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# Quantum Entanglements, Part 3 | Lecture 4

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Roy this program is brought to you by Stanford University please visit us at stanford . EDU let's review now I am
not going to be consistent about choice of the speed of light sometimes I would put it in in quality other times I will just call the 1 I will try to tell you when I do and I don't see that's just formulas get a dupe complicated when you put this season and you all should be able to figure out if I give you a formula which the speed of light is equal to 1 you should be able to figure out by dimensional analysis where the speed of light with Lucy's go our but here and they're all overseas buckle when it's important when I think that's a relevant and interesting to reduce our otherwise I will choose the speed of light to be 1 which means we working units like light years three-year-old like seconds per 2nd quarter hour or such Sojourner artillery review quickly we just start with Galileo relativity be Galileo in 3 Einstein no Newton Galileo relationship between reference frames moving stationary reference frames it is given by the simple equations which it called Galileo and both were going transformation Our for example from motion along a one-dimensional lined a year are relationship between the call admits in moving railroad trains and the stationary stations are acts prime is equal to X minus the velocity of the train times AT & T it is equal to take time is the same for everybody and the difference between X coordinates is just Vera because we move the origin of the molding frame is moving with velocity v relative to the stationary train to draw up a picture of the drug picture of what these transformations look like if this is often go myself as a moving train think of you with the stationary our our observers arrest or speak of media the moving as big of you that's arrest of course I think I'm at rest and I think you are moving as of course what relativity is all about but let's a wicked Galileo refers Galway relativity if I were the plot the kind access and the X-axis and then on the same Gaurav draw he prime acts as the steep rise in taxes which stands for the location the locus of points of origin Of coordinates for the prime moving observer would look at an angle like self and the slope of Bangalore angle would be controlled by the velocity of the moving reference be that blind would be given by X just given by X equals city which is the same as X prime equals 0 sewn back so much Our moving frame boats but then ballot the plot leader lines of constant time as seen in the reference frame their exactly the same as the lines of constant time in Kneeland Prime reference for so they look like that you don't have to
draw to actually a space but you do have to draw on the key axes and this will become the tee time Texas after the moving reference reference removing court so if we now remain agreed at this would be the grade for the moving observer an agreement for the stationary observer would be similar except airlines review vertical it is the basic equations that govern be relationship between observers America our Delaware relativity notice that if you solve a few and the equations they simply look like this they like X equals X. prime plus V E but I can also write that is meeting Prime TNT same thing and he is equal to T the relationship between the prime prime coordinates is exactly the same as the relationship the inverted relations between on prime the prime Courtnage with 1 exception you change the sign of the velocity that represents the fact that used moving down the track with velocity plus VAT ICU moving backward in the opposite direction with velocity might so that's how Galileo if you written the equations would have thought about relativity it is awaiting Newton for about it it's the way everybody else did until the advent of Our new discovery that the speed of light is the the same in every reference frame those equations are not consistent with the speed of light being the same every reference frame they have to be corrected and the last time we were about what these rules about life in imply about the corrections to those equations of so we write them down for you to remind you of course the low rents transformations and they look like X primacy Quebec's miners bt exactly as no and Galileo might've written it but then the square root of 1 minus V square downstairs now that if you want to put back the speed of light see equals 1 sometimes you want to put back the speed of light this equation doesn't make good dimensional sense of velocity has units of length per unit time you want 1 has no units at all so what you really want to do is survived by a speed of light Square squared oversee squared and of course if the velocity of the moving reference frame the train in this case is ordinary velocities 50 miles an hour 100 miles an hour thousand miles an hour or even a million miles an hour that's quite
101 miners 1 might be divided by square warmer squares what's that worker over here one-liners upstairs and then the square root of 1 mind square downstairs that's the same as the square root of 1 minus V times 1 minus that is what writers Drury row warmer divided by the square root of 1 mind this plans 1 plus former lenders we squared as 1 might be times 1 plus the 1 canceled and all of this is just equal to the square root of 1 minus V of 1 class OK so 1st of all we rewrite this equation says that X prime plus 2 Rory is equal to X plus Tracy time something which depends on the velocity depends on the velocity as 1 might be all 1 plus nuts look at X prime prime must look at X prime might Steve prime instead of X plus he keep called prime plus deprived Matsuo we get X Prize Money a steep price subtract them within a given X minus from here we're going to get a plus vii because was subtracting arms expert nor do I just for lawmaker small changes I wanna right says he prime plus X prime same thing I haven't changed anything plus it just because it agrees notes in our like this 1 keep prime Maringa 6 Brian keep price minus X Prize Texas track this from that's where we get there and we get team ideas X times 1 plus B divided by the square root of 1 mind as square 1 plus V because of the opposite side of appears to get T minus X and they will get miners V Comdex might that all a up 2 201 1 plus divided by squared away might be squared but if you go through the same exercise here 1 class 1 plus 1 plus with you get is square of 1 Class B over 1 it looks almost the same as the the equation fact prime plus prime but with a slight difference the square root that appears here the square root of 1 plus or former industry now but stop Maryland look at what we have is that I'm working with ex ante I've worked with no coordinates no coordinates X plus and X minus t no didn't think of those as sheriff Mrs. CTN this is acts and 45 degrees remember represents the motion of libraries and ex- prime plus elect Class C is a coordinate along that axis X might be is accorded along this axis This is the X plus T coordinate and this is the X other T minus X coordinate what this is telling us is that X plots tea when it's Lorraine's just get multiplied by by something that depends on the velocity and at the same time he might miss X also just gets multiplied by something but that something is the inverse the something here is 1 over the something here so 1 picture of what is mathematical transformations light is that keep plots X depending on whether these positive or negative he X gets stretched its workers is begets stretched or squeeze depending on what it is these positive and one-liners B is less than 1 Brooks I would say that this transformation squeezes keep prospects people aren't perfect prime p plus X number less than 1 so it squeezes 1 axis and stretches the other axis axes by the inverse amount so for example Lorentz transformation would be appropriate velocity would squeeze this access by a factor of 2 0 and double this axis by a factor of 2 that's the nature of Lorentz transformations there really quite simple their stretching along 1 axis 1 45 deg axes and squeezing along the other 45 degree at notice important thing is they don't change the 45 sees the 45th 5 degree taxis Stacy 45 degrees wise that important because that you that light rays continued to move with the speed of light in any reference frame city learns transformations are really quite simple there stretching along 1 axis squeezing along the other taxes but they preserve the 45 degrees taxi they don't preserved other lines here other lines will get stretch this way and squeeze the other direction but the ones which are along 45 degree taxis do are meeting so that's nature of Lorentz transformations squashing a 1 like like direction and expansion of the other like like their actions are along the direction of motion of the relic of relative motion to reference for bribery questions about the structure of Lorentz transformations before we go on at they are now route OK said but there was to make some assumptions but the assumption is but if To what objects quit and instead of space time if 2 events are at the same instant of space time and now
implies that the distance from the origin of the squalor of sicker square of the distance from the origin Claude Esquerre that's equal to X squared plus Y squared it must also be equal to X's prime squared plus Y prime squared another words the quantity X squared plus Y squared is reined in any coordinates system ordinary coordinate system involving the rotation of court something like that is truly its transformation something slightly different a little slightly different mathematical expression is X she and this is some point here With coordinates XT all our X Prize keep prime depending on which frame of reference I'm looking at that he said the statement which would work out from this equation here I would do it with Stewart let's take the point X which could also be labeled by the prime coordinates worked out the quantity X prime squared not people square plus sex prime squared but keep prime squared minus X Prize square a little bit of a glitch from what we would do just rotating quark gay pride squared minus X prime square let's right right here 1st of all the denominator will get 1 might be squared because we've square mile note the next prime square that has X squared it has Class B squared he squared and that it has money this tool X we see bats all X X prime square have that's that comes in with a minus sign but then let's and keep prime square feet high squared has he squared plus squared x squared minus 2 x also provided by 1 squares I know this is T prime squared this is next prime squared and we subtracted the 1st thing that we Norris is that these things these crossed terms cancel we subtract beast To the crust terms canceled now be X squares don't cancel because Seeley have these square X squared minus X squared so of minus 1 might be squared x squared them what about 50 square Terrence C squared here my ears are square T squares so that's he squared times 1 might be squared all over 1 might be square to make a long story short bus right about keep squared minus X prime squared equal 50 squared minus X square just work out if you didn't follow algebra he does work out square this square this subtract in in your way and you'll find out that keep prime squared minus X prime squared is the same squared exit manner 6 quarters very very much like when you rotate coordinator the square of the distance stays the same square of the distance from the origins stays the same now a physicist would say that means that the quantity he squared minus X squared is invariant it's the same no matter whether you look at it from the point of view of the prime frame or the young prawns and has a name because it invariant did support anything which is invariant which is the same for all observers must they have a special report it's something which everybody will agree upon and so we ought to give it a name the name for it is proper time the proper time between this point and this point proper time the proper time between the origin and the point X is by definition a proper time equal to the square root he squared minus 6 square 1 square root well EDS's is very similar to definition of course it's a definition we could work with definition with the proper time was just uh D squared minus X squared but it's a definition the square root but is very much like looking at a distance between 2 points in using Pythagoras around the Yout Pythagorean theorem says the distance between the point is the square root of X squared plus Y squared the parallel with that we we call this a proper time between origin and the point X T and that says definition but a highly useful definition will see why it's useful go along the main thing about it is that everybody agrees about it OK 1st of all let's look at some possible values that the proper time can have various . 68 if he is bigger annex where the mean 52 annex In magnitude of T squares bulimic square 50 square is bigger than X quarters that means we're talking about points which is a wedge here with time with time component is larger than the x component T-square is larger than x squared all in here but let the particular stay in the upper half from space these squared is begin next quarter if you within this wedge the his alike are why should it it in essential if something and if we're working in 3 dimensions not sorry 3 spatial dimensions not 1 spatial dimension that we should also a minus why square minors a square he miners why Prime square miners prime square the a that makes sure that our definitions don't depend on direction space as long as we know there were only talking about Lorentz transformations along a given access we don't really have the worry about wine but wanted the transformations along some other axis 1 day and y z we would note that this whole quantity keep prime squared minus 6 women's wife women zinc prime squared is invariant why is that incidentally well remember the zinc prime why prime don't transform where we go from 1 frame of reference along the x-axis torn up so that doesn't make any difference but in prime is a prime abuts a convenience store edit and then we have an expression which doesn't depend on directions space
and it's just use proper distance proper distance means he squared minus X squared the square of the proper distance Peace Corps X women's tees with because now 1 of the other and you the fact that did the same in the 2 reference frames so let's think this point over here what is its proper distance of all call a proper time because it's not time like but instead of looking at he squared minus X Quebecor of excrement these where the what is invariant distance four-dimensional space time invariant distance between his point in that point well on the warning and if I work it out in the current frame of reference it's just keep trying squared minus X prime squares that's a square of this proper interval here where this Point has coordinates T prime what what he primary this point the prime of this point 0 0 This is the surface keep prime equal 0 this whole surface corresponds to to the origin of time coordinates for the moving observer so long hair prime is equal to 0 and the proper distance from the origin is just what square the proper distance is just minus X prime square on the other hand let's find the on Prime's coordinates of the sport this was supposed to be an 1 meter long so this corresponds to X equals 1 equals 1 and stationary reference frame the stationery stake in the stationary reference frame is exactly 1 needle of this point corresponds to X equals 1 this fall trajectory here what about that way about the height of this point facts take right that's T equals X. all along this trajectory so long hair TV is equal to 0 does was the expert lexical 1 and so on here this point 4 years x equals 1 and T equals I'm storm left inside here we have the proper distance expressed in terms of prime Court lets expressed it now in terms of the aren't prime coordinates that's going to be he squared which is B squared minus X squared which is what so I find out we discovered that kind of space coordinator of this point over here is X Prize is equal to the square root of 1 mind is a square I just Jesus sorry and take a square root Watts that removing the meterstick which is arrested you offering reference in mind frame of reference appears to have waned square root of 1 mind square another words coordinate distance between the 2 ends of the meter stick in the mold reference frame areas square root of 1 might be square next prime this . 2 0 x prime this point square warmer be squares so you see what that says that says that stationery meterstick is seen in the moving reference frame as having a shorter life again and application of these variants of a proper distance he prime square mile sex prime square is equal to T-square minus X of fraud you just go through it again this point or a 1st offered 1st point of the left here everything is equal to 0 XTX Brian primarily 0 and that's trivial but this point on the 1 hand he Prime is equal to 0 so across this out on the other hand X is equal to 1 and was equal to V so that gives someone might be square over here and tells us the next prime is equal to squared warmer squares so the comparison of the weight of the meters stakes in the tow reference frames says that when meters thick it at rest a moment observer being shorter request a is also Treblinka could be worked out a similar way removing what I we said stationery meterstick looks short the moving observer need moldings meterstick looks short the stationary observer that's the reciprocal character of a of Lorentz transformations OK any questions about the length contraction kind their waste in our Lorentz transformations or any of the geometry that we the and not boat yes revenue we not only do we know it world we certainly know it within the framework of the general theory of relativity yet yes within the framework of the general theory of relativity yes do we know as experimental facts we really have never directly seen gravitational wave would like to see gravitational waves but what we have seen is that the effects of gravitational waves coming from binary pulses ours and what's known about that tends to suggest not terribly strongly the polite thing gravity travel with the speed so now has no direct experimental evidence that gravitational waves travel would be electromagnetic waves on the other hand Einstein would have had to be very wrong about 4 not a threat to be untrue the problem but they I'm worried about not every accelerate fasteners be like Bosnia but that's a lot here are others yes From there it a lesser for years a stroke yes that's true and the latest that OK so here's a moving observer it is woman observer he is star No. 1 year's star No. 2 and that's for light away or we wanna find out how long's not in power stationary reference frame it takes to get here but how long those these moving observers say that it takes and that we have to calculate the proper time between here and here we the point is that the proper time at this point is point maybe she an
Ex the point is the proper time which is he squared the square the proper time is T-square minus X square meters only smaller than just T-square by itself so the proper time between here and here is less than the actual coordinate time so for that reason yes you've actually write this personal but he would say that it takes less time than then former light years away several would be at the stationary observer would the robbers that but the lawyer start fast well what what the only observer would say Is that he would order mowing observer would save for himself was on frame of reference and then you would look at the 2 starters and he was ceded to 2 stars were a lot closer together than what a stationary observers said so you would say it took a short amount of time to go from 1 to the other another words the moving observer would see the low rents contracted distance between the stars and so it would conclude that it took a small amount of time to go from 1 start to the other it would be a good buy you know nor Norway the stars would not appeared to be travel faster than the speed of light it what appeared to be traveling with exactly the opposite velocity the stars see the astronaut traveling with a certain velocity and the astronauts star traveling with exactly the negative velocity vector the yes but after S 1 is moving How far apart disease stars moving with close to the speed of light that see the stars as being not 5 right use of 4 light years over with said before a light years Times Square 1 might be square and that his reference frame would be the description Sarwar difficult fighters but right is With sorry set for the hour OK like like earlier this year you're correct but then we have to analyze a more complicated problem complicated problem and starts with an observer which is arrests and may again be observer starts to accelerate and has to accelerate up to the structure is velocity and we have to take into account the effects of acceleration the simple a calculation to do is to imagine an astronaut there was always forever and ever moving with uniform velocity remember the special theory of relativity just like Newton's theory only makes sense inertial reference to find out what happens among commercial reference from we can answer the question we can answer the question but the question will not be intrigued by application of exactly the same formulas that would reduce the social let yet laughter a guy with a limit talking about it an inertial reference the study none the reference frames we have to begin all over again and as you correctly point out there was obviously don't be some effects of acceleration which we have to take into account for that there is little right but Rafael at yeah not erased sorry yes the show violence yeah yup yup native was born Jew was born moving with a larger losses if West that right revenue still the same figure still calculated total proper time along trajectory so the proper time have contributions from different velocities only those out there so blue workout but if this right now we still want when twins born with the huge velocity in Ireland has left off the the poor the poor woman could be it works a mail order all Rock said but that well we that time is not a special prize our yet we tell EU right in the back of our minds we can we can't isolate time special but the lesson the special relativity is time is not that special power there were over the real question is can we recover all of the physics of slow velocities that we normally known learned in love from our Darwinian circuits that we that we inherited from from all the years that nothing ever moved faster than 50 miles an hour recovered by taking the formulas from special relativity and armed specializing in the case of very slow velocities of course we can but I also got but where as far as they that's as well yes as necessary not only unnecessary Einstein never thought about that he said time was the 4th dimension is not necessary to think of times the 4th dimension but once you start mixing it up with space where are Lorentz transformations here with an awful lot Of the physics which looks very symmetric between space and time for SunOS symmetric key squared minus X square were square mile squared they don't get mixed up with each other under a under Lorentz transformations in a way that they wouldn't under these Delhi and transferred so he's seeing Galileo kind really just his time whereas in Einstein or low-rent time really does a period of time really does get mixed up with space In a way that makes you not want tool our think of it as are fundamentally separate them and be couples geometrically part of a whole lot which is bigger than either of her young at each other but that not think of the money needed nearly million plant I wish down her so the preceding
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