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Quantum Entanglements, Part 3 | Lecture 5

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this program is brought to you by Stanford University please visit us Ed Standford . EDU With solution or about 4
of the complexity of it a stable last shit shit br yacht arm us right here is a picture of the you have a 5th dimension and think of the 5th Dimension existing between a Florida loss of is a gravitational field pulls everything down her flora yeah particles or strings whatever they are which can live in this 5 dimensional world of sentiment strings and that tend the fall fault about a year the the bottom they hate drugs protons neutrons arms and so forth that they get stuck at the top and they can stick to assume figures that the ceiling if start-up how they are Gravitt found both more elementary particles bar not not head runs a drums of big fat composite objects on when you collide to Hager that such colliding particles are on the floor show Clive make a big splash of energy and the big splash of energy is apparently nicely described well described by saying that a black hole forms of black holes a big puddle of energy a big puddle of energy are a lot a lot With edge the edges the the horizon and Michael black clothes they have temperature of entropy thermodynamic properties of black holes and this object is a kind of black hole it sitting down on the floor it's not the kind of black holes four-dimensional calls black holes that we think of as astronomical black holes for kind of black holes we would make by scattering particles of enormous energies above the plants get those kind of black holes also have horizons they're very similar except they are droplets hanging from the ceiling heading this way hanging from the ceiling patch of the ceiling and their stuck to the and a sticky Stewart otherwise there the same kind of thing with the same kind of thing in 5 dimensions the colors on the black holes that correspond Verizon's and the correspond to all the things that that Stephen Hawking and others have studied over the years now if High Energy collision happened was an enormous lump of energy remote Pappas particles energy they get collapsed like that pancakes by Lawrence contraction what happens is to he takes direction and this direction that like pancakes so they are out of drove exactly Bob whatever they are there's begin in this direction and they come in and Nikolai collide with the enormous amount of energy and make a great Pavel of energy like that but that puddle of energy is more or less like a lot of viscous fluid and not so biscuits grab on this case a lot the fluid and deposited in this gravitational field the gravitational book pupils it down spreads it up and spread around on telecoms some equilibrium like a puddle on the ground that would stay that way for except for 1 thing and the 1 thing is Hawking radiation Hawking radiation starts causing radiation from our office then what is that radiation that radiation is the ordinary particles it we see protons neutrons mayors on and so forth other we see coming out of this rule when he he showed us some of these pictures of these events it was 7 thousand particles coming out of a little 5 year 0 7 thousand particles a nest picture they're essentially the Hawking radiation boiled off the other parts of the black hole here arm as I said whether you should really think of it as a 5th dimension and real honest-to-goodness black hole sitting on the floor of the 5th Dimension here or whether you should just think of it as a mathematical analogy at this point I would say it is up to you after I think in time we will understand better the connection between these ideas and gravity guests Is they will be connected in the deep way and that that we will learnt understand this picture as part of a bigger things part of as would think theory will be quantum gravity would be severe we haven't seen yet but he I think the 5th Dimension is quite real it was a 3 Rio it means it fits together with the other dimensions of a mathematical structure that Obama Breyer her 3rd jet that makes the laws of physics or look simple providing this into real space our fictitious space in treating the fictitious space completely different seems to make the equations more complicated than if you think I love if few really literally think of this the 5th dimension but this fundamental structure here really is nothing but quantum chromodynamics QCD theory of quirks and sold has as a say at the moment that's kind of up to you with think of it as having any reality or you just wanna think of it as a mathematical analogy that allows you to
difficult problems easily but yes but a line drawn out city and which line command alliance this fund that's a special mention yards based and everything space here well this would co-sponsor horizons now I am it's not well Her baseline the attitude the are old right now the moral Aindrea well yes the details of it there the devil is in the details this prediction about the viscosity was a stunner nobody expected that sir but was purely the only explanation of that we only quantitative explanation of comes from the peril of black holes are on beat that was a stunner that was not expected that other interesting things other interesting kind of experiments way particles move through this plasma volume the details of it may be better described by thinking of it as a black hole and thinking of it as well thinking of it in terms of quantum chromodynamics and doing quantum chromodynamics correctly and calculating it honestly if we know how to do that should give us the same answer is not that this series is replacing theory believed to be a mathematically equivalent description but 1 that useful in context where we simply can't calculate car more dynamics of that we've never been able recalculate quantum chromodynamics very effectively acceptance of very very special situations that right now that's right the shot so far well in the end it's 5 dimensional incarnation it does have gravity and does however horizons and warned them into versions this is just a viable quote without gravity look but young right there is viscosity can always be described in principle by understanding how quarks and gluons but when I say in principle nobody has now has learned how to do that can't you appearing is a difficult subject nobody knows how to calculate those sorts of things the equivalent with gravity gravity is also ordered but something so easy so things are easier as it happens calculating viscosity of our horizons if out bodies not West Rail said modified seeds is not policies that long said that it is not so wild Our didn't included in tonight's lecture it very well could have been but let me tell you the answer is right something called center of mass energy which is the energy of the system was viewers from the center of mass frame and other word overtook particles of colliding with equal opposite momentum the center of mass energy is called naturally and dumb that's equal roughly speaking in units in which sees equal to 1 it's just equaled is very relativistic particles relativistic particles have an energy which is equal to their momentum and so it's just equal to twice the momentum calf twice the momentum of these particles and fact and that can be very large millions if you slammed the fast particle into a standing still particles and you ask what the center of mass energy of that removes what would be the energy if you went to the center of mass of its energy would be much smaller much much smaller it wouldn't be twice the momentum that it would be something like that of the square root of the mass of this article turned the momentum of the fast particle here that's much much smaller Carlos deal this is much smaller than this the moment as large so the point is that you'll lows as North America affected energy by letting the particle standstill if you could accelerate the same speed the other 1 is moving that doesn't give you twice the energy in effect it gives you the square of the energy basically our served a much more energetic collision Pavin collide head-on and we could do we will be able do that will be in position to do this detailed but there are not but that's reason energy there is a place where it has received a lot of it is best stay that way on you electric and magnetic fields could only be so big for a road before they can be supported by steel and concrete and things of all the accelerated together are you could only create magnetic fields which are 10 Kessler or something like that and it's a magnetic fields accelerating the particles and so you can't accelerator probably somebody that some manner of in principle it's a matter of strength of materials all sorts of engineering things and the ability to create huge magnetic fields that limit the ability to accelerate particles over short distances of space is simply can't create big enough fields accelerate particle relativistic energies of broke onto a highly relativistic energy over a meter of the more you wanna accelerated
the more rural indeed because of all the acceleration is Somoza longer push to give a push to get a push give the putsch and so you gotta given a lot of questions and that's all it is basically the energy of the accelerators proportional to Woods Lane because it just accelerates uh with more or less uniform energy per room per meter they had a a a a wall of all well then I think you would simply run in early hiring cyclotron I think it's simply can't create big enough magnetic fields lasting long enough to of other very important for it the other the the other very important point has to do with energy loss when you accelerated particle charge 1st of all you can only accelerate charged particles you can accelerate neutral particles that charged particles when they move in circles accelerate and the smaller the except a small the circle forgiven velocity the bigger the acceleration particle moving in a tight little circles is highly accelerated acceleration means radiation and radiation means energy loss was very very important technological point here that the circle is too small and will simply lose energy at a rate which is as big as it dumping energy and you're limited in the size of the accelerator by not wanting it to lose energy where radiation to a straight accelerated dozens of Article moving in a straight line with uniform velocity doesn't accelerated does a radio so give a push approach push push well a little bit of acceleration alright which causes it to radiate making it turned our around the circle really causes big problems are those problems can sometimes be advantages for example of a slack they run electrons of circles and they take off synchrotron radiation radiation that comes off and they use it for other purposes but point is to get a lot of energy into your charged particles you want to accelerate the Administrator line possible be costs in that way you are you eliminate the tendency of particles radio away energy E C is a lot less unless they can be young young arm electrons are are a much lighter than there are so to get a given its energy that's important your dumped energy into a small volume and see what comes out now of electrons are very much lighter and Proton so principal you gotta get them up to or larger velocity for energy our you gotta go Motor Lodge a velocity and getting them to larger velocity that also means for a given circle a larger acceleration so if you were to accelerate electrons In a joint collider they would be subject and try to get them up to the same energy they would have a much larger acceleration a much larger energy lost so again it's energy lost alike particle needs to go fast the the same energy faster means more acceleration Monaco's circle acceleration if they wanted if what 1 half hour you could do you could do a Johnson career accelerator former after drums and accelerated you you get to a few get to accelerated Mini Mini times as it goes around the circle going around in a circle and accelerating it is equivalent having Newton honestly big accelerator eager to Persian Persian Persian Persian budget so you yet you don't know me accelerator reject the linear accelerator area much bigger E for your good long wait to do it in but you don't want to much radiation the Ahrens be compromised stab a big circle and to keep accelerating along the route until its limited by a by the expiry radiation will be limited by the radiation at the same velocity the electrons will be limited by the same velocity but that means much more energy you're for protons up OK if less us if that's get back to the special theory her wrote Hillary is just fun let's do 1 puzzle all problems in special relativity or puzzles and pressure relativity usually saying they have to do with 1 basic flawed concept that would be that we carry around those evolutionary baggage confuses us every time we have some relatively problem thing to do is to draw good diagram and to remember in detail what what they're diagrams telling us is a puzzle of things puzzle is usually phrase in terms of a limousine trying to get in dog garage arises size at 12 feet 12 feet away 12 13 feet blank along murder arises from water but that just big enough that an ordinary car complete into it a day stretch limousine wants to get into it will Of course this way stretch limousine is 20 feet
long Quanzhou into the garage salt drivers as a piano the special theory of relativity speed up my car and whoever is watching from the rest frame will watch this somebody watching from the rest frame or watch this and according to his calculations my car will be Lawrence contracted in the frame of reference of the garage will be learns contracted African women's contracted enough making it move fast enough I can get it into the garage getting into liberalize means the front end of the back end of both within the garage simultaneously for me other the other Moorad but the car standing still and the garages bullet tore the car and so from a frame of reference in which the cars standing still we garage looks Lawrence contracted and therefore is much smaller than the size of a or it's quite clear that the kid didn't grow so we have a puzzling have a paradox can limousine gains or cable interrupt which I said what I means limousine the beat in the garage the front end and the back and are simultaneously inside the garage and the nasty concept that we inherited and we have a difficult time getting rid of is the concept of Siam alternate what it means for the 2 ends of the car to be simultaneously inside the garage that's the rub that simultaneous committee is not the same in all reference frame so the antidote to understand this particular problem will 1st of all we open the door to a garage full of the Korean War but just for safety sake I'm also going to open the back door I will simply him a car and Eliza car ever simultaneously all of it inside the garage which we won't worry about what happens if if it gets in the garage who won't worry too much about what happens Woods decelerated by the back wall of the garage and later Lloyd that is simply open up the back door to return with Jewish purpose chocolate OK so how Analyze This is always the 1st thing to do is the joys of picture Carver drive suffer picture and get everything laid out on the picture vertical axis is time this is all problem that takes place and 1 mentioned wonderment so we don't have to worry that much of a white z pretty much it is we have to worry about 5 and the 1st thing we drawing this diagram is like call the trajectory of light rays is just Lawrence Ellison reminds us what very very fast means the speed of light so we draw him like like trajectory for and then we begin through Oct then the next thing is to put in Otto observers 1 of observer a busy observer at rest and coordinates ex ante just look like to draw the 2nd observer's moving wrote to refers to a velocity and so his axis looks like best be called pride but put a little bit bigger angle the bigger the angle of faster is moving I want to exaggerate the motion for a way out here with something like half the speed of light or something like that then we also have to remember that axis new x-axis theater climax His counted relative to the X-axis meaning that weapon or observer course simultaneous is different than what the old observer corps the old observer calls everything on the horizontal axis simultaneous leaders to say that happens at the same time the no observer the moving observer calls everything all along L order Burrall will drawn 1 more try I was taught when you draw pictures like this in your neck straight lines take your eye off the chalk and and only look at the end point never works look at me and . her sister from melt adds spent over here to March about acceleration but not that this is not a moving fingers now OK now leg Yao they are they are yes they are equal is coordinates in which the speed of light is 1 you don't try to draw their summits in the coordinates which the speed of light as a 186 thousand miles per 2nd beers and this is practically horizontal annual BOC anything actor coordinates him with speed of light warned the picture looks like now alleged role in the garage 1st of all there is of course car front and of the front end of a garage and the back end of row at the front end of the garage stand still an our frame of reference and that means looks that the vertical lines the the same is true of the back end of barrage it's also all of this and here that's also a vertical line would search were so big we did here between 2 red lines with just 5 conditions but with a white the region between little red lines is the interior of the barrage now about the limousine busy it moving tho and it's not only with velocity of the other frame of reference the frame of reference is the frame of reference of limousines he has the backing of limiting the back in the limousine or moves and at this point Estrada green in the back in the limousine is here the front end of a limousine is over here withdrawn OK I I will draw on the front end of a limousine the front end of the limousine and growing poets sea come back our for were here there's a front end the limousine movie with the same speed of course is the back in now I the back end here the back so losing goes right through it doesn't mention the 5 disposed of an art question is is there any instead of time at which the entire limousine and awards distance from the back going for the front ring or from the rear of the car the front of the portion of the limousine between the rear end of the front there is a time at which it altogether simultaneously inside the red now we have to decide what we mean by simultaneously if we mean simultaneously from the point of view of the stationary observer that's asking the question is there a horizontal line you could draw here there which both the back end and the front-end will both peer inside the rock yes there are hear the backing in India front end is still inside is right here the front-end his weather front and went into the garage front and lost the beginning of the here and the front end went out the rear door of here the back end where did the liberalized a here and comes out over here so here is a timeline of India when the back end is just getting in and the front end has not yet gone outside the Iran so yes there is In a time when the entire cars inside the garage but that's from the perspective of the stationary observer What about the moving observer for the motor observer lines like missed by simultaneous violated thank point and draw the appropriate line would look like that From the moment observers point of view of this point at that point thus far more paying notice at this point and this point on unmarked both side of the road
for time the backing gets in the front the valve so from the point of view of the moving observer not the whole card never was inside the garage is not paradox is no contradiction the contradiction of the apparent paradox is simply by is simply too not distinguishing what we mean by sigh moping mirrors in the toe friends a reference irked different yard realized said yes it is a plan that would be a big game for us this is not a member of the House has passed that they want what that blind is that line has a link which is the low rents contracted length of the a limousine this is the low-rent contracted length of the limousine Florence contracted because of the length of the limousine and seen from a stationary reference if I were to calculate the proper distance from here to here that would represent Sri Lanka limousine and the limousines reference right now but that's a that's a little puzzle at some of the pillar of Austria of a large number of puzzles and relativity theory and usually the answer is to remind yourself that simultaneously is a relative concept and almost always with you analyze the experiment in these kind of puzzles is a hidden in use of the word simultaneous and that's where we the paradox comes in OK now we want to get back to Lorentz transformations to the mathematics of Lorentz transformations and the summer want to move on and understand the laws of physics as they are described in relativity theory the same laws that for example Newton's laws of forest Newton's laws of motion so for what happens to them we will want to relativity we need some mathematics most of mathematics is notation not much more notation 1st of all we have 4 coordinates X Y and Z and the 4th coordinator see it's a convenient location of the lump them altogether P X Y and Zidane but they usually pretty and and call them all by the name X X 1 X 2 X 3 and x 0 1 0 instead of 4 I don't know if that's a standard notation maybe I should put 6 0 at the beginning but threw out my notes I seemed to her output the x 0 it in there over the end of line 0 X 1 X 2 X 3 those correspond x y z and T so the odd man out here is X naught that's the 1 that is kind of the other 3 are spatial axes altogether we also want them altogether and call them X New New Greek letter new coming from the rear end of the alphabet represents Air indices 1 2 3 and 4 nuking go from 1 9 1 2 3 4 1 2 3 0 0 X yellow eagled the 1 yard we are setting Siegel 1 and when it's interesting to do so I'll put the seize back only wanted interesting to do so for example take nonrelativistic limits of situations we want to see how things behave was slowly moving objects no more conventional our situation then it's convenient but the seas back into the sea to see how we approach the year ordinary physics yard Odie CT right now would be CT Zachary now there's a mother show that which is X will allow a meal new in this expression goes upstairs is a incident recalled cities across Contra Marion indices was together which runs which when upstairs core call was low and country is kind of a physicist go upstairs and downstairs we never use that language we talk about upstairs and downstairs I will look at this again which is if because of work so these are the call variance components these components Our eh of an object of that you will see are but there are 2 ways to describe just don't this different descriptions of the same thing and difference in the description is that in these variant components are the same vector you're right them as minus X minus why miners E & T another way take the same object and you just consider the space components made negative if I give your point X Y Z c and tell you about are those of positions at a certain point and you what other cold variant indices of that point where the coal area description of that point so the country you just multiply the space components by minus 1 that definition you simply change the signing of the 1st 3 components Why do you do that you do this only foreign need notation Australia with the notation tho With it is a subscript is subscript no now OK weaker right this is he the suspects warned this is equal to X 1 X 2 X 3 X Joel X 1 with lower index is minus the X form of to the upper there was nothing there's nothing fancy going on here you just take the 1st 3 entries and change their size and that gives you a nor object which is called X sub so for example have I picked the point Mexico's 3 equals 2 0 's equals flora and T equals Taylor member the standard express India would just be free to roam for came about RBX mere works meal where equal 3 tool for 10 but this will be my nursery minus minus 4 contend that some different point that's the same point of purchase of different notation from the same point they said same point differed different notation for it just remember when you see an object of coal marine industries like this remember if you wanna know exactly where it is space time change the of these core whitey do this well makes a nice makes a nice notation makes a notation which systematically makes elegant and pretty equations and helps to keep track of various might decide their card relativity OK let's let's consider the following object some additional new summation of a new means used some of 1 2 3 0 X new tax knew what is the subject by Sunday new means X 1 X 1 1 that was X 1 X 1 X 1 of lower index is just the same as X for upper index except changes sign so this is my nest X 1 squared which is minus square what about X 2 X 2 extra Cho Dexter was wired and next was wiII so the mooring Beck
representation we change signed in this becomes minus Y and then fuzzy it's the same thing might square but then plus he squared wiped vastly square feet because the lorry next and the upper index correspond to exactly the same sign for the time component this just X 1 X 1 Lollar index of upward next but that's minus X squared plus next to a lower next to up the square next to upper but minus why squares that's 1 of my C squared plus square foot that's exactly the the opposite of the that's exactly the object that we called the proper distance from the origin of the proper time from the origin of the point x y z so we have known notation rotational tracks so far you don't see what the trick is but you find out that the trick is quite useful that we can write a proper distance from the origin to a point just form summation of new X Mueller was 1 index and 1 index lower now own an all over an all over again and in relativity theory you be doing some like we use some of the 1st direction the 2nd direction of their direction of what direction with the 1st 3 of them having the opposite sign from the last war that happened over and over again in fact it happens all over and over so many kinds of Einstein got sick of writing the summation signed here and it just left that out he said rule whenever you see an expression like this with an operating next summer Lauren Jackson the upper index no index of a saying it means some although the industry call the Einstein summation convention it was probably not Einstein's most brilliant accomplishment but it was a pretty big and smart so X new X knew this kind of expression Oregon next will be repeated once lower and once opera and although earlier this 1 supper and wants slower and use you remember to somewhat over the index what do you get you get the proper distance of a proper time from the origin of the point x y z so this is indeed notation was a floor it's another patient a combination he squared minus 6 square miners why square miners E. squares of voters the and Tower was a status symbol of sometimes called S them but gaps that was a standard is it is a state of civil proper time I will use that over and over it gets tiresome to write down exclaimed he's who X quipped these women's women as wise Gramercy squared off and this is the notation that we that's 1 piece mathematical notation among core mathematics it just notation that was very very convenient relativity and that it would be very difficult to write the equations of relativity without this is notational device booking next let's talk about transformation laws in general and let's talk about not Lorentz transformations laments transformations are things which makes time and space together for example if you consider Eddie up to what observers which are moving relative to each other along the the x-axis then we have a range transformations which makes X tinny leave wines E. along also Lorentz transformations Our Aaron principle of relativity is a principle of symmetry that says the equations of our theories should be symmetric should not change we make allowance transformation while the other symmetries the more familiar with the simplest of which is as rotation of coordinates space ordinary spatial rotate car accord the rotation patients are in fact is fully combined the 2 together arrange transformation spatial rotation as we get a large number of transformations which represents basically the entire set of symmetries wrote physics almost partnership but he is point here is that Straw 1 more is next he is he has not said drawn y somewhere in year in the 4th dimension we can't see it but it's also very now I've taught you how do Lawrence transformations along the x-axis Meridith but you had a compared be coordinates a moving observer was moving along the x-axis with the coordinates of the stationary I don't think it would be through hard work and remember what does it mixes ex ante together but leaves y alone is why alone does not transform the coordinate perpendicular to the motion relative motion so everybody agrees about wiII but they disagree about teen but nevertheless transform into each other ambulance transformation is now I think you could probably figure out if instead of thinking of 2 observers moving relative to each other along the x-axis suppose we had the 2nd observer moving along the y-axis either you could probably figure out how to do the transformation instead of writing things like X prime is equal to X minus he oversaw corners of 1 mind squared Be Kind equals he minus X of square 1 might be squared and why crying Nichols Wii and also chronicles a heat over you really have to do His interchange X right that white Prime is equal to 1 bt keep trying his he might be but X Prize musical tax that would merit hardly just realized that what you originally called the X-axis has now become the Y axis and simply make that replaced but supposing you want to do a Lorentz transformation under the words you want to consider an observer who was moving along some funny access by their X the wine but along some funny angle that nothing could do would be before you make Lorentz transformation do what a rotation of coordinates in space find a newly coordinate X prime and new coordinate why primed for judges rotated in space as they do Lieurance transformation along with the new Klein by compounding together rotations of space with Lieurance transformations along 1 axis you can construct all of the possible frames of reference moving relative to each other with uniform velocity if there if they're moving along some not all along 1 of the original sees all you do is rotate coordinates from the or rotate spatial coordinates until you have your axis on the other axis of motion aligned with 1 of the Nowak and been to Lorentz transformation along so the other thing that we have to know is how to transform coordinates on the rotations of space that is something that has nothing to do what Einstein goes back probably the Euclid that work was the first one who figured out how to rotate coordinates today and off you could use coordinates of vectors and I don't think he did political back they find much of it because finally they cars young so calm but many case it goes way way back past Einstein pensions of corn at every location in space has an exodus we can pick up access would pick busy access to rotate around it we picked access to rotate around then only X white transformed stays the same so rotating about z-axis is vertical Anwar rotating
about that taxes and so XMY get mixed up with each other but right down I don't approve takes a little bit trigonometry to figure out what these transformations are let stratiotes Plains this way he is Texas his wife Harris next time he is why Prime the angle of rotation is favor and some vector With coordinates x y the questionnaires and scored his ex-wife means that if you drop of perpendicular the X-axis at point X and draw a drop perpendicular who will give you walk now you want the prime coordinates after retention do the same thing you drop a perpendicular to the X prime taxes you drop of perpendicular the White prime exodus and you look at the coordinates and the X Prize why Prime Max so that the geometry of the geometry of the Cold War and the transformation from XMY 8 X Prize in white it's just a matter of dropping perpendicular to the appropriate actually give you the formula FX Prime Leipheimer terms of acts of it depends on the angle theta and you could check this it's a little bit of trigonometry element is to trigonometry next time is equal to X's cosigned data plus Y signed of data MY Prime is equal Tom minus X signed made a plus cosigned you could check that for example when they is 0 degrees were the same data 0 degrees side of played 0 Cosira is 1 so it's as extra equal to X premises why can't cyclical why that's got a reputation by 0 degrees the rotational now trial rotation by 90 degrees new fever also works and that's a general rotation of court now this transformation is something which is left invariant by this transformation is a distant point y play left invariant I mean that the distance the squalor of the distance for the point x y what the square of the distance is X where plus Y squared Our if you move the player you move a point out of the plane you might also a Lindsay squared off foot said but since a Z Prime's equal cuisine you rotating about the z-axis so nothing happens busy but what this transformation leaders invariant is X squared plus Y square Posey score and that's we do just to have start with sex workers 1 or did that means that if you made the transformation X square was why squared same thing hands at Prime square was why prime square well geometrically the reason is because it doesn't matter whether you calculated distance by referring to leave original coordinates of the rotating distant must be the same distance is a measure would take measure and tape measure are doesn't know what your way your coordinates are so the answer has to be the same X squared plus Y squared must be Quebec's arms quibbles why prime square but you could check that it is checked that for example X prime square meters call co-starring see signed beta-test right so much ex prime square with text prime squared at x squared C squared plus Y squared squared plus twice XYE X squared C squared plus Y squared as squared plus place across that 6 times square for about why Prime's scored that's why try and squared his X squared has squared from here plus Y squared C squared the 2nd and Martinez figures as a minus sign he might have to X Y CS now it together right and left they inside you just get X prime square plus 1 prime squared off but on the right inside you get X squared times cosine square was signed square cosigned squared plus sign square is 1 right has just X or Y squared you also have as Plassey squared that is why squared and the crust earlier just cancels let's YCS and my historic goes away so approving what's obvious anyway that expression for the distance doesn't depend exactly how you already in the act sees it's always just export why squared every alien Zeke pride in enclosed sea since Equus same Izzy Prime would also say that the distance in 3 dimensions is unchanged z is the same as E prime serve was nothing new there the rotation mathematically are transformations on the components which preserve the light which keep going on changed that's the mathematical definition of rotation was right now there's another way write such a transformation I want people seem to be walking in confused but is storm resist war-worn terms for 1 OK so that's the idea of rotations if you combine the rotations with Iran's transformation another words after having me the transformation that's probably why Prime and also cheap prime equal to if you like if you wanna think about the full 4 dimensions our then you can go and do a new low rents transformation along the X Prize keep prime axes and that way below the observer who was moving along the new X Prize axis rather than the old combining rotations and Lorentz transformations basically gives you a wide class of different transformations which correspond to the rotations of all kinds and Lawrence transformations along all the possible acts sees that you could but you could do em right now more notation more notation rotations boring but once you get it you've got a birth it makes equations much easier simpler very easy to manipulate everybody I know a little bit of linear algebra and that's the wrong question as anybody and not money around Everybody knows linear algebra goods we don't need that we don't need to worth OK go that's excellent I know that there are 2 ways that we can write these these equation is legislature stickers were here wicker rioted in matrix form and Prime white Prime is equal to some matrix times ex-wife The Matrix wooden factors be co-starring later signed fade a minus sign Bader karzai data we could also write in another form of weaker right X and with gossip but it will support is prime Zeke Ryan Z and then matrix would be a freebie 3 matrix 0 0 1 the Matrix would then act among the XY components theater mix them up leaving Seattle picture altogether and then they were back on Zepa gives you Prime which were just busy cozy prior so every such transformation has a matrix associated with it on end is another later writers who we could also like another former I try now I is only going from 1 of 3 so far I have mixed in time for next time I I could be 1 2 or 3 representing x
y z is equal am I J E XJU where MIJ just represent the components Of the Matrix is the same expression after multiply and then something over jet as always when we see repeated index we some of reject in this case since is only got to do with space in time no different world where my story about Upstairs Downstairs indices it's only when time comes into it that we worry about that our this is a way of writing a vector is able to matrix times actor and we can write it this way he both exactly the same expression look any question about that because you know if you know this and we can move on from Iran's transformations rent transformations also have a similar form there also matrices multiplied by components so let's work let's work that out in detail but the matrix associated the matrix associated with the rotation is in the most general rotation is the most general rotation about z-axis with the rotated out some other axis get more complicated but this is typical although revision occurred bird website transformation acts trying equal X miners VT all the square root of 1 mind be squared white Prime is equal physical Hawaii zinc Zedekiah musical dizzy and prying physical T minus the X divided by square 1 might be square just gold warrants transformation except for the Y and Z doing nothing saying the same thing we could also write In terms of reform by 4 matrix before a for matrix equation if is decks OK X works but what's label the entries X Y Z T X 1 X 2 X 3 X X not X wife Z but entries are the coefficient in next time the coefficient multiplying x is 1 square root of 1 might be squared the coefficient multiply and wider 0 0 yeah and uh minus V all square 1 miners square 0 1 0 0 0 0 0 0 1 0 0 and all the square root of 4 miners V squared 0 0 0 0 Viola square 1 square architecture let's see what this I maintained their take the Commerce acts prime Prime's there 70 equal to this matrix times a column X Y Z but that really is and transformation let's was it said it says that X prime to forget so is the next forum is equal to enter wrong not now were not yet for 4 years while I so says that X prime is equal X divided by square 1 might be squared minus VT over square warmer and be squared with nothing mixing in from 1 that's correct it also says that while Prime is equal white zinc kind physical disease and Prime is minus V over square 1 might be squared times plaster while the square 1 might be squared a sea of us less equation here so once again although events transformation can be written as a matrix multiply the column XYC rotation could be written as a rise a matrix multiplying column ex-wife GTE but usually think of rotations as having to do anything with Katie Bowie Jacinto rotation as a special rotation but that just means the keep Prime is equal to T when we do a rotation so I have formed for matrices which correspond to rotations we call them are now we have followed by 4 matrices which correspond tolerance transformations we can call them Hill what if we want to do a transformational Arens transformation along some other acts what we do know is we 1st use the rotation matrix throughout taxis in the end we use Owens transformation matrix to boost auto give us some velocity along the new X Prize taxes another words what it comes down to is you multiply rotation matrices by Laurent matrices you 1st rotate 1st red tape and then you learn transformer you multiply the matrices together defined what action would be for a low rents transformation along some of angle relief work out 1 useful application of matrix multiplication Motor which rotations and game now when I can't do much with rotations of space and Lorentz transformations simultaneously as enough to know that our laws of physics are invariant under rotations and Lorentz transformations along 1 single access because their invariant the rotations and Lorentz transformations along an axis then they will be variant of the Lorentz transformation in the symmetry on the rotation what's easy will have a think about that tree under Lorentz transformations are interested well I work out using matrix multiplication it is the compound Bingham velocities if I am moving relative to you with velocities and a 3rd observer was moving relative to meet with velocity you V lets me relative to you you'll that's him relative to me said b mail after being me relative to you you'll him wrote that made is a clear the for the year at then what transformation between him and you think what's the velocity of that you will see him move with relative to you get me out of the picture eliminate that velocity will call double your but the trick is defined W in terms of being you know it fashion pretty relativistic physics and so was his velocity is V plus your I'm moving with velocity V is me with velocity you'll the natural expectations in ordinary Newtonian physics would be that him wrote to you is velocity V plus you so this is a formula for free relativistic physics Einstein physics W is Eagle need plus you it can't be the right thing and we know that relativity theory it's impossible to exceed the speed of light Have I can move with three-quarters of the speed of light relative to to you and he could with three-quarters of the speed of light relative to me that we use this formula he is remotely relative to to you with 1 and a half times speed of light to some growers can figure that correctly or we need to do with the compound tularensis transformation 1 with velocity v and 1 with velocity you and that corresponds to matrix multiplication let us we don't need wife Kinsey in here let's write down matrix of transformation between you and me but that's just 1 over 2 rows of 12 square root 1 might be squared Be of the square root one-liners these
square he always square root miners might be over square 1 square 1 always square 1 miners I just threw away br everything but we execute part of this transformation that's the the matrix that takes us let us say from York for Mike Ward Valdez a 2nd French former Schirmer which takes From Michael that coordinates and that transformation has performed along with scores X double prime and he double prime at his coordinates and that's equal to 1 over 1 might use square square root minus you will always square root of 1 might pursue squared you'll Over square 1 might as you squared minus sign and they won over square kilometers 2 square that the the transformation forums ex prime prime so this takes us from meeting here M and this takes us from you to me what is the transformation which takes you to him all we have to do is plug in for next time here this matrix finds XT and so we see that X double prime he double crime is people a puke we have the multiply these 2 matrices together defined the transformation between you and him OK this is not so much to do so etc. multiplier transformation but do it 1st and 5 minutes doing it let's say we have this find this
mind this time is that gonna be warned plus you think of after the poor over here once pursue via from respect and celebrate 1 times 1 mom right this my plus you what about the next corner over here that's 1 firms miners V and my Desue city and I think it might be Myrna Steelers arrived a bit today I think it's also minus V minus you down here and I think it's warrantless UVO here now the question is this hand performer if where what form the reworked have 1 0 0 1 minus W squared square root W. all the square root of 1 mind W. squared minus W square warm Ernest W. square 1 way or another does it actually itself have the form of a Lorentz transformation and if so what is the W OK the answer is yes it does have a former Lorentz transformation but I'm not going to work the whole and not the through rule this whole matrix here has more we would do instead is just show you how to tackle off the value of W value of W is easy to to pick off you know it is it's just the race all of this the value of home if this truly has a form of Lorentz transformation limits easy to pick off the value of W it's just the ratio of the component of India the component of the heel of the minus sign right minus the ratio of this this is just W so followed do is fickle off the velocity of the compounds transformation and not bother trying to prove that actually has form which it does although they have to do is look at the ratio of this component of this component and that must be W. saw the careered off W. equal your classmate 8 divided by 1 plus you see that's it that's what we have to do to figure out what b relative city between you and him his compound the transformation go back later and proof that this really has this form our but if all we want to do is figure out the velocity of relative to you or to you only have to do is worked out with the ratio of this component of this component Miners remind of poor vetoes you divided by 1 UVA that's the velocity W. now are we could put some money we could put some speed of light in Paris they units of you wouldn't be of velocity and even some W. also supposed to be velocity so that looks pretty good the only thing wrong with it is you can't hear the which has velocity squared 2 wanted from Leicester 1st divided by the speed of light square so inclined the speed of light W is equal to you'll plus VAT that's what know and would have said they would suggest you but Einstein comes in and throws the include the nominated he city both you would be a positive imagine you would be a positive for all going down block the X-axis were positive velocity he's going down relative to me with positive velocity then 1 plus UV is bigger than 1 and double will be less than you placidity in fact this formula will never get bigger than the speed of light and fact let's see what it does do here from let's say take go crazy limitless but they're crazy limit which you wouldn't be both very close the speed of light take them very very close to the speed of light equal to the speed of light that what does W well that be comes just twice the speed of light and the bar as bad speed of light but the bond you won't be are both equal to the speed of light so you may overseas where 1 1 plus 1 is but it hasn't gotten bigger rumors fetal life even a your compounding total loss use each of which is exceedingly close to the speed of light so close that it is a speed of light I add them up idea something which is no bigger than the speed of light which is just right our it's not hard to prove that this expression never gets given the speed of light as long as you and me are not here as long as you and they not bigger than the speed of light the compound here the composite velocity cannot be and that's easy to prove coach but more us clears the absolute limit where you would be years because possible I still haven't gotten bigger the speed of light as pretty clear that I can't exceed the speed of light by compounding any number of Lorentz transformation the result will always Comerica lessons be life so this is an exercise in matrix multiplication club which is neat told this particular kind of operation cried at next wants to come through the concept of velocity and relativity which will even talk about velocity let's talk about 4 vectors it's an then Matt the coordinates I point can be thought of as the components of defectors
X Y and Z components of the displacement vector from the origin but X Y Z he can be thought of as the components of a four-dimensional record including trying was components are these objects X new X new components of 4 vector for former rector is 1 which includes not just 3 spatial components but time component also we can have more general object just as we can have all kinds of vectors in space velocity vectors acceleration vectors of electric field vectors all kinds of vectors not only X new itself we can have a general concept back Quebec there is a thing which friends forms under Lorentz transformations and rotation is the same way that X new transform it has 4 components and if you wanna know how how its value in any coordinates spam you simply transformer from coordinate system coordinate system using the same Lorentz transformations according transform X where I won't bother writing that down but if you like down the Lorentz transformation Exs and simply substitute in Media instead of axes he's that is a way for vectors transform the other 4 vectors 0 before we talk about 4 vectors transformed let's also talk about the Contra variant called very This is the Contra variant form of the vector and consists of 4 4 1 2 0 0 3 before there is also the the call variant form of which is just the occasional device exactly the same thing was reformer except the 3 space components have minus sign you simply change the sign of the space component and that's equal be warned tool 3 4 not 4 North sometimes alright forms that not bum be just as there's a concept proper distance vector X there is the concept of a proper length of any victory and that is just given by a V New mural is the length of 80 square disclose square it's analogous to X Mueller it is equal to power square where towel would be the proper distance from 1 end to the other end of the vector here it's just the general location representing no concept look concept it can be white definition this is definition of be lady of 4 vector Was it means it means being a lot square minus view 1 square mile 2 square mile 3 square exactly alike ordinary vectors except for this extra minus sign not squared minus V 1 square might be 2 square mile 3 square is called the properly of a vector PT Aneka may in this form will lower index of an upper index implicitly some Over pharmacy a useful fact you could Gold will be further unisys opposing you'll have to vectors end of the W then you can invent the product of the 2 vectors and think of that is analogous to about product of 2 vectors remote Court this is this is what's called WVU but didn't think of it as being an analogous to the dot product of 2 vectors in ordinary space W. terms it represents a kind of a product of the 2 vectors and important point is that this quantity is invariant if you change frames of reference that changes the components a book W but does not change W. your quantities like that are invariant they don't change endurance transformations and they don't but they don't depend on what family of coordinate you OK that's so that's the idea a before vector but we particularly for the particular for that is the velocity for vector of particles obviously do not talk about things like velocity and acceleration in Newtonian physics velocity is a but a 3 vector 3 components it doesn't have a 4th component in relativity theory anything that has 3 components will have a 4th component and velocity is a quantity which has 4 components follows 4 components well let's start with the space components of velocity is what I usually mean Molossidae DXT see that's the x component of velocity with that is Waikiki he sorry this is we were concerned DX this is Hawaii and likewise easy this ordinary concept the velocity the time rate of change of the position of but what would be the time component are Milosevic TT deeply as 1 a doesn't make sense we have a object which really doesn't have 4 components 1 of the components here just completely trivial this can't b 1 term from good relative this would think of as the velocity so I explained to you the way relativity theory views the velocity of a particle and
is a very good definition of velocity it's just not a thing which transforms nicely and the Lorentz transformations it doesn't have a nice were our way of transforming when you transformed court hears the definition of velocity that turns out to be more useful to start with your coordinates will working with particular coordinates ex ante and Winans AT T X every point of space time is labeled with a set of coordinates X T X 4 x 1 now particle moved here and along the trajectory longer world line of the article which also be moving in wines moving out of a blackboard along the trajectory particle balloon displacement is called BXT X New BX could represent the displacement along the x-axis the axis but it can also represented displacement along with time so DX not represents the displacement along with time exodus BX represents misplaced and along the x-axis and BY busy similar we represent a displaced so I have bought them from giving me a of for ruled differential distances along the world worldwide the particle now or never I would divide these by D take but think it is time as seen in a stationary reference frame installed I'm going to divide it by she is the frame of reference of the moving particle sewers and a moving political moves along it has a watch takes off time just how much has X displacement how much hello and moved in a X in a given amount the proper time how much will the Hawaii in a given at the proper time how much of a moved in Z in a given amount proper time now proper time is invariant it doesn't change from coordinate frame coordinate frame so if we defined DX meal by deep how would Terrell is the proper time from year to year that is a former which will transform the same way as extra X mule although differences annexed to transform with Lorentz transformations conventional arms transfer divided by D he doesn't change that because geeky doesn't change under Lorentz transformation of DXT XTT becomes a force it's got 4 components and what those for components are in terms I the old-fashioned velocity the old-fashioned velocity BXV 1 Starview XVY easy 2 late these things the old-fashioned kind of velocities and settling this is called you from call the proper velocity but proper here it means that kind of being used to differentiate with is the proper time but still you figure out what it is like to write West just take a let's take for example the X by town that's equal to X by D. fans DT by D. towel likewise for why why by account BY by IDT and B by D. and so forth that we can either write down that DG by deep sorry yes deep-sea by now that's the 4th component of the velocity that the decay but it does just peachy but cannot so I know what DT by account was I would not go back and forth between the old-fashioned kind of velocity BXT and this new kind of relativistic velocities you let's figure out what DP by deep hours not difficult 1st of all right d Palace Square is equal to the key squared minus the X squared minus D Y squared might use square that's just the definition of a small proper time interval here it has a exit residue White was easy and has a deep she and the properly of that little vector Is he squared minus DX square mile square mile square now let's divide this geeky squared well this is just war is my Ness however says this is V sub at this is the XTP squared the SVX squared this is BY His Veazie square what we wept was once quipped was squared this is the magnitude of the velocity of the particles that the magnitude of the ORB ordinary velocity the magnitude of the ordinary velocity and so this is just 1 might be squared with this year represents full velocity of a particle that's equal to D. Carol square or better yet Kalb ID is equal to the square root of 1 mind be square that's great because I bt but can always just 1 all the others so deeply by Harold is equal to 1 divided by squared a warmer square that tells us that I was right
to obstruct and the equations U.S. X is equal to the X by TVA's VX plans to debate a panel of squirrel woman was a square likewise with you want using wide provided us with a woman's risk where you z z quotas to using over square of Warner's scored well about the 4th component for 4th component is just DT but because PT but it cannot be X but it can be 1 the county but down from the deep BP buddy tower that is just you Zero which is equal to 1 divided by squared one-liners we squares so him work component of the velocity of the relativistic velocity vector in terms of the ordinary velocity notice that the losses this really should be as always see square down here he scored oversee square and this therefore velocity is small by comparison with the speed of light in the thing these the commentators as 1 of your Lexus same as VXU why is the same as why uses the same as easy as long as the velocity is small but by person with the speed of light and use euros is 1 of just 1 defeat by DT facts if the velocity assault born the velocity gets fast is appreciable differences between you and me and fact you'll is typically let's see one-liners these is bigger than 1 that you would is typically less than is right now viewers typically bigger Veneridae not you was bigger than it was typically bigger than to our 1 might usefully I knew was bigger than an undertaking the U 0 0 is nowhere near 1 it is huge and this is the concept of the relativistic for vector of velocity of removing particles Ms. a 4 vector it transforms if you know it in 1 frame of reference Ucon computed in any other frame of reference just by transforming it the same way as x y z he transformed end that makes it much easier to work with vendors the ordinary velocity which has a very complicated transformation property now some of us stop soon but rather just tell you another way of thinking about this for vector velocity has an analog in ordinary you could in geometry and it's the pinger tractors tore her if you have a worldwide Britain this notion Our for velocity is basically here it is is basically the key injured vector of the trajectory of along the trajectory but let me show you about engine factors in ordinary geometry nonrelativistic many Austin Kearns based two-dimensional space three-dimensional space doesn't matter much I want to construct the components Of the Tianjin that the occasion vector means a vector along the curve while the unit Lane a vector unit vector which is change the at this point is well aware of differential is displacement here DX now aren't talking only about ordinary space is obviously pointing along with Clinton that is just as the sector over here from year-to-year DX points along the Cajun vector but it's not a unit vector it's a very small where to make a unit vector and hopefully we won't do we divide the X by dissidents along the curve of yours we take as little differential displacement that gives us DXT wiII and Daisy now we divided by the distance along there was a distance along their and yes for Leonard is equal to DX post BY square post easy squared and we simply defying the engine that there could be D X by Best be XQY ideas and easy by as for ordinary geometry no time just ordinary space we defined these engine factor incidentally could you see why this tangent vectors unit vector is because the sums of the squares of each component recalled 1 why is that because via squares DX square trustee once quipped was busy square so a square up the components and add them together the BX Web post square post easy squared is yes squared so this is a unit vector just gotten by taking the differential displacement along with courage and dividing it by the small distance along differential displacement and then letting it does not shrink 0 ordinary calculus this is they might be a here instead of dividing by the ordinary distance you divide by proper time between points it will be acts and divide by the proper time so this is an alarm this is the relativistic analog of the he injured sector alone would be along the world line of the article as I said its space components of a particle moving slowly space components of the ordinary velocity and it's time coordinate is just 1 of particles moving faster than these are appreciably different Avenue nonrelativistic analogs they I 2 yet at yes he
loved her Our about proper time is usually something measured along prime like trajectory Our if C is equal to 1 apart from the funding signed its appear yes you can't I think he's Siegel Towanda caterer a minus sign so I'm sure what you mean by saying is proper time the same although I'm sorry meaning is in the car here I was simply talking about ordinary Euclidean geometry distances dissidents there's no question a proper time because time is nearing come into it proper distance and I'd call that X a relativity the analog along here along a trajectory is decoupled recall a proper time yeah discord dissidents a proper time to the call a proper distance if you like but doesn't Edvard Munch there's just distance measure by a tape measure along a curve yeah it's also just distance but distance measured by clerk or so Oh it's basically the same concept of vector the velocity that now of course you could differentiate the velocity that with respect the proper time alone there and that gives you the proper acceleration factor why would they be interested in the proper acceleration vector because the analog of Newton's equations traffic equals becomes a statement about proper acceleration for proper acceleration it is not the 2nd derivative contracts with respect he the 2nd derivative of X with respect to distance 0 0 both respect the proper time along curve say or through want to understand moons equations change we want to understand proper velocity proper acceleration another example but say I think we can show stop Hiroshima should I go find another 10 minutes right will run on 10 minutes and just illustrate the concept of momentum momentum is an important concept in mechanics wary that the mechanics yet but as you might imagine we will have a relativistic concept from moment mementos of vector In ordinary
physics In relativistic physics part of for sector so let's go through that in ordinary Newtonian mechanics envy is equal to the momentum quicker but some indices on here X component of velocity y component of velocity oligarchs poor little of Vector index it torso talking about that there was some momentum the velocity of vectors messes just the number of incidents in the modern terminology 1 doesn't speak about romance changing were velocity man's is simply a characteristic of a particle that Ford used to be called rest when 1 speaks of man's at a number that attached a particle it is what used to be called mass when the particles arrest see what's connected by adjusting commands as something associated with the particle and not with the velocity if you look up a massive an electron and a cable or masses of particles it won't tell you the mass as a function of velocity just tell you the of electrons blah blah blah blah blah peer kilograms and it doesn't change with velocity Sabri the number the server to patch of electrons OK analog of momentum that the momentum of a particle in relativistic theory is not gotten by multiplying the man's by the ordinary velocity rather demands by the relativistic velocities so right definition and relativity theory is that he knew what components is equal to the pions your new best for components momentum in nonrelativistic physics only had 3 components but it's interesting to ask what the 4th component figures up tell you next time how you use of momentum you use the momentum because it could serve if you want to know what happens to particles with Nikolai particles when they decay and so forth the basic tool is conservation of momentum but not conservation of 3 components of momentum but concentration of all 4 components Our so we ought to find out what the 4th component of momentum is 1 of the 1st 3 components he acts like that and times which is and plans VX divided by square of 1 mind square oversee squared OK everybody is small 1 might be squared overseas where is basically just 1 and so for small velocity the momentum is just and me as a velocity gets bigger and bigger the momentum the momentum starts changing in deviating away from just bigger than just but small loss cities is just a girl momentum what about the time this is something new or is it about the time component of our equivalently 0 1 its Energy Energy is also concerned and the 4th component of the momentum vectors is energy let civilians see why that's true 1st of all it's not provided by the square root of 1 might be squared and let's put in the speed of light now now I lied to you a little bit I said that PT is energy that's not quite true you have the multiplied by by the squeeze of speed of light squared energy the main point is that it's consider whether or not you multiplied by the speed of light squared is a matter of convention and a matter of making it consistent with older choices of units energy has certain units but was energy will units of energy units of energy on mass times velocity squared when I get back from 1 he FMV squares the kinetic energy of a particle an ordinary ordinary relativistic physics so if has anything to do with energy better multiplied by C squared in order to get units of energy but apart from units it's just PT apart from putting in the appropriate powers of give units that were familiar with whom what we talk about energy Pt it is the energy let's say Let's go will further let's work this out ceded its energy see the events related nonrelativistic energy kinetic energy 1st of all this is MC squared times 1 of square overseas squared minus 1 half however because of the denominator to a negative power and because it's a square root its Apollo half for now I wanna work this
out for small velocities velocities very much smaller than the speed of light to do that we use the binomial the binomial theorem says the main thing for opera the bar films it is that 1 minors a small number the core of epsilon any power P E is approximately equal told 1 minus P epsilon at the binomial fear Neukom for example there are higher orders in Epsilon plus things of camps on square on cue whatever higher orders of epsilon before example 1 minus epsilon squared is 1 my to epsilon plus epsilon square epsilon squares very very small amount of tourists what about 1 minus epsilon fuel as 1 minus 3 epsilon plus things which are water epsilon square at epsilon squares is very small and 1 minus epsilon tour power is just 1 minus the power times Epsilon you know that that's what the binomial Murad tells us about general powers a warm research or by now these square overseas squared is small for slowly moving particle be square overseas where is truly very small forward Mario velocities and so I can apply this rule that says that this is and C squared times 1 now we have a mom might from here but we have another miners from here on plus 1 half these squared overseas square plus additional things which are very very much smaller than square overseas squared the next 1 will be 1 of you for water that possibly small for any ordinary flu velocity so this is basically a good approximation was slowly moving objects and Square times 1 force 1 of these squared overseas and now the rest is familiar and C squared but slow before but that's only the energy of the object is a rest periods moving his mother eternal which is 1 half and the sea squares canceled these square this is the ordinary Tony and kinetic energy of a moving particle and you and that of a constant and squared that's what the 4th component of momentum it for lease a slowly moving nonrelativistic particles the 4th component of momentum is they can't stand which is characteristic of the particle and C squared plus the ordinary Newtonian kinetic energy in a constant for the energy of a particle particle was not allowed to to change its name and Newtonian physics particles never change their man Our of particle cannot change its man if you you have any number of particles all of this adds up to some can't stand an additional can't stand the energy doesn't make any difference of important Colony particles of moving slowly this is 1 half square that's attorney energy which depends on the velocity it's what we call kinetic energy so that's a that's the floor vector of momentum includes momentum and energy conservation of the 4 vector momentum is the conservation of momentum and energy they go together even more when you arrange transformed from 1 frame to another momentum and energy get mixed up with each other because the components of former vector so what is energy in 1 frame becomes a combination of energy and momentum in another frame but In a every frame the momentum is conserved that's the important thing momentum concerned every reference which means all 4 components energy and momentum of a isolated system harvesting before and after collision will work out some examples and use it next time and then I wanna go 1 2 no news equations the concept of acceleration but then wave equation wave equations because regard once that electricity and magnetism socialite very light light waves and so forth and so will want to understand we relativity and help particle motion relative particle motion also called articles and hoping that the next time the question we all my life I am
more her Ray the
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Band <Textilien>
Weiß
Motor
Großtransformator
Matrize <Drucktechnik>
Jahr
Ersatzteil
Ringgeflecht
Anstellwinkel
Borosilicatglas
Potenzialausgleich
Verbunddampfmaschine
Direkte Messung
Mechanik
Bergmann
Erwärmung <Meteorologie>
Lichtgeschwindigkeit
Erdefunkstelle
Auslenkung
Begrenzerschaltung
Leistungssteuerung
Computeranimation
Teilchen
Rotationszustand
Zylinderblock
Bildfrequenz
Fahrgeschwindigkeit
Brechzahl
Reisewagen
Multiplizität
Array
Kaltumformen
Relativistische Mechanik
Beschusszeichen
Längenmessung
Elektronisches Bauelement
Eisenbahnbetrieb
Wirtsgitter
Rootsgebläse
Elektronische Medien
Werkzeug
Tonbandgerät
Satzspiegel
Fernbedienung
Großtransformator
Knüppel <Halbzeug>
Matrize <Drucktechnik>
Lineal
Drosselklappe
Direkte Messung
Intervall
Lichtgeschwindigkeit
Zugmaschine
Radioaktiver Zerfall
Auslenkung
Armbanduhr
Satz <Drucktechnik>
Teilchen
Radialgebläse
Hochspannungsmast
Schalttafel
Bildfrequenz
Messschieber
Fahrgeschwindigkeit
Gasballon
Stunde
Array
Relativistische Mechanik
Rückstreuung
Waffentechnik
Elektronisches Bauelement
Längenmessung
Diesellokomotive Baureihe 219
Rauschzahl
Übertragungsverhalten
Rootsgebläse
Druckkraft
Trajektorie <Meteorologie>
Sturmgewehr
Weiß
Motor
Großtransformator
Bestrahlungsstärke
Jahr
Scheinbare Helligkeit
Analogsignal
Unterwasserfahrzeug
Klassische Elektronentheorie
LUNA <Teilchenbeschleuniger>
Messung
Masse <Physik>
Magnetisches Dipolmoment
Direkte Messung
Mechanik
Kraftfahrzeugexport
Mechanikerin
Radioaktiver Zerfall
Minute
Leistungssteuerung
Teilchen
Kopfstütze
Gasturbine
Videokassette
Fahrgeschwindigkeit
Patch-Antenne
Brechzahl
Negativ <Photographie>
Pion
Kontraktion
GIRL <Weltraumteleskop>
Array
Konzentrator <Nachrichtentechnik>
Kaltumformen
Relativistische Mechanik
Längenmessung
Elektronisches Bauelement
Energieeinsparung
Niederspannungskabel
Urkilogramm
Rauschzahl
Rootsgebläse
Elektronik
Trajektorie <Meteorologie>
Werkzeug
Ersatzteil
Analogsignal
Unterwasserfahrzeug
Drosselklappe
Mechanik
Bergmann
Lichtgeschwindigkeit
Impulsübertragung
Leistungssteuerung
Computeranimation
Teilchen
Feldhäcksler
Umlaufzeit
Bildfrequenz
Kopfstütze
Fahrgeschwindigkeit
Array
Eisenkern
Relativistische Mechanik
Optischer Schalter
Licht
Elektronisches Bauelement
Kombinationskraftwerk
Elektrizität
Energieeinsparung
Druckkraft
Magnetische Kraft
Nassdampfturbine
Flüssiger Brennstoff
Lineal
Drosselklappe
Computeranimation

Metadaten

Formale Metadaten

Titel Quantum Entanglements, Part 3 | Lecture 5
Serientitel Lecture Collection | Quantum Entanglements: Part 3
Teil 5
Anzahl der Teile 9
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/15126
Herausgeber Stanford University
Erscheinungsjahr 2008
Sprache Englisch
Produktionsjahr 2007

Inhaltliche Metadaten

Fachgebiet Physik

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