General Relativity | Lecture 2

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General Relativity | Lecture 2
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(October 1, 2012) Leonard Susskind introduces some of the building blocks of general relativity including proper notation and tensor analysis. This series is the fourth installment of a six-quarter series that explore the foundations of modern physics. In this quarter, Susskind focuses on Einstein's General Theory of Relativity.
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Sherwood beer any questions before being quick worms as I was saying a good notation look Terry long what notation is good it just start of automatically tells you what to do next ninja the physics completely mindless way just you know suckling pig it's pretty clear which end the stick cast to go into the has gone through a thing with the whole you can try putting off Poland or Hall of forcing a stake in tow a stick His only 1 thing you could do it from but the stick into the hall and in the other end of the stick to go another Holik a small holes it with sticks and do earned the North Station a general abuse much like that if you follow the rules you almost can't make mistakes book you have to learn Robles who also rules of cancer analysis of algebra cancer analysis and but these questions which way aiming at now is to understand the about analysis matrix could be able to distinguish a flat geometry from among flat geometry that seems awfully simple frat flat means like the plane on flat means of bumps and lumps in it and you would think we could tell the difference very easily sometimes it's not so easy for example as I mentioned last time I think this page that pages flat and page like this it looks curved but it's not Kerr it's exactly the same page the relationship between the parts of the page the distances between the letters on my page and so forth they don't change at least not the distances between the letters measured along the page don't change when I to various things of the page so a folded pager of our core occurred because current would be the wrong thing but works right word for was well it's not really no matter what the were deformed just that curled court curled we don't stretch it and you don't perform it that it is not introducing curvature and was not technically it introduces what called extrinsic curvature extrinsic curvature has to do with the way of space in this case the page is embedded In a higher dimensional space three-dimensional space of this role the page when the pages laid out flap
like rabbits and bedded the embedding space in 1 way it's curled like this it embedded in the same space in another way and 1 says there is extrinsic curvature be extrinsic has to do with the fact that it's In an X-terminal space and has to do with how the space probe of pages embedded space but it has nothing to do with intrinsic geometry you I can't think of the intrinsic geometry as the geometry of a tiny little Bob that moves along the surface cannot look out of the surface only works along the surface crawls along the surface he may have our surveying instruments by which you can measure distances along the surface draw triangles measure the angles of the triangles within the surface and do all kinds of interesting geometric studies but never looked out of the surface and therefore never did have notice is the fact that might them In the higher dimensional space in different ways just learns about intrinsic geometry intrinsic geometry meaning geometry that is independent of the way from surface is a matter of general relativity it is about and Riemannian geometry geometry is all about the intrinsic properties of the geometry it doesn't have to be two-dimensional can be any number of dimensions and the basic thing which defines the geometry distinguishes it from other geometries is move imagined sprinkling amount of points out or room my pager with the back of a page sprinkle much appoints draw lines between them so triangulate the space and then with the distance between every pair of neighboring points specifying those distances specifies a geometry sometimes bad geometry can be flattened out on the page without changing the lengths of any of these links summer give your example Armed if I drove onto triangles which I represent go lengths I'm not doing very well the triangular lattice it's a triangle lattice buildup of triangles and every triangle issues an equilateral triangle this equal the or go there by a man at the at the nodes here we can you know what's a Werder Dilek hinges stay hinged they did you will allow to sort folders but as long as we keep going the same and don't change their making all equilateral triangles this can always be laid out on the desk as flat on the other hand if we were to say take several of these links although ones coming into some point earlier this when this when this when this 1 this 1 this 1 6 of them and would the double the size of their than this point we have no choice but come up out of black warming to the blackboard there would be a bold about Corey and yet can't push that bulge back into the black for flight out without changing the elites on the length a current space is basically 1 which cannot be flattened out without distorting it in fact about starting at intrinsic property of the spaces not expert OK so our our both of which is I said last time matches closely the question of whether there is a real gravitational field president or whether the graphic detailing gravitational field is just due to an artifact of funny coordinates funny space-time coordinates the question of whether a space is really flat although whether or whether it's really flat really curved all-weather which just we presented that faced with curved coordinates those 2 mathematical problems is really a gravitational field is just an artifact of curved space-time coordinates all it is the space of the top of the table top really flat even the it may look courage because I introduce would draw on the table but the truth coordinates of various kinds the question is how do you tell relief space flat what you get typically given the metric cancer but before we go in for that question and it sets a hard question it's gonna take us it probably probably won't get completely to the point of a only answering questions from our youth not as that care we don't define we're not trying to get any mathematical set a tribunal tuition Japan while round table because as the good enough I'll tell you exactly where I could for the last time before you again are what it means is the metric tensor can be chosen to be just a chronic Delta's symbol come right before a CompuAdd we need to to understand a little bit about cancers we talked about the little bit but I was formalized it tonight arm Prince dealers and vectors scalar vectors a special cases of cancers and cancers of the general category of objects they have industries and they transform the most important thing is that they transform when you go from 1 quarter system to an hour In particular we're going to be interested in spaces with quantities fields on them so that every point in space there may be some quantities associated with at that point and of quantities will be considered them because there were other kinds of quantities also that will not be but particularly interested in cancer field the field a thing which can vary from data from point to point and the simplest kind of cancer is scale a scalar passive acts as a quantity at each point of space where everybody in all coordinates agrees about value so the transformation properties and goalie let's say from the white from the X coordinate Y X court next Emma 2 wide and changing coordinates the value of the scalar field at the actual point of space space time we'll get space time the value of the scalar field does not change so we can write about as best trying on why equals as directs the prime here simply denotes the fact that I'm talking about a quantity in the quarter system y without trying it will refer to the court system X to coordinate systems it the x coordinate system these of all lines of constant X over surfaces of constant X and then every point is
labeled by a collection of Exeter's many exorcism wise but depends on the middle of the nite of the bases one-dimensional more anxious 1 coordinate label where you are and what its two-dimensional to coordinate sex sex next to or why 1 white three-dimensional 3 recall Preval recorded 1 x 2 x 3 1 1 1 2 . 3 but only X cool admits or the white that's a different but we assume as a correspondent there if you know the value of the x is for particular pouring in the wise so it's right that y and not way and runs from 1 till however many is not always function To be known function Of the Exs I this this stands for now the whole set of Exeter's X 1 X 2 X 3 X 4 were many end that we can in verdict the relationship so that we know the value Hawaii point we can also know what Exs assets according transformation as some kind and that coordinate transformation can be pretty complicated will assume its was so Mips continuous will assume we can differentiate only need to differentiate but um nothing more special Ramat poker isoscalar was transformed trivially if you know the value of S at a point you know it no matter what coordinate system you you next year of vectors vectors and that 2 kinds of vectors we spoke about them West Timor will tell you know about the geometry which means there are countries vary vectors which have an index upstairs and there recalled Marion vectors I shouldn't really same symbol because they are not the same thing at the moment but was just do the the same symbol of VV for vector With the next downstairs these rules will write down will see the transformation the a moment of amid 1st tell you a little bit about the the difference between coal variant and Contra variant intuitively work so let's opposed for moment worst located right at the point of the year we have some coordinates these could be the X coordinates would draw them straight lines for the moment because at the moment I'm not interested in the fact that the the coordinates workers and varied directions from 1st place from place to place and mostly interested in the fact the court would taxis may not be perpendicular we may not be perpendicular and the or what the implication of perpendicular clarity is these quarters furthermore distance scoreless streak callers X 1 callers X Q and not perpendicular relative to each other they could be perpendicular to each other but not necessarily and actual physical distance let's say measured in meters from x equals X from his X 1 equals 0 0 he is X 1 equals 1 is X 1 X X Y equals Julia's X 1 Eagles 3 or 4 years X tool equals 1 actual physical distance let's say measured in meters a centimeters or of a units are between x equals 0 equals 1 is not necessarily 1 unit and perpendicular torch other nor do these labels represent actual physical distance along taxis they just of point is the point x equals 0 0 and so forth and so on OK don't introduce some vectors tool vectors is not necessarily tool hero jar two-dimensional space there could be more taxis for each access I will introduce a vector which is basically a vector X standing In 1 EU leaders of coordinated space along with the particular access when give sector and need that extends from this point of this point Mexico 0 x equals 1 and work E 1 the 1 here stands for this 1 x 1 and not X to another 3 in other words along the x y axis and there's another vector along the X 2 access which are calling each of these electors these sectors think of them as physical little arrows and of course as does third-quarter net to point out a blackboard evil that so for a while in fact weakened label these east of all items he's survivor will vectors and as Hi goes from 1 to 3 a wonderful or whatever it is is go over various direction so that now next issue taken arbitrary vector taking arbitrary vector let's call quality and arbitrary vector for example former we can be expanded as a sore arm of coefficients cars he won but other coefficients currency to put more coefficients and 3 other words they can be expanded as a sum of terrorist threats having new 1 and the coefficient of 1 will call V 1 plus tool a tool for Class V 3 3 now little sectors and formula I think he's gravitas are actually numbers but that components of these vector components of these and they tell you how much of each kind of 1 of these factors are present in the sun be 181 Posey to lead to a 3 3 or 4 these coefficients these coefficients part called the Contra variant components of the vector Vt sister name there is nothing in what I did that required me to put the 1 he downstairs in upstairs this from downstairs it's a convention to write this form but 1st of all you see what the country bearing components are there the expansion coefficients numbers that you have to put in front of these 3 vectors to expand given back the next day let's define projections of a given vector onto the X-axis in the Y axis I we do that its definition it is not product of the vector vie with the vectors EU worry too but think about the next thing would be the dot product of vectors vector vie with distillate No . product would let's say you want start with he warned now if we would just using conventional Cartesian coordinate perpendicular to each other and if these really were unit vectors of the distance representing each coordinate separation he was 1 unit whatever the unit for dealing with them these coefficients he would be the same disease that product not products and the Corps officials would be the however we may have a peculiar Courtnage system with angles and with None units separations between year of the success of coordinator lines here this is not true so let's see if we can work out with VW 1 incidentally v . 81 this called of 1 where they call variance index I notice nothing so that together we can also write this year as V an EM summation convention Einstein summation convention which means V 1 you want was too weak to oppose 3 3 3 3 Notice concocted things so that upper index always goes with the in Texas with comeback that's
call contraction of indices but has noticed I've done it in such a way that the summation always involves an upper index for next and by definition this is the Contra variance components Of the call Mariam components are the dot products of the vector With the base of the cold is the basis vectors rumor Maine physical the bases vectors and Rico variant components of about products will be so what do we get to figure out but plug-in V its expansion he and they think it is not product would be a sequel to this is equal to a V and and from here and now I got it until he warned but while 61 has just let's just go over good a right now general the general case the dot product of would be in basis vectors and this will be E sub he end but he end is something no it's something new Let's isolated because 2 indices all that's the metric tensor it is the Magic cancer let's let's go further to see where connection with metric tensor is a layer of vector is product of the vector with itself let's calculate the lady of these shows in various ways we can write it's v . ie now VW it's calculated E . 3 from here we go right the 1st V is V. I. M E M that's for 1st but then let's take it is not proper secondly 2nd really is Vienna EN I have to use a different index for this omission from a summer Schirmer most mix of starvation receives severe him means we 131 Placido you we 3 forums we warn want free through its for when I tried this summer former Vietnam VI N columns the quantity eat . EN lets callers quantity and that he and much call GMN finale ever Formula V M V N GE and this is the character of the metric tensor server metric tensor tells you have a computer link 4 vector could be just dx dx an NDX end and that we would just be computing the length of a little interval between 2 neighboring points GM OK so well that I just put this on the blackboard here To give you a little picture of the difference between call variant and Contra varied Country variant indices other things that used to construct a vector out of a bases that sectors called variant indices are the top products with a with these With the basis sectors with different geometric things but they would be the same if talking about ordinary Cartesian coordinates that's Bell are just asserted that discussions you to give you some current geometric idea called variant and Contra very means that also what the metric cancers symmetric cancerous constructed out of these bases vectors by thinking about art both back young at this was just for you you know U.S. means they press but I assumes redress of the media so in a way that Jesus was set of final and unit PARIS separation between the 2 sets of court it further look that not in Karelia coordinates typically in general recorded directions on Nov of viral Twitchell recorded the directions but the direction along which 1 of the coordinates increases keeping near the ones sex that him out typically knocked orthogonal whichever resigned as president would follow it said if the farm quite is also the question of the link between neighboring quarter Cartesian poured some would mean that their orthogonal but also means that the separation of the same here here here Miss this only dream example where the accord taxis are whichever ball with a separation from office in polar coordinates when Apollo coronach me on lines a little along which the radio coordinated increases would just be the radial lines and lines of with angular coordinates circles and clearly at every point circle this is perpendicular to the line passing of the standpoint of these coordinates are all 5 of I'd rather rare world but these coordinates are orthogonal to each other tank but the separation distance between the close of trading equals 0 over here and let's a equals 1 over here the separation between these 2 . is not the same as a separation between these 2 . is not the same as a separation between points so not Cartesian coordinates and the according to a rough definition I gave of these EU the ease along the way that direction would be increasingly Elaine as you went up here with more than just perpendicularity perpendicularity and quality of the distant so you move along as you go affixed separation next more than that In defining Cartesian coordinate OK so now let's come through analysis the race over a up above us all before the metric cancers introduced we just have abstract Courtnage vectors there are co-vary in vectors and comfort factors in a different things DX would be a kind of that there and defied by DX were fire would be some scalar that would be a call very different kinds of things and I will say he wants the metric tensor is introduced I've I've introduced it here by introducing user unit vectors heroes vectors once they once they once the matter cancers introduced it possible to make a correspondence between the Cauvery vectors contraband prevent such a way as to say every Vector B E the representatives called variant all by all countries will come to that will come on at the moment I should be only talking about Kohl Contra variant components of this language here we have 1 vector and we had a score variant Contra bearing components and that's components stores so about comport but may sometimes lapse in the calling them co-vary our country vectors by which I mean the country variant components of the vector OK now or objects which are characterized the thing which characterizes I can't make that is the way that it transforms on the caught in a we talked about this a little bit and go back I want to do again b is some vector it has contrary components and has Contra
bearing components and the white coordinates and X compile the accord corner charmed here keeping the vector affects not change of there but change I will change its components clearly so how do they call variant contrary component change when I change court hears oral misses and component of a primed with deep prime means the components of V in the white caught trying means y I'm trying means acts might be in the court next were unknown basically reviewing what we said last time is given in terms of the on try coordinates by DYE and might be X and he and this is the object on prime coordinates awards X coordinates and got by multiplying by BY by BX terms an example example is d try and just D Y E D Y Kim that would be the prime version of a little interval is obviously equal the a D Y and my ID X and times DX and so this is sort of a archetype of a of a Contra variant vector component and you could see that In fact forms exactly this way the call Marion components call variant component of a vector would be for example if you have a scalar and you differentiated with respect Hawaii or X Y and this becomes the prime component of a call the area vector for the kind called variant component of the gradient sector Miss the gradient the and differentiator with respect the White Nile X this a prime component and this is equal to be but that this would be best by B X end these and x 10 bodies these y damn notice the difference notice how these notations carries you will walk the left hand side I have a wife with a super index upstairs index and on the right hand side I also have all white with a super index boomerang inside the exercise of course on every Buicks and VX and upstairs has to be balanced by X and downstairs you get familiar with you're allowed to contract indices 1 upstairs and 1 downstairs and you could see the pattern you could see the symmetry of this relationship has likewise here here we have a lower index corresponding Hawaii the index In on the right-hand side I have something below the fraction are also be white him and then above the fracture I have X and but that compensated for by another BXN downstairs he begins so the the notation pretty much carries you along this is a standard form for the transformation property of Contra variant components This is standard form for transformation properties of coal variant and is an example of contrary now looks like it is considered a dimensionally but what direct your represent receipt the 1 the units are distance program what is it a 5th of general but I get but pickup point that I think I know it's not true in general V could for example be a velocity in which case it would be a link per unit time or it could be a force component of forests in which case it wouldn't be a close call but army units have the match so here we have for example this could be a force on the right of force on the left and a wire wire by VX which is dimensionless perhaps of measures but say we're not so sure they'll do what but the parcel differentials quite partial the partial because there are several quantity of Sichuan direct for and this is a derivative of 1 of the wives with respect to the is keeping all the other suspects quite so this is a transformation properties of Andy Cole variant object and the corresponding thing for Contra variants would be an order of this W. prime end about sorry yelled from reprise em but back and using WNV to worth just to give the weather's things this is equal to they X in Bari D Y M these Sublett WON the same pattern was already yet again the X buddy warnings and so forth with yacht when you look at say would is a lower In next year type white white white because trying genes wife right prior is associated with whitey is a in next year In which is of a white wide-eyed was gotta be a lower index below the fracture Morris type him on the side of the equation equation about you get a feel for these things at the way 1 but right now about 8 cancer of higher rank cancer of higher rank simply means cancer with more industries and the simplest example of tensions with more indices I just prolapse of vectors so what's taken object move WO not W. now our stake cancer W. babyish accord fiasco W W trying another warrants in the y coordinates it has so indices alienating and unlimited 1 of those indices the br upstairs and maximum number to be abounds stirs Contra and quotas on a pensioner like this could just be the product of 2 vectors 1 with the country during the next 1 with a call X 1 right now but what makes this thing a cancer visage transformation properties show the transformation properties are the prime components again each index for each index was the same kind of pattern this is a prime component of the index and must be of BYD and upstairs writers of DD X P downstairs now about this index here this is a white index because prime downstairs so there must be a DYE downstairs and therefore the Guardian DD X of caught Q upstairs now we multiply that we multiply that by by W. cancers prime frame P
downstairs has gotta be a P upstairs here is a killer upstairs there's gotta be acute down so this tells me cancer rank to 0 with 1 country variant and 1 called variant index transformer for each index a BY by DX RDX by BY and simply try equity indices go an upstairs room upstairs and downstairs and announced there were avid W. here on prime W. it's gonna have some components Watch column P and Q and the appeasing Jews have about either as a very general if I take a cancer with any number of indices always had let me do what other example supposing we had a pencil with tool cuts of areas in see also make it took all areas and that ended just as another example a cancer with Tool Co variant indices how does a transform a again there are downstairs y is white and another downstairs wine and imminent gold worry work for upstairs well on their Izak's calls from close when fuel and now this is W P Q prices transformation property of anything with 2 indices purely call and until pattern is the same for every index you do the same thing either of the toward wide might be excellent year would be expired and some over repeated index in half and is not repeated indices there on the side they also appear on the side but P & P R repeated and queuing kill REP so this is a bubble sort of bubble some PDQ because peeing cure repeated why this is a basic um notational device who invented by Einstein was or drop the summation symbol curves received you you remark douse was in there and the whole inventor of the patient care of its very systematic but 10 show it is an object which is characterized by its transformation from now knows something about cancers if they uh 0 all in 1 frame but some the scale scale is 0 and 1 frame at 0 0 in every frame if we're equal now surpassing a Vector 0 0 in some frame of B X Frank for that had to be 0 it doesn't mean some component is equal to 0 means all of its components are equal 0 vectors only 0 if all of its components of 0 of all of the components of the Contra variant stone vector 0 then obviously the transformation properties Dutch that all the components of the private sector 0 0 likewise with any concern but as with many cancer if all of its components or 0 1 1 frame and warned court system than all of its coordinates components of 0 in every phrase that means that once you've written that equation cancel former could always of course transfer everything to the left side of the equation equal to 0 . fm equation of that type is true in some frame Ramage true in every frame and that's the basic value of they allow you to express equations of various kinds equations of motion equations whatever happens to be in a form where are the same exact equation will be true in any court system From and that's cost a deep advantage to thinking about centers there are other objects which are not the way we come across some of them which I'm not shares which have the property that they may be 0 and some frames and 0 another thing right Frame scores in its history so cancers have a certain invariants event that components are not variant that composed change from 1 friend to another but the statement that 1 the cancer is equal to another cancer cancer W P Q is equal to a lessee keep P Q always said they would you write a century equation the components have about it doesn't make sense for right in equation like W P Q equals P P killed all go right Yukon right all you want but this left-side transforms differently than the right side and even if this were true 0 in 1 coordinate system it would not be true in another court system the transformation properties of the 2 sides of different heights of normally we would override equations like this we might we might say in some particular quarter system Cohen this system of the blob lob this might be trouble but then if you change coordinates a won't be true the kinds of equations which a true in every frame always in which the indices balance transformed the same way strong 1 Koyama truth of the equation if this is true in 1 equation in 1 1 coordinate system that it will be true in every court system these days but magnitude will be the same but the on the individual components in different coordinate systems work we can always right W minus W minus equals 0 at every component of his W minus she object is 0 minutes joint every reference no it's not meant to effect is the the victor itself it's 1 thing it's true that the magnitude of Vector 0 0 in ordinary geometry the vector itself was there will not be truly in relativity is not not from magnitude vector being 0 the sectors of the magnitude of the vector and the vector itself of 2 different quantities the magnitude of the vectors a scalar we have vector itself is a complex thing points in the direction to say that 2 vectors are equal means that the directions are the same and magnitudes make cancer of higher rank is more complicated object which points in several directions it's got some aspect of it . 1 direction rigor come what they have toward their geometries like soon enough but for the moment that defined by the transformation properties OK as I said the importance of cancers is that we're can't equations drew frame Mitchell an early next operations on cancers things you could do it cancers that make no tents when not point introduced into interesting things that you could do Cantor's which make other kinds of objects which I'm not cancers were interested in things we can do a cancer operation can them which will make no attempt and that way we can make a collection of things I which we can build equations the equations being the same in every reference so let's let's right there some set of operations in there or go through what they are and how you do that the very simple but most of them are simple last war is not if you put book but 1st of all you Kamoke applied cancer by number I didn't write standard you could take any cancer and multiplied by numerical number is still a
cancer armored Barber without 1 additional centers as the 1st operational talk about an additional cost includes also subtraction if you multiply cancer by a negative number and then edit subtraction multiplication of tenses mixed sensors 3 contracts but tell you what that means you may or may not know the word at this point but will not word so attraction and former but differentiation of cancers are not ordinary differentiation coal variants differentiation or we will define that I think can I call variant differentiation of cancers 5 think those of the 4 basic process he's due attention differentiation with respect what Fulp differentiation with respect to position these tensions might be things would vary from place to place live at a point value at point from next point different value with the next point very different value and learning to differentiate them is going to be a fund and the heart now will OK getting cancers your only and cancers if their indices match and kind for example if you have cancer T him with a bunch of other more upstairs and this is contrary this season maybe a collection 1st of downstairs indices you have another cancer of the same kind plus S & S does not stand for scale the scalar here but not lot of about about PE and other words their indices are exactly the same kind of might be and and Lawrence off for this could be PDQ wherever they can safely in this match in Europe and then construct a new cancer which are just called keep losses S M Q dot about about Bob peak is the cancer the thing which is just some of the indices defines note Center which is the sum of the 2 countries it's obvious that this transforms the right way 50 transforms by multiplying it by a bunch of VX by wise in the wide by axes and s transforms the same way that you just factor video transformation coefficient DXT wizened axes then you could see easily that sum is also a cancer forms of cancer are they do same thing with minus incidentally difference also minus says attend said and this is the basis the saying the cancer equations of same every reference frame T minus a sequel Zero is a cancer equation equation that attends team and assesses equipped 0 OK next multiplication of cancers now like addition multiplication of cancers can be done with cancers any rank Frank means a number of indices and independently woes indices are upstairs and downstairs and he is aware works star with an example multiplying 2 vectors supposing we have that there with the Cold War the contrary index now we multiplied by some other factors to make your life a little more complicated let's multiplied by a vector With a call variant index this is a cancer it's a cancer With warned upstairs index and and 1 downstairs index In you have to remember which 1 is which do not cross this is and this was a member of the swans and that ones and so forth this it is any cancer With 1 call variant sorry 1 call variant indexed and warned contrary to set of values the set of values 1 value of this thing for each engine and the fine the components of the cancer with 2 industries you could not the same thing with some other that their continue called W 1 with another upstairs index and this word than some other cancer was 1 upstairs and another upstairs index is upstairs and downstairs because I can't go after remind myself which one's called variant which one's country variant upstairs and downstairs easier to think of birth of the center in usually demand tho buy a product signed in here show like just to keep track of affected to multiplying what's sets we know this is not done in vectors this is making we talk about the Mormon this it is making a cancer of different rank high rank just by juxtaposing these together how many industry harmony a component of the subject cat well what let's work and for the way going to be interested in four-dimensional space-time let's say the number of indices for em runs of the number of values and can take on 4 the 3 dimensions it would be 3 but 4 dimensions as will have 16 independent component 16 independent components for Ferguson for for that this would be a 16 Component Object it is not the talked about product only as 1 component the of numbers sometimes called the out of priority Ford purchased a pensive problem just depends the product of 2 tensions and makes another Penta typically the cancer of different remained than either 1 of the only way you could make a cancer of the same rank is for 1 of the factors to be ask a is a cancer and you can always multiply cancer by a scalar pick any scalar multiplier by V M and it's a cancer was 1 index upstairs Europe's multiplying cancer by a scalar gives back cans of the seem climb up the same cancer multiplied by of the same height generic types a number of indices in the same place as the only situations where when you multiply tantrum by some by some other tends to you get backup cancer of the same type generally you get back a cancer of higher rank with more indices obviously each where these cancers will come in will find however coming soon enough but so far this is just the notational vise so far think of it as a notation device everybody happy with it it's really easy Gerhard make mistakes like that BDAT yacht well OK let's be careful now let's take go at stake this 1 with 2 upstairs be W it is equal to W In Vienna but it is not equal to V and W GM firms W it is the same as the year of the event as a matter which sometimes restrict I'm inquire mechanics every multiplication of numbers and components on numbers components of numerical
values so are on the other hand when you write this as a cancer you must remember the 1st index refers to a V and the 2nd index refers to the end you must remember your convention that the 1st index here was associated with me and the 2nd next year was associated with W but the transformation properties are just the transformation properties other things with 2 Contra variant indices yes there's nothing abnormal about multiplying components of vectors are just numbers when you multiply 2 numbers Yukon multiply them in any order you might say with getting Oh Carey Goethe incidentally how early prove that a thing like this is a cancer well we just right down the transformation property of V in the transformation property of W and that tells us what the transformation property of becomes w where's list manifest their earlier today some examples of it he said that products like theirs continued to be cancer cancers would be appropriate index structure so that's good we have addition we're multiplication Nellie have contraction contraction is also very easy easy algebraic process by rules that contraction lead cancers we knew train you Minor theorem no mathematician recalled during course of year may be at most a landmark OK here's what Sears says remarks consider object XBM mostly used an innocent teasing Q the overseas on the site using a isn't easy just aren't enough letters of the alphabet to take them all from the same range in the alphabet take X B by GUY am multiply that by in by D X does and place it in this formula is a over wife some over camera possibly this means summer let him was the subject of an order subject this is a change in NXB when you change bit finds a change in 1 AM when you change X a little bit summer where you change why 1 a little bit change White to a little bit was at the occur words losers I knew would probably right where arrived our slightly more general formula let's take D F of X or DFT F F as both a function of X and Y remained a function of X Bacall sex depends on why it's also a functional white PS but D Y N D Y and my ID X a notices change in F when you change wire little bit further change and why would you change extra little bit residue it gives you the FTX with contracts gives you f by D X this the change in effort to change X a little bit of captivated by some of steps were 1st to change while I want a little bit and you change why 1 in response to a change in excess of any change White to a little bit and responds to a change in excess and that gives you the effort now what next month F 1 of best happens to be XB van carrying up a formula tells me that's the derivative of expertise with respect to x that looks like a stupid things that in what's the derivative of at BU with respect to x 8 what's the next 1 with respect X 1 the next 1 will detect X 1 1 with severe Exxon respect textual 0 0 so this thing here is just a chronic Delta Delta a not know it's true fur any set of mood don't drew Freddy cord Norris has 1 index upstairs and 1 index downstairs Delta BA would not find out that dealt the BA by itself Soviet cancer because we're just a set of numbers but cancer 1 opera more modern will probably come around to it eventually OK that's a little glamor that we need in order to understand index contraction so that's still example of index contraction and then the fine and more generally I love you Garnett I have my limit on the blackboard that these are OK let's see but story over here but but as an example let's thank you at cancer which is composed Arafat VI N W Ayer has a Contra index and W has a cold area Beck's Excuse me end him this is the TEAM and construction Is it means take any of operating next in any lawyer index combined them together identified them there no and set them set the number equal to each other in other words V and W O and this means V 1 W 1 was veto W-2 was a 3 W 3 and so for you identified a lot of upper index with the law and not allowed to do this with 2 upper indices you know what to do with 2 Laura indices but you could take an upper index and alone Rex and lets us color forms part Her 1st put right down the transformation properties before we set the indices equal to each other V. V. WOV M W trying climbed version of it is white dinner by D X A E D X B by White end times V now works a W b checkers sea of this equation next send the transformation properties as always have for each index the cancer 1 of these D Y by the excesses of by is on the left hand side we have wife y components of something new we have upstairs index and so we have a D Y and by DXA here they wiII indexes all Lawler index was 1 of Lawlor DXB by BY and then we multiplied by the same cancer except the cancer In these and on trial components ABB AB components this is the transformation property of the ranks to attend with 1 index upstairs in 1 index down prime version of new prime version now let's set an equal to in the contract industry contract means identify
improper upper lower index and some older there so we now have a V M W and Billy indices is a subject that kind of object escaped with 2 scalar has no way of knowing this sees what about these indices these indices a Sun Belt is over there this is not the 1 component of something of a tow component that's dislike about proper 1 1 component to tow component 3 3 component together the result has no components some of our pilots see what it is all we have to do the prime component of it it is equal all we have to do is identify him in a N D Y N like D X a DXB body now DYE and ended in an identified VAT WB mountaineers comes in handy DXB by D. White and this thing over here the down opposite here is VXP by D Y and his DXP by BY and here's D Y M by DXA used PY MID except so I have exactly this combination appearing and that's just built there may be some this monstrosity over here is just dealt a B and don't they beat is simply the instruction set a equal to beat that's what will machine which has set a goal and the result is that this thing is equal to V a W a Lewis says they will be OK I said a group B said sorry said beat equal to a lower have left on the left
I have the contraction of upper index will Lauren vector doesn't have a a that's not what I have the contraction of the upper limits DECserver Laura next to neon try and frames and that's equal the corresponding quantity in the prime rate cause In the firelight summation that doesn't care what call was a steady
Mississippi I object that I've made has no components and it has the same value every frame for you call caught cancer escape coruscate cell by contracting to indices and make another cancer in this case pass it's easy the probe you could do this yourself to take any cancer with a bunch of indices Salem said In a M R P Q S argue at any number of them take 1 index from the upper in this it an index from the Lauren disease set them equal to each other the has end In borrower key aura dec's homily In this subject have we're here 1 2 0 3 4 5 6 is a really 6 and that subject not because of some of this is that aura is not really in next anymore it's been some of the services of index subject I've taken 16 next object in the words of cancer ranks 6th and by contracting opera with lawyer index I have read I have lowered the rank of the cancer by 2 . 1 but I can't the video OK what you would find that so let's do that let's start with this expression over here them that is both have upper indices of this has to be DYE In that by X b and then we would have a right to recognize that this has to upper indices this has to operate in this series is a prime component and in both cases we have D Y by DXT exert now supposing I said nearly equal and in some misses of illegitimate notation alleged that would enable this project is not subject this hazard DXT YDY by DX massive thing which is dealt a this hazard BY XTY by guest is nothing in particular that settling up just crocodile that may be sold since this is not how account doubt be missed was not become VAT transformation property of this thing over here with tool upper indices contracted like that is just some bastardized thing with no particular no no no it a product of 2 vectors is 1 offered 1 Lawler there is a vast generalization Buna part of the record Rod but for most of the country where are we doing in moment introduced symmetric cancel talk about a proper vectors and also said yes that were in some some space of the given dimensionality Loranger values of the index runs over dimensionality of the I as President Ezer continue while others would measured if you look at what which is set me 0 0 but there is way adding that when you do that if say that 0 the John betraying him but it is a pack he is deed and from he lowers by linear you mean that um guess it depends on the Daily Mirror shut judge isn't it is correct OK we've defined ballots the metric tensor metric tensor plays a big role here I've illustrated by a particular construction of the way MEN Is that a cancer but its define song terms abstractly forget things it done before but let's don't again OK the doesn't acidimetric cancer this if we have a differential length element DX where just represents the components I AT displacement vector let's code displacement go a point in the space called X X 1 2 X and this place it and call out little vector DX it has components Contra variant components VX and the contrary components that you wanna remember what they mean geometrically what they mean disease expansion coefficients in terms of some basis vectors but easier want to get the hang of it just heard this to proceed with these notation DX and there is a Contra variant components over spice and vector as was the link of that vector what I haven't told you enough to to tell you what the that this could be some arbitrarily shaped complicated the space and specifying what the space In Is it affects specifying what awaits of blue elements I'm so we take the DX em and let's take the squared life it's always easier to start with a score of Pythagoras but Pythagoras a becomes the more and in coordinates which I'm not orthogonal Pythagoras is theory takes a more complicated for me it's still quite Dragon be Exs From evolves a DX and DX and a quantity M said Judy and head the OK In general the GM and will depend on where you are so it depends on acts and that they killers have very complicated curved space of some sort with curved coordinates and you pick some little DX over here than the length of that will not only depend on the the axes but will also depend on where you are misplaced so this is basically the more general thing Yukon right down which is quadratic let's that I'm going to stick with the case of 4 dimensions just because I have firmly planted on my head for the dimensions and keep with that but how many independent components are there GM and now German see
wife but to begin with a 16 therefore exodus multiply anyone by any other 1 so you start with 16 but the X 1 times DX 2 is exactly the same as the X 2 times DX 1 so was no point in having a separate G 1 2 and G 2 1 set equal to each other and then if you count your capita attendance and awards Greece puppetry free space at 6 . 4 3 space for three-dimensional space 6 arm of over account X 1 X 2 X 3 X 4 X 1 X 2 X 3 Texas for him because our matrix 16 entries want chose 3 4 5 6 7 8 9 10 11 12 13 14 16 OK but the 1 hears Earth it's too early is a 1 1 element or 2 elements and so forth you might as well take the want to element to 1 element to be the same in both multiplied DX 1 times DXT so wherever elements that he had just crossed an element here element here on the here and now here and DX ones where it is to square his 3 squares squares we have a 1 2 elements of the 1 3 in the 1 4 the 2 3 2 2 4 3 4 but we've all but we don't need to camp separately you want to 2 1 so we are left for everything any new ones down here from any other ones too 3 4 5 6 7 8 9 and 10 independent components of the G and came as a say in 3 dimensions it would be sex about 2 dimensions of the diagonal elements and one-off bag were tested which you could take to be the same as the OK sorrow Tainan independent elements of the metric tensor so far we have proved the look at co-anchor metric tensor but let's now prove jets attention GM but now the basic guiding principle is that the link of the vectors a scalar River will vector somewhere everybody agrees on a link of it although they don't agree on its component that's the underlying principle about a metric space that the length of a vector is a skill that everybody agrees about going to the back of a let's go from X coordinates would have come forward remove this is not from a good movie but she may have seen that before but we don't fire going WBX that their arrears alive again square square GE in the end of X BOX M P X but now let's go to the White quarters of the White coordinates the prior quarter this must be equal the gene the force birth what's ecology Of course GP user same Index GP Q Y BUY P BY Q there was something wrong with the inquire right that's because the should be the kind components of the cancer the cancer and these are the prime coordinates and the white port wine OK let's rewrite various by writing that D M it is equal partial of X ever With respect to Oh Y P. D. Ho Y Pete who will different element halls extra separation the river acts with respect Hawaii Pines little change and wife let's do the same things should be X B X and is equal the parcel of X and with respect white Cube why Q and now plum to expressions in 50 going so well we get forget that this is a photo G. M In all X Man partial of X and with respect P partial of X dinner with respect to white Q BYP DYQ but at so some of at side of the here and look at the size over here Mr. BY PDY killed the says divide she's PYQ this subject over here must be G prime the play the same role was a set of objects PQ objects which you multiply by DYP UYQ defined length of the vector so we found the transformation properties the metric concerned the time frame is given by the metric tensor and they aren't trying frame funds are good old friends the X by White times DX by why this is just exactly the transformation properties of a cancer with tool variant indices so we discover that they met cancer is indeed really of cancer fast the 1st fact about that country really we can't it transforms as a cancer In has there are many fabrications any questions about a magical Ewing more about with them OK Lemieux warned with the magic cancer tool lower indices and that Quonset multiplies these differential displacements which have operated now the method cancer is also just a matrix it's a matrix with fervor in the in indices this is just the matrix and G 1 1 due out Of 3 dimensional space G 3 3 G 1 2 G 1 2 G 1 3 G 3 love to 3 as and furthermore symmetric matrix he went to and you want to Fuji 1 3 and G 1 3 G 2 3
over here ecology 3 1 but it was be taken the Beagle G 1 3 because the X 1 times reasons DX 3 times the X 1 so you could just take they could take it to be a symmetric makers barely about symmetric matrices as 1 more factor the matrix thought as a matrix it has eigenvalue the Eigen values are never 0 and the reason agonized members euros because 0 Eigen value would correspond to a little vector of 0 length and the or no vector 0 Lane NATO for every direction has a associated with what is known about matrices which symmetric don't have 0 eigenvectors used in only 1 they are listen but they also invertible Nova ways they have in inverse let the cancer has an inverse so that means that as a matrix which when you multiply by the metric tensor metric matrix gives you the unit matrix right equation down and then we'll see what the immersed matrix in Morris matrix the metric tensor is also called G but you put the indices upstairs G an end upstairs GM and with indices upstairs it is also a cancer and it's the following by the property by defining property that as a matrix it's the inverse of the Matrix with ToolBook so how do you write that is the lesson the heady right the fact that we have 2 matrices that's just take 2 matrices AT&T is called a and and and the other is called a um B P Q had only multiply 2 matrices together with multiply 2 matrices together by identifying in index In and something over the index and that gives us a product of the Matrix and Hughes Prof product number ranked equation and I think you will recognize but not back to it GE M entering time is GE end up squalid P some contracted this is a legitimate expression here is G P is truly GNP is truly a cancer would upstairs index and this is a legitimate product of 210 shoes coach would be index a contracted can you guess what they answers for this product equals Delta M P E was Dr. MP as a matrix was a matrix that's the yuan Turks the identity matrix the identity matrix were busy Quayle said this is the product of the matrix GM with the matrix G & P with upper industries the unit matrix it identifies this subject here busy in 1st matrix to this this is called the metric tensor with copper variant of the seas to Contra variant sees this as the metric tensor with took all variant of this is tensor 1 country the coloring but about but In any case the definition of this matrix over here is just be inverse and inverse is such that when you multiply With the original matrix you get the identity matrix but this will play important role of fact that there is a mecca cancer with upstairs indices downstairs and the season will come through it I think would quit for tonight I think we've done enough for tonight will talk just a little more about the metric tons next time and then we'll go on the subject of curvature the subject of parallel transport curvature differentiation of so far everything or do is easy it's just getting the location and following the rotation the idea are they call variant of the there's a little more complicated not much energy but essential they have to know how differentiate things in the space of do anything useful in particular for the study with the spaces glad we have to know how things vary from point to point the question of whether spaces flat has to do with fundamentally has to do with derivatives of the metric tensor and the character and nature of the derivatives of metric tensor so the next time will talk a little bit about tensor calculus differentiation of cancers and and especially notion of curvature hope we get the curvature the young of that Dad tastes grew up for young and that they warrant former air there will be a fat puts it His the really said she wrote but both said I live in that there what if functions a position in an hour and half their but bad guy but remember that they have this basic they live and has the same dimensionality as the number of indices went to 3 is the mention about this space saying his number what but the work that is brought a death while review National you've college Euclid Euclid but these are Ottawa over me give you another chance were space these things with daily here's the current on curved space that maybe that Everything is a function and every point on that basis resort call the engines with Kansas space exactly the same dimensionality as the space itself roughly speaking you can think of it as space of flat planes which are connected to every point at every point there's Tianjin with tangent space the same dimensionality base itself mathematician would say that these cancers live in the Tianjin space that helped you not that does OK to be considers vectors and forth live In the pigeons space they have components in the tangent space arms and backs that's mathematical way to describe the space as yeah sure they'll pull but gave us out of this equation last which can you eat it this is since is not this is a symbol which is 0 could offenders not equal pay and
1 them does equal the equation makes cents for every eminent pay and pay for example it says GE 1 G To tool is equal to 0 it says the G 1 in GE and 1 corpse Warren equals 1 Service an equation for each M and some of them say that the right hand side 0 summer but say that the right hand side is 1 has lost it end all that only that gave also not known along this whole thing is the identity matrix it's not particular component Brazil identity matrix was hauled Is the matrix it's right about that tried out for tool by 2 2 by 2 two-dimensional space this would be G observed what the right-hand side is matrix 1 0 0 0 1 train research we do it OK what to do this is in pain and neck but let's right out we start with a matrix which is G 1 warned your indices GE wanted to gee up 0 2 1 G 2 1 is saying that you want to and G 2 2 matrix now will multiply it by another matrix who's indices are upstairs and this it G 1 Warren G. true G 2 1 G 2 2 bucks multiplied but on Tuesday that a piano what is ripples pay down but you can have hear viewers at here in this way with up left to right if OK not that I had a great idea reported here of free space for favorite based that led major reason that leisurely about right among women this equation full-blown glory 1 0 0 0 1 not only that but so were they say it G 1 1 EG G 1 1 G 1 1 plus G 1 2 G 2 0 1 this forces terms this is equal toward it's a but writer Olivetti it's as G 1 1 G 1 to force G 1 2 G Tutu is equal 0 is for equations here 1 for each entry for equation part of this times this is equaled the product of this times this is equal 0 and supplement natural oils short Europe what Europe she which is and everything's and Bayern attracts the ball and what mean said there is a place where year dementia the same Evans I think they feel for half a century eternal before the things that make it just that it is asymmetric space 3 of the Center for about 3 matrix work there not added the same database of a the that was the only 1 of 3 every single 1 of them runs from In this case from wanted to tool and three-dimensional case they would run from 1 to 3 each of these will be square matrices In Bayern square matrices yes result thank poking through the role square matrices by matrices right but because of 4 distinct equation our up a school and some number of distinct equation some number of distinct equations each independent component got wind of the others Syria death me Melody mean a positive the Eigen values for but OK good for a conventional space with a positive net we're all distances are positive the answer is there has to be positive test In that context that we will be coming to relativity it will not be positive and why act because if you remember from special relativity the distance the space-time distance between 2 points Is he squared minus X squared minus 1 square mile squared or depending on convention at school EPA's why square Posay squared minus T square so the answer in relativity in general relativity is that it's not positive definite talking real money talking about a conventional geometry in which all distances of positive then metric would be passed his order asking young but we are at the moment which is doing ordinary geometry how we will come to Lawrence geometry the geometry is nothing but throwing lecture starring will come from it's worth learning 1st about differential geometry and hours more conventional situation like that I came here are part of your life there that why what not only Eigen values of 0 9 of the eigenvalues as 0 that's what allows you to invert the Matrix is that start it all that under the aegis of young water can offer of 10 independent components right what is absolutely said last year that it is at referee broker connection of fans can occur broker to remember that but trailer but it had been a lot of money on me but not give us a little bit of lot like a box mom argued Becker by eigenvectors Electric Co metric tensor for example and the two-dimensional space would be 1 minus 1 and that has eigenvalues 1 and minus 1 volume 0 for Lula
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