1. Introduction and the geometric viewpoint on physics.

the key textbooks for this class is general relativity it's now almost twenty years old I would say I think it's listed on the website as required I would actually call it sort of semi required it is where I will tend to post most of the readings to the course it is a it's it's a really good complete textbook for a one semester course which is what we have okay and I will not be going through the entire thing you can't in a one semester course and I will from time to time there will be a

few topics that I cannot go as in into as deeply as I would like if we had two semesters maybe I'll be able to do a little bit more and so for those who are interested I will suggest readings in this my personal favorite supplements to this is the textbook by Bernhard Shutts a first course in general relativity okay so these are all things where the MIT bookstore I'm not sure how much they carry they're all available through Amazon you can definitely find these kind of things if you get shuts it's textbook definitely get the

second edition the first edition contains errors okay there is actually a very important geometric object that we are going to introduce in a couple of weeks that his textbook has a really clever derivation of them or seeing the thinking wow that's really clever and I was like the reason you would do it so simply is because it's wrong it's corrected in the second edition another one is gravitation by Misner thorne and wheeler this is sort of a bit of personal history for me I was thorns graduate student and I used this textbook when I learn

this subject originally I frankly do not recommend this textbook to somebody who is learning the subject for the first time okay it's a good place for a reference for certain things it's available in the reading room nobody can get a copy and pick it up so first of all picking it up kind of a it's good exercise okay it's actually like a you know it's a huge book it gravitates so it's a great read friends and has a couple of good sections than it for new students but I will indicate from time to time especially

some of the stuff we do in the second half of the class I my lectures are kind of inspired by mr. thorn and dealer wheeler known as MTW um and for those of you who have sort of a more mathematically minded approach to things general relativity by walled in the course syllabus I call this the UB book I mean this really is their the self-contained book that really you know pins down the subject very very well it's quite terse and formal but also very very clear if you are a mathematically minded thinker you will find

this to be a really good textbook to refer to there is one particular derivation then I'm gonna do in about a month and a half that I essentially take from Wald's textbook because it's just beautiful other things in it you know to me it's a little too terse for my own personal taste but others may find it to be pretty good okay so two other quick things so there are eleven problem sets and your grade is determined entirely on these problem sets okay the syllabus gives the schedule for when they will be posted and when

their to be handed in 11 does not divide evenly into a hundred so what we do is we have ten that are worth nine percent of your grade and the eleventh is worth ten percent of your grade okay is that eleventh one really one percent longer you know it's this again I'm taking the viewpoint that you are graduate students or at least you're playing one for the next hour and a half and if you're going to sweat the difference of one percentage point on that thing come on it's about learning the material don't worry about

those little details that much okay so that is where all all of your assessment is going to come from I once the very first time I lectured this course I had a final exam and it just turned to be a complete waste of my time and the students time you really cannot write effect you write problems that are so easy that you can do them in your sleep or they're so difficult you can't do them in the time period of an exam so we're just gonna stick with problem set and that's fine okay let's move

into then the way the course is going to be structured so my presentation of this material the first half of the course basically it up to Spring Break is essentially the mathematical foundations of general relativity okay there are several choices that need to be made when you're doing this this often in some universities they have multiple semester sequences but I'm gonna cover in its first half of our semester in some places goes for a full semester and what this means there's a couple of things that I just cannot cover in quite as much depth those

will be things where for those you were interested in it have that kind of a mathematical thinking of things happy to push you to additional readings we can dive in and look at in a bit more depth my goal is to give you just enough stuff that we can do the most important applications of this subject and I'm an astrophysicist it'll be blunt most of my brilliant arresting applications tend to be things that have to do with things like cosmology black holes dense stars and things like that and so I want to get enough formalism

together that we can get to that part of things and so the goal of this is that by the week right before spring break what we will do is derive I put that in quotes you'll see why a little bit later the Einstein field equations that govern gravity in general relativity okay so there are several things that we could do that are not strictly necessary to get there and just because of time limitations that's I'm going to choose to sort of elide a few of these these topics the second half will then be applications okay

we will use everything we drive in the first half to see how general relativity gives us a relativistic theory of gravity we will begin applying it we'll see how Newton's law is encoded in these field equations we'll see how we go beyond Newton's law get some of the classic tests of general relativity and then start looking at solving it for more interesting systems looking at the evolution in the universe as a whole looking at the behavior of black holes looking at gravitational waves constructing the space-time of neutron stars things like that so it's a fun

it's a fun semester it sort of works well to fit these two things in like this and for those of you who are interested in taking things more deeply there's a lot of room to grow after this and it does look like so people who particularly would like to go a little bit more detail on some of the mass if you've looked at the syllabus I have one of my absolute favorite quotes from a course evaluation and I is put on there where a student and like 2007 or so wrote you know the course was

fine as it was but professor Hughes as an astrophysicist tended to focus on really mundane topics like cosmology and black holes and if you think those are mundane topics what can I say guilty as charged but flows you who do want to take a different approach these sorts of things we will probably also Nate lecturing this course in the future between someone from the ctp who works more in quantum gravity and things related to that lecturing and that'll be meta Engelhardt in spring of 2021 all right so we're now ready to start talking about after

doing all this sort of prep we can actually talk about some of the foundations of the theory so before I dive in are there any questions all right so what we're going to begin doing for the first couple of weeks I'm not really first of week the first couple of lectures is we're going to begin by discussing special relativity but we are going to do special relativity using mathematical language that emphasizes the geometric nature of this form of relativity what this does is it allows us to introduce the form basically the the formalism the notation

all the different tools that are important for when things get more complicated okay when we do when we apply a lot of these tools to special relativity like we will be doing the first three or so lectures of this course you know it's kind of like swatting a mosquito with a sledgehammer you really don't need that much mathematical structure to discuss special relativity but you're gonna be grateful for that sledgehammer when we start talking about strong field orbits of rotating black holes right and so the whole idea of this is to develop the framework in

terms of a physical system where it's simple to understand what is going on okay so these are sort of the way to introduce the mathematical tools in a place where the physics is straightforward and then kind of carry forward from there I will caution that as a consequence of this many students find these first three lectures to be on the dull side so it's just some stuff that we kind of have to get out there and then as we generalize to more interesting mathematical objects more interesting physical settings it gets more interesting all right so

let's dive in so this setting for everything that we will be doing is a geometric concept known as space-time so we give you a precise mathematical definition of space-time so a spacetime is a manifold of events that is endowed with a metric okay it's a wonderful mathematical definition and I've written it in a way that requires me to carefully define three additional terms so the quant the concepts that I've underlined here I've not to find them yet precisely exactly what I mean by them so let's go over to the sideboard talk about what exactly these

are so a manifold this is if you are a mathematician you might twitch a little bit about the way that I am going to define this and I will point you to a reading that does it a little bit more precisely but for the purposes of our class a manifold is essentially just a set or a collection of points with well understood connectedness properties okay so it's essentially what I mean by that is I'm going to talk about manifolds of space and time okay and so I haven't defined an event yet but I'm about to

but I'm gonna say that you know there's a bunch of events that happened at this place and at this time a bunch of events that happen at this place and at this time and the manifold of space-time gives me some notion of how I connect the events over here to the events over here there really it's a manifold is a topological concept it's all about how one connects one region to another one okay so you know if you're working on a manifold that lives on the surface of a doughnut you have a particular topology associated

that if it lives on the surface of a sphere e of a different topology associated with it if you would like to see more careful and more rigorous discussion of this this is one of the places where Carol is very good so go into Carol and at least in the Edition that I have it slim pages 54 to 62 it goes into you know he introduces a bit of additional mathematical machinery discuss things with a little more rigor than that I'm doing here rigorous at all so it significantly more rigor than I'm doing here those

of you who are into that that should be something that you enjoy so in a vent this is when and where something happens okay could be anything from our point of view the event essentially is going to be the fundamental notion of a coordinate in space-time okay we will actually in many cases that's bad that's bad word choice I should say coordinates are actually sort of labels that we attach to events okay we are gonna be free to adjust those labels but the underlying geometric idea that the event is here that's independent of the coordinates

that we choose so we will label these things with coordinates but the event itself exists independent of these labels okay so you know just to give an example there might be one event which is I punch myself in the head okay and so I'm very egotistical say this event happened at time zero X Y & Z equals zero because I define this corner of my skull as the origin the coordinate system and I always think whatever's happening right now is the origin of time okay those you out in the room are also egocentric and you

would say whatever I'm gonna call that you know let's say you were at Y of three meters and I'm gonna put the floor as the origin of my z-axis so Z of 1.7 meters or whatever and you will come with your own independent labeling of these things okay so you're all familiar with the idea that we can just change coordinate systems I'm gonna harp on this a bit though because there's gonna be a really important distinction we make between geometrical objects that live in this manifold of space-time and how we represent them using labels that

might be attached to coordinate systems okay and I'm going to come back to this when we start talking about some additional geometric objects in just a couple minutes so the last object that I have introduced into here is one that we will begin talking about in a lot more detail in the next lecture but let me put it into here right away and so that is the metric metric comes from a root meaning to measure and what this is is this something that gives me a notion of distance between events in the manifold okay for

the physics to work this has to exist right but it's worth knowing that the the idea of a manifold is in some way more primitive than this okay you can have a manifold without any notion of a metric attached to it and if that's the case you know people like to joke that if you don't know what a the difference is between a metric with a manifold and without a manner to me a manifold with a metric and without a metric you know feel free to drink coffee out of a doughnut okay because topologically those

are the exact same thing but they're geometry which is encoded in the metric which tells me how the different points on that manifold are arranged are rather different so what this basically does is it's going to give me a mathematical object that enforces R really conveys the idea that different events in this manifold have a particular distance between them so without this a manifold has no notion of distance encoded in it so the two things together really make this this concept come come to life you can see a lot more like I said so talked

about you can get more information about many these concepts from the the readings that in Carroll Wald's textbook also goes into quite a lot of detail about this stuff okay so this is the venue this is the setting in which we are going to talk about things and just cutting forward roughly two and a half months worth of lectures what we're going to find is that part of I'm Stein's genius is that it turns out that this notion of the metric ends up encoding gravity okay and so that's kind of where we're gonna end up

going with things the idea that the mathematical structure that tells me how far apart two events are is intimately connected to the properties of gravity it's pretty cool and it is something that you know physics is an experimental science all of our measurements are consistent with it so that's cool all right so everything I've said so far nothing but mass okay nothing but definitions so let's start working with a particular form of physics so we are going to begin as I said with special relativity this is the simplest theory of space-time that is compatible with

physics as we know it not fully compatible but does a pretty good job and we'll see that it turns out to correspond to general relativity when there is no gravity ok so to set this up we need to have some kind of a way of labeling our events and so I'm gonna introduce kind of a conceptual can almost think of it as like scaffolding which we're going to use to sort of build a lot of our concepts around and in this one you know I I mean that in a kind of an abstract sense that's

gonna be sort of like the edifice that we use to help us build a building that's going to be the mathematics of general relativity but this one really is kind of like a scaffolding because what I want us to introduce here is a notion of what is called an inertial reference frame okay so I'm going to sketch this quickly and I'm going to post to the course website a chapter from an early draft textbook all right Roger Blandford in kit form so when I talk about the inertial reference frame I want you to sort of

visualize in your head a lattice of clocks clocks and measuring rods that allows us to label in other words to assign coordinates any event that happens in space-time okay so I just sort of in your head imagine that there's this grid of little clocks and measuring and measuring rods and you know sceeto lands on your head in theirs it's right near a particular rod in a particular clock it bites you that's an event you slap it that's another event and the measuring rods and the events are what allows you to sort of keep track of

the ordering of those events and where they happen in this four dimensional manifold of space-time so I'm going to require this lattice to have a certain set of properties first I'm gonna say that this lattice moves freely through space-time what do I mean by moving freely I mean no forces act on it it does not rotate it is inertial every clock and every measuring rod has no inertia no force is acting on it at all okay you look at that you might think to yourself why don't you just make it at rest well I did

it's at rest its back to someone okay but we might have a different observer who's coming along who has no forces acting on her and she's moving relative to me at 3/4 of the speed of light okay it's not at rest with respect to that observer and that's actually kind of the key here okay so this inertial reference frame is at rest with respect to someone who feels no forces but knots to all observers okay I'm gonna require that my measuring rods are orthogonal to each other so they define an orthogonal coordinate system and I

am also going to require that the little markings on them that tell me where things happen are uniformly ticked okay in other words I'm going to just make sure is that the spacing between tick marks here is exactly the same as the space between tick marks here okay you may sort of think well that's you know a result or an idea worthy of the journal duh but you know it's important to specify this do you want to make sure that the standard you are using to define length is the same in this region of space-time

as it is over in this region of space-time when we start getting into general relativity we start to see if there can be concerns about this coming about so it's worth spelling it out and making it clear at the beginning I'm also going to require that my clocks tick uniformly okay we're gonna make this lattice that fills all of space time using the best thing that Swiss engineers can make for us we want to make sure that one second an interval of one second is the same here in this classroom as it is somewhere off

in the Andromeda galaxy okay we want to make sure that there is no evolution to the time standard when we do this finally I'm going to synchronize all these clocks with each other in the following way this is going to use the I'm Stein synchronization procedure this is the first place where a little bit of physics is actually beginning to finally enter our discussion okay so I'll comment that I'm gonna go through what this procedure is in just a second ah but in Easter Egg here whenever you see a name that has Einstein in it

your ears should perk up a little bit cuz it probably means there's something important okay quarters in relativity and that tends to be you know that the things that end up mattering even when they end up being really kind of easy to think we look at now and kind of see as fairly obvious it's important okay and we often attach Einstein's name to this so this is fine synchronization procedure this takes advantage of the fact and by the way we are not going to teach special relativity in this course eight 962 I assume you've already

studied special relativity and you're all experts in this and so I can freely borrow from it's important results this procedure takes advantage of the fact then the speed of light is the same to all observers no matter what inertial reference frame they might be in okay so the speed of light is a key invariant okay it connects you know because it's a speed connect space and time and because it is the same to all observers it defines a particular standard for relating space and time that is going to have important invariant meaning associated with it

okay so it's to remind you what this means so you don't have a laser pointer with me I might don't worry about it let's have a pretend laser pointer my chalk some laser pointer so you know I point my laser pointer at the wall and you all see it dashing across the room at 300,000 kilometers per second I then start jogging at half the speed of light as is my want and continue to point that you guys measure the light going across the room you still measure 300,000 kilometers per second I on the other hand

measure the light coming out of my laser pointer and I get 300,000 kilometers per second okay so just because my laser pointer is moving at half the speed of light according to you it doesn't mean the light that's coming out of it is boosted to a higher speed if you study special relativity you'll know that it's energy is boosted to a higher energy level but the speed of light is always just C so we're going to take advantage of that to come up with a way of synchronizing our clocks so the way it works is

this so let's look at a two-dimensional slice of my lattice here it's time and let's make this be the x-axis and so I will have let's say this is where o'clock one exists and this is where o'clock two exists let's go into the reference frame that is that rest with respect to this lattice we want to synchronize clock one with clock two so as time marches on these guys stand still so here's the path and space-time the world line traced out by clock one here's the clock path and space time the world line traced out

by clock - so let's say let's call this event let's say that this event happens at a time t1 e t1 is when o'clock one emits a pulse of light okay so this light will just follow a a little trajectory of it through space-time this goes out and strikes clock - at which point it is bounced back to clock one let's ignore this point for just a second so let's just say for the moment that this is then reflected back okay and then is received back at clock one at a time t1 t1 our clock

one receives the reflected pulse so the moment at which it bounces we'll call that T to be that is the moment at which the light bounces off of clock too and the way we synchronize our clocks is just by a requiring that this be equal to the average of the emission and the reception time totally trivial idea right all I'm saying is I'm gonna just require that in order for clock one and clock two to be synchronized to one another let's make sure that when I bounce light between any two pairs of clocks they are

set such that when the light bounces it's the midway point between the total light truck it's the halfway of the total light travel time that the light light moves along very very simple concept okay so there's nothing particularly deep here but notice I'm Stein's name is attached to this okay and I don't say that to be so I'm not being sarcastic I mean there really is it really points to the fact that we're using the fundamental one of the most fundamental results of special relativity in designing how these clocks work in this inertial reference frame

believe it or not this thing you know that there's this really simple concept it comes back to sort of bite us on the butt a little bit later in this course because when we throw gravity into the mix we're gonna learn that gravity impacts the way light travels through space time okay we're gonna get to some objects where gravity is so strong that light cannot escape them and we're gonna find that the way our perhaps most naive ways of labeling time in the space time of such objects kind of goes completely haywire fundamentally the reason

why time is going haywire when you have really strong gravity is because we use light as our tool for synchronizing all of our clocks and if the way that light moves in your space-time is affected the way you're gonna label time is gonna be affected so this is a very simple concept again I sort of emphasize these first couple lectures you're swatting a mosquito with a sledgehammer but we're setting up this edifice because this we'll come back and it's important to bear this in mind when things get a little more interesting later in the course

alright so let's start setting up some geometrical objects here oh pardon me let me do one other things really quickly before I start setting up some geometric objects so when I sketched this so called this things on a spacetime diagram here I should have talked a little bit about the units that I'm going to use to describe the ticking of my clocks and the spacing of tick marks on my measuring rods what we will generally do in this course is choose the basic unit of length to be the distance light travels in your basic unit

of time okay while you slightly Jabberwock in my apologies while you a parse that sentence what that's basically saying is suppose I set my clock so that they tick every second well if my clocks tick once per second then my basic unit of length will be the light second okay if you want to put this into more familiar units that's about 300,000 kilometers one of my personal favorites if the time unit is one nanosecond the length unit is of course one I will call it LNS light nanosecond students who are in 803 three with me

are not allowed to answer this question does anyone know what one light nanosecond is actually there is this is a little bit ridiculous but to within far greater than a percent accuracy it is one foot okay the English unit that comes on these you know asinine roar is that those of us educate in the United States learned and all of our European friends sneer at us about the speed of light is to incredible precision one foot per nanosecond to be fair let's make that a wiggly equal so what this means is that in the units

that I'm gonna be working with the speed of if I then wants to express the speed of light in these units so C is one light time unit per time unit which we are just going to call one okay so we will generally set the speed of light equal to one just bear in mind what this essentially means is that you can think of if you want to then convert to your favorite you know meters per second furlongs per fortnight whatever it is that you're most comfortable with C is effectively a conversion factor then okay

and so what this means is that when we do this all philosophies that we measure are going to be dimensionless okay really what we're doing is we're measuring them as fractions of the speed of light okay so now with the system of units defined let's talk about a geometric object so let's imagine that o is an observer in the inertial reference frame that I defined a few moments ago this is the mouthful to say it's even more of a mouthful to write so I'm going to typically abbreviate this IRF so-oh observes two events which i

will label p and q [Music] okay so let's say here in space-time so imagine I've got coordinate axes that have been drawn it's gonna be three dimensional for simplicity I'm not gonna actually write them out let's say I've got event P here and event Q over here now if we were just doing you know Euclidean geometry in 3-space you guys have all known that once you've got two events written down on a plane or in a three-dimensional space something like that you can define the displacement vector from one to the other we're gonna do the

same thing in space-time okay so let's call Delta X I'm gonna make a comment on notation in just a moment this is the displacement in space-time from P to Q okay and we're going to define the components of this displacement vector as seen by o so when I write equals with a dot on it that means the geometric object that I've written on the left hand side is given according to the specified observer by the following set of complex which I'm about to write down so this looks like some okay so a couple things do

I want to emphasize that I'm introducing here bits of notation that we're gonna use over and over again in this term notice I am using an over arrow to denote a vector in space-time okay different texts different professors use slightly different notations for this those of you who took a was three three its MIT with some of the Tal a he preferred to write a little under tilde when he wrote that for us working in four-dimensional space-time is going to be a paramount important to us and so we're gonna use this over arrow which you

probably have all seen for ordinary three-dimensional vectors for us it's gonna represent a four-dimensional vector now in truth we're not gonna use it all that much after the first couple weeks of the class our first couple of lectures even caging will bust it out but we will tend to use a more compact notation in which we say Delta X that displacement vector has the components Delta X mu where mu lies in either T X Y & Z or 0 1 2 3 okay when we set up a problem we need to sort of make a

mapping to what the numerical correspondence is between you know I need to tell you that mu equals 0 corresponds the time you equals 1 corresponds to X we'll switch to other coordinate systems and I'll have to be careful let's say almost always new equals 0 will be time but what the other three correspond to that depends on coordinate system there might be a radius might be different angles things like that ok so again just sort of being a little overly cautious and careful defining these I will note though that generally Greek in the seas in

most textbooks they tend to be used to label space-time indices and then there are times when you might want you to sort of imagine you've chosen a particular moment in time and you want to look at what space looks like at that time and so you might then go down to Latin indices to pick out spatial components at some moment in time we'll see that come up from time the time just want you to be aware there is this distinction and you know as you read other textbooks there are a few others that are used

always just check usually in some of the introductory chapters a textbook they will define these things very carefully Walled is an example of someone who actually does something a little bit different he tends to use lowercase Latin letters from the sort of the top of the alphabet to denote space-time indices and those from sort of ijk he uses them to denote Latin indices if you're old enough to get this this is often called by some of us who grew up in the Dark Ages and sometimes called the Fortran convention if you've ever programmed in Fortran

you know why that is if you didn't please don't bother learning it it's really not worth the brain cells it would take okay so we've got this geometric object that is viewed by observer oh let's now think about what this looks like from the viewpoint of a different a different observer a different inertial observer so let's say somebody comes dashing through the room here and observer o sees them running across the room at something like eighty-seven percent of the speed of light okay you know since as I have assumed you were all experts in special

relativity that they will measure intervals of time intervals of space differently than observer owed us so here's a vent p here's a vent Q here is Delta X this is all as measured by observer o bar okay something which I really want to strongly emphasize at this point is that this P this Q in this Delta X notice I haven't put bars on any of them okay I haven't put primes or anything like that it is the exact same P and Q and Delta X as this over here that is because P Q and Delta

X they are geometric objects whose meaning transcends the particular inertial reference frame used to define the coordinates at which P exists which Q exists and that then define the Delta X these geometric objects exist independent of the representation okay if I can use sort of a an intuitive example you know if I take and I hold careful the pose I knew with this let's say I stick my arm out you know I say that my my arm is pointing to the left right um you guys would look at this and say your arm is pointing

to the right cuz you're using a slightly different system of coordinates to to orient yourself in this room we're both right okay we have represented this geometric object my arm in different ways but me calling this point in left and you clawing at point to the right doesn't change the basic nature of my arm okay you know it doesn't mean that you know my blood cells changed you know because of something like this happening this has an independent existence in the same way this Delta X it is the displacement these two events you know this

might be mosquito lands on my head this might be me smacking it with my hand or flying in my head to me smacking with my hand those are events that exist independent of how we choose to represent them so the key thing is we preserve that notion of the geometric objects independent existence what does change is the representation that the two observers use so I'm gonna jump to this Greek index notation and so what I'm gonna say is that according to observer o'Barr they are going to represent this object by a collection of components that

are not the same as the components that are used by observer oh okay and to keep my notation concision let's put a little Oh underneath this arrow okay so this just sort of shorthand for Delta X is represented according to oh by those components Delta X's represent according to o'Barr by these components and again since I'm assuming y'all are experts in special relativity we already know how to relate the bard components to the unbarred complex they are related by a Lorentz transformation okay so what we would say is Delta X's 0 bar component or t

bar if you prefer just given by gamma I will define gamma in just a moment you can probably guess so I'm imagining an observer that it's just moving along the x axis or the coordinate 1 axis now what went on in that transformation I just wrote down is o'Barr moves with me along spatial axis one with speed V as seen by oh and of course gamma is 1 divided by a square root of 1 minus V squared remember speed of light is 1 ok we don't want to be writing this crap out every time we

have to sort of transform different representations so we're going to introduce more compact notation for this so we're going to say Delta X mu bar what I get when I sum over index nu from 0 to 3 of lambda mu bar nu Delta X nu so that's defining a matrix multiplication and you can read out the components of this lambda matrix from what I've got over there and even better yet Lamba new bar new Delta X so on this last line if you haven't seen this before I am using Einstein summation convention if I have

an index that appears in one geometric object in the downstairs position in an adjacent geometric object in the upstairs position and it's repeated so repeated indices in the upstairs and downstairs position are assumed to be summed over there full range from 0 to 3 we're going to talk about this a little bit more what's going on with this after I've introduced build up a little bit more of the mathematical structure in particular what is the distinction between sort of the upstairs and downstairs positions some of you might be saying well isn't one way of running

it what we call a covariant component one a contravariant component if you know those terms mazelTov they're not actually a really helpful and so I kind of deliberately like to use this more primitive wording of calling it just upstairs and downstairs because what we're going to find the goal of physics is to understand the universe in a way that allows us to connect this understanding to measurements and measurements don't care about contravariant versus covariant and all these things are essentially just ways of representing objects with our mathematics that sort of a go-between from some of

our physical ideas to what can eventually be measured okay so covariant contravariant whatever at the end of the day we're gonna see as we put these sort of things together it's how these terms connect to one another that matters okay the name is not that important I do want to make one little point about this as I move forward it is sort of worth noting that if I think of how I relate the displacement components according to my barred observer relative to those as measured by my unbarred observer I can think of this Lorentz transformation

matrix as what I get when I differentiate one representations coordinates with respect to the other representations coordinates okay kind of trivial in this case and we have a when we are doing special relativity there is a particular form the Lorentz transformation that we tend to use but I just want to highlight this because this relationship between two different representations of a reference frame is going to come up over and over and over again this is a more general form this idea that you're essentially taking the derivative you're looking at how one representation varies according to

the other representation so using that to think about how to move between one inertial reference frame to another it's going to be very important to us so this is a more general form let's see so I want to make one further point and then I will introduce sort of a more careful definition of what is meant by a a vector and I think that'll be a good place for us to stop so when I look at this particular form so let's look at the last thing I wrote there where I use the Einstein summation convention

this is just a chance for me to introduce a little bit more terminology and notation so when I wrote down the relationship Delta X mu bar was lambda mu bar nu Delta X nu how i labeled the index that I was summing over it's kind of irrelevant okay this is exactly the same as lambda nu bar Alpha Delta X alpha okay I can switch to something else if you have enough fonts available you can use smiley faces is your index whatever the key bit is that as long as you are summing over it it's kind

of irrelevant how you label it [Applause] when I have an index like that that is being summed over its going to sort of disappear at the end of my analysis it's only role is to kind of serve as a placeholder it allows you to keep things sort of lined up properly so that I can do a particular mathematical operation so in this equation nu or alpha is called a dummy index now when I'm doing it with something like this where I'm just relating one set of one index objects to another set of one the next

objects and it's kind of trivial to move these things around we're gonna make some much more complicated equations later in this class and we'll find in those cases that sometimes it's actually really useful to have the freedom to relay below our dummy in the C it allows us to sort of pick out patterns that might exist among things and see how to simplify a relationship in a really useful way on the other hand I do not have the freedom to change that mu bar that appears there right I have to have that being the same

on both sides of the equation okay because you know I I'm free to sort of mix around things where it's just gonna be summed over and kind of not play a role but in this my mu bar apart me for a second mu bar is not a dummy index and so I do not have that freedom okay we sometimes call this the free index as I write that down that seems like a bit of a strange name actually cuz it's really not free you're actually constrained it would it can be what can I say probably

or some history there that I don't know about alright so let's do our last concepts for the day so let's carefully define a space-time vector so a space-time vector is going to be any quartet of numbers those numbers we will call components which transforms between inertial reference frames like the displacement vector so if I represent some space-time vector a as some collection of numbers that has observed by O has components a 0 a 1 a 2 a 3 if a second inertial observer relates their components to these by if that describes the components for observer

o'Barr then it's a vector okay if you have taken any any mathematics they carefully defines vector spaces this should be familiar to you okay it's a very similar operation to what is done in a lot of other in a lot of other kinds of analysis the key to making this notion of a vector a sensible one is this transformation law okay so if you know I had a quartz head of numbers which is say the number of batteries in my pocket the number of times my dogs needs this morning how many toes I have on

my left foot and I'm sick of counting so let just say zero for the fourth component that does not transform between reference frames by a Lorentz transformation okay it's just a collection of random numbers okay so not any old collect quartz head of numbers will constitute the components of a vector it has to be things that have a physical meaning that you connect to what you measure by Lorentz transformation for it to be a good for it to be a good vector it also has to that a has to obey the various linearity laws that

define a vector space so if I have two vectors and I add them together then their sum is a vector if I have a vector and I multiply it by some scalar by the way it'd be a little bit careful when I say scalar here because you know you might think to yourself something like ah you know the mass of my shoe that's a scalar but be careful when you're talking about things in relativity whether the scale you're dealing with is actually a quantity that is Lorentz invariance okay with quantity like mass if you're talking

about rest mass we'll get into the distinction among these things a bit later okay you're good you know just yet be careful to pick out something that actually is the same according to all observers so when I say scaler this means same to all observers if this is the case then I can define D to be that scalar times the vector and it is also a vector question Oh stretching okay all right um I think actually I'm going to stop there there is one topic that yeah I just don't feel like I can get enough

of it for it to be useful right now so I think yes I'm gonna stop there for today when we pick it up on Thursday we're gonna sort of wrap up this discussion of vectors again I want to kind of emphasize I can already sort of see in some places you're getting sort of a thousand yards there there's no question this is we're being excessively careful with some of these conditions they are very straightforward there's nothing challenging here but there will be a payoff when we do get to times where the analyses the geometries we're

looking at get pretty messed up having this formal foundation very carefully laid will help us significantly so alright I'm gonna stop there for today and we will pick it up on Thursday