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# General Relativity | Lecture 3

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#### Automatisierte Medienanalyse

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Steiger University general
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
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
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