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General Relativity | Lecture 3

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wrote but it has a reputation for being very difficult having to reason is that it's very difficult it's kind of calculational intend In this city's symbols tha away these people haven't been too expressed things in some more convinced very very neat notations but just learning them itself is it passed that would take some kind of things right fear mind forms spinners twisters and all sorts of different kinds of mathematical objects are you could call many of them notation notational if you like and they do simplified equations so I sometimes feel as if our presenting these things the way I do that sort of like Maxwell who were were wrote every single equation of his company of family Maxwell equations are but in 8 8 8 altogether we don't usually right all components of the equator ends we put them together into vector notation so forth and therefore smart we can even avoid indices by inventing symbols like Corrow the same thing can be done to some extent general appeared ready but as I said media the computational methods and techniques are harder I'm not mix remember no means of extra remember both of you know the subject not to use it but quite honestly when I got to do something using Jarreau typically I go back to the indices article by can follow somebody else uses the fancy but keep myself as much as they can do most of the natural me I wind-up going back Hussein things that would doing here indices are derivatives the whole works my feeling news Is this the right way to see for the 1st time but it is in no sense so this is all over the place of stations all over the place too much of it give helpers so will some aspect of it we you have to get all over that form of cancer patient indices 4 and then it goes easier than both more smoothly until you come to do a calculation and right back the mess again them if you're really interested in doing computations in Jarreau living there are packages there are no you just put Ian now we're metric as a function of position the metric cancer and Our deals that various cancers and their Remora 10 service this kind of cancer that kind of answer Einstein answers and then you could say without even looking at the result you could say OK please Mr. computer set the real content set the then set the Einstein tensor equal to the energy momentum tensor and tell me what comes out so yacht computers go better but promise as I said General reduce heart and part of the harvest is just a the year the complexity of the notation is necessary is a subject that was starting up from around his money in geometry reminding geometry the geometry of curves spaces geometry of smooth curved metric spaces that called I suppose we have remind the blame for the year of the complexity of the station's actually simplified Einstein's simplified further but there are nevertheless it is notation in computationally intensive the basic concepts I'm really not so hard but we have to go through a mess before we get to the concept before we can express those concepts a start-up questions at a young age I know very well well several herself on the most active areas of research the 1 to which are really varied very strongly active among theoretical physics product is the quantization of general relativity the year quantization a black hole more probably the most active area Barack astrophysical areas were again black holes important after a physical areas are entirely classical general wrote the numerical relativity numerical relativity means an easy really silly for the purposes of solving if at the physical problems numerically pouring in various initial situations a pair of black holes orbiting around each other and simply running the computer to find out what happens by solving Einstein's equations and from entering into very subtle businesses I'm really is not is not that you just put the equations and growing it much more supple America but and the numerical relativity is also very hard subjects the purpose for example being defined our How much gravitational radiation are the kinds of things is entirely classical gravitational radiation is classical radiation went to black holes falling to which other thanks a lot of gravitational radiation beneficial way why we interested their wealth among other reasons because we have a chance of being kept their anything you could be packed a window and phenomenon that you'll ordered you over gravitational wave detectors may be able to detect gravitational waves from the collision of 2 black calls that his estimated that there are a few coalitions every of black holes that can be seen by gravitational radiation so there's a lot of different areas extremely active area on the equations of general relativity just equations of general the quake by called have been found to be useful extremely useful and condensed matter physics and narrow fluid dynamics having nothing to do with black holes in gravity saying equation solver very heavy amount of activity in areas of applying the equations of general relativity the other areas of probably would probably most factories in length of the last over money career in physics from being a real backwater physics nobody's study general relativity verum March 19 sixties when I was a student except except for the people who were really especially interested and there were focused on questions of general affinity the bulk of theoretical physicists were arrested start become interesting In astrophysics and eventually show supporter on questions of fundamental theoretical physics white because of a clash of gravity in general committee anyway for whatever reason it is probably now have more are reported in the day-to-day workings of of theoretical physicist almost the
subject madam quantum field theory and the connection between OK let's turn try today will be the last on which were will be studying remind geometry of such without really discussing gravity next time we will really get into gravity what does all this have to do with gravity I tell you a little bit in the 1st lectures that the problem of finding out if there's a real gravitational field as opposed to just some funny money coordinates in which you're expressing space and time which leads to curve trajectories but not really curved trajectories the problem of finding out there really is a gravitational field Our is a problem which is mathematically I would say very similar but I think I would think they would go for further win that I would say is I'd denticle Philip problem of finding out if a certain metric geometry geometry characterized by is flat or not geometry which is flat but two-dimensional geometry to begin with surface of some sort a flat geometry is 1 for which Euclid's postulates a correct the power lines although all the stuff you learned in 9 Euclidean geometry is called characterized in terms of attempts by the mistake not that the metric has a very special simple form that's not the right state special form being the chronic rebelled that's not the right statement that the metric former the right statement is the core board which can be found such that With those coordinates the rhetoric as a simple platform Our but more generally just given an arbitrary metric cancer was varies from place to place the the mood has good differential properties what the problem of determining if that's is flat or not this is a somewhat difficult problem what you have to do look at sigh yes there the media of the back of the problem a better way to do the problem is to search the rule all possible coordinates firms transformed the metric from the given formal whatever our roads given presented to transform the metric new coordinate system due of constructed calculate no magic and ask whether it looks like a flat net now of course you go we that will take it in the amount of time you never get their city better technique and the better technique is to search for diagnostic quantity quantities which are built that of the metric that can derivatives and so full of things you can calculate which each if they are zeroing everywhere throughout the space that the basis flat if the Annunzio or any place it says that spaces curve the diagnostic quantity that said is called the curvature tensor today's goal is to get to the curvature tensor covering Monday geometry left and a complicated goal I want you to go pay attention also fall asleep this is this is a very stopper reflect a subject I'll try from time to time to tell a joke or whatever he would hit but the other hand if you followed you learned a lot and it is very interested we could find very they are brought make it war that at it the former friend Morocco bomber never apologize and here I am apologizing array of being born OK start with we start with a space and space it means 1st of all a number of dimensions the number of dimensions money in geometry can be but In principle you could even have 0 dimensional space but that pregame boring just a point a one-dimensional spaces either an infinite wiring or closed curve the Clifford's closed curve was characterized by its characterized by only 1 thing topple length of the curve every curve is equivalent to every of occurred on the same morning another words just think about from moment take a wrote that closes on itself has a certain way wiggle it in any kind of ladies curve any kind ways it can always be map or put on top of another piece of led a rope of exactly the same plane In a precise way without one-dimensional spaces on all equivalent to each other not quite there characterize only by their way through the infinite or have only a and every 1 of them of given Lee Elise identical to every other warned in its intrinsic properties above walking along those a rope but cannot distinguish 1 geometry from the other geometry all they can do was count the number of steps the victims around the world with democracy someplace goes around the rope tow comes back democracy council members steps that's the only thing about aspects two-dimensional spaces two-dimensional space and begin become more complicated but your flat ones curved ones plate curved Laurent Kabila's fear could be a justice based on the surface of the earth with including the mountains and valleys and so forth they can even have we're a topology the surface of the not a Taurus Warwick you don't talk another hole in the forest and then make a Taurus with 2 holes what so by the time you get to two-dimensional spaces that pretty complicated nevertheless they are County characterized by the method and the question of whether they're flat is a computation was 1st questioned find something which distinguishes were therefore I find the quantity Genie those 2 . 15 new GM and the metric pence if it is not equal that is not equal to 0 that does not mean the space is not flat it just means you may you may be in the wrong coordinates so you can't diagnosis space by asking whether GM and 0 or you want defying the quantity let us call are at the moment I'm not going to tell you how many indices it has some quantity best best it should be a cancer because of its had and it's nonzero it's none 0 and every frame of reference we like to find some quantity called our students from Ramonda incidentally among was the 1st person who enters the question and answer that love we want to find a thing called remark such that every month is equal to 0 that implies that the spaces for that and that there exists coordinates sucks that start ICG amend Article 2 0 0 and about equal to the mn GMN as Aziz as indentured equals 0 was an interesting space it's not very big enough to Yemeni called the band members will remain pot the
matter search for the curvature tensor attempt to reduce remind you about a few properties of the metric and certain words were to do it at 1st of all GM then is by definition of thing at position in general Evelyn now functional position incidentally if it was a constant matrix constant coefficients very easy defined coordinates awards GM and His Delta you simply have to straighten out the angles our so as GM it is the pending thumb position a set of coefficients numerical coefficients and the space is definitely for right stone GMN annexes the metric tensor in he delivered it take a little differential distance DX and multiplied by DX end his Ewell differential displacement characterized by a set of differential coordinate shifts PXM BXN multiply again and that gives you square of the link between the tale of under family Head of the year of the vector that's equal to DES square s standing for spatial distant supposed but now the metric tensor is a cancer I think we proved that last time and the way you see that is just by saying S quartered that that's invariant quantities all call all coordinate their what's called then poured systems all coordinate system should read about the length of a given little vector and we take that thing Square as rioted in white word of export that Boeing has to be the same and that sufficient to prove that GM really is a concern that transforms architecture OK next properly GM end is isn't has an inverse as a matrix to make sex and has an inverse the inverse satisfies the following equation GE written again as a matrix when I say an inverse I mean it has an inverse a matrix GM and the matrix of entries and it has a in inverse G. M and which written with upper indices they are in fact Contra variant industries and the properties of GM GMN with Opel industries as such if you multiply by Jean Claude In all and some of but the some of and because Einstein tells me that the Coppernoll index like this are automatically some over some this actually corresponds to taking the Matrix GM and multiplying it by the matrix GE In the aura and result in matrix His deal maker why because I thought Virginia In the 1st matrix a 1 him and the had the product and the identity matrix and the identity matrix is a chronic at Delta the crime the Delta in general relativity only money geometry is always written as a thing with 1 upper index In 1 Laura index at least that is being treated as a cancer chronic Delta a think approved proved that if you take the Konica Delta new transform it pretending it's a cancer you just get back a crowd of things Yukon proved so tensor equations this is the meaning of saying the GM and his In the 1st matrix of G in and once you have that all is another thing that is important in remand geometry as a Einstein geometry and Riemannian geometry all lengths are positive all links a positive also the squares falling more important squares of only a positive Norway were also part of the square of links positive investors something about the proper use of the metric tensor says it is that there are no direction along with which to evaluate this you get anything which is that's equivalent to a statement about the Eigen values of GM it all of the eigenvalues of about positive but for our purposes now all links a positive the same thing but it does say something about it says it's positive that that's the same thing as saying all of its eigenvalues past that will change when we got to a theory real gravity Delgado go buying in geometry by Magic allows you to do something in allows you to raise and lower indices if you'll have a controversy in vector or better yet another sector which has controversy in components of the M contrary means upstairs you can't put it into correspondence with another actor with call area indices I was really the same vectors we talked about this a little last time I told you the difference between country very very industries and the way you construct be call variant components component of for coal bearing components I'm vector giving its Contra variance components is just a multiplier but the metric tensor G and and books about 40 AM in now and some already and this is now G sub and a visa and likewise is something to prove given me and with upper indices this if you like them because the definition of the Lord of the year of covariance component Rebecca how do you go back well Solis equations but very easy to solve solution if you know the inverse matrix would upper namely v AMI is equal to G. M In V and so if you know the opportunities if you know the call variant of the Contra variant a component of a vector this is the way you build call components and another call varying components is the way you build controversy there's a little bit of a Wilbur approved dual namely there are you could see what we have to to do you have to check the beast or compatible with each other but they are OK but for what then can you force metric could look like a flat what extent the sooner the method is not necessarily a flat rate a flat record I will mean not only as a space flat but that them but the 1 are such that GM is equally if that's the case if that's the case this is bases definitely flat but don't want stand given any space can you find coordinates such that GM in b for example the chronic Delta over some limited region or approximately over some limited region equal to Delta and ISO his appearance there are coordinates every point is coordinates aura coordinates which are good in the immediate vicinity of point they're not global coordinates they don't cover the whole space just God within the immediate vicinity of a point which means they good up to some order in little displacements away from pouring in the 2nd quarter Courtnage called gal see a normal coordinates is you do just as 1st think of those single intuitively we have some curved surface we gotta some specific point and we draw coordinate actually we drove forward back season we keep those caught Nowak sees as straight as we possibly can later learned that the terms frustrated Estrada's possibly can curves a
geodesic but you make a straight curves you can above move above both on the surface and a nose following you know nose who's going straight ahead of what can and that defines 1 coordinate axis that it comes back the point and goes often right angles but observing the figure out where right angles was the off from another direction as straight as you can and then build scored based on the series said is the 1st of all at every point at every point you could chill Court Mips said goes normal for such that 1st of all act GM end is equal to Delta mn act x equals at 0 . 6 0 give a that more than 1 way if you find coordinator with his stroke you could rotate the coordinates you could rotate the court and if you rotate them just corresponds to a rotating the cordon different different back poor so the more than 1 week next you could lose your coordinates Yukon straight up these lines to a degree with derivatives on the Met trick With respect directions with butter X RCS any direction derivative of component of electric with respect that indeed directions can be set equal to 0 that duo by choosing these curves To be Estrada's possible Alta April the actually very easy it's just accounting problem of counting how many variables auroral equations roasters OK finally yacht that said not only are at Mexico 0 at certain articles 0 could be a two-goal at x equals at 0 if you try expand that think away from that point no matter how you try to its extend them in general on West the spaces flat you try to extend the cordon away from that point you'll find out that removes the higher derivatives GM and are not equal to 0 and General Foods for the general generic situation the 2nd derivatives let's say with respect to some X foreign some other X as those in general cannot all be chosen equals 0 except of course of the spaces for water adding pouring point there is no content really In the saying that the metric could be chosen to flat light what's calls flat like even opt to 1st derivatives always be dark Italy's 2nd derivatives that somehow the fact that the spaces in the flat or not start to shore up our personal prove this will this is actually hard let's take up point their of interest to be the origin was just take it to be the origin but 0 supposing we have some general metric and some coordinates Hawaii symmetric has some some form knock of this former here let's look for some Exs which will never bill which earned be functions of the white let's Susan following at the place where X is equal to 0 in other words of our origin over here let's also assume that wives equal 0 so the deceptive coordinator of the same origin that is it X will start our our we just tickled white X M Eagle told why area and then in the next order will have some sort of coefficients I'm simply I'm simply expanding powers of wine the next water will be something quadratic wife sort of something quadratic Hawaii collapsed seat in last but here in the in R and put X In the X as the most general quadratic thing that I can write books stark white white white and white they things may it more complicated later Brokers commonly such what slots take a number of dimensions of space but pick a number of images but the 4 dimensions of space how many such coefficients are well I'm really combinations why In the Oliveira ensues can try counting why 1 squared White to squared while 3 squared wife Foursquare why 1 1 0 0 1 1 1 3 1 1 1 4 a White tool wife 3 White to white former wife 3 . 4 the every possible thing here which is quadratic why 1 2 3 4 5 security can so they attend possibilities here many of the components of the chain wiII why hours so that means in an hour bye give you 10 possibilities but then each is hard you have and that there are 40 independent coefficients here 40 independent coefficient the most general exit Ucon below the quadratic water has 40 independent coefficients now I'm early in the parlor independent components of Jr that dancers can serve only equations Aziz correspond to components and setting all of their derivatives equal to 0 means that this is 40 equation 40 equations 40 unknowns and that allows you to be sure you could solve these equations the quadratic or a sobbing into quadratic order is saying the same thing that you can solve these equations but your failed the next order fail at the next order and his the next order so as always possible to choose a metric that has the property that To quadratic order in other words to the 1st derivative Of the record physical 2 0 and the record is will fly at a point in a case that has left more than that under of show the not that that's according the transformation now that this is a quarterly transformation which were found Tuesday sees in such a way as to make the derivatives of metric 0 I always do it so every metric in that sensors flap over sufficiently small that's saying its locally flat that that the statement that slow gradually being awkward flat until it is really just a smooth surface could be approximated over a small distance fire by plane now it's tangent to it for buffer that couldn't you're on side yup Syrian Yaphank in this case to can I think in this case you can Yahoo young guys in this case you can is drive do it those with that by 2 matrices stood in two-dimensional case and there's nothing special about 4 dimensions you find it right after March under a rock or any case it's interfere with
Delcine normal coordinates exists at every point reporting and they have the proper like the reason we introduce the because you want to get some means to derivatives of vectors we talk about some of the Member of of cancers and general drew of cancers with talk about various procedures various operations we don't want to make begin a had them would subtract them with the multiply them Bolick and all kinds of things can we differentiate them in particular if we have please concern which is a function on position that means all of its components of functions of position can we construct another Center which is some appropriate the derivative with respect the position of the tension that we have sorry share why this is a little bit tricky it is a sailor OK take 3 components of the I pick the components of the cancer looks very components and differentiate them that gives you a new set of components which of the derivatives of the components that gives you the derivative of that it sure you could do that but what you get b a cancer and I'm sure you worry that was yet to be set free mills 1st overtures Mr. this is 1st lover 0 4 yet but it is not the point of view of both terms it's just I said let's choose the origins of the 2 court systems to be quite so that says wherever wiser grew 0 x physical was arbitrary but it of doesn't cost let's not let's look at a vector field vector fewer vector every position and starlet coordinates which either of the spaces flat In which case I'll just use ordinary Cartesian coordinates bases curve in would face these coordinates here represent gaps in normal court as Street as possible now let's take a another system superimposed on top of that is not Garcia normal coordinates they may be the cordon so we started with our represent those in greens and curved relative to these coordinates for example blinds might look like that 1 of the cordon of lines everyone might look like yes wherever but now before we ask how we differentiate a vector 4-year-old Butterfield a veteran every position here but says put means would mean for the vector field to be constant inspect the farm he was because the space is curved it becomes harder compared vector at 1 point the Member point the coordinates if they can't be chosen to be everywhere flat than what exactly do you mean by saying of that over here is equal to a vector over here it doesn't really mean anything because arms to compare vector over here with vector here unless you have a nice flat quarter there is no unique way to do that on so what talker for more money if the place is flat really that I think I know what it means for a vector over here to be the same ought to be equal to a vector over here that means equal points in the same direction and has the same way on the other hand does it a I'm thinking about the Greenport now think about the Green caught in the black coordinates the components Of this black actor in this black clearly nice Cartesian but what about the coordinates of this vectors here this vector over here along the Green ax cities what's even works special case of state is could be vertically up so that we know it only has all white component in the flat court let's assume that this sector over here is equal to sector points in the same direction in flat space arts coordinates alone the greenback series on May the same as the coordinates along the the Black Cats not this coordinate along the green actress here is different than the corridor along the Black access here In fact this sector we have both X and why components along the Green taxis wherever the road reactors have the as book kinds of comply so it's clear that even tho the vectors on the same the components are not the same they cannot judge when you have curvy linear coordinates in general will have curvy accord that you can judge whether the components see the paintball Contra variant you can't judge whether the vectors constant in space another later say the same thing is that in 1 0 coordinate system the derivatives say with respect tool some coordinate X are part of me and component of that there might be zeroing will coordinate system and not 0 another coordinate system it may be the case that all of the derivatives of VM or 0 1 1 coordinate system and 0 only other coordinate system because the court want chef because the court in direction because the sector is changing but because appointments changing so this statement but the derivatives of a vector are equal to 0 it is not equation which is likely to be the same in every reference frame therefore it cannot be a cancer equation it's not cancer origin and the derivative of a vector the components of vector With respect accords are not themselves components of the cancer component of a cancer because they could be 0 all of them in 1 frame of reference and not them if they want components of the concerned we would think of this as a rank to take certain M index of Lauren I said look at this b cancer tease em off well if the cancer mr the end of 0 1 frame or court system 0 in every quarter but that's simply not true of the year of the writ of these derivatives Of course Holmes records not because the vector they change from point to point but because the cord that still orientation of accords exchange saw we need a better definition we need a better definition of the derivative of a vector just differentiating its component we need something which is 0 1 1 frame 0 1 in every 4 so here's all of their is how the fine derivative 1st of all to define derivative at a point you only need to look at nearby points strike you only need to look at nearby so the 1st thing you'll do is to construct galaxy in normal coordinates near that point Garcia normal course straight as possible but the specter of a bit closer just to indicate that the day and pretending that the gal seemed normal coordinates are really nice flat course shift this vector so that its origin is the same as the starting vector over another words show keeping its components fixed and constant even Garcia normal court gas in normal coordinates I keep the components of the vector and I compare it with vector over here shifted down here and the difference between these show is the thing that I will use to define the director of derivative will be these difference between these 2 provided by the separation along the sacks another word to differentiate vector differentiated in calcium normal court they are the ones which are closest of all the Cartesian court derivatives In the gas normal coordinates define the derivative OK now when you differentiate sector in general coordinates you get to
2 terms calm 1 one-term columns because of that there may be changing and the other turned columns because look Ward changing course that's a rotating out from under you you have the vector doesn't change if you work out this prescription take the derivatives just ordinary derivative Sidalcea normal coordinates and then we express it an arbitrary coordinates he is what you will find you will find that so the derivatives which will now be called capital D along we are axis of the components of the vector V M In arbitrary quarter I'm not just the derivative with respect or are of Vienna but there's another turn the other Turner has to do with the way the coordinates themselves or changing from point to point what looks like something is proportional to the beach if double the size of a double the stern here obviously after proportion of the year will have revealed that it does not have a derivative because this trend doesn't come from the fact that good the vector is changing its coming from the fact that the coordinates of changing work search put something this season travesty see what kind of thing it could be my have key component here on the left hand side we have indices following and that means we have to have some kind of object of India has index on air just to match the indices on the left-hand side on the other hand it's going to be it's gonna be proportional to the components themselves use perhaps a V T this equation doesn't quite make sense they're directions here 3 years you gotta soak them up a half to put another index into this object here at a fact the standard conventions from right side this in fact is what you will get a few take a vector differentiated in Galle seemed normal coordinates and they and transform it to other court in any other coordinate system for what you get is a derivative Martinez I think of called the moment components of view themselves as to some Doherty the industry index structure on the left-hand side of Oregon yacht there also that holds no From accorded system but I'm tho do Gamez are working on like that Out no Garcia normal coordinates justice like again now we get we are right will find that began as a proportional derivatives of met they have to do with the metric changing from place to place going to proportional to the rivers of metric in Delcine normal coordinates the rivers symmetrical 0 At the trick use Garcia normal coordinates to differentiate Bacall lost the 1st order of the derivatives of metric 0 0 wanted differentiated Bucanan transform the object as authority cancer to any frame of reference and the formal always be like always be like contain indeed a derivative but also contain another term which will not be derailed differentiated has to do with this is is said it has to do with the fact that there is changing from point to point but the fact that coordinates a change from point these objects at various names is call connection it connects neighboring points tells you how to differentiate them something from 1 point to another cousin basically compare vectors different point it tells you prompted compare Our vectors even those coordinate system may be changing misses call the variant derivative not because it indexes downstairs at cortical variant derivatives the of just because of called derivative for the core area derivative and it it is a tensor it to pencil in that if it is your only frame 0 in every 4 Hart said generalization that if you don't think it is never thought about that we think it is your roots connected with connected derivative a R no doubt about that Our keiretsu Pam think about it but certainly co-vary derivatives and fluid flow right OK so now this is comical variant derivative and these symbols here stormed the 2 names usually there either called connection coefficients or the crisp offal from Kristofer are truth they also known on occasions Christ awful symbols they are pretty well enough that complicated but complicated enough on likable OK now 1 back about them that follows and that don't approve everything goes and I say because it is too many little pieces but they're easy to check it follows from the definition namely the definition is you differentiate in Delcine normal coordinates and a new treaty object is a cancer and cancer with 2 wins a follows as theorem easy theorem but it's a little too intricate daunted by word that this Crist offal symbol has a symmetry the symmetry of the Christoval symbol is gamut T horror and equal the gamut T M R doesn't matter which order you put an arm yeah that follows the arm generalized me money in geometry called geometries with caution geometries torsion the symmetry it is not true but those geometries I'm not widely in use in the ordinary gravitational theory Mary gravitational theory Riemannian geometry of follows from the principles laid out and gamma Oregon is equal to get them you don't yeah the Gamma Gamma Gamma yet you know Billy Middle River in In these now seems normal coordinates which almost flat fires could be in treating it as an object Dial Tom right is is very very similar to what we do gravitational theory when we evaluate something in a freely falling frame the end for example in a freely falling frame we calculated how like moved across an elevator and then we transformed it the frame of reference in which the elevator was accelerating bomb that's closely related to the operations doing here we calculate something because we normally do it in coordinates which are as far as possible that would be a freely falling frame in general relativity and then we transform attack any quarter which we like accelerated court anything we write we translate to stake from 1 court system for another year the notion of variation of a vector from point-to-point is done 1st announced in normal coordinates and then it's transformed and that this is the form that you get for of the corresponding and I would have gone repressed awarded will more but but right now the higher derivative will leaders tonight suppose sorry far a higher rank cancers supposing we have a cancer with more than 1 index M led say end and we want to differentiated call variably differentiated again in this case along a faxes I'm a wire to resist all are
used on a warrant differentiated along the S acts as a I know what I mean these are here that derivative with respect X derivative with respect to end all are these a bomb at noon yes not not at any court system and any coordinate system Amar will always mean be DXR this like place which went in the water this 1 here not that any court system and Garcia normal course there Garcia normal coordinates you just differentiate dousing normal corps everything is easy an ordinary a derivative vector is just the delivery In the arbitrary coordinates is of the humor that out by transforming his but doing transformations yard sale scheduled for a vote a bad fall yes yes yes it's only at that point right up no it not tensor this is it this is intended to be a cancer former tried to choose Gamma so this is a cancer of this we know is not tents and therefore this cannot be but yeah I was on my and in fact they built up battered the rivers of the metric in a court-ordered system in which the derivatives of magic of Zero Christoval symbols 0 but cancer of the 0 and 1 coordinate system 0 in every quarter systems so I can't be beat answers when it up OK let's just go before we go and figure out what began as all work amount that that but work out the track let's right now it's they now lot this except for every index in the cancer there is a term like the every index after our works there is a crisp off will turn every index fell for example we start out by hiding in a tight end and pretend that this was just a vector Ben Wheeler right this is derivative of tea With respect the X as of the 1st turn here and then we would put a turned Gamma S M were used here Chris errant tee he added but now expose here and put on both sides but not finished I have to do exactly the same thing covering up a cover-up In try to it by our another turn he where is passive in is active and so we get Gamba S S in T T and see whether the former always you might decide might assigns a wrong said bring science as here not us effect cover about if I covered up the 2nd index you em in if I covered up every place 2nd index it would look exactly like the called it a derivative of the vector if I covered up the 1st index it would look exactly like the covariance derivative of that city and that's should be M a R T guilt yacht paralleling this here I notice in 1 of the terms he occurs in the 1st place and the 2nd tournament occurs In the 2nd place so that's that's the form the call very derivative cancer would take the rule of the same you go back and gas normal coordinates and you just differentiated cancer just derivatives the number of of his job and then you transformer the arbitrary coordinates assuming that the results here is a cancer and this is what you get so allows you to differentiate any cancer the moment which is dealing with cancers with called variant indices will come into contrary to some points but in particular let's apply because to the metric tensor itself but something special about that 10 in DRC in normal coordinates its derivatives a role 0 now this touches like but what area it's differentiating things that's what comparing things at different points you use is what field equations field equate the field equations are gonna be differential equations which represent our fields of change from 1 place to another we want them to be the same equations in every reference and we don't want to write down equations which are special to some peculiar reference frame we want to write down equations which a general the patrol in 1 frame the betrothal frames and they have to be depend to equations and that means we have the know-how to differentiate pensions get evidence of such as this is a partial derivatives this is a partial derivative with respect to coordinate X X shut its what a yes yes absolutely absolutely DX if you want a normal thing changes from 1 point to a neighboring point you adjust multiplied by the access what no no no no no put I say it again point is the key components of the vector may change even if the vector does 8 wanted right we have to keep them my services boot will see some examples of our Our Fortier now as a Catholic that's right out of black distortion of the accord there even then flat space to choose funny quarters but important point it's there even in flat space if you choose 1 quarter of choose any coordinates at which the derivatives of the GM ends 0 in which the recorders your vary from point to point the funny ways this will be there is not really a feature necessarily of curved spaces it too feature of crude coordinates OK so because we metric cancer is constant Easter 0 derivative In Delcine normal coordinates and the definition of coal variant derivatives to differentiate 1st in normal coordinates of antiquity get treated the answer it must mean that the call derivative of the metric tensor 0 at such good or allow us to computer class the Crist symbols right so let's write down here is the form for the different derivative called area derivative other and with lower industry so let's apply it to the metric tensor itself VS G M ended it people 1st Terrace G. M. a body be minus gamma tea G you In my Gamma T S N G M T for guns this equation and now head medical variant derivative of the metric cancers equal to 0 White because he ordinary derivative 0 downstream Court Delshire normal right now let's write the same quite except Munich Re industry the changing & and so forth this is the equation for differentiating with
respect to s of GM by inspection Why am I doing this Boeing this because I want to get as much information out here about things like this as I can and then tried receivers anyway that I can manipulate the equations forget that isolate and figure out with gamma so that people or extraneous at the sort of things look at it I know what to do if you don't know what to do what we would do the once and then when your friend calms unless you honey find Chris powerful symbol design or experienced OK do same thing except now undergoing the interchange S and am just from U.S. them and that means we have a right down M G S and I do it I just do it my In the changing or replacing S I am MIPS and that gives me derivative of Jedi but s with respect to X M mightiness now changing S & M here doesn't change anything because S & M from love From the symmetry here was the same thing Demarest guaranteed G see no adjusted the changing S & M G T and and that over here and the changing S & M services gamut am end T. G S see the cost for exactly the same reason this is also recalled 0 finally has won the last possibility here as is the odd man out emanating go together it was the odd man out and then go together and the last 1 D and 0 G S AM ends the street fairs as copied equals 3 by D X end G guests M minors gamma-ray bursts N T Z T N G T Miners C and C G S T and equal to 0 now you stare at this for a few minutes How can I these track the feasible something clever that will isolate only 1 of these terms of began theirs so I don't have a horrible bunch of equations but basically only want a quick saute wanted to call this equation 3 scoreless equations to and Corliss equation 1 end end equation 3 2 tool and subtract Kwan falls to act to 1st of all of course we will get these derivatives this derivative forces derivative miners is derivative so let's put them down on 1 side of the equation and transport transport all of his job the oversized objects on the left side of the equation we will get he is a man G S M plus D M G S Emma for D & M Street attack that c G as in M. this warm Norwest warms my years GM and derivative GM said that's this class this minors that's our bit plus this might not maybe some things which canceled but filters things which canceled this comes with a signed this comes plus sign this cover plus I sell this and this of the same but they canceled because of the officers surprise were T S N R this 1 and this 1 opposite side so they have gotten will last with 2 terms but at this stage so luck we can light this is equal to 0 twice we need all the right hand side twice yeah image now T G S well we still have Restormel Lewis from up there we would like to get gamma emitting by itself trying to find out what began with the Christoval symbols or so go figure out what they are in terms of the rivers that were almost 1st of all we see that of all the derivatives metric because 0 the Christoval symbol must be 0 but how would immediately of his genius the answer is as he has an inverse show basically you can't just take this all that the other side were replaced by the inverse matrix of 1st of all it's for a 1 half and now the bracket around this and take his GST over to the other side except in its in verse form and this is the correct formula a formula for Microsoft Chrysophyllum symbol they looked pretty simple who will gather for a moment on B in the name of symmetric and and interchange fees to a symmetric again as 1 negative turn looks down 1 negative to positive terms not very complicated the problem is is a boatload of them when you think about four-dimensional space Inuit S & T E and Surrey England and let all of these coefficients range over from a from 1 of 4 is just a lot of the stuff that makes Calif doing calculations in general relativity of very tedious visitors Our intrinsically there's nothing hard about it but in fact doing a calculation and some Giro Tiffany context usually fills page if the page after page of nothing more complicated than just computing the rivers of putting together now like OK I'll say it again G is constant has 0 derivatives in downstream normal course at all at that point I'm not but want a chore want born as short every point to derive it at a point you go together seemed normal at that point say I want them I want a formula appropriate to a particular point I go to that point use gas normal court and find out what can you want to know pouring it's true at every point this is a connection between Obama and the a derivatives of G at every point the thing is that this is not a cancer cancer and so it can be zeroing in 1 frame of reference and not 0 another for example and Garcia normal coordinates act the point at that point at that point this is equal to 0 but in some other coordinate system it's not are even more take nearly flat in flat space you confined coordinates which the G are constant everywhere flat space but you could also find the the coordinates and not cars there polar Courtnage Fiji's of constant Cartesian coordinates they that can't stand the Crist awful symbols can be non-zero even if the basis for actor generally public order yup just what is is the why not too but them by former forward you're for that all time
no it's in my my got me in the cities and towns and now that's an might invite him but this some symmetries whose 40 components to it it has 40 components bathing always 40 mn if you fixed feet fixed fee then down a man who has been comport saying came components for symmetric cancer and then you like run 4 components and that is you fought over a 40 independent Veracruz stuff know what all what for a while she no no no no no no if you have some coordinates Twitter general coordinates all over the place and I want to know how to differentiate at that point at that point I go to normal coordinates a Duma differentiating and then I related back to look wordnets the general coordinates which cover the whole space at each point at each point this is what you do there where this formula Chu no you take it current coordinates wherever they are they say I want to know how the compare vectors of neighboring points so are you going to I'm going to now seemed normal Oakwood and that point tells me how the compare them at that point his tells me to do it we are not the associated with any coordinates with Delcine normal coordinates these assume they're all 0 at that point the rivers or call 0 at that point these are the coordinates the 2 began when you transformed together coordinates derided all of this but they knew finished work for the for the gas in normal court officially gas normal coordinates and you go on is adjusted trekked the guest in accordance with just a trek through to eliminate locally at each point the variation of the metric tensor with position for the way it is that it because what is a no no no no Lewis does not tell you spaces this is just Russia stars were down if of gamma is I assure you of gamma 0 everywhere it not only means a is flat but also means you chose Cartesian quarter he said the code that wearing to remove metric tons of 0 8 0 0 that's office a missing metric is His constant over space metric cancer a statement that the metric tensor itself is constant would be the ordinary derivatives is Contra 0 right at the call variant derivative the rhetoric is always do White because coordinates 0 cancer 0 on 1 corner frame 0 in every quarter frame and 1 more thing that call varying derivatives Kansas services the since is concept which retreat from the end of the baby boom role as calculator could bottles of bowls replaced ordinary derivatives many contexts replaced ordinary derivatives with covariant derivatives Yukon right equations in Garcia normal coordinates images Our ordinary derivatives but if you want the same equations In general coordinator the new replaced ordinary derivatives by you by covert derivatives that's the rule think about I looked as requires sitting down thinking about it understanding what we're doing we're not that far richer yet so it's try to get to curvature here we have the definition of let's talk about what curvature areas it's easiest to start with two-dimensional curvature but we pulled the curvature well let's show let's struck would begin by drawing space which is Curtis fears obviously but I want I don't want to deal with Sephira or deal with coal will not comment but a call with around point at the top the top of a mountain but with besides being nice and quiet so that if you're away from the top of the year of talking up here just around it call homicides here it looks comb-like the carpets smooth author and shaved off from Iran look at 1st aid is such a column his flat everywhere except near the top of the scene that just take the same space down here but continue it so that it really does former a genuine called case that we have a only differs from the space that we're talking about but this facility appear not take that call slice slice it Darren here and open it up this is a car owned slice apparent here in open up the clearly flat but clearly space but it's a kind of question open up and was look-alike it looks like depending on the angle of the common ruck here it looks at a flat space but when you cut it and opened up it was a PC cut out of a missing from this increase on that have a scissor issued brought but they anticipate cut cut out a wage of a given angle glued together and you make a call and the cone has kept up here the tip is clearly not flat but the rest of the column is flat you cannot fly sentences for like a coffee you can slap down the pace table without tearing it but if you can't get anywhere if you cut it open CompuCom layout make a flat I'm Oh Scissor day the cover notes the dead have dissipated amid paper we convert but cut out a certain angle and there when I saw them together or blow them together in this case it made a kind of flat color could wear the hat OK if I come out of vigor angle carmaker more acute angles at the top of the a airport about a larger cut out a lot wrangled now and I saw together if it makes a point be called a to it because 50 get died that's a point here call so that they really angle of thank you the bigger the angle you cut out begin the deficit call the deficit angle comical deficit the begin the conical deficit appointee of the call is a cargo deficit is very small acute-angled means the column very flat on top Dave right now let's think about let's think about 80 vector of vector field on the space heater which is constant but thinking field which is constant where the need of a constant it means that pointing in the same direction everywhere for simplicity let's take Italy
pointing in this direction so funny over here only Hewitt's along this line over here practically perpendicular to the soaring remember when this when they when medium when he cut his identified like the lines in 1 of the reports take a vector pointing over here here yeah here and here from we saw it together we discovered that a funny jumped in this direction pointing along the cut pointing some funny direction here even the only open up the wrong you see that was hired as you go around starting year pointing in the same direction until you get around to the other side of the car which case you find that not in the same direction as as the was over here nevertheless y open about the role start the tip of the continent it is such that if you carry a vector around when you get around to the other side has been rotated has been rotated dueled through a comical deficit at the par-4 exactly the same will be true In this geometry over here if I take a vector on a given direction and I carry it around the in such a way that it's always pointing in the same direction if I opened up by the time I get back to the other side will be pointing in some of our actions that is Bacau the top of the column is curved that's the effect of curvature effect of curvature of a region of curvature has a property that if you take a vector pointing in the same direction as much as you can to keep pointing in the same direction and taken around the region of curvature vector will rotate will rotate when it goes around any region of curvature even tho you took P E is to make sure that every point you are moving it parallel to itself another words you took pains to make sure that they call them derivative that sector was 0 but nevertheless when you came around the side and shifted Armed theirs another way to say this which is actually more useful rule it is the other way to say that it's equivalent to quote if you have a read if you have a curved stasis and curvature here curvature he knew taking vector field and differentiated along 1 axis in a move it and differentiated could take differentiate along 1 axis and then differentiated along a 2nd axis take a vector displays a little bit and this place along some along some 2nd axis we will the vector will change and how the vector change it will change typically buy differentiating the vector along these Tilak seas In see quite 1st differentiate the vector of along 1 axis and then differentiated along the 2nd axes and is a small change in the vector due to these 2 derivatives are go wrong he and other words of I take the difference Of the vector over here over here they consist of 2 changes change to go from year to year the change to go from here and that change would total change vector is proportional was 2nd derivatives that would really be true in any coordinates will be Anthony coordinates to carried a vector from here to here and then from here to here and compared with vector at this point over here would you would
be calculating is the 2nd partial derivative of vector derivative of V end X s directions and the X on their action in this would be the which the river would be it would be a derivative with respect would be call variant derivative with respect the aura of the coal variant derivative with respect to of the vector unless end component of vector this operations says you move the vector a little bit along the S direction and this woman says you take the different derivative and would only art direction and the net result is change in the vector going from here he don't call valiantly which means that you could have done the same thing in Garcia normal coordinates with just ordinary derivatives Garcia normal coordinates you would just be doing this to reverse course toward Mary derivatives you could also gone the other direction you could let us to say you could've gone 1st came be on direction and in ES direction and calculated the wave vector changes from here to here that would be the opposite order of differentiation ordinarily and flat based in general these 2 were equal to each other this is just the fact that derivatives it doesn't matter which order you take them but it's not true curved space current space this meant the difference between these 2 operations can be thought of as taking the vector around the closed loop the difference between these 2 operations taking the vector from here and them from here he might the opposite direction is essentially taking the vector around the loop closed taking Iran will close look we discovered that when this curvature at the tip of the Kong here you don't get back to work he started current space called variant derivatives are not interchangeable and flax space they are interchangeable that's the way we test for forever for whether spaces flat we test whether differentiating cancers particular vectors in the opposite order gives the answer is yes everywhere in the space for any vector spaces flat if we discover that there are places in the space where the order of differentiation gives different and then we know the space has some kind of defect in it all has curvature has curvature our use of it Top of the year Top amount Storm always have to do Oh it's a little bit complicated but it's not that complicated his computer the 2nd derivative of the a vector 2nd call variant of the fact that opposite orders and compare them not so you ought to do euro durum occurred through a door on too late but easy acidic require it mechanical McCarrick offers they were steps are not the answer is entirely mechanical or just pure plug-ins too late to carry about the yes what interested in is taking over a vector Vienna the components scoring from forms of Computer Inc the call area derivative of the art direction and in differentiating Iniesta action later on we'll interchanges in Oran subtract from let's just do this this is equal to D S now this subject over here it is a cancer it's a cancer and we have the rules for calculating the derivative of the tensor where a contribution from each index there write down the answer them all Webb said yes DR DOS both Dr about pop up the is minus gamma aura in T V T are still stuck in here the call variant derivative of Vienna that's what's in the box here just this thing over here now after differentiated again using this formula to next object and receive I wavered to you to calculate received calm write down all right down further equation defects of DRG and scare Merv key Pulver knowledge getting to late or Frederick panel reliever for you to do just pure blood your plug-in you plug in the CMA this object over here right into the air for right calculate the 2nd derivative and then compute the following STAR of vegan minus V R D S that corresponds the difference between the 2 ways to be different he had by going through this direction or that their actions about your to get but but you get something called Auber it's a tent worth the recovery in and this 1 controversy and received index and multiplies Vt where permissive pretty easy honors a little bit complicated R N T equals GER a N T s Jaromir aura in city plus against a P P R C Meyer Swanwater Fairmount Park in P Jaromir Peter S. C 2 turns involving derivatives of the press powerful symbols and 2 terms involving products of estoppel symbols is a some more here are some over here industry take a look at this match it's a complicated expression and it's even more complicated when you remember what began as again as
themselves are complicated derivatives of genes the curvature tensor curvature tensor the remodel curvature tensor was remind curvature tensor is itself another derivative and also product of Chris powerful symbols hazard when you different the Gestapo symbol of involves a derivative of G-8 differentiating began his 2nd derivatives of remember the 2nd derivative of GE other things that you can generally set equal to 0 1st over the G confined frame of reference it was 2nd so by the time you finish calculating this thing here 2nd derivatives of Geof and it is testing out and feeling out the geometry a little more thoroughly than the than the 1st derivatives would would feel about feeling of 2nd secondary derivatives doing out the recording systems the quadratic order store has 2nd derivatives powerful symbols and where quadratic things involving 1st derivatives it's a complicated thing I would actually write it in terms of the crisp off or symbols or what to try the calculated for a given record against Philip pages and it will talk pages but conceptually words doing is just calculating the difference in a vector if you transported around here keeping a parallel to itself until it comes around Our all ways around a little change in the vector and going a little this is being curvature tensor the curvature tensor as a sets complicate calculate but you can't calculated to complaint computer and UPS the computers 0 back in the computer roof and it's good to have a few have a computer that can do algebra and if you have the metric and some algebraic formerly have the metric some some mathematical form you could Dooley's operations and then passed out whether this curvature tensor 0 everywhere if the curvature tensor 0 everywhere all components of the spaces flat now will will study the curvature tensor more but as I said it's a complicated thing but the main thing about it is that it tells you whether the spaces flat and if not how on fly year's arms it is closely related to quantities gravitational physics guess what it is it's a local quantity which tells you that space is not flat so it must be something Inc tell knew what is really a gravitational field president or not tidal forces tidal forces exactly related tidal things with the gravitational field squeeze 1 way or the other way what these things are in in fact they really are the tidal forces cargo forces represented by the curvature tensor you could sort of see something like that if you were to imagine we Guelph imagine you have again is kind of conical geometry does not become geometry but something which is flat faraway 80 yards so it's flat faraway 5 following somewhere the center it has a bold so that curve T now it takes an object out here it's a thing living in flat and two-dimensional depicts an object that might be composed of a bunch of links might be budget triangles together links were and they start moving as you move in the flat region nothing happens till it perfectly happy it doesn't it stretched the doesn't get distorted the doesn't get performed the doesn't get squeeze 1 way or the other way when I want happens when you try and into the Kurdish Richard botanical moving into the curved region it simply can't follow the curvature without having to stretch its flanks 5 it has the following the metric properties of the curved space and in particular if you go around a little closed loop in this space of links here where did doing this sampling Inc this double call variant derivative here what you unifying now is that when you go around little closed loops of angle shifts from what would be in a flat the angle shift the late shift when you try and push up here it gets stressed it gets performed the get stressed to get stretched so full of and a measure of how much it gets stressed locally is exactly the curvature curvature tensor is telling you how the system performed as it moves up into the region curvature that flights connected with With tidal forces so it really is an important property if you're in a region where this curvature you feel you feel as a distortion of if you're in a region of gravitational field looks a uniform gravitational field uniform gravitational field don't have curvature and that's why in free fall perfectly uniform gravitational field simply feel nothing so the next time we moved into the gravity land will talk about what you have to do to change from Riemannian geometry Einstein geometry and then we will study a particularly example the particular example of this what show geometry short show geometries geometry of a black hole star gravitating may to allure we left out 1 subject to resume my notes tonight and it has to do with you but we're ideas will have to do that but will do that after we start thinking about gravity real OK How I wish I had a way to make this book fewer indices but I don't know what is it a lot after all Bibbins between looking at object is free fall from an accelerated who when the system are accelerated frame of reference and not already part or looking at a system which has a genuine gravitational field simulate what looked like a gravitational field many purposes but just accelerated frame of reference but it won't cause tidal forces her real gravitational field may buy real matter out of our than produces tidal forces tidal forces means that there was something real going on not just a year coordinator at the Gap no well 1st so I don't really them a uniform gravitational field Rio gravitational field
of Eros looks like my uniform produces tidal forces varies from place to place no such thing as very real uniform gravitational field His approximation approximation near the surface of a very big planted over 80 relatively small solemn angle the gravitational field looks approximately uniform there is another thing to say about tidal forces have much less effect on small objects from large objects you could see that also is true of curvature if I have a very very small objects and move it around if it's much much smaller than the scale on which things vary but it won't experience any great formation if you have a large object and you move it around it will experience D formation because different parts of it get stressed differently but to saying that tidal forces have very small effect on very small objects 8 bacteria falling under the surface of Europe to experience essentially long before we experienced no tidal force were speaking about where the 2 thousand mile man known would be crushed by private force has cost for beer so when you speak about the strength of tidal forces you have to discuss how big an object talking about the same thing is true here we speak about the effect of curvature on object moving around in this space again it depends on the size of the object to crop up and this Coors Carson hundreds of blew a 4 yet is but you very often very often you be presented with metric some analytic form the uniform for the formula you can differentiate the formula and everything you have become was not some some analytic functions that calculate look at 0 0 you're right if you adjust given numerical metric cancer which is given a bunch of numbers at every point in our huge b Gough our spreadsheet filled with loads and loads of numbers you would have to check the curvature tensor at every point the most complicated as anything else so where this is useful this situations where you know the metric and analytic sends which you can calculate derivatives track and the walls thing to do and they look at me however cancer and CVN or 0 but so in that sense there is a problem it's not as bad as having look at every possible metric and see what you can make it every possible metricate every possible point that this is this is this is better vows part every possible that every possible quarter systems where every possible court system of every possible point until you find work which expresses the metric is being flat this is used to calculate was 1 thing and you compare a point-to-point focus for Moor please visit
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Titel General Relativity | Lecture 3
Serientitel Lecture Collection | General Relativity
Teil 3
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/15041
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

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Fachgebiet Physik
Abstract (October 8, 2012) Leonard Susskind continues his discussion of Riemannian geometry and uses it as a foundation for general relativity. 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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