 So this course is on Einstein's general theory of relativity, which remains almost 100 years after it was invented, the best classical theory of relativity we have. So the point of this course is many-fold. And the most basic level when we want to understand it is the theory of gravity. We want to understand the course of gravity. The force of gravity, we know something about it. Newton's laws of gravitational motion in an approximate sense. A lot of you are very familiar with the Newtonian theory of gravity. Newtonian theory of gravity is not very correct. It was a beautiful theory and has a large rate of validity. But it is not completely correct as it has now been seen in many ways from experiment. And its lack of correctness is also clear. Also clear given Einstein's special theory of gravity, if we believe the relativity principle is correct, that what really makes the Newtonian theory of gravity sort of already weakens. Because for instance, the law of universal attraction, which tells you that the force between two objects is inversely proportional to the square of the distance between the two objects, you could ask the square of the distance in which frame. Or the square of the distance, you know, measured along what line of side of the thing. Which is a more or less important question. Now then you could also ask, is it true or is it plausible? Okay, so this is getting too crowded. Yeah, what do we do about this? Yeah, there are a couple of seats in front. So please, two people at least come in. I have the suspicion that this is the first class crowd. So we'll try to avoid moving the class to a bigger room. We'll do it if three classes down the road we still have a cultural space. Okay, great. Okay, so maybe actually there are very practical things that we should do before we start. So why don't you guys write your name and email address in the sheet. Also, to Umesh, who as a classical organic speaker knows that you do. Well, it continues to be the different discourse. Okay, now we start an email list to Umesh. Sorry, any practical issues we should discuss over here. By the way, are the lecture timings okay? Okay, this is not a Thursday or the afternoon. Now there's one small problem. This room may or may not be available Thursday afternoon. There's some confusion regarding that. If it's not available, we'll send you guys an email telling you where Thursday is. Okay, fine. So let me just continue. So the new journey theory for everything, which was approximately great, sounds at the very least ambiguous given the formulation of general special relativity. Okay, for instance as we said, the other question is distance in which frame. Another question is, doesn't sound really reasonable that you've got a force which is sort of like electromagnetism. But it has no analog of, there is no analog of forces that are affected by the motion of the masses. None of this sounds like it's likely to be able to be made consistent with the principle of special relativity. Okay, so Einstein was led to search for a better theory of gravitation. A theory of gravitation that would be consistent with the big principles of general special relativity. For obvious reasons. You know special relativity is correct. You've got experimental evidence for that. It's a nice dynamical framework. You know gravity is correct. Okay, you've got experimental evidence for that. You've got experimental evidence for that. Okay, gravity exists. You need a theory that makes the two consistent. And Einstein said about trying to find such a theory. And in doing so, Einstein realized that he had to modify, or maybe more accurately, in large the theoretical framework of special relativity. And come up with an entirely new theory of gravitation. You know not something that takes different laws and tricks of it. But come up with an entirely new theory of gravitation. A theory of gravitation that is qualitatively different from Newton's theory in the sense that it has phenomena that are qualitatively new phenomena that are not present in Newton's theorem. For instance, in the theory of electromagnetic netism, which governs the interaction between two charges. There's a force that governs the interaction between two charges. But there are also excitations carried by the force being the waves in the electromagnetic waves. In Newton's theory of gravity, that's nothing like that. Force between two objects. The correct theory of gravity that we will study in this course has waves, impures waves just like electromagnetic waves, waves of the force in the carrier of gravitation, which are called gravitation. And this is a consequence of Einstein's theory. So Einstein's theory of gravity modifies Newton's theory in some ways, but also gives you qualitatively new phenomena. So the first motivation for studying general relativity is to find the correct theory of gravity. Okay, but suppose you are condensed, suppose you are a statistician who feels he is unlikely to encounter strong gravitational effects. You know, maybe you've got, say, a biophysicist. I don't know, a physicist is not particularly likely to encounter strong gravitational effects. And if I think, well, that then is a modification of the laws of gravity in slightly strange circumstances. That's not a great relevance to me. So for me, why should I study Einstein's theory of gravity? Okay, I personally feel that there's good motivation for everybody to study Einstein's theory of gravity. And let me tell you why I think so. You know, I think Einstein's theory is a paradigm for good physical theories. You know, it's a theory that, as I mentioned, was invented in order to resolve a contradiction between two different elements of known structures of physics. And just by the desire to resolve this contradiction, Einstein through 10 years of intense and frustrating labor was eventually led to a new framework that completely overturned many of the conceptual underpinnings of earlier classic theories of physics that predicts a wide host of new phenomena. Okay? And that has a great degree of theoretical elegance and beauty. So theory with no parameters or no new parameters predicts a wide range of new phenomena, has a large amount of context with beautiful mathematics. It's a really beautiful theory. And was invented purely with the desire, you know, from starting with the conviction that there must be a way of overcoming the apparent contradiction between two theoretical notions, both of which are known to be grounded in reality. I want to emphasize here that almost, as far as I'm aware, almost uniquely in the history of, you know, physics of this high level, the invention of general relativity was guided by experimentally and very legitimately. You know, Einstein, of course, knew that there was gravity. He knew that special relativity was there. But he wasn't trying to fit the data of a particular experiment or a class of experiments. He was trying to find consistency of theoretical structures. And in doing that, he came upon a new theory of tremendous beauty, but also enormous predictive power that then has been successfully tested by zillions of experiments. So it's in the fact, in the way that it's a theory that sort of, that was allowed, that allowed itself to be reduced by very little information, just by calling it deductive logic. And then allowed the description of a wide variety of physics phenomena including, you know, evolution of the universe. We can see some of this. Because it's really, in some sense, a paradigm for great physics, for great theory in physics. At least for some of us. It's like a great intellectual victory for humanity to be able to do the same thing. Okay? And for physicists of my international history first, it's the inspiration to hope that other great contradictions in physics, meaning the apparent mismatch between quantum mechanics and quantum physics, could also lead to beautiful new results that have a large number of experimental predictions, but are guided largely by structural theoretical considerations. Okay? So, I feel that, so, I said all this because I think that the study of general relativity is important, both because it's a beautiful theory of nature that describes a wide phenomenon, nature. But also as a paradigmatic, as the best, the example of perhaps what I think is the best theory of physics. That's the most beautiful theory of physics. Okay? Just as an example of, you know, if you're going to try in your career to make a good theory of physics, hope this up as an example. Okay, if you can do it up, you know, if you can back out, it's nice to be done. Okay? Okay. Great. So, so now that we've had this introductory, a introductory spale, let's start the course. Okay? So, I'm going to start the course. Because by reminding you of something we discussed with this classical mechanics, many of you took the classical mechanics course last year, and we started a discussion of classical mechanics by saying that, look, maybe assume that motion, classical motion, the motion of classical particles was discovered by Lagrangian. And we were also going to assume that this Lagrangian was a function only of positions in the laws of physics. So that the equation of motion would be second order in time. Okay? And the first thing we tried to do in that course was to try to understand what Lagrangian governed free motion of particle. Okay? And in order to try to do that, we assumed the principle of Galilean in that case. That is, that two observers that were related to each other by constant velocities saw the same laws of the equation of motion. We used the same Lagrangian to describe the equation of motion. That describes a particular free motion. Okay? So, so then we said, look, we were interested in Lagrangian to describe the free motion of a particle. Let's say that the coordinates of that particle were x dot, sorry, were x, x dot and x dot are these. Okay? Since we wanted translational invariance for free motion, the Lagrangian had to be a function of x dot in general Lagrangian function of x and x dot. But translational invariance requires this particular Lagrangian to be a function of x dot. Then we wanted rotational invariance. So, this thing was going to be a function of x dot dot x dot x dot square. So, then we said, well, the most general Lagrangian would be some function of x dot times x dot. Translational invariance, rotational invariance, all satisfied with such a Lagrangian. But this leaves about a freedom, a freedom of all functions, half of a real Lagrangian. That wasn't good enough. Then we applied that Galilean principle. We wanted this Lagrangian to be invariant up to possible total derivative of those under Galilean boosts. That is, x dot was x dot responsible. And you remember under x dot goes to x dot plus a constant. Let's make this constant infinity. This Lagrangian transforms up. Well, clearly, the Lagrangian, let's look at the change in the action. So, delta s, which is equal to integral delta l is equal to f prime of x dot t to the square times the change in x dot square, which is x dot times dot absolute. This is probably my question. This guy, in general, is nothing nice. Prime of x dot is a constant. Then this is the total derivative of the dot and the x dot. And that, we argue, is more or less the only situation in which it would be, for arbitrary motion, would be a total derivative. So that tells us that f prime of x dot squared had to be constant. We call it the constant. The constant was related to what we call the last particle. So that tells us that f was a linear function of x dot. The constant part was irrelevant. So we conclude it from there that the Lagrangian free motion was m x dot squared by 2. For this course, we're interested in more, we're going to be interested in motion of things in a relativistic city. So the first question I'm going to ask is, why is the motion of a free particle in the blue? What is the Lagrangian for the motion of a free particle within the relativistic city? I'm going to address, I remind you of this because I'm going to address the question in an entire reason. So as before, we want our Lagrangian to be a function only of first derivatives of position. First derivatives with respect to time of position. So now you might think, well look, what we're interested in is some sort of action which is L, D, D, which is invariant of the Lorentz transformations and the Petronian first derivatives in time of positions. We see this as a very basic way to think of Lagrangian to be able to start doing Lagrangian transformations because in this Lagrangian time appears in a special way to start with just FDT. And that makes life messy because you want to be able to rotate space and stuff. So instead of thinking this way, let's try this. Suppose I've got a little particle and it's moving around and what I'm going to do is parameterize its motion by a false coordinate, by a fake coordinate that we call it. And instead of thinking of the motion as being described by xi as a function of x0, we will think of the motion as being given by xmu as a function of lambda. And no xmu is a function of lambda. And it has reasonable properties of 1 to 1 in the right way. x0 is a function of lambda. So 1 to 1 functions. Then I can solve for lambda as a function of x0 and then plug in that solution to obtain xi as a function of x0. So knowing this is equivalent to knowing the motion of the particle space. What I'm saying is that there's a curl of in space nine and I'm parameterizing that curl by some sort of fake coordinate on the path. Now, why if I knew this, I could find xi as a function of x0? This is nothing good. I cannot find this function from knowing xi as a function of x0 because this function is a function of some fake parameter about which I said nothing. You choose one parameter to parameterize the path. I choose another single-value function of your parameter that equally where parameterizes the path. Introducing this, this function has a large degree of ambiguity. It takes the same physical curl of space time and parameterizes it in different ways. Different parameter choices of parameter come out to different parameterization by this fake parameter. They will keep us in mind. Nonetheless, what we're going to do is this. Let's imagine that we that we have such a description of this particle and what we'll do is instead of right, we take this Lagrangian area dT and what we will then do is to write this as integral l dT by d lambda. d lambda. That was it. This lambda has nothing to do with the range transformations. The requirement of Lorentz and Bayes is this action here. It's just the requirement of this object so let me call this guy l tilde which was l dT by d lambda. This quantity I'm going to try to make Lorentz and Bayes. You know, like before I'm going to try to make some function of dx mu this time by d lambda. So what function of dx mu by d lambda? No x is allowed because of translation. Once again, we've got only dx mu by d lambda. And what function of dx mu by d lambda is Lorentz and Bayes? But that's clear. It's e times mu nu dx mu by d lambda. dx mu by d lambda. This is nice Lorentz and Bayes object. And it's the only Lorentz and Bayes object that you can build out of dx mu by d lambda. Yes. So it's possible that we can find a function of shift by a total derivative of function of what? I mean this l dT by d lambda. Yes. It might change by a total derivative of what? But it's not possible. Yeah. We will allow that possibility. You know, at the moment I'm saying that whatever l is, oh, you mean that it's not invariable. It's fine. But changes under a total derivative. Good. Good. Good. I need to, I need to, it's a good question. It's a good question. So what I'm assuming here is that the Lagrangian itself is invariant under a total derivative. I suppose I made that same assumption that the Lagrangian was invariant under rotations. Yeah. Okay. So I should do better if I want to address this. Well, let me think about it. Let's make the assumption, however, that we're looking for an object okay, excellent. So we've got this object here. But, so now, we want some function of this. But remember that this function should actually be dependent on this fake parameter we do. It shouldn't have the property that the action is invariant if you take lambda as a function of that. But the action has held till the lambda, like the e lambda, correct? The only way that this thing is invariant invariant is if the homogeneity of this function in lambda is minus one. Because then the change of variables will cancel between the change of the measure of the lambda and the change of the rule in the lambda. Okay, minus one in lambda which means minus, sorry, which means half in this object. It will be like to the Lagrangian L is equal to minus square root of minus dx x mu by d lambda dx nu by d lambda eta mu nu d lambda. With precisely the square root the change of variables the change of variables that we, if you change lambda to some function of lambda the change of variables from here will cancel when you get to the measure, the middle change. Because you get two factors in the denominator of this square. Is this correct? So just by the demand of the lens in variables is your basic general. So it will be narratively to this action apart from Hatshaw's question. Allow me to put it up to there to check if you can think about it. Okay, sorry. It's okay. I put a minus sign inside the bracket. Okay, good. Because I should have said this he might, in this course I think the problem is lambda we should use the opposite dimension. Okay, what would that be? Maybe I should just switch to lambda. Yeah, okay. So minus sign or not depends on what kind of dimension it is. Okay, so let me once and for all bite the bullet and adopt lambda which is stupid invention. Okay. Being up one minus one minus one. Lambda which is classical theory of feelings will be the main text for the first part of the course. So the course I should have said the course is planned in the following fashion. We're going to have about a couple of months on basic general relativity which is roughly pages 250 to the end of this book. Okay. And I'm hoping we will have time three weeks on black hole physics and three weeks of cosmology. Okay, so the idea of course is to cover basic general relativity and then talk about black hole physics which is very exciting cosmology is the two main current research applications. So now of course we do this then we could get a class because you know a particle moves in along some sort of time like trajectory. You don't want to get square root of the interval. Okay. With this collection of eta this guy is just positive and then we when we still need the minus. We need this minus time to be how do I identify this parameter here identify this parameter in the following way. Okay. See, this collection here has been written in terms of the lambda. I could choose lambda to be anything I want. As one example I could choose lambda to be time. Okay. So in the case lambda as equal to it's not sorry. High Sunday High Sunday can be written in 4 or 7 so if I'm working on it I could make something like that. Okay. Okay. Okay. So suppose I choose a lambda to be x0 then my assumption becomes minus m. Let's write that really explicitly. This is dt x0 to 60 and this thing here becomes 1 dx0 by d lambda squared minus xi dot squared. Of course I'm working on speed of light. Okay. Now I take this guy and I expand it in small velocity. This becomes l is equal to minus m plus m by 2x i dot squared which is the non-religious non-emotional regression from other sorry. I've got a name over all of it. Okay. Of course. If there's a name over all of it it will appear everywhere. It's the same thing. There are two ways. Here. Minus m integral this is the one I take the square root. Square root of 1 minus a is 1 minus a squared by 2. Okay. This now is a constant. It's totaled out. It has nothing to do with the motion. This now gives you the non-religious regression. So I identify the fact that we have to make contact with the non-religious regression. So that would be the same mass that we refer to as mass in non-religious regression. Now I have the Lagrangian for non-religious regression but it's one sentence. My son has been picked up for one make sure. It's a Lagrangian that describes the motion of a free particle in relativistic physics. I'm really motivated by the introduction of our study of general relativity by the following consideration. Look. What is special relativity? It's a beautiful framework for business. There was always something a little around. A little unsatisfying about special relativity. And that was the arbitrary specification of which frame is an inertial frame. We start in special relativity by saying there exists these special frames. Inertial frames. Once you know one frame is inertial you know others are those that are related to a bad boost. But these frames are somehow different from all other frames. They're different from accelerated frames. The laws of physics take one form and the other forms in other frames. This seems somehow a lot. Who decides which is the inertial frame? This question will get partly but not completely cleared up in the study of general relativity as we'll see. But is this discomfort that might lead you to ask look, suppose I wasn't using coordinate X mu to be appropriate to an inertial frame. So suppose I was using arbitrary coordinates to describe the motion of a tree particle in special relativity. Then what would my Lagrangian do? Lagrangian mechanics is very well suited to answering such questions because it's very easy to make changes to the variables within the Lagrangian features. Substitution. Let's make the substitution as an arbitrary function. Of some new coordinates we can go back to. Now what happens to this Lagrangian? Well, clearly you get the following. Lagrangian goes to minus m integral d alpha beta d y alpha by d y d y by d lambda d lambda with g alpha beta is equal to is equal to w x mu by d y alpha using the chain rule of differentiations. But, under the condition that x mu is differentiated from y alpha with all other y's. And I get this by using the chain rule of differentiations. Substituting d x mu by d lambda by the variables of the derivatives d y by d lambda by using the chain rule. This is scary. Is the Lagrangian now for relativistic motion in an arbitrary coordinate system? This is already useful, by the way. You know, suppose we were testing special method and you wanted to understand how the work in a coordinate system adapted to your problem. Some rotating coordinates. This is the Lagrangian in that coordinate system. You can derive the equations of motion straight away in those coordinates without having to ever rely on inertia coordinates. Now, looking at this Lagrangian you might think the following. You might think, wow, this is quite a nice Lagrangian. And you might wonder, no, this Lagrangian depends on this object g alpha beta. It's a symmetric object. This object is clearly symmetric because one multiplies the symmetric so any anti-symmetric basically. But it's also symmetric to definition. Because this is symmetric. This is symmetric. And you might wonder, look why doing all these coordinates by choosing while, far, in an arbitrary way can I generate all possible symmetric g alpha pages? So this is, this object at any given point in space time is a matrix. It's a four cross four matrix. It's indices run over four where it goes here and again. And while term is a constant it doesn't depend on where you are in space time. This object depends on where you are in space time. Because these derivatives depend on where you are. Because ds alpha by dy mu is a function of y. So this object here is what's called a symmetric two tensor. It's a tensor field. You might wonder, can I generate all possible tensor fields by doing coordinate transformations on eta? So this is a better way to state what what what the laws of physics are. Because it doesn't make we can't let's first discuss this question. So can somebody tell me can one make every g alpha beta by starting from eta and doing this coordinate transformation? No, you can. That's correct. Tell me my flat. You guys aren't using knowledge that we're not supposed to have. It's true. It's because it's flat. But if something wasn't if you're looking at this problem for the first time you've never heard of flatness of curvature. It's still obvious to me. What? You're still using knowledge that we don't have. I'm sorry. It's fine to use knowledge that we don't have yet. But there's something very simple to say. What you're asking is can you make an arbitrary set of 10 functions? Just count. How many functions are there in g alpha? We're counting how many components there are in the matrix. You've got four components that have 10 components. Can somebody tell me what formula gives 10 out of four components? What is the difference? How many components are there in g alpha? That's one. That's one. That's two. And if we split four into five by two, we can take functions in four dimensions. We're trying to generate arbitrary sets of 10 functions. Starting from a node tensor. And multiplying by objects built out of four functions. We're just demanding our degrees. We can't make arbitrary sets of 10 functions using arbitrary four arbitrary functions. So this construction doesn't give you an arbitrary metric. It doesn't give you an arbitrary g alpha meter function. Why? It gives you a four meter set of the 10 barometer set of possibilities. This is inaccurately set because we actually took no functions. It gives you a four into infinity barometer set out of 10 into infinity. Possibilities. Mathematics says you four into infinity is the same as 10 into infinity. It's just not true. Fine. So fine. So this Lagrangian that we've written here cannot, I mean it's not that there's no coordinate system. You cannot, you know, if I give you an arbitrary g alpha meter, you will never be able to find a coordinate system in which the original Lagrangian, this guy, takes this form. But having the Lagrangian of this form you might wonder, could it be that there is a role in physics? Could it be that there is a role in physics for this Lagrangian? Four arbitrary functions of g alpha. Although from special relativity, although from special relativity, we only get this four values. Could it be that somehow that's because special relativity it's not got everything right? Could it be that there's a role in physics for particle motions government Lagrangian for arbitrary functions of physics? Special relativity may not it may not have got everything right but it's really got a lot of things right. Okay? So if you look at small deviations from those metrics that we've formed from it you'll get a small change to the equations of motion of free motion from special relativity. That small change to the equation of motion means a small is what we think of as a small force. So if there is a role in physics for motions of this particle in an arbitrary g alpha beta this must be associated with what in special relativistic language we thought of as a force. What can we say about the force in question? That at the moment the crazy situation is the body. Suppose the power of the forces of nature can be described by just allowing g alpha beta to be an arbitrary function not necessarily a function transformed by coordinate transforms from this sounds crazy it sounds like there's no motivation so far but let's take this thought and infer it's supposed to be true what can we say about what can we say about this force? Now look why you cannot get arbitrary functions g alpha beta? Why you cannot get arbitrary functions g alpha beta by doing these coordinate changes? Something we will prove once we start doing this doing all the transformations in the letter is that at any given point in space you can always take an arbitrary g alpha beta and make a coordinate transform to bring it to each other or vice versa and at one point you can make any g alpha beta you want actually you can make any g alpha beta and any derivative first derivative this is something we will prove once we start the first statement is also basically a statement that I'll prove it later the point is that you can use a coordinate transformation to diagonalize this and then this coordinate transformation isn't quite a similarity transformation it's m m transpose so you can use that to set the eigenvalues to whatever you want okay this is actually very similar to something we did in diagonalizing quadratical branches but we will prove this but just not to distract let me just assert and even point in space we can produce any g alpha beta we want from e now by an appropriate coordinate transform and actually it's true that even the first derivative can be set by by an appropriate coordinate now what does that mean I could say that g cannot be set by a coordinate transform had we been were we able to set g to eta everywhere we would be back to free motion so we would be setting the force to z okay so the fact that we cannot set g to 0 everywhere is something that we need to go beyond local to z locally you can use g to eta at a point including first derivative that's pretty straight argument setting g to eta by coordinate transform you know coordinate transform is moving to a frame of reference and you choose coordinates in which you are setting but I say accelerated compared to the important one so what you are saying is that this force whatever it is is a force that locally can always be set to 0 in a little region and be set to 0 by moving to the right frame so it is a force that in a little region of space time looks like no force in the right frame or in other words in an arbitrary frame looks like the sooner force from and being in a non-inertial frame look around and ask now they sound like a lot of words so you can ask is there a consequence of these the consequence of these that's the major consequence the major consequence is that these pseudo forces by being frame change things don't care about which particle you are looking at if you have 7 different particles they all were initially in identical motion the force of them these frame change pseudo forces is always the same independent of any other properties of these particles in the whole movement if it's true that there is a role in Asia for making G alpha beta arbitrary that is if it's true that there is some force that is described by making G alpha beta arbitrary these force must obey the principle of equivalence the principle of equivalence which states that in a little local region you can choose coordinates to get rid of forces not and these coordinates not particle by particle you can choose coordinates for all particles in that region to get rid of all forces if it's true that this G alpha beta has been a certain force then this force obeys this beautiful principle that it is possible locally to move through a coordinate system in which it looks like there are no forces this is Einstein's famous principle equivalence and now you can look around nature and ask how many forces that obey this principle in particular it implies that two particles with the identity the initial conditions follow identical trajectories irrespective of all particle properties there is one force of nature that has this property it's gravitation two particles, Galileo's famous thing you take a vacuum you take a feather, you take the straw you drop them together and you follow it at the same time if it doesn't have this property if you take an electron and a positron it becomes different if you move to a coordinate system and reach the forces, gauge it away it's taken away from the electron the positron will be moving in the opposite direction yes, the electromagnetic density doesn't have no other fundamental force nor that we know of but gravitation does ok, so if our crazy idea is correct if our crazy idea is correct we can get some sort of force of nature by making this g alpha beta an arbitrary function then the only candidate that could be getting as force of nature is the force of gravity of course I would be saying all this at the beginning of the semester long course and it was actually true it's true the correct formulation of the theory of gravity is in terms is completely different from the way the correct formulation as far as we know is today of theory of gravity in terms of completely different nucleus the basic objects of this theory are the trick this quantity that we are going to call the trick g alpha beta g alpha beta in special relativity in special coordinates is this object in arbitrary coordinates we can get from this object by coordinate transformations but in general relativity it is an absolutely free function of coordinates we have 10 absolutely free functions of our people no constraint that they be legible that they be quadratransformable to this form and when you have a matrix it cannot be quadratransformed to eta in such situations particles moving around the experience with special relativistic forms of force and that force will be the forms of gravity so this is this is what we are going to try to do we are going to make a theory of gravity we are going to make a theory of gravity in which there are two components the first component is of our theory is how how so the first claim is that there are new fields in our theory there are new fields there are new dynamical fields just like in electromagnetism there are new fields aiming there are new dynamical fields that have not been properly appreciate for Maxwell in gravity there are new fields now just like in electromagnetism moving charges around what the charges are doing what other things are doing determines what a new will do we need an equivalent rule for that for the rule for what is the rules of dynamics for the new field I will try to find this rule with much of our effort in the next few weeks we need a Lagrange for the metric once we have this Lagrange once we have a particular metric test particles moving around in it experience an effective force and the rule for that effect the force we have already interned at least as far as test particles go this is a Lagrange so what we are going to do now in the next couple of lectures firstly I am going to take now I am going to take I am going to take a couple of lecture interview on talking about some the mathematics of coordinates that we will need through the rest the rest of this course and then we will talk about the properties of particle motions coming from such a dimension this we can do without even knowing the rules for what the evolution of the matrix is and then about four lectures from now we will either start getting to the real part of generating which is the theory of the dynamics of geometry okay so now I am going to pause the questions and after you after the over we start talking about mathematics of coordinates okay now just why are we interested in mathematics of coordinate transpose we are going to emphasize this there are two basic ideas that go into general rather than here a basic physical method one is this principle of equivalence that we are trying to write down a force which can locally be transformed away by going to an appropriate components on all particles such as these but you have to the idea that forces are obtained just by taking this Lagrangian the second thing the thing that is motivated always you know we started this discussion by saying look this business about working with coordinates and inertial frames a little artificial let's free ourselves from this okay so the second basic principle that you know is practical like more operational than the hard work you are doing is the principle of coordinate that is we are going to try to write down a Lagrangian for our system that is correct in every coordinate system we are not going to single out some coordinates as special because they are inertial or some coordinates are special because they we like them to some other reason our Lagrangian we will demand is corrected in every coordinate system now particular coordinates may be well suited to studying particular problems of physics in the universe but that's another matter the form of the Lagrangian will not discriminate will not specify this is the Lagrangian in this coordinate system and it's more complicated in some other coordinates as far as the basic equations of dynamics go we will be completely democratic in coordinate systems that is the second basic idea underlying this whole way of thinking it's the principle of coordinate in mathematics now because we want this is what we want to do we want to try to write down some sort of Lagrangian it's completely coordinate in there it's clear that before we start doing physics we need to take a mathematical interview we need to understand how to make coordinate transformations that's what we are going to be doing starting now after questions no no this is an independent question our question is about this it's the idea of what we are trying to do the motivation for what we are trying to do is that clear ok correct now let us so we come back to this Lagrangian in the couple of lectures in the couple of lectures and study its dynamics in some decay but before we do that just because the objects that will enter in the study of the dynamics of this Lagrangian there is lots to say about these objects we are going to do some mathematics and the mathematics we are going to do is a mathematics of coordinate transformations ok but the one thing I want to say about this Lagrangian before we do it is that look we wrote down the Lagrangian this way and you see these two factors square root d lambda are really cancelled so another way of writing this Lagrangian is you take the square root g alpha d tau dx alpha dx d y d tau without talking about lambda at all just of course satisfying because we know that changes the variable the principle that determines what we need ok now this quantity in general general relativity has a name special relativity has a name it's the textbook side it's the invariant distance in special relativity ok and this quantity this little quantity or more physically maybe it is the proper time the square root of this is the proper time experienced by an observer who is moving along this trajectory particular trajectory ok now this g alpha beta this g alpha beta here has determining distances on a space life ok so this quantity here the square root of this quantity it's just a it's just a name but the important thing is that this quantity itself is coordinate invariant by definition ok when we start with the eta mu nu dx mu dx and then we change both things of the quantity right let me remind you we started with eta mu nu dx mu dx mu let's call this ds there and then what we did is this is eta mu nu del x mu by del y alpha del x mu by del y beta dx alpha d y beta is equal to whatever it was because g alpha g alpha beta d y alpha d y beta ok so if I put y here and use g here g appropriate to y I guess the same quantity has put in x's here and put in the g appropriate to x so this quantity is an invariant coordinate invariant quantity ok which we will refer to just as a name as the distance square doesn't mean that many senses will be analogous to a distance physics and this quantity is a problematic the thing that the job is distance yes is there any constriction of beta to be something like having the ideal determinant or anything yes because it can be g alpha beta yes yes g alpha beta you can always diagonalize g alpha and the diagonal object has to have one negative 3 possibilities when you bring it to diagonal form it should one value should be negative one of the diagonal elements should be negative and the other three in particular the terms now we will study this that's a good question ok right so now now we did the study of the mathematics of quantum transforms mathematics or some part of the mathematics of metaphors we are interested in this space time because a mathematician might call space time a manifold ok so we don't give much about that and we put some coordinates of space on our manifold let's say I put on the text your friend doesn't like x mu coordinates he decides to use y mu coordinates such that x mu is some known function of y y ok and what we want to do is to relate the description of all quantities of interest that you use to the description of all quantities of interest that your friend that is our program you want to understand how to transform between things standing in the x coordinate and things standing in the x coordinate so let's start really soon first, suppose you've got a function on your so you've got a function that you call phi of x now you're off what is phi of x? it's a number associated to any point of your math given the position x ok now your friend that also wants to describe the same function but the same point in space time the same physical point in space time he wants to associate the same time now is he going to use the same function of his variables y? yes or no no say because I mean let's say that my function let's say the simplicity is one coordinate so my function phi is equal to x and by x is equal to y squared clearly your friend's function is y squared function of one variable, it's a different function it's a parabola in our history ok so the first question that we want is what is what is the function suppose your friend wants to associate the function of phi tilde of y he wants to write what phi tilde of y is what is the function of his y coordinate what is that? the answer of course phi tilde of y is equal to phi of x of y so they're at the same space time point you get the same value this looks like a trivial, it's not at all trivial you know just as functions these are different and one must keep this in mind so we're going to be doing fields here so we're going to be studying functions of position and coordinate transformations change these functions in a way so as to keep the value of this field same at the same space time point but as functions of coordinates they're different so this is the first thing that we let me have ok, now let's look at something a little more complicated suppose you were interested in the derivative of it and look at the field del mu phi and I'll be in that my expression let me say because we're working partial derivative keeping all other axis constant this thing gives us four new functions of space time now you might ask your friend might say this guy thinks he's smart and I can do the same I can make del by del y by mu by phi tilde of y these are my functions that's because your functions fine the question we're going to ask is knowing one how to begin to do so for starters we found the formula what is the equation of formula for these derivatives so you can say look I know that phi tilde of y was equal to phi of x of y del by del y of y of x of y what is that by the chain rule is equal to del by del y mu of x mu times del by del x mu of phi of x and then once we've got this we've taken the derivative we substitute x goes to x of y because we want the answer to be a function of y but in taking the derivative we adopt this as a function of x now let's begin this kind of thing let me call this object a mu of x let me call this object a tilde mu this quantity here is a tilde mu of y so what we've derived a tilde mu of y is equal to del x mu by del y mu into a mu of a change of in this dictionary element there are two things there's the fact that the argument has changed that we see here that's the same as in the scalar function but there's something new there were indices in these functions and they get mixed around linear derivatives of scalar function structure and the coordinate changes because we'll have use for this later on I'm going to actually just work this out for a special case the special case that the coordinate changes in infinitesimal change suppose x mu was equal to y mu plus epsilon mu of y and we're going to work with the first order okay then what does this formula become in that case where do we get two terms first we get the term from here and in this part okay so a tilde mu of y minus a mu of y is equal to let me write this down and you see the grid the second term is epsilon dot del into a mu y and the second term is plus del epsilon mu by del of y mu by mu into a mu so in epsilon this is how the function a tilde of y is different from the function a of y a mistake that is very common to make when the first is studying these things because textbooks sometimes say this is so bad is to keep this term but to forget this term just as difference is in function this came from expanding the dependence in the coordinate you must remember that that's how it looks okay as difference is in functions there's the term that exists even for scalars scalars are invariant but as functions they're not the same that's very important to keep okay and these two terms are the same for time okay excellent so that's how derivatives of scalars are transformed now we're going to make a definition any object that transforms in this way according to the law of transformation of derivatives object that transforms in that manner is referred to as a contravariant vector and a tensor with one index down go ahead go ahead so is a tilde and a they are evaluating the same spacetime point of you've got a function a of a they evaluate the same physical same a tilde no no we expanded the difference now in this difference we're not evaluating the same spacetime we're not calculating we are looking at the difference between the two functions okay two functions the same values of one yes the same value of argument that's the reason because excellent okay so let us continue so an index down okay right so any object that transforms like a derivative it's called the covariance factor okay and the prototype of this kind of object is the inference has been displaced yes look at that my man I've got a point I've got a point and I take a at each point I draw a little vector by which I mean drop a little arrow on my hand very little inference the inference has no object I choose differently a different point of space who can stop me from doing this so I do it so I've got a vector field of these little arrows as a function of where I am spaced coordinates are very competitive as you draw these arrows I'll draw my own arrows so he draws a dy mu um of why I'll is how do we go from your description to your friend's description of isobus for these little arrows that we draw on all of these is this clear is this dx clear clear dx mu is just the difference in coordinates between the tip of the arrow and the base of the arrow where this arrow is different and this shows it completely arbitrary as an arbitrary function of space what we will later on call the vector field something with an index see indices up are called vectors this is a vector field in space what I'm trying to understand is what is the law of transformations for vector fields so now that's very easy it's very easy for all of the values so what we know is that that x is equal to x y to imply very big up so we get dx mu is equal to dx mu del x mu by del y alpha so this was at position x I'm going to make everything as a function of y of y so in order to to get what your friend wants what your friend should do we have to invert this matrix now the inverse of this matrix is the other one I will ask somebody to prove it to me but I'll just use this for a moment so dx mu of y is equal to del y mu by del x alpha dx alpha but I will use the fact that del x by del y the matrix del x by del y is the inverse of the matrix del y by del x okay so let's not face that the fact that these two matrices are the inverses of each other you probably know but anyway it's obvious from the following observations if you write dx mu is equal to del x mu by del y alpha d y alpha so you can also write d y alpha is equal to del y alpha by del x theta then plug into it so take this relation plug in here we have to get d y alpha as d y alpha but that's problem these two matrices that's the codler state sorry sorry about that now any object that transforms according to this transformation rule is not a vector field and what is the transformation rule let's write it down the transformation rule is a del del mu of y your friends vector field is d y mu by del x um a alpha of x something about this transformation rule there is less natural than the transformation rule for um than the transformation rule for um for the object to go in next step and that thing is this look we are so far saying that you had a function x as a function of y in the rule for objects done everything that we used is just a function x as a function this derivative of computing of x as a function of y and this defect is a sphere of x as a function here we need y as a function of x to take this derivative this is the x as a function of y but this is both y as a function of x now as long as you take a coordinate change which is inverted and you know through much of our discussion however there are some situations in which you want to go you know in which and we will be dealing with this soon in which you are interested in some sort of submanifold of all space time and let's say that y go at this level positions on that submanifold this formula still makes sense but this one doesn't what we concluded is that given a one form an object with one index leads to a natural formula for a one form on submanifolds an object with one index down and submanifolds but the same thing is not true for vector fields this is a throwaway comment which is confusing you forget about it we come back to it when we need it the reason is of course that the vector field may have a component that's not in your submanifold then you have to determine a rule for projection and that involves more information than what we want but one form on the submanifolds this dissension has fancy game in mathematics called a pullback it's a nice thing pullback okay good fine so far we understood so that was the throwaway comment don't worry about it okay so far what we've done is determined the rules for transformation of vector indices and other things downstairs indices go very well downstairs indices okay now I'm going to make a definition as we will see in the study of general relativity we will need vector fields we will need one form of fields but we will need not many different kinds of fields arbitrary whatever we call arbitrary change so an object A with indices A and Mu1 to MuN Mu1 to MuN has functioned away it's called the tensor field and I have some name let's just call it with N upper and N lower indices with N upper and N lower indices provided that the rule the rule for coordinate transformation of this tensor field is given by and it's right the rule for coordinate transformation is given by until the alpha Y let's call it alpha 1 N beta 1 beta M is equal to del Y alpha 1 del del x Mu1 del Y alpha N del x MuN and then del Y Mu1 and let's call it beta 1 and Mu1 del x MuM del Y beta M and now with an episode multiplying A of Mu1 MuN MuN Mu1 MuN of N of Y so let's go over mouse field this field avoids this transformation the rule we call the field a tensor field with N upper and N lower indices as you can see what we've done basically is give one factor for transformation matrix for every upper index and one factor for this transformation matrix for every lower index so in particular we had just one lower index we recovered one more transformation if we had just one upper index we recovered the vector transformation but this is the natural generalization so now it's the same in particular if you've got a product to do vector as an N server with two upper indices can you see this could someone how would you prove this to me if I would ask you to what would you do to study these steps it's totally obvious I agree but just here you say so what you would do is take A till now the first vector field right in terms of A of the first vector field A till now the second vector field right in terms of each of these transformations has one factor of this del y by del x so in general you get two factors in del y by del x smaller than the product so this rule is generated so as to ensure that product of tensor fields are tensor fields with the indices just added you take a product of a vector field you will get a tensor who is indexed with one index up next up so you'll call it to this definition is the converse rule means all tensor fields can be represented as the products of vector signers okay as sums of products of tensor fields but not a product let me ask you this question about matrices okay any matrix can be thought of as an outer product I mean it's like a robot a product of a robot and I call them as a matrix can every matrix be thought of clearly not right because only matrices can be thought of but anything can be thought of as a sum of such things good other questions excellent so in the case of we are dealing with as he said that the determinants is minus one so general coordinate transformations are not but not minus one we are doing a general coordinate transformation yes and the Jacobian should be positive yes the Jacobian should be positive you see the determinant transforms the two factors all we require is that the coordinate transformation is one two with a positive negative negative should not have that fine our question is lost okay we are almost at the end of our time give me one more I should just clear okay the last thing I should say is that with these rules okay let me say two things with these rules the object how we define tensor now there are interesting objects which are called invariant tensor invariant tensor has the property that it makes the same functional form in every coordinate system the tensor in one coordinate system can be transformed into another coordinate system nothing special but as we see for instance in the infinitesimal transformation law for the vector field for the one guy down the stairs the one form field the nature of the field functions in view of exchange we can do this for the transformation now a field some sort of tensor field such that you get the same functions of x and you have functions of y such an object is called an invariant tensor that is that you do the coordinate transformation property and the answer as functions is the same okay now I emphasize this as functions with the simple example we have trivial dependence let me give you an example suppose I gave you an object this object a field a tensor field t are the value of which is this it's a chronicle delta and it's 1 only when the derivative is 3 let us find the rule for the new the transformed chronicle okay so this was the tensor field in your coordinate system friends very thinks this is a cool tensor he wants to know what he should do in order to get your chronicle okay so he says well that's easy I have this transformation though so let me apply it okay now finally this transformation you can always trigger it because the x of y of this is triggered because it's another function of x so you just have to worry about this part so delta of like this alpha beta is equal to del y alpha by del x mu delta mu del x nu by del y beta nu so the thing that he will get by transforming your tensor field to his coordinate system is this okay so now delta mu just identify the indices so that's because these indices should be identified inside but that means multiplication like a matrix but we've already seen that these two are inverses of each other you make it fun so if your friend takes your tensor field if your friend takes your tensor field and apply the transformation rule which was this apply the transformation rule he gets a tensor field which has the same rule in his coordinate system so the tensor field in his coordinate system is the same but given the same formula as function of coordinates as was true in your case such a thing is called an invariant tensor okay this is a nice and very effective thing the tensor field of the coordinate delta but that is very interesting it's also clear from the fact that a mu is equal to delta mu mu a mu this equation is true in every coordinate system this transform factor this transform factor this is a better transform like an answer given something I could not say the second thing and this is really important is that if you take an upper index and a lower index and some set those equal identify them at some overall values so I take let's say I've got and I've got a vector a mu is a vector field theta mu is one form field theta mu is a vector field now I probably in the following object sum over mu a mu what is this transform answer transforms in the scalar like a scalar proof a delta mu is equal to del by del y mu theta a theta of x of y delta mu is equal to del y mu by del x of beta beta of x of y now I'm going to find these two put them in here so what I've got is 70 since these mu's are equal in here and summed over everything and just sum over this index here then I use the fact that these two are inverses of each other so I get a factor of delta theta beta so we could feel that a theta mu is equal to sum over mu is equal to sum over theta theta of x of y theta of y