion of particles to have a theory of the motion of particles we have to have notions such as velocity position of course momentum energy kinetic energy whatever we're moving toward a first of all just a motion of particles and then tour the dynamics of how particles move a the generalization if you like of F equals MA will have a dead notion of acceleration all the things that that Newton had except the relativistic generalizations of them and they will be in terms of four vectors okay let's see if we can grab let's see I have ten to eight is so who's right that clocks right we better better get yourself a good seat for more please visit us at stanford.edu

Channel: Stanford

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Keywords: mathematics, science, mechanics, field theory, quantum physics, classical mechanics, space, time

Id: qfTJP7Soto4

Length: 54min 0sec (3240 seconds)

Published: Wed May 09 2012