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"2017-10-09 22:16:30"
Hillary Clinton speaks at launch of Stanford’s new digital policy program
\\for university now you're all familiar with the fact that silicon valley's played a central role in launching many of the digital technologies that revolutionized our world the applications of those technologies of course raise a variety of important issues but societies around the world are now confronting privacy freedom of expression the quality of discourse needed for the functioning of democracies the impacts of digital technology on national security in Germany members of parliament have been hacked by Russians in France microns campaign was hit by a massive cyber attack just before their presidential election Russian agents pushed incendiary social media posts last weekend during the Catalonian independence referendum which is rocking Spain their weapon of choice is not tanks or missiles but let's not mince words this is a new kind of Cold War and it is just getting started we need to get serious about cybersecurity government and the private sector need to work together to improve our defenses against future cyber attacks including by making necessary investments to protect our networks and national infrastructure corporate America needs to see this as an urgent imperative because government cannot do this alone this is a perilous time at home and abroad and like a lot of people I'm deeply worried but I'm also confident that with American innovation and patriotism with the insight and leadership of the far sighted thinkers in this room in rooms like it across our country we can show decisively that extremism and authoritarianism are no match for democracy and free thinking people I have for more please visit us at Stanford.EDU //
"2008-07-15 23:34:05"
Lecture 1 | Modern Physics: Special Relativity (Stanford)
\\I'm this program is brought to you by Stanford University please visit us at Stanford.EDU it's quarter we're going to learn about field theory classical field theory fields such as the electromagnetic field gravitational field other fields in nature which I won't name right now propagate which means they change according to rules which give them old wave like character moving through space and one of the fundamental principles of feel free fact more broadly nature in general it is the principle of relativity principals special print versus the have the principle of special relativity in this particular case the principle of special relativity well let's just call it the principle of relativity goes way back there was not an invention of Einstein's I am not absolutely sure when it was first announced or articulated in the form which I'll spell it out I don't know where those Galileo or Newton or those who came after them but those early pioneers certainly have the right idea begins with the idea of inertial reference frame no no so reference frame this is a bit something a bit tautological about our inertial reference frame Newton's equations F. eagles MA are satisfied in an inertial reference frame what is an inertial reference frame it's a frame of reference in which Newton's equations are satisfied I'm not going to explain any further what inertial reference frame is except to say that the idea of an inertial reference frame is by no means unique a reference frame first was a reference frame entail offer a reference frame first of all in tales are sort of coordinate axes in ordinary space X. Y. and Z. when you know how to think about those but also entails the idea that the coordinate system may be moving or not moving relative to home relative to whomever we sitting here are you sitting here in this our classroom here define the frame of reference we can pick the vertical direction to be the Z. axis the horizontal direction along my arms he it to be the X. axis X. plus that way X. my exit minus in that direction and which one of my left out I've left out ... the Y. axis which points toward you from me so there are some coordinate axes for space Hawaiian Z. and I think this in addition to specify a frame of reference one also imagines that this entire coordinate system is moving in some way relative to you sitting there presumably with the uniform velocity in a definite direction if your frame of reference is an inertial frame of reference another words if when you throw balls around go jungle or do whatever you suppose to do an inertial frame of reference if you find yourself in an inertial frame of reference then every other frame of reference that's moving with the uniform velocity relative to you know remember what uniform velocity means it doesn't just mean with uniform speed it means we'll uniform speed in an unchanging direction such a frame of reference is also inertial if it's accelerated Arafat starts standing still and then suddenly picks up some speed then it's not an inertial frame of reference all inertial frames of reference according to Newton and also I think also Galileo Galileo is often credited with the idea but I never read enough of Galileo to know whether we actually have it or not ... not that I read enough of new they both wrote in languages that I don't understand arm what was I saying oh yes right according to both Newton and anybody else who are thought about it very hard the laws of physics are the same in all inertial reference frame laws of physics meaning F. equals are many forces between objects all the things that we would normally call laws of nature or laws of physics don't distinguish between one frame of reference Hoover and in other if you want they kind of pictorial example that I like to use a lot when I'm explaining this ... through arm ... to children or to grown ups I like to think about the laws of juggling they're very definite procedures that you train your body to do ... in order to be able to juggle balls correct very now you can imagine yourself being in a railroad car moving with perfectly uniform velocity down the X. axis and trying to juggle we'll have to compensate for the fact that the train is moving and for particular when you throw a ball up into the at the you have to reach over to the right to compensate for the fact that the train is moving to the left my left your right the answer is no you don't the laws of juggling are the same in every reference frame when every inertial reference frame whatever you do in one reference frame you do exactly the same thing and you'll succeed or fail depending on whether you're good juggler are not but it will not depend on whether your moving with uniform velocity so laws of juggling of the same in every inertial reference frame the laws of mechanics of the same in every inertial reference frame the laws Newtonian laws of gravity of the same in every inertial frame according to new what about the laws I will like trickle phenomena well there there was a clash the clash had to do with Maxwell's equations Maxwell's equations with the field equations the field theory that governed the electro magnetic field and the way that it propagated in sent waves electromagnetic waves that we ordinarily call light or radio waves or so forth the fundamental dilemma as you all know I'm sure you all know are the fundamental dilemma are was both according to well he was the dilemma Maxwell's equations said like moves with F. certain velocity if you take the various constants that appear in Maxwell's equations and put them together in the right way you get the velocity of waves moving down and access and then velocity comes out to be a certain number out of Maxwell's equations you have 2 choices one is to believe that Maxwell's equations are true laws of nature as good as any other laws of nature in which case principle of relativity says they should be the same in every reference frame but if it follows from Maxwell's equations that the speed of light is 3 times 10 to the eighth metres per second which is about what it is ever follows for Maxwell's equations that light moves that fast and if Maxwell's equations all laws of physics fundamental laws of physics and if the laws of physics are the same in every reference frame then the speed of light must be the same in every reference frame but that was very hard to swallow because of a light beam is going down that axis and you chase it run along with it that let's say 3 quarters of the speed of light then you want to see that light ray moving much more slowly read 3 times 10 to the eighth metres per second relative to you on the other hand a light ray going in the other direction since you're so running into it you should see going even faster so all these possibilities could not simultaneously be correct that the laws of nature are the same in every reference frame and that Maxwell's equations are laws of physics in the same sense that Newton's laws of physics namely the same in every reference from something I have to give well point was of course that they were good laws of nature and that they were the same in every reference frame the thing that I have to give as our concepts of velocity space and time and how we measure velocity especially velocities were up richer up near the speed of light now I'm not going to spend the full amount of time that I did previously on the special theory of relativity that can be found on lectures from how long ago are in there there on the internet I believe relativity and electro magnetism I think that was maybe about 3 quarters ago and lost track yeah they're up there there on the net and the other the lectures on relativity special relativity and electromagnetic theory we're just gonna cut through it real fast when I got through the basic ideas of relativity a little more mathematically then I would do if I were teaching it for the first time package of the first time I tend to feature the way ... Einstein first conceived of it how do you measure distances how do you measure velocities how do how does the propagation of light influences things then I'm going to pick a more mathematical view of it and think about the properties of various kinds of ... coordinate transformation coordinates now consist not only of X. Y. and Z. but also time to eat so imagine every event in the world is characterized by just like every particle would be characterized by a position X. Y. and Z. every event taking place in space time it's characterized by 4 coordinates X. Y. Z. in 3 let's suppress for the moment why in the city let's just pick up and forget them for the moment and concentrate on ex ante that would be appropriate if we were mainly interested in motion along one axis let's focus on that motion along the X. axis let's suppose there is no motion along why NZ then we can forget why NZ for the moment momentarily we'll come back to them and think of motion along XMP and the various reference frames that might be moving along the X. axis alright here's yes prime vertically here's space horizontally this is just always draw space horizontally time vertically I found out that mathematicians or at least I'm certain computer scientists always drive time going horizontally I didn't know that and I got into an enormous argument with the quantum computer scientist which was ultimately resolved by the fact that he had time going horizontally and I had it going vertically ... is a traditions I guess traditions grow up around subjects but time is north and ex is east I guess or at least times upward yeah yeah that's what that that that's the point that is the point yes they're thinking of time is the independent variable and everybody knows that it's a law of nature that the independent variable should be horizontal okay alright now let's and let's imagine a moving observer moving down the X. axis with the velocity V. let's take his origin of spatial coordinates here's our agenda spatial coordinates at prime P. equals 0 is just the same let's assume that my arm I'm I'll be the moving observer I moved down the X. axis I am my own origin there is nobody who is your origin that seat is vacant over there so that absence of ... Nov ... human over there is the center of the X. coordinates in your frame I'm the expert prime coordinates and of course I being very ego centric will take Mari X. acts my ex our origin to be where I am there I do I move down the X. axis we pass each other our origins pass each other a T. equals 0 so that means a teak was 0 your axis in my axes are the same or your origin my origins San but then as I move down the X. axis my court my coordinates center moves to the right most of the right that's supposed to be a straight line that's it was good as I can doing that the circumstances straight line and it's moving with Hoss city V. which means it's ex prime equals assortment X. equals VT but it's also that's the way you describe it in terms of your coordinates my center you described by saying exit was meet the how do I describe it I just say ex prime my coordinate ex prime 0 ex prime equals 0 is the same as X. sickles VT our what's the relationship between ex prime an ex ante well it's easy to work out you believe this picture the ex prime coordinate is the distance from my origin the X. coordinate is the distance from your argument so one of these is ex the other is ex prime the upper one here is ex prime loan here is ex and the relationship between them is that they differ by an amount VT particular X. is equal to X. prime mine is VT our ex prime is equal to X. plus VP grab a wrong yes I do ex prime is X. minus BT an ex is ex prime plus Ricky yeah I think I have that that's correct now but what about time itself well according to Newton and according to Galileo and according to everybody who came after would up until Einstein time is just time is just time is just part there was no notion that time might be different in different reference frames Newton had the idea of a universal time sort of god's time god that up on its cloud that ticking off with user with his super accurate watch and that time was universal for everybody no matter how they were moving and so everybody would agree on what on the time of any given event in this map of space and time via and so the other equation that went with this is that cheap prime is equal 50 let's forget the pop equation here service you have it one might say that this was the Newtonian or the Galilean transformation properties between ex ante your coordinates and their coordinates that I ascribe to a point in space time now let's examine all right ray moving down the plus sex axis if it starts at the origin here then it moves along the trajectory which is X. equals CP you see being the speed of light now shortly I'm going to set sequel to warn we're gonna work in units in which see is equal to warm but not quite yet this is simply once you understand a bit of relativity working in coordinates in which see is not equal to one is about as stupid as using different units for Exxon why our if we use the yards for X. and feet for why the then we will have all kinds of funny factors in our equations which would be conversion factors from X. which is measured in feet to why which is measured in ... our yard's you know those ideal has its uses log scale has its uses running Mohawks killing through large scale well dressed like a compound interest not Russia okay okay I am just saying it is quite often practical circumstance that one uses difference yeah if you sometimes you might there might be a good reason I me arm it wouldn't be totally unreasonable for a sailor to use different ... units are for horizontal direction in vertical direction and I mean he's used to moving around horizontally might use ... what miles ... miles versus fathoms or something nautical miles versus fathom yeah it is that I'm not sure if this question is relevant but some of your balance the frame of reference you need to specify a period of time because obviously goes at 15000000000 years Christmas yeah we're ignoring now the fact that the universe began at some time and ... we're imagining now as Newton did and that's the early Einstein did that the universe has just been here for ever and ever and ever on changing totally static and space and time have properties which don't change with time now of course that's incorrect in the real world and at some point we will take up the subject of cosmology and find that's not right right but as long as we're interested in time intervals which I'm not I I suspect is what you're getting at as long as we're interested in prime it the wheels which I'm not too long in particular time it doubles over which the universe doesn't expand very much and so forth we can mainly say the properties of space don't change over a period of time and so everything I just ... stays the same as it always was exhibitor asking one it seems that that this is lotion if it is me placed fortress driving so rockfish the question is would not mention lead to Saddam ... point is it doesn't lead it doesn't lead to what I'm describing with his room for different ... formulas here this is a formula which is based on the assumption the assumption being that time is universal that's what Einstein found was wrong basically what he found is that when you are in a moving frame of reference 2 different observe observers will not agree about what time a particular event takes place this is the culprit here this one and some modifications to this one but any case to see what's wrong let's go to Maxwell's equations Maxwell's equations say that light always moves with this velocity scene being some numbers in meters per second okay 3 times 1038 meters per second we will later over said Stacey because one let's imagine a light beam moving down the X. axis let's describe how ex prime sees it in other words you see the light move this way to the right how do I see the light well let's see what I see let's just work it out ex prime would be ex which is seeking for that library minus VT which is the same Missy minus VT all this says is that I see the light moving with a diminished velocity of loss to the C. miners V. why is that because I'm moving along with the lights are naturally assume moves slowly the slow compared to what you see it what about the like going in the other direction supposing was a light beam going in the other direction then how would you describe it you would describe it as an ex equals minus CT and I do exactly the same thing I will find it ex prime is equal to X. Mina CT minus V. T. which is the same as minus see plus to be prime city so what this says is that I will see the light moving also in the negative direction it's the minus sign but I'll see it moving with enhanced paucity see plus 50 if this were the right story and he's all right transformation laws for space and time many could not be the case that Maxwell's equations are laws of physics all laws of nature and the sense that they were true in every reference frame they will have to be corrected in moving frame just like a juggler who had to reach to the right who didn't actually but who thought he had to reach the right to collect the ball when train is moving the physicist interested in light beams would have to correct things for the motion of his reference frame now it's an experimental fact that this is not the case that you don't have to correct promotion famous Michelson Morley experiment Einstein he just rejected the ... he just felt this can't be right Maxwell's equations were much too beautiful to be relegated to the approximate thought of the ... contingent on which reference frame and so he set about to find a framework in which the speed of light would be the same in every reference frame any basically focused on these equations and after various very very beautiful Gedanken experiment so thought experiments about light then about measuring and so forth he came to a set of formulas are called the Lorentz transformations I'm going to explain the Lawrence transformations arm in a more mathematical way not fancy mathematics but just for get it we want to get right to the heart of it than not spend ... 3 weeks doing it the best way is to set up a mathematical problem but before I do let me set up of different mathematical problem ... which is most of you seen me do this before but nonetheless let's go through it again a problem of rotation of coordinates are gonna do this quickly let's just take spatial coordinates now for the moment 2 dimensional spatial Quora coordinates let's forget accent tea and just concentrate on X. and Y. 2 coordinates in space instead of invention space time concentrate on a point in space a point in space has coordinates and we can determine those coordinates the X. Y. coordinates just by dropping perpendiculars to the X. axis and the Y. axis no we would describe this point is the point act position or it's just car X. Y. now there's nothing sacred about horizontal and vertical so somebody else may come along some crazy mathematician really 91 who wants to use coordinates which are at an angle relative to the vertical maybe a couple of beers and you don't know the difference between ... vertical and ... or what what what we should give this direction a name public alright the oblique observer the blue observer can blow be seen everybody can see blow okay god are the blue observe also characterizes points by coordinates which he calls ex prime and why prime the ex prime in the White prime coordinates are found by dropping perpendiculars to the ex prime in the White prime axis so his ex prime is why prime when given a point X. why the there's a rule it must be a rule if you know the value of X. in Warri you should be able to deduce the value of X. prime and why prime if you know the angle tween the 2 coordinates between the X. coordinate in the ex prime coordinate the formula simple proved we've used it Naisten these classes many times I'll just remind you what it is ex prime is equal to X. times cosine of the angle between the 2 frames between the tool or coordinate systems minus why prime sign of the angle and why prime is equal to minus or plus I think X. sign of failure Los why cosigned beta I just want to remind you about a little bit of trigonometry all of trigonometry is encoded into a very simple formulas I've used them this signs on the signs of all right that's a hell of at times bigger than small so the songs here are wrong so states are exploring it's bigger than others yeah let's say I mean I have a small if you rotated to the next so that ... why is quite rightly 0 it's for the record ex prime well right back with yet what chart again I'm not gonna fit normally her ... so I would say this picture links picture I was the little perpendicular there no I crimes last life trying my friends this is Wierd crime here right right that's why crime can be X. prime the speaker that plus you know it's probably good acts with very ... yeah ex prime is bigger acts yeah experiment began axle truck that's probably right probably sign but then this one must be men negative yeah okay this is an easy way through our correct for it another way to correct for it just call this angle mining this data that would also do the trick because cosine of mine the state it was the same as cosigned a fader and sighing changes sign when you change stayed at the minus the ada so instead of calling this anger fade or I called of mine this later than my previous formulas would be right true true but there is an excuse alright what do we know about signing cosine it's important to understand Simon cosine everything you have learned about trigonometry can be codified into very simple formulas if you know about complex numbers the 2 very simple formulas are bad co starring our faith there is a need to be I fade out plus ether remind us I fade over to a sign of failure is E. through the I. fade out minus E. prova minor sci feda over to Heidi those 2 formulas contain everything about trigonometry you don't have to know any of the formulas other than these for example I will assign you the homework problem of using these 2 formulas the find cosine of the sum of 2 angles but the way you would do it is just right the sum of 2 angles in here and then re express the exponentials in terms of co sign in sign that's easy to do Ethan the I. fate are is equal to co sign of failure plus I sign theta and he took a minor sci fi there is cosine of fate a minus sigh sighing fate of so work through these formulas get familiar with them they're extremely useful formulas once you know them you will never have to remember any trigonometric formulas again on the other thing to know is that each of the I. theta times each of the mind the sight pater is one are each of the anything times each have a minus the same thing is one those things Carrick characterize all trigonometric formulas in particular as was explained to me by Michael a number of times if we multiply eaten the I. fate of primes heat from the minus side beta we will get one on this side but on this side we will get cosine squared of fate up plus soaring squared of theta not minus sign square book plus 9 square cosine squared and then ice minus I squared sign squared that gives us cosine squared plus 9 squared cosine squared theta plus signs quickly that's equivalent of the fact that he to the I. fate at times it to the minus side favors one alright now the most important fact that again follows from the simple trigonometry is that when you make the change of coordinates from ex wife the ex prime why prime something is left unchanged namely distance from the origin to the point decks why that's something which is you know you you count the number of the molecules along the blackboard from here to here and that doesn't change when I change coordinates so the distance from the origin to the point sex why has to be the same independent of which coordinate axes we use well let's take a square about distance the square of that distance we know what it is a text McCormack S. squared not sure why I use S. but S. for distance S. S. resistance S. for space I think it must be for space that I'm using it for the space suits for the spatial distance from the origin to the point X. why we know what that is it's perfect versus theorem ex squared plus Y. squared but as I said there's nothing special about the X. Y. axes we also want to be able to calculate it as ex prime squared plus Y. prime squared well it's not too hard to work out that ex prime squared plus Y. prime squared his ex squared plus Y. squared it's easy to use do our ex prime squared plus Y. prime squared will have exquisite cosine squared fader it will also have exquisite sighing square thing you know when you add them you get X. quit post why screw you know you know they're the rigmarole ... so it follows from cosine squared plus sign squared equals one that expert I'm squared plus Y. prime squared equals also equal area is equal to X. quipus once quit work that out make sure that you have this on the control that you understand why from the trigonometry not from the that the basic a physics of it of the basic geometry of it is clear make sure that you understand that you can see that from the trigonometry okay one last thing about signs in cosines if I plot are on the blackboard for every angle if I plot sign or co sign along the horizontal axis supposing a plot cosign the fade along the horizontal axis sign of pain along the vertical axis then if I plop all possible angles they will correspond to a bunch a points that lie on the unit circle why on the unit circle because sign squared plus cosine squared equals one so one might call the properties of sine and cosine the properties of circular functions circular in that they're convenient for rotating the convenient for describing unit circles points on unit circles of described in terms of coordinates which I cosines and sines of angles and so forth it's natural to call them circular functions these are these are not the functions they come in to the transformation a new transformation properties first of all these are wrong among other were use X. so yeah not just it wrong no you have it wrong no no gallery or whoever was posturing who postulated Einstein modified it now we're gonna have to make sure that my Einstein's modification doesn't change things in situations where Newton knew it when Newton's equations were good approximations the situations where I mean Stein's modifications are important is when we're talking about frames of reference moving very rapidly up near the speed of light before the twentieth century nobody or nothing had ever moved faster than ... than 100 miles an hour probably well of course something's did like did but for all practical purposes like didn't travel at all it's just that when you turn on the switch to light just went on ... so like didn't travel nothing in anybody's experience direct experience travel faster than 100 or 200 miles an hour and it will I should say nothing travels faster than 100 miles an hour and then lived to tell about it so all of experience was about very slow velocities on the scale of the speed of light on the scale of such velocities Newton's formulas must be correct they work they're they're very useful way work ... Newton got away with it so they must be ... good approximations it had better be that whatever Einstein did to the equations particular for these 2 equations here had been reduced to Newton's equations in the appropriate limit okay let's come back now polite light according to the Newton formulas doesn't always moved with the speed of light but ... let so let's try to figure out what it would mean of a better formula of a replacement for various but like always moves with the speed of light first of all let's check the speed of light equal to warm that's a choice of units particular it's a choice are the relation between space units and prime units if we work in arm light years force but for distance and years for prime than light moves one light year per year the speed of light is one if we use seconds of light seconds it's also one whatever whatever scare we use for space if we use for prime the time that it takes light to go that distance one unit of space if we use that for time units then the speed of light is equal to one now from the ordinary point of your very slowly moving things those arguments but if we were electrons within neutrinos him whizzing around like folk terms they would be the natural units for us ... speed of light equals one so let's set the speed of light equal to one as I said it's just a choice of units and then a light ray moving to the right just Mosey along the trajectory X. equals T. see it's difficult to warm a light rain moving to the left his ex eagles minus TV how can we take both of these equations in putting together sorry exit was minus 3 can I write a single equation which of its satisfied is a light ray either moving to the left or to the right yes using equation XQuery equals T. squared it has 2 solutions X. equals T. and Mexico's minus 3 the 2 square roots all X. where it was P. square is equivalent to either X. equals T. or X. equals minus 3 another words this equation Hannah as the necessary and sufficient condition for describing the motion of a light ray I not tell right walk to the left supposing we found a replacement for this equation which have the following interesting property that when ever I let's let's write it this way X. square minus 3 squared equals 0 zine betta proper persons XQuery minus 3 squared equals 0 that's the necessary and sufficient condition to describe the motion of a library opposing we found a new set of rules and new set of transformation properties which which arm had the property that if X. squared minus 3 squared is equal to 0 then we would find that ex prime squared minus keep prime squared is equal to 0 another words supposing this implied this and vice versa then it would follow that what the un primed observer you in your seats sea of alight ready the prime observer me moving along also see as a light ray both of us agreeing that light rays move with unit velocity now this doesn't work for Newton's formula here it just doesn't work if X. is equal to tee it does not follow the next prime is equal to the prime in fact that says something quite different so a form of these equations must be wrong let's look for some better equations now at this point or let's say in fact what's even be a little bit more ambitious it turns out being a little bit more ambitious actually simplifies things let's not only say that were next square minus the squared is equal to 0 then ex prime squared minus the prime squared is equal to 0 let's say something even bolder let's say the relation between X. T. and X. prime P. prime is such that X. squared minus T. squared is equal to X. school prime squared minus T. prime squared another words pick any ex and any tea and calculate XQuery minus 3 squared then take the same pouring except reckoned in the prime coordinates in other words we pick a certain event the light bulb goes off someplace you say that corresponds to ex ante I say it corresponds to ex prime NP prime but let's require just giant out see if we can do it let's look for transformations so that XQuery minus 3 squared will always be equal to X. prime squared minus these prime squared that would be enough to ensure that everybody will agree about the speed of light why F. X. square minus the squared nichols'ex prime minus the prime squared for all X. T. and so forth then one X. when minus the squared equals 0 ex prime minus the prime squared will be 0 and then if this is a light rain so was this a light ready everybody get the logic okay good so let's assume now now let's ask can we find transformations which have this particular property no it's not so different from looking for transformations which preserve XQuery plus why squared equals X. prime squared plus Y. prime squared it's just a little minus sign other than the minus sign here X. where minus 3 squared look of these 2 is very similar and the mathematics quite similar here are the transformations which preserve ex squared plus Y. squared one of the transformations which preserve X. squared minus the squared well they are the Lawrence transformation where are the fundamental transformations of the special theory of relativity then not this but they are closely related or perhaps one should say closely analogous do these equations here but we have to substitute for a circular trigonometry hyperbolic trigonometry so let's go back and remember a little bit about hyperbolic functions instead of circular functions where are the mores that alright these are the basic rules governing circular functions co starring fader the vests sign Belarus equipped for this and the ... either the I. theta in terms of co sign in sign let's see if we have a now we do have a ... blank blackboard here I mean right whoops what I do here race something I do mean Tories that's what I believe is everybody see how I got this side from the side just add and subtract the equations appropriately and ... you isolated to the I. fade anytime a minus I think that's element to exercise alright hyperbolic functions whether hyperbolic function right those are functions of the form hyperbolic cosine Koch hyperbolic cosine first of all the angle fader is replaced by a variable called omega which I will call omega omega is called a hyperbolic angle it doesn't go from 0 to 2 pie in then wind around on the circle it goes from minus infinity to infinity from minus infinity to infinity so it's a variable interest extends over the entire real axis but it's the following the manner fairly similar to co sign in sign cost omega is by definition you not allowed to ask why this is definition each of the omega plus eat come among us omega over to all we do is substitute for feda or for omega fader I theater substitute omega and that gives you a hyperbolic function likewise are similarly ms the hyperbolic sign and that's given by each of the omega minus ether Amarna so may go over to essentially you throw away all our zone of that formula of of top formulas just throw a all star all our he's I the equations on the right hand side become each of the omega it was hyperbolic Kash mega plus central mega and aether minus omega equals kasha omega minus in Jamaica I think that's right is it right Koch minus soon it is yeah it is right okay now what about the analog of cosine squared plus 9 squared equals one that's simply came by multiplying this one by this one so it's still the same operation multiplying need to the omega by each by each of the minus omega gives one and now that gives Kash squared minus singe squared you see we're getting a minus what we want we want that minus the minus is important we want to well somewhere is under here was a formula with a minus sign yeah we wanna get that minus into play here that's because show may get squared not not coast but Kash square omega minus since square omega so it's very similar everything you want to know about hyperbolic trigonometry and the theory of these functions is called hyperbolic trigonometry everything you ever want to know is codified in these simple formulas reasoning is and then more or less definitions but they're rather useful definitions now if you upgrade yup not only is it worth mentioning I was just about to mention it phone up experiments Weiss grades one hyperbole yeah right exactly so if I want to play the same game that I did here namely plot on the horizontal and vertical axis the values not of cosine of beta assigned of beta but costarring coast during college effect of omega and since omega no the words on the X. axis now we're gonna plot custom made here and the Y. axis cinch omega then this is a hyperbola not a circle but a hyperbola and it's a hyperbola where the asymptotes that are had 45 degrees you can see author Michelle you why why the asymptotes or 45 degrees when omega is very large when omega is very large then Ethan the minus omega is very small right when omega is very large eat of a minus omega is very small and that means both cosh and sinh are both essentially equal to eat for the plus omega another was when omega gets very big Kash and since become equal to each other and that's this line here Kash equal search along this morning here so maybe it's very large the curve asymptotes through our to a curve which is a 45 degrees it's not hard to see that in the other direction when omega is very negative that ... that asymptotes to the other asymptotic warning here so that's why it's called hyperbolic geometry it that I'm really ... the hyperbolic angled the hyperbolic angles the kochs the cinches play the same role relative to hyperbolas as signs in cosines do the circles any questions simply no so it comes omega equals 0 however if you plot that Purple Heart from a man Oka square menacing squared equals 0 no no cost where medicines quit equals one in the same sense that science quipus cosines we were inimical 0 I think what I think you want ask a different question I think well well since I'm a vehicle zeros homes on blacks Jews because you make equal 0 is a very broad brush right okay well this is 3 out of stacks intercept yeah it's right it's the vertex yeah I just think that here's 1.out on the from that I the X. intercept there was one yeah because because so many Koch of 0 is one see that just plug one there are 0 in here one plus one divided by 2 is one at least it was yesterday stores okay so now we we were sort of starting to cook a little bit too stunned to see something that has that nice minus sign in it but what's that got to do with ex into the next prime anti prime we're now set up to make let's call it a gas what it suggests which is based on the extreme similarity between hyperbolas in circles because Susan cosines and so forth he is against America no we'll check it we'll see if it does the thing we want to do my formula instead of being this is guy went and we're now going to have instead of Exxon why we're gonna have ex ante time an ex later on we'll put back wian Z. we're gonna have to put back why NZ but they're very easy okay so let's start with at prime ex prime is to coordinate given to a point of space time by the moving observer namely me and I'm honored guests but it's some combination of Ericsson T. not too different but not the same as where is it ex prime nichols'ex minds VT I'm going to try Kash omega X. that's right ex Concho mega minus he Central America sort of in parallel with this I could put a plus sign here but you can go back and forth between the plus and the minus by changing the sign of omega just as you did here so this let's do it this way X. because show may go minus T. Central America and the prime the look similar but without the extra minus sign here this when you know the relation between signs cosines unconsciousness in shoes is one of just leaving out an ID you go from signs and cosines for consciousness inches by leaving out the army well if you track it through carefully you'll find that this minus sign was really an eye squared it's it's not gonna matter much I would just tell you is really came from some I squared and if you leave out I I squared just becomes one squared is no minus sign so here's the gas for the formula connecting ex prime P. prime with ex ante it equals let's say axe Centro mega minus no miners plus T. Koch omega in this case there are 2 minus signs in this case there was only one minus no but let's check what do we want to check we wanna check that explains squared minus T. prime squared is equal to X. squared minus 3 squared your ass you're probably asking yourself what is this omega what does it have to do with moving reference frames I'll tell you right now what omega is it's a stand in for the velocity between the frames were gonna find the relationship between omega and the relative velocity of the ... reference frames in a moment there has to be a parameter in the long run these little orange in these awful Lawrence transformations connecting 2 frames of reference in Orange transformations a parameter it's the velocity the relative velocity that perimeter has been replaced by omega it's a kind of angle relating the 2 frames a hyperbolic angle but we'll we'll come back to that for the moment let's prove that with this transformation law here the next prime squared minus the prime squared is equal to 0 drop is equal to X. when minus the squared I'm getting to that point in the evening when I'm gonna make mistakes but this is easy you just work it out you use all you have to use is that cosine squared minus sign squared is one you can work that out by yourself but we can just see little pieces of it here ex prime squared will have exquisite Carr square omega keep prime square will have XQuery since square omega if I take the difference between them I'll get a term with an ex squared plans cost squared minus since squared but Khan squared minus since squared is one on so we'll find the term with an ex squared when we square take the square of the did the difference between the squares of this and this I likewise will also find that the squared across term when you square ex prime you'll have X. T. cautious stage when you square T. prime you'll have X. T. Kash singe when you subtract them they will cancel and it's easy to track a term basically one liner the show that with this transformation here expand squared minus tease prime squared is X. squared minus 3 squared which is exactly what we're looking for let me remind you are looking for if we find the transformation for which the left hand side on the right hand side are equal then if X. squared equals the squared another words of the right hand side is 0 the left hand side will also be 0 but XQuery but X. equals T. that's the same as something moving with the speed of light in the X. frame of reference if this being 0 is quick moment to the left hand side being 0 it says that in both frames of reference the light rays move with the same velocity so that's the basic that's the basic tool that we're using here ex prime squared minus the prime squares Equifax squared minus 3 squared I thought those follow by a couple of lines using cost squared minus and square it was one but what I want to do let's take another couple of minutes now let's take a break for 5 minutes and then come back and connect these variables omega with the velocity of the moving frame of reference somebody asked me a question about ... the ether and what was that people were thinking somehow Einstein never got trapped into this mode of thinking on well what we're thinking about when they were thinking about that either what exactly was the Michelson Morley experiment well I'll just spend a minute or 2 are ... mentioning it certainly Maxwell understood that his equations were not consistent where farm without a Newtonian relativity he understood that but his image of what was going on is that the propagation of light was very similar to the propagation of sound in the material or water waves propagating on water and of course it is true that if you move relative through the atmosphere or move relative to the substance that sounds propagating in you'll see sound more with different philosophies depending on your motion if your at rest in a gas of material isn't is a natural sense in which is a particular rest frame the rest frame is the frame in which on the average the molecules have 0 velocity if you're in that reference frame then first of all I have the same velocity that way as that way number one in the house of a lost city that's determined by the properties of the fluid that the sound is moving in okay Maxwell moral less fox vat light with the same kind of thing that there was a material and the material I had a rest frame and that particular rest frame was the frame in which light would move with the same velocity couple left us to the right and he thought that he was working out the mechanics or the behavior of this particular material and that we were pretty much at rest relative to this material and that's why we saw light moving the same way of to the left of the right one would have to say then that Maxwell did not believe that his equations were a universal set of laws of physics bought a vat they would change when you move from frame to frame just happened by some lock we happen to be more or less at rest relative to the ether to this strange material arm because you could do an experiment with sound if you're moving through the sound you can check the velocity in different directions is different you do there had let's not proud to their worry exactly how you do that that's what the Michelson Morley experiment was Michael Sam and more we are suppose said look the earth is going around in orbit maybe at one season of the year we just happen to be at rest relative to the ... ether by accident had some of the season 6 months later we're gonna be moving in the opposite direction //

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