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General Relativity | Lecture 1

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activity G R for some reason that I don't know special road to never called Sr their wrote to is often called G R my guess is a good many of you know a little bit about Gennaro Tivoli I'm starting at the beginning of the end my own taste in the matter is to start preen much where Einstein started that's not that I can't get past my love affair with that that it just seems to me the right way to to approach the subject is about to begin with the equivalence principle begins with the basics simplest facts are probably said this before but it was a pattern of Einstein's thinking about what we really really simple elementary facts that almost a child Columbus and induced is incredibly far-reaching consequences of and personally I think that so that's the way to go about teaching at the start of the simplest things and deduce the consequences are so I will begin with the equivalent principle was the equivalence principle the equivalent principles the principle that says that gravity is in some sense the same thing is acceleration so I get back on the table sale of given an example of till about how Einstein used it and then go on from there to move asking what kind of mathematical structure with a fury half the ham in order that the equivalence principle be true for where are these kinds of mathematics of have described I'm sure most of you know that general relativity theory not only about gravity it's a theory about geometry it kind of interesting to start at the beginning and ask what is it that led the man who were to say that gravity has something to do with geometry OK sold gravity equals acceleration what does that mean that I take back will go through 0 a very elementary derivation of what that means it's not that this derivation is important but the but it's it's worth the 4th its worth formalizing formalizing means making equations that say the words you all all know that if you're an accelerated frame of reference elevator accelerating upward downward you feel the effective gravitational field children progress because they feel it on you know that you could probably even explain to me why that Stroh but worth making a little bit of overkill but making mathematics out of it and then seeing what that mathematics is talking about OK so let's Einstein for experiment somebody in the elevator the elevator got promoted a rocket ship and later textbooks but I've never been a rocket ship I have been in elevated and so I know what it feels like to be in elevator end all that elevator is moving upward it may be moving downward in which cases velocity vector here that the negative love Estrada's velocity vector this is not the cable pulley Opel or it could be cable pulling up but it's just a velocity vector velocity vectors up and there are to coordinate frame 1 coordinate frame is it rests on the earth rocks called Z a user's E. coordinator of the vertical coordinator space and here some places equal 0 could the service of Europe matter now really elevators also a coordinate system and that coordinate system is being measured arbitrary but will do it anyway it's being measured from the floor of the elevator and saw the same physical point a point over here as a Z coordinate and it has as you and coordinate and zinc Prime is measured from the floor now the height of the floor of the of elephant the elephant the elevator the floor of the elevator its height relative is equal 0 tyranny score a L & L is clearly a function of time will be adjusted the following question if we know the laws of physics in the frenzy surely we can figure out than in the frames Prime do that now 1 warning warning but 1 fact about today's lecture at least in the start I'm going to ignore special roasted this is tantamount the saying that were pretending that the speed of light is infinite all are we talking about motions which are so small that the you on the speed of light to be regarded as infinitely fast track you might say in general a generalizations special relativity have Einstein get away with starting thinking without special relativity and is something like this on special relativity has to do with very high velocity gravity has do with heavy NASA's is a range of situations where gravity is important where high velocities so I started out thinking about gravity was lost in the men's combined it with special relativity think about combination of fast velocities and gravity and that's the general theory but let's see what we know just from from slow
velocities for French let's begin with inertial reference works opposes a primal Manzi prime vessel reference frame and that means they related by uniform velocity uniform velocity and that means that L of just be taken to be Vt the height of the elevator relative to the floor relative to another floor but the relative to the ground is just given by a uniform velocity v times I've chosen the coordinates such that a T equals 0 they line up at people 0 both Zia's primary 0 and now the elevator was rising and the height is proportional idle floors proportional to VT which I write down what the connection between privacy Zia Prime Minister since it is just a Z miners LAL as a function of time myself take if we know what else is this is a kind of coordinate transformation why not he prime what about kind in the elevator would also like to ask how kind behaves in the elevator if don't to forget special relativity then we can just say keep trying until the same thing we don't have to think about rents transformations in that sort of thing so the other half of the coordinate transformation would be key prime is equal to T justice and thing and finally we could also put another coordinating he later on will be interested in the coordinate x going this way also warned coming out of a black wine and is low the elevator was not sliding horizontally and ex- prime an ax to be taken to be the same it's true so offset the elevator relative to the taxes but I didn't have to do that I could put z-axis right down the middle here it which case I would say that X prime is his acts that's at a quarter the transformation coordinate transformation of the space-time coordinates its rather trivial only 1 coordinating Lizzie is involved in an interesting way we can a free like proprietor here because deprived of the same thing OK now I set of please equaled the VT and this is very simple plans they got Malatesta about or physics but take a lot physics Newton's laws of motion she wore is it a 80 is Z doubled not a 2nd time derivative of with that kind of score the acceleration the vertical acceleration is of course is young it is busy component of forests but forget about BX component of Y component of force them out and the 1st thing in this context whatever force is exerted no this is the formula In Z frame Susie we can imagine it a frame that there are forces acting on a particles from a particle of forces acting on it where most forces do do too well there could be some charging here exerting forces are on charged object in here they could just be a force due to lower a rope hanging out of the particles from the ceiling any number of different kinds of forces could be acting on the particle over here we know that equation of motion in New Riegel frame of reference is just f people's and double not 1 of the equation of motion in the prime frame Willis is very easy always have to do is figure out what the original acceleration is in terms of the prime acceleration was a prime acceleration the prime acceleration is obviously Z prime doubled off and prime double got his Yukon acceleration is just gotten by differentiating this twice but of a differentiated Vt twice a year nothing that's linearity differentiated twice to get nothing is simplifying Z double prime prime double bop Egizi doubled up the acceleration the 2 frames of reference of the same nite million others say no you know about this but I wanted to know formalized to bring out some points in particular the point that would do we Nick Ward transformation and were asking how laws of physics change in going from 1 friend to another so now what can we say about they know it is locked in a prime frame of reference well we can now just substitute equals and disease doubled that Prime if I we find that a lot of Newton's law in the prime frame is exactly the same as as long prime that's not surprising the moving with uniform velocity relative to each other if 1 of them is a social frame of the residual frame us that the laws of physics are the same in all commercial Frank so that's a sort of the formalization of the are now let's go to an accelerated reference frame accelerated reference frame Elliff will be increasing in an
accelerated way accelerated way would be to put here 1 half the acceleration times tee times squared Excuse me let's call the acceleration G G for obvious reasons for acceleration GE fuel for the obvious reason is defined by now this is a uniform acceleration I differentiate L twice the 1st time with the differentiator once I get yelled back and that's just Jedi Norris exists as the velocity of
the floor increases would kind are steady velocity and doubled prime bubble . is just equal to g service elevator was uniformly accelerated upward and the equation connecting coordinate transformation of different greatest one-half G square so that's I knew will coordinate transformation to represent the relationship between coordinates which are accelerated relative to each other talk until 2 assumed that and Z coordinates here the laws of physics are exactly what new Torah let's ask what the laws of physics are now in the prime frame of reference we have to do this operation all over again but we know the answer is we differentiate twice here we get this thing differentiated waste plus or minus this case just G a 2nd time to remove this it is just mind G right of the client acceleration and try and acceleration differ by Ramsey gene and now we can write news equations in prime frame of reference z double dot Swedish transpose Gedaliah beside G plus there's Onarga substitute them says that equal AU forces E planting double plus and Jake so defined as expected of chorus the finger resort renewed expected that in the prime frame of reference it looks like we'll go actually want somehow I think I have the wrong side arm my sister was to my list but the disorder on Thursday priority is busy double by using double prime . plus Cheney is at right yacht from thank right well sorry havoc I have that I want to shift it's the other side of the equation of course minus energy on the other side of the equation shifted to the other side because I want to make it look like a Newton equations man's times acceleration is equal to something and that's something I call the forest In the prime frame of reference and you notice as expected before us in the primed frame of reference has an extra term sort of fictitious territory which is just man's Pansy acceleration of the year of the euro elevator noted DOS thing about it of course it would watch industry about looks exactly like the force a particle the surface of the earth or the the surface of any kind of gravitating systems uniform gravitational field Waseca uniform gravitational field but spell out what looks like gravity the special feature of gravity is the the gravitational forces are proportional commands that has deep implications the deep implications since the equations of water ethical ceremony and if a force itself somehow proportional amassed the man's canceled and the acceleration problem motion of the system of object doesn't depend on up a characteristic of gravitational forces that an object moving gravitational force small object gravitational force his motion doesn't depend on mail an example for exile would just be a motion of the Earth about son is independent of the mass of the Earth if you know where the earth is its image continue our how fast moving then you could predict how moves without regard for the last spot it is an example of a fictitious force if if you like if you that mimicking the effect that I think most people before Einstein considered this largely accident just the Babel nor would they certainly knows that the effect of acceleration mimicked the effect of gravity but they didn't pay much attention it was Einstein who said Look this is a principle is a deep principle of nature that gravitational forces cannot be distinguished from the effect of an accelerated reference for the effect of looking at physics and accelerated reference for car that physically they are the same thing they are indistinguishable now that that's what we're after discussed at a little bit worried believe that quite as much detail as I said but before we do that we draw some pictures of workers varies coordinate their transformation look like Let's 1st take a case where is just to illegals Vt that was ex prime physical the X Reedy was progress Zia Maine Z it will zip Rod my fear is gone vertically plot tie-in horizontally I plot Z a Z Nancy Prime put in some guidelines used cluster vertical axis is equal 0 years equals 1 His Yukos tool and so forth physical strain now let's put in here of course is a sequel 0 foot is a prime equals 0 Zinc prime equals 0 is the the same as equals Vt Zeal's see is a trajectory that were would go right looks like there's a buzz Prime 1 as it chronicles 1 looks like this still looks like St . 3 looks like this and this is the nature of according transformation that relates the primed and Leon Prime reference frame every point has a coordinate T and Z pair audits teensy also an ex incidentally drop at a tea and as these white and busy Prime and Zeke Ward related this way keep primacy quantity of failed to find so that's called linear track transformation straight lines go to straight line not surprising Newton tells us that particles will in straight line is friend reference and was a straight line in 1 frame and therefore better be straight line before they move in straight lines not only in space but space time OK now but still the same thing after the accelerator coordinate system if you like where is it Z Z is equal Z minus 1 FGT square so let's look at Z. prime equals 0 the trajectory of origin on busy prime coordinates that's the same disease Eagles won him Jedi square parabola OK it's a parabola lying on its side y because because because parabola line outside of work Friday that's the trajectory it's accelerating its going faster and faster and faster incidentally I could run is back into the past a uniform accelerated acceleration upward forever and ever we correspond to something moving downward decelerating in going up your back up so they want to continue this To make a full of uniformly accelerated trajectory look like Z comes down thing goes back up and down our about time this is it this now is Zeke prior Nichols 0 what about price equals 1 well that's just the shift of the parabola which I always have trouble drawing always always get cramped somehow Z to look like this these parabola seemed to be getting narrower and narrower
but that's only because a good drawing them each 1 is just shifted relative to the previous 1 by Warne unit end the point is Of course not surprisingly straight lines in 1 frame are not straight lines and the other framed their curved lines the song would say Vera command that was awarded duty square GE T-square older to best serve court system here on this is a curvy linear coordinate transformation so something Einstein on the very early is is a connection between gravity and curvy linear coordinate transformations of based on that that special relativity was all about linear transformations transformations which took uniform Molossidae uniform velocity Lorentz transformations are of that nature picture Lawrence space 2 straight lines of space time but if we want a mark up gravitational fields their effect on after the effects of acceleration we're really talking about transformations of the coordinates of space time in which a curvy linear until Einstein realizes that if you that don't that now I serve to again when they just give you 1 example the sound all of a sudden extremely trivial but have to normal and
I think every physicist when Einstein said it probably also knew it said Ali our big deal but Einstein was very clever he realized this could answer questions it nobody answers give you 1 example users the class number of times that of the example of a question that Einstein cancer dissuade the question had to do was what is the influence of gravity on a light now that time and rarely 1907 or something on my 1st thought about the 1905 start thinking about gravity most physicists were answered There is no effect of gravity on light writers like gravity is gravity lightwave moving the earlier gravitating object just was a straight line that's a lot of light that was a straight lines and there was no reason to think that gravity has any effect by censored now if it is true that this a principal the equivalence between acceleration in gravity the acceleration of gravity will affect light they argue was very simple again 1 of his arguments which are you could almost explained to a child let's start out with a light waves or white deemed wherever it is X now not disease is used vertical let's start out with a white bean that moving horizontally starting out at he goes 0 equals 0 at equals 0 the velocity of the elevator 0 equals 0 the velocity begins to build up from equals 0 At equals 0 2 frames of reference over matched that both at rest relative to each other people 0 and a flashlight beings a beam of light this way back in lead trying frame of reference Z accuracy frame of reference we now have a lightweight novel what we signed said In in the frame of reference our without any gravity is that gravity and talking about how are we shouldn't be too for our purposes now the earth is is the virus's us light the doesn't a making gravity although the gravity is due to the acceleration of gravity is due to the acceleration or apparent gravity said well in the frame of reference at rest where there is no gravity we know what the light lightwave does polite way just
moves horizontally across Ivica right down again I want to write the equations just below the illustrate was the equation of the slight being well as 1st of all a horizontal it is equal to the speed of light transit time make this distance here Taylor different L different 5 X is equal to CT that's our light most and also Z physical 0 Ozzie doesn't change in the year ending on prime frame of reference just most horizontally face about sets the equation of white but now let's plug in the prime frame of reference and the prime frame of reference Zee equal to see Prime I 1st TG squared so we have there Zeit private plus GET square is equal to 0 that's the equation light beam Exs still just equaled the CT an asset so here we see what state what they like being is going in the right frame of reference namely its moving on a curve trajectory if we were to calculate its vertical component of velocity we would discover looks like that I say In the prime trim reference gravity is going beans now you say no no it's just that they accelerated that the elevators accelerating upward and that makes it look like
being was of curve trajectory and Einstein said the same thing this proved to him that the effect of a gravitational field on a white really was to close it occurs the course of the day per cent broke 3rd was something that nor the physicist car so clever OK but after what we've learned things when I while abstract matters is that it's interesting to think about curvy coordinate transformations when you do think about linear coordinate transformations Newton's laws do change of the form of Newton's laws change and 1 of the things that happened is apparent gravitational fields materialized that are physically indistinguishable from ordinary gravitational fields well are they really physically indistinguishable not really Our let's talk about real gravitational fields namely gravitational field of gravitational gravitating objects like son of beer it is a son of Edith and the gravitational acceleration doesn't point vertically on the blackboard points center so there's gravitational Fields Point being radially inward it's pretty obvious that there's no the way that you could do a coordinate transformation like theirs which will remove the effect of the gravitational field yes if you're in a small laboratory here and that laboratory is allowed to simply toward the sun or whatever happens be then you will think In that laboratory but there is no gravitational field but there is no way globally to introduce a coordinate transformation which is going to get rid of the fact that as a gravitational field pointing inward toward the center if you make a transformation like this it might get rid of the gravity of here for this falling object of the same transformation on this side with just please gravitational field so now there is no way to get rid of it we know there's no way to get rid of it and 1 way to I understand why you can't get rid of it is because if you think of an object which is not very very small gravitational field suppose Her my favorite the 2 thousand mile man but Our over 2 thousand mile man falling in the gravitational field was put Switzer started out differently 1st with star about 4 feet 1st day because is large laughter point mass the different parts of the feel different gravitational fields the further away you are this week of gravitational field is so was head feels a week gravitational field of seat His feeder being pulled harder than his head gets what he feels is being stretched freely falling he can't distinguish it from a stretching of feeling but he knows that there's a gravitating object that gravitating object cannot be removed by these dissent discomfort that it has cannot be removed by going toe falling reference is of course tidal forces tidal forces are cannot be removed prior year it is very small of course is very small fetus had feel the same gravitational force what happens if he's a sideways while look at this gravitational field you can easily say that is a component of the gravitational acceleration gravitational force on his feet which is pointing upward here just because has
upward component and this 1 has a ban would component social experiences of compression and that since a compression is again not something that you could be moved by car Bart Court by any court transformation the invariant back being squashed invariant it's not something you could make Golway but due to coordinate transformation cell Our so it is not true that gravity is equivalent to going through an accelerated reference rate I was due he was no fool you this so we really said that small objects for a small point of time cannot tell the difference between no gravitational field story friend reference but that raises the question if I presented you with a force field for example the force field associated with uniform acceleration is just a vertical force field pointing downward and uniform everywhere it's a consequence of this non-linear coordinate transformation but never behaves 4 eggs were um observer of using these coordinates like a gravitational field but with but kind of cool wouldn't transformations you couldn't make the gravitational field look more complicated and other kinds of coordinate transformations for example transformations which effect X coordinate also for example they can make the gravitational field being towed the X-axis opposing you simultaneously accelerate along the z-axis while oscillating back and forth in the the X-axis frame of reference is oscillating backward and forward along the x-axis an accelerating along the z-axis what kind of gravitational field you say a very complicated 1 that's got a vertical component that has a time-dependent by oscillating component along the x-axis so by properly coordinate transformations you could make some pretty complicated gravitational apparent gravitational fields if I tell you what the gravitational field is everywhere how do I determine if just a sort of fake gravitational field coming from doing a quad the transformation using a frame of reference With with various kinds of accelerations in is it a real JUY warning gravitational field as well as assistant Tony gravity this it easy way you just calculated tidal forces you calculate whether it that gravitational field will have an effect on an object which will cause it to squeeze if it does then to real gravitational field if you discover that that gravitational field has no such effect abject Noel title for snow no tendency to distort a freely falling system a system which is in free fall then and no matter where it stopped here dropped but he teardrop he will be drop it doesn't have such a cargo effect then it's not a real gravitational field young athletes it exist but arrives they the pleasure to him has us the Committee on time of reference there was no gravitational field so there's no way that there can be they have a distorting effect on you are anybody else because you calculated the prime there was no gravitational so the question then is how do you know what it's possible to make according the transformation that removed the gravitational field how do you know where is just an artifact of court Mich it's a real physical things due to some gravitating massed for example that's a question not this has been delayed now in hand is he had won by a that you want you go to I more than half the the death of course that we don't even need have a gravitating object we have no gravitating object in god Apollo coordinates figure to Jakarta preparing now that means we have to coordinate foreign data so funny we did you know that these polar coordinates always knew what was it was accorded are in accord with data an ideal I told you how a particle moves they then an offer as a function of time How would particle moved say what with figure of C and or see told they behaved as a function of tar remote plaque that I look at it and see what kind of motion the particle had played an hour a function of time so let's get what's get our butts oppose a particle moves across here what happens well Our starts decreasing decreasing decreasing in insolently there is increasing when I get back its decreasing going this way they is evolving and meanwhile at the same time aura used decreasing decreasing decreasing delegates that here and the Nichols backed out this person who was using polar coordinates and in no way with say there must be a repulsive repelling the particle away from Oracle 0 the and that repulsive force as we know what it is and tropical force Texas introuvable force of having some angular momentum but nevertheless if you didn't know where you got these coordinates from the same I got reports happening coming in and going back up effectively there is some kind of a repulsive force that repulsive force would be proportional to the mass introuvable forces proportional commands and this is an interesting situation as a radially outward gravitational field in every respect it would behave as if there was a gravitational field pushing things outward from the center maybe you might think this it's a matter they that wouldn't be right but a ring of matter pulling it that would be right but you have discovered a force on an object which pose out pulls out proportional of Massey assess found no discovered ahead of it by gravity bully away from the center so basically any kind of coordinate transformation you do if the cordon the transformations nonlinear there's declined 4 lines will produce effectively some kind effective fictitious forests where are the fictitious forces here tropical forests Coriolis force those kinds of things they all have the character of gravitational forces on but again forget somebody in a small laboratory moving along here would not experience any sort of distorting effect clearly because we can always a study of Anthony and vessel frame the linear very nicely reference frame and the Ronald distorting it follows that by cordoned the transformation you can get rid of these fictitious forces the question is but just give you a gravitational field our know I get rid of them what we had to calculate the things life at is half but not at all a cow and up to her
is yes so that what Mama low so I yet so there is a she also is not only a gravitational field here this is just an artifact of a quarter net system have to it was war what rarer about while making a gravitational field will will bigger tidal forces for something falling in Oregon this past now talk here creating a gravitational field and 1 of them again somebody gets close to 1 of them they can forget the other ones in no experience a trial lawyers tough what do you do you try all possible coordinate transformations coordinate transformations needing means coordinate transformations of space and time and rewrite the equations of motion and see if you confine the coordinate transformation that eliminates all of the fictitious forces if you can't say there's no real gravitational field there but you can't say yes there was a real gravitational field could not Peter women male friends dying not was Over work take a chunk about a massive some kind of an object of crystal and you put the crystal into the gravitational field at all and you ask whether there are stresses and strains on the crystal whichever which squeeze a one-way stressed the other way and it'll be detectable thing that that is rocket this is not America around the photos took up the use of coordinates which were not there were over so there are seated that I won't worry came only a day the OK nor from where I say there is no effective drama kind force I mean for a freely falling object freely falling object is not being held in place by ropes of Foley's a anything else it's just allowed the free fall then no no and if there were no if there was a carousel rotating Inc rotating and if somebody is in free fall they will appear to move very oddly from the point of view of the person on the cameras but in the reference frame of media over at rest militant 2 years they move in a straight lines and they feel no force of you calculated the effective the tidal force in either afraid you would find 0 tidal force means effectively distorting stresses the stroke restoring stresses on matters that squeezes public our simplified doesn't give you a headache doesn't give you a headache the answer is if you were in free fall if you if you're stuck on the bid is stuck on me and Mama Cassell you might get a headache but do not stuck on the carousel you moving in free fall you get the firm said gap with you not they have not hot work that it be for a time when we wait but far less OK this is not just 2 thousand are rare he is also a thousand miles wide but From could be distrait ends use stretched thin squeeze them is simply no way to get rid of it food already enough but here that followed for a year or well means us restated carefully means it has to be studied carefully and own stating it carefully would say something like if you take a sufficiently small system smaller need that the gravity of the gravitational field across it doesn't vary much of the gravitational field of small enough for the gravitational field doesn't vary between 1 part of the system and another that when New year equivalent principle holds with accuracy I thank you all that year and so we ask themselves the question What kind fanatics goes in to try dances kind of question now none the other elements that came into where it was special relativity and Newell from the work of the Task D special relativity had a geometry associated with the geometry had the property that had a kind of links associated with the length was cold proper time proper time of proper distance and discover remind you about that very quickly in special relativity theory is an invariant notion of these space time difference between a pair of points so apparent neighboring points for example we're taking overtaking arrest now from gravity and recurring back a special relativity but and the only thing on the use of a special relativity is that space-time has a geometry geometry is geometry discovered by Minkowski and it has a link element length element means any pair of points where X and tea and will use X and he now of course is also toward accord that but let's concentrate on and the space difference or distance between 2 points called D S squared 5 depending on who riding the equations they will be the right Diaz squared or squared they only differ by assigned cow score is whether they take a square mile BX were just lettuces a speeder like going to this if we wanted to put in the DX where overseas where Soviet Square Times Square but In that case it was fun always get rid of this war was square by redefining yet the of call-in X codecs overseas
and has become Dec squared distances of X X overseas and that in itself was accorded the transformation sorrow you go always bring this coordinated double this fall was being called proper crime and some physicists like Hughes be as a which is exactly the same except by sign square erred I thought I about this geometry and he realized that this question of what kind of cool are they coordinate transformations which can remove the effect of tidal forces was very similar to a certain mathematics problems that had been studied at greatly length by reminded they wheat the question of deciding whether a geometry was flat on rocks geometry of a page is flat geometry of the hemisphere's not flat sphere is not flat and he realized that it was a great deal of similarity in the tow questions of our whether geometries flat whether a time has gravity we'll gravitational field and they were very similar so let's talk about that bit on this might sign he reminded had never dreamt about real had not thought about geometries which have a minus sign he said he was thinking about geometries which was similar to doyou clean geometry what's the right formula nuclear geometry and think we want to start with the mathematics of you could cover of money in geometry when I can spend a huge amount of time it but with a little bit of time but all of but what's the corresponding thing for you cloning geometry analogist say a bunch of axes this could not be X 1 is could be next tool there could be another X 3 coming out here and supposing there's a little interval he characterized by a DD X and let's call it an alien represents whether it's X 1 X 2 X 3 is not 1 squares is X 1 X 2 1 X ray and dumb if I prescribed for you a set out in this case 3 D X 2 3 components 3 components of a Little Chef What's the difference the distance between these 2 points over all knowledge just turned Pythagoras Bureau a number of the agency saying the square of the distance scores now the square Of the spatial distance between those 2 points is just equal to be X 1 squared plus DX squared plus about about that however many are the spaces three-dimensional and they were free of the spaces two-dimensional Mr. spaces 26 that 26 of them and so forth that is the formula are you clearly in distance EU Quinney and quickly in space for the distance between 2 points now Dallas had a ready understood that aren't curved surfaces the formula for the distance between 2 points was more complicated in general and that's a two-dimensional surfaces and rain on generalized bestow arbitrary dimensions however many dimensions you have and what was known from the fury of surfaces it that the right formula for the distance between 2 points of reversible before we talk let's suppose we have some kind of curved surface will we might be incurred is not clear yet but some sort of arbitrary surface would looks like current has no particular shape to but the but we have every reason to think it's got something that we might call curvature OK so the 1st thing we do it is we put some coordinates on it we want to be able to quantify various statements and so we want to introduce and when it's on it we just way out some kind of coordinates we don't worry all about whether the corn taxis a straight lines are not because for all we know what the services of real curved surface they probably won't even be things that we can cost data lines so we just lay out some coordinates call an exit now we take to neighboring pouring used to neighboring . Barragan related by Of the court this shift over here is again called DX area but now for an arbitrary curved surface with arbitrary coordinates Cohen it's an arbitrary kind of complexity the formula was not as simple as that it has similarities with it about it looks but With dealt with any geometry curved or otherwise it is gently this formula also applies the flat geometries if you take curved coordinates to take a flat geometry like surface on the blackboard and you introduce some curved coordinates curvilinear coordinates and ask what is a distance between 2 points in general the distance because square the distance between 2 points will be of quadratic form In these little the separation quadratic form means that you have a matrix GM and and the could run from 1 3 for example and you just combined together GM and just as every year you've seen these kind of things before heading general G will depend on position you know example distanced end another
distance on the the surface of the earth between 2 points characterized by longitude and latitude below the formula for the distance between 2 points to have class but don't have yes squared they which rose with charmed her lighter favors allowed to squared glare of the radius of the earth later square reward Civil War Nevada new flows of defy squared but I can't just be defy squared joined try Gaza sway there goes this way a given amount of define could be a big distance a few near the equator and the same amount of defies a small distance near the North Pole the South Pole so the something else across India which has to do with how close to a lot of stuff we will it it I squared not sign a favor square besides corner of is an example is an example of a distance not just being beta squared plus defy squared but having some coefficient functions in front on this case be interesting coefficient functions assigned data square in general if you use arbitrary coordinates on you'll discover a much much more complicated formula but it will always be something which will involve the coordinates quadratic D squared terms a defy square terms a B dictated kinds defied terms it will never be defeated Cuba terms it will never be things linear everything will be quadratic like this and get some general formula which will involve a matrix and moved it's easier for last year's they realize it would only latitude from the quake delays nuclear all I'm maybe but say on the idea of a politician measure latitude from the equator because square was our duty yup that's correct if latitude measured from the equator its cosigned squared measured from the North Pole who besides quit report your right but the important point is here but the formula for the distance Our differential separation involves functions which sit in front of these quadratic expressions in terms of the differential distances and I said this is just a very simple example more general examples would have more complicated functions there might be a complicated numbers a complicated by a function squared it might be a function defined squared and may also be a Decatur defined all possibilities of possibilities are included by saying the general matrix here and in the end matrix with John multiplies the differentials this flight to us that this is not a good place for it which 1 you're on no applies voted no reason doesn't apply for arbitrary use the because you have to specify what point Fabio talking about and is only good for distances India by that point we looking at the so is in general for curvy linear coordinates it only applies if you want know 1st of all if you want to know the distance between 2 points you better defined take some arbitrary curved surface that address comes bumps hills at all kinds of things what deal even needed by talking about the distance between this point that point over here well you could mean something you could mean the shortest distance you could mean put up pegging here and peg here and polar strangers tight as you can between the 2 points that would define 1 notion of a distance between points of cluster might be talking inches 1 Michael around the Hill 1 way 1 like the hilly of so in in general but think about it from various even if it is only 1 answer unique answer it's complicated you have to know the geometry everywhere in between you have to know the curves in the hills on the holiday calculated distance but the nor actually where the place a string of but put another he'll in over here the strain will have to go differently so the notion of the distance between 2 Point is a complicated 1 of the notion of this between 2 neighboring points is not so complicated all our offer small distances between points so basically would you do to imagine doing is your building a geometry us of a bunch of Tinker Toys Erector sets thinker Tories or whatever a bunch of little units of given a lane and piecing together at the Michael Slater a geometry if will pieces putting together are correct correctly chosen you might be able to put that doesn't mean the world equaling but you might be able put together into a flat plain it doesn't mean all the same link defected to compromise figure for flat where you are barred if they pieces are chosen some particular way over with the length of the pieces which is equivalent to
saying that the metric cancer this is pretense the metric tensor has a particular property it may be that those pieces Camille laid out the differential elements can be laid out and what nonpool flat Blank connecting geometries said to be flat will define Mara more carefully later but it may also be that the pieces are chosen awaited kid Slater Malloy were flat maybe the pieces might be a little elements that you would get by so triangulate in sphere and some then they would be in no way given the only given you know what they like is going to each 1 of you and you know that they can be laid out on a sphere without stretching them without breaking them then Is it it's true that you cannot lay them out on a place in such subsurface is not flat so the question is if I gave you the length of each will sticky without building saying how could you tell me whether it's a flat space or whether it's intrinsically current space that cannot be flattened out and laid out on a flat plate let's say it more precisely given G N N of X given G and then X given to give its name it's called metric tensor as a function of position in some set of coordinates keep in mind their mini set of coordinates and every set of coordinates the metric tensor look different may have different components but I give you 1 set of coordinates and I give you 1 metric tensor I tell you what it is in effect told you the distance between every pair of neighboring points and I ask you is this flat reserve software that worries me by asking a tad bit odd a bit odd rap I know what you mean that's good that's very good for two-dimensional surface stuff so great Ferrer for higher dimensional surfaces yuck that's that's 1 way of doing it on does now I said it's not so easy to do with these differential little things here because so they're so small that deviations from is behind a lot but but what the mathematics of taking the metric tensor asking spaces when wasn't for it to be flat or what it means to be flat Is it to confine the coordinate transformation of different set of coordinates In which the interval distance formula is just the X 1 square was X 2 square X 3 squares a would-be nuclear geometry that's the definition of flat space the space is flat not a G M is just a number of zeros and ones as a forward to the Pythagorean theorem but if of course would mean a transformation to make it look like that so in that sense it has a they similarity it turned out not to be a big similarity all close parallel between the question of whether you can find the coordinate transformation which removes gravitational fields question is can you find a the transformation which removes the curvy character of the metric tensor here ought to know what they answer that question the geometric question here have to do some mathematics and mathematics is central to relativity we cannot get away without doing it because I understand without and mathematics is cancer House concerned houses are a little bit of differential geometry it if annoying it's annoying because these indices floating around the and different coordinate system so mildly annoying but once you get used to it it's really very simple and really really simple was actually made simpler by Einstein the famous Einstein's summation convention which means that the rights of nations of the place years you worry about that said that that yacht Yao would that Treadway we assumed that small enough that's that's exactly right in both money in geometry in theory of surfaces is always assumed if you look at a small enough peace I was small enough piece looks flat that the technical statement as a small piece located at some point can be well approximated by that tangent space into space will be the flat cans in the you're they said that trend but it's a question of waters of differentials to question our orders of different differentials Doran really Lane still limits of small and they're defined in such a way that the square of the small things 0 so visited there was a technical mathematics small differentials the the state has a yacht was it was us that lot and or to me America crevice yet rewrite bomb nor the magic always be taken to be continuous so often derivatives of and which are not contiguous but stone or crevice means the crevice does not mean increased of course Greece this space here is as flat as the space package exactly the same space for forearm in terms of
differential geometry the distance but be careful with say exactly what would be the distance between any 2 points born here between 2 neighboring point doesn't mean the distance are between these points if you go through room and a the page it means the distance evened the surface this in the surfers and that distance between these 2 points doesn't offend unworthy of fall the pager marked exactly the same with the page fault as with the page on the right terminology is interested in these intrinsic geometry of the intrinsic geometry means those aspects of the geometry which only depend on what's going on inside the surface and not how the surfaces in baby in 3 dimensions these are different and betting of the same surface all sorts of different settings of the same surface but the intrinsic relationships between the point on the surface are independent of how we have been so we're talking about intrinsic geometry geometry that would be measurable by a little creature that only gets to move around on the surface and doesn't get to look out what thought that a lot of were barred the you can't bagman general you can buy noise everywhere simultaneously look start the bag do we find recorded God roars yeah you can't do you could do that at a point points you can diagonalize it but you can't diagonalize everywhere simultaneously but that's a point where we can tell by looking at a point where surfaces flat our working appointed always look flat you say it may be that the surely should face lesser to sailors were 608 afraid delayed again to get away with dropping question about gravity for the mom I started by explaining to resist mathematical problem of deciding how to find out is a gravitational field of the gravitational field community move by coordinate transformation now we're talking about a problem with sounds similar factors is similar how do you tell given space where it is flat and beef the answer is to look for a coordinate transformation which turns G end in into into what what what GMN Ferrer a Cartesian coordinates chaos however state at all they owe what this is a tragedy matrix young or the crime occurred Delta similarly us remember chronic adult the adult man this is a formula for the interval in Cartesian coordinates for a flap geometry them without their mn is there and is just the matrix diagonal matrix everywhere is equal to 1 off diagonal everywhere equal 0 so 0 will we would say is that flaps based in Cartesian coordinates the metric tensor is just a Konica curvy accorded sets not curvilinear coordinates this could be more complicated so I give you a metric I give you a metric tensor and I ask you can look buy coordinate transformations be reduced the lives of yes spaces calls flat if no the spaces called curves of Class a space could have some portions which a flat bearing on those portions it could be that there's a set of coordinates were in that region the metric tensor to be chosen to be the crocodile but called flat everywhere is flat everywhere but that's another problem given Aegean and X how decided coordinate transformation which will change it into the Chronicle symbol the instead that we have the understand better how things transformed when you make coordinate transformations and that's the subject of cancer analysis this analogy between
tidal forces and curvature is not analogy it's a very precise equivalent that in general theory of relativity way
that you diagnosed tidal forces is by calculating the curvature tensor the way you define a flat space is faced with a curvature tensor 0 everywhere and so it's it's a very precise our correspondence gravity really is curvature Martha will come through it as we get it OK so obviously the 1st question we would like to gaps in trying to determine whether we can transform a lady Our GM and next turned it into the
trivial dealt Rex the 1st question to ask is How does G of X transformed when you change coordinates said that we have to introduce notions of cancer analysis which are rather
easy sometimes a bit of a notational Lucent's because of all the indices and you can get confused by them but I hope your OK so let's begin it approach With a simple things GM and X let's just start was supposed to sets of course which is a set of core X In the end I could call them the coordinates x crime but then and have a horrible notations where will have at a time in the old boy who looks like X a X Prime 1 would be x 11 let's keep away from that do that and let's call it took 5th recorded X and why it's not XYZ pipe wise just X 1 X 2 X 3 for however many NY 1 1 2 1 3 1 4 2 sets of coordinates X and Y end of course they're related because of you know the coordinates of a point in 1 set of coordinates that principle you know the way the point is and therefore you know it's coordinates and the other from the other court system we might write that X coordinates x In our functions of while am what I mean that function of wine a year as I mean functions of all of the components of white and likewise Y and are assumed to be known functions on the exit X and Y end of means and of X 1 X 2 X 3 adjust or write all of the year of labels in them we have to coordinate systems each 1 being a function of the other Our and let's ask the question how these differential elements here transformed another words giving the X end that means give it up terror points given a fair point neighboring to each other there will be X and differential distance between and differential separation between what's the corresponding differential separation of the y coordinates this is differential separation of the X coordinates what's differential separation of alike were wounded calculus of and deal why and his people the change and why and when you change any 1 of us X coordinates by a little bit of scored BXP plans DXP analysis is 1st of all this formula is Saarland old peak of it this just shows that the change in Wadi from particular compartment is a of changes in why if you change X 1 a little bit and change in X 1 must change extra will be change DX too and so for is a standard calculus formula the differential change quantity our are when you change the corridor from 1 set of coordinates went up now right here we have the 1st example of the transformation of the cancer D X NDY where they represent a resent the fact that send an infinitesimal vector in the sense but they represent a little vector of vector going from 1 point to a neighboring point located at some point in the space located at some point the space a factor this is the transformation properties of the components of the sector this is the transformation properties Of the components of what is called a Contra variant vector Contra very invective simply means a thing which transforms like this I will now give you the definition of a general Contra variant nectar it's a thing that has components Lee M. which In the trying frame West think about the prime frame from speaking about the wife the White court let us go back for 2nd where we talk here we have 2 sets of corn stretched as draw 1 set of coordinates we have a little vector someplace and we are interested in me components of that that there will only act sees prescribed by quarter those coordinates have to be were interested in components of these little infinitesimal has more vector along specific acts easily axes which are locally the ax associated with the court those are these be wise and the exit if I change coordinates a change coordinates at this point here there might be some of the court but and the components of exactly the same vector In no coordinates will be different sustained vector but different components white just because the ax OK so this is telling you if I tell you what the DX is our 1st Pacific vector only this tells you what you waters are is the transformation properties of the components of a vector if was not too How America's war with each of those lines is that all of you were series that he said those lines represent lines are they represent surfaces strictly speaking they represent surface strictly speaking lines surfaces of constant grinder another but on a two-dimensional surface just like gap so if you change coordinates you change the components of a vector even billion not changing the vectors the vectors very definite vector you're changing its components because you changing the coordinates but you relating to the vector to update a thing which transforms in this way I will now a riot in the prime coordinates and his people the BY and RDX P times V he Mississippi component of a vector this is and component of vector in the prime reference frame this is the transformation property some Nova P this is the so-called transformation property of a Contra variant sectors we'll see why there where we have to caught country a logical country area as opposed to look on something else but will see below 2 different kinds of vector indices OK so there and a litany things which would transform this way for example if this were just ordinary space of some sort talking about the velocity of a particle on the surface but opposes was surfaced when talking about a component of velocity along the surface the components of velocity along with surface would transform this way if you know the velocity components and 1 set of coordinates this would tell you what the velocity was Of course it's so this is an example of transformation properties now the 1st thing that banks from my sons 1 1 1 0 I send great discoveries was particularly about the summation signed here and people figure it out by looking at the equation and say Oh wait a minute this side has no P this side has peas this equation doesn't make sense for do it well it
really means is some of a pig whenever there's a repeated index like that that doesn't make sense from the left side of the equation appears on the right side of the equation it really stands for some additional but not all of were stature or orally more or they were the I have assumed that can just assume that of coordinates which are functionally related to each other I and buy you something some things we're assuming the degree of continuity you're assuming that functions a smooth enough that they could be differentiated but aside from that no we're not making any deeper assumptions about the nature of his court fight so this is a transformation proper friend down over here put up the quarterback Contra variant transformations contrarian vectors are defying the defined by the transformation properties here it is prime AMD or tried this people why M buddy VAX please V. BP XVIII on Trident coordinates This is in the prime of coordinates wiII is really the prime coordinate an X is on trial court as I said I've called Hawaii instead of ex- prime just to avoid a miserable location with Prime next indices appears as country that that grip now let's talk about call Marion vectors mother creature completely different the Grady and the gradient if you have a function of suppose we have a function think was a scalar function function our rapporteur doesn't have to be transformed just has some meaningful value at every point of space Court s then the gradient is also a kind of vector wears a gradient the gradient is descent things of components derivative with respect to x end of guests were S is some scalar function paranoia scale areas scalar was a thing which doesn't transformative value at every point of a well-defined value at every point it could be some like temperature it could be something like arm arm of the Higgs field where it is something which our love doesn't have components because as just a number every point the
gradient of a scalar is also objected to gradient sector but it doesn't transform the same way as a D Exs doesn't transformers a Contra vector would seem figure out how it does transform but says give in supposing we know that can we compute what the S leggy wife scores pedia coast to a computer work be derivative of With respect for the wife again it's Elementary Calculus problem Elementary calculus problem is that the change in when you change white Inmos bit keeping the other ones 6 to be all the other components vexed is just derivative vests with respect acts and is a derivative of X P with respect to why and where this was as rule called says some of the chain if you want something changes when you change while better than a 1st change X 1 a little bit and see how much X 1 changes when you change Tech change X to a little bit see how much X 2 changes when you change 1 of the center 3 and again this means some Over P. I'm going to write some overpay anymore if his repeated index on this side which just doesn't show up on the side it must mean that some of us about Einstein summation convention terror may life a lot easier for printers and 4 other publishers who didn't have put submissions I don't right so let's see how does it compare with this this is the price is now some kind
of variable is now in the prime court system let's call called the gradient here let's call a W W sub and on the left side of W sub and on the left-hand side but I'm putting index downstairs this index was upstairs samplings from downstairs why a mixed my sense of this index downstairs wise downstairs from downstairs prior why and is equal till now X key by white and this is the prime wife is the prime gradient derivative respect a wife a sequel to the X P thereby BY M times was a sin over here this is W P C under Prime Computer Brady sea
on trial gradient because differentiated with respect to the untried variable WP over here here we have D X P by Y him and here we have the climb gradient this is different instead this similar but trying contrary vector has components proportional to the aren't trying but would be y by X here we have by BY BY by dx dx wife of official which transforms this way transforms means that if you could calculated that 1 framed his value calculated the other frame a film which transforms this way is cold 8 coal variance vector of prepared in here object to transform this way now I can tell you right now that if uncomfortable with this and it doesn't ring a bell present there are a dozen CO good to you this it is what you need the state could really is completely central Cairo subject of General Committee where the index's go where the indices go and how they transform that's what relatively is all about it's all about the transformation properties of different kinds of objects of vector a special case of the cancer choose are things which are defined by the way that transform on the way that they transformed means the way that they change when you go from 1 set of coordinates toward is so that is what you have to get down if you your in general this transformation properties vectors and other objects but here are the tools Kohl variant and contrary to that death for a short end up yeah look a 7 down and this is labeled a different coordinates and NOTICES Norway has India which I've told you I've told you whether this is two-dimensional space three-dimensional space four-dimensional space could Ava the dimensionality of the space of course is the number of different components Pico run from 1 4 it run from 1 of 3 people from wanted to but the formulas are exactly the same in every dimensionality formulas for transformation properties OK so this is the notion of coal variant and country variant victory indices factors let's not talk about tensions now for more complicated kinds of object is obvious complicated but there we have the goal will be our 1st finish we can't there cold nothing yet nothing yet nothing yet with 2 different kinds of objects now later on we will find ways of taking any variant object or any country variant object and constructing from call them by objective prior one-to-one correspondence that way but for the moment the just different kinds of objects basically the gradient operations the this is characteristic of velocities this is characteristic of gradients the rate of change things so for the moment the difference is last days of the war I am so far but added that the moment which dealing with Ukrainians without dealing with a relativity young just coordinates in an ordinary space will come to that signed change when we yeah so is followed by a that EU should not be a reason for very long I'm here well if you know what the connection is yes you can but at the moment now will find a way to relate every corner contrary to you could think of basically of vector as having you co-vary their country representation of the Vicomte next let's talk about the tensions with more than 1 index now the pattern for potentials of more than 1 index is just talk imagine products but
suppose there are true contrary infected such start with 2 different country variant that that's Cohen and I suppose ready call it and you don't differ warns aspirin next to each other and take me component of end in component of you and let's just call this T M N I think just for the sake of argument let's suppose we're talking about three-dimensional space so that in the end the inland although from a 1 3 commonly components as VM have great comedy components as UN have sorry how many components as he and had not right notice and matters whether of you put an end or in in the different things V M and is not the same as we can and if you Normie Seoul where you put next matters but the think called TDMA CNN is a special case of a cancer of rank to rank to means has to industries like that in this case too would be 9 components for the mentions it would be 16 components two-dimensional before component of soulful OK how does this thing transformer for example Vienna Newland could be the components of Burmese vectors in on trying frame of reference Figure 1 PRI called CNN services key in prime frame of reference and it's composed out of the components of the individual sectors also prime frame of reference well since I not individual components transformed I can't tell you how he transformed so let's see if the figure out what he former I'm in years well 1st of all what is he trying to meet Prime is over here equal to 0 GUY by DD X P times EPA that's Vienna while trying versions be trying can know about you Prime you try and saying kind of formula partial of Wii end With respect to allege college X Q. What you'll kill does look at to pieces separately you BY and by 6 p times VP that's V em deep climate BY hidden by D X Q terms you kill that is you prime In just done this operation sane operations but I'm done among the separate components but now I can just take us apart but his VP over here but however is object or CPU killed Mrs. TPQ prime frame so I found out how to be transformed not surprising because it was just a combination of tour 2 vectors and basically each Hey next year forms with a with a D Y by DX type thing here a thing which transforms like this is called a coal variant cancer of rank too the each index there's an operation for the ending next there's operations BY and BXP fully and indexes a D Y N by X Q and You multiplied by PPQ this is common this is contrary misses a visit cancer with 2 contrary industries and indices appear and and I'm not ah PQ comes at Q use are in and then L L and they had no but comes before P. Norris a repeat Q are and what are missing missing deal Y L body D X all are this would be the transformation property of a rank 3 contrary cancer what kind of things are there which of these kind of cancers many many
things but a particular just products of vectors parts of components of vectors will see that this metric thing he is cancer but a cancer with co-vary overseas so obsessed how things of call variant indices transformed here His away a single call variant index transforms go back then take a product of 2 of them again 2 different coal variants cancers Y W Prime times w Prime if With variant called means that the downed downstairs here how does this subject transform we not be interviewed not W W letter not accorded busy but components is a W primed and is equal to the X P IDY currents WP Z trial in saying game D X Q by a D Y a N kind Z Q Chrysler here I've discovered now a new transformation property of thing with tool called variant indices to downstairs and let's call it he's an antidote to different objects diffusing Kiefer cancer cancer with 2 lower it transforms this way but but but but now visitors transformation properties of thing with 2 call them or Leavitt you try to guess what transformation might be of his 1 opera in 1 lower Rametta use tonight are but let's above finished work think I think I'll just do 1 more simple thing here before we finish the question is how would the metric tensor transforms and this is easy supposing we want the same interval except expressed in terms of the white would all we have to do is write Dad GDP rock the squares of GM end of X Suplee led what I have here but was and in terms of what partial of X M With respect to why I have he said thailand's DYP no Ungarie retired I think we do next time where appoint however is when we worked out the transformation properties GE will find out that it's of cancer with 2 core areas received and been comes a question given that this is the transformation property of duty can wheel can we not wanting they coordinate transformation which will turn GM and into the open as the mathematics question is a very hard mathematics question in general but will will find the coup will find the conditions for more please
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Metadaten

Formale Metadaten

Titel General Relativity | Lecture 1
Serientitel Lecture Collection | General Relativity
Teil 1
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/15032
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract September 24, 2012) Leonard Susskind gives a broad introduction to general relativity, touching upon the equivalence principle. This series is the fourth installment of a six-quarter series that explore the foundations of modern physics. In this quarter, Leonard Susskind focuses on Einstein's General Theory of Relativity.

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