Cosmology | Lecture 1

this program is brought to you by Stanford University please visit us at stanford.edu all right I am NOT going to tell you the entire history of cosmology we're going to get straight to the point the point is the expanding universe and the geometry of the expanding universe well let's let's back up for a minute and discuss what it is where it is we want to go I want to teach you the basics of modern cosmology and the basics of modern cosmology consists of a handful of things not a lot but of course the things which went into the discoveries yes especially the observational discoveries were a lot more than I'm going to tell you but basically we're going to take you know go to the go to the end of the story to a large extent and say what is it that we know now um we're going to start with the basic Big Bang Theory we're not going to go back in time before there was a big bang theory we're going to start with a basic Big Bang Theory the three possible versions of it we're going to do it from two points of view and then match them and see how they relate the first is just thinking about geometry the general theory of relativity that we discussed last year now we're not going to have to know very much about the general theory of relativity just some very very simple things and we're going to talk about expanding space we're going to talk about the hubble law the hubble law of recession the relation between velocity and distance and so forth well then derive that from some simple principles well state or the simple principles of cosmology are or Big Bang cosmology and we'll work out some examples of expanding universes the geometry of them not the physics of them this is the geometry of them the physics is driven by energy and momentum and other dynamical things they get a handle on that I don't know if we'll do it this week but next week either this week or next week we'll move on to the Tonie --n f equals ma cosmology F equals MA cosmology takes us a long way and we're going to see some connections some connections between the geometry and the F equals MA version of cosmology Newton what Newton would have understood as cosmology then we're going to put them together into a single package called Friedman robertson-walker frw cosmology that will take us a couple of weeks from there we'll address some other questions what is inflation what is the inflationary theory of cosmology what is dark energy what does dark matter how are they different a cosmological constant which is dark energy the current accelerated expansion of the universe and then perhaps we'll move on to some more modern esoteric strucked subjects such as why do physicists think perhaps there's much more to the universe than can ever better than we can see but more than we ever can see and the idea of a multiverse and I'm not sure what else we'll fill in some details as we go along that there are particular things you want me to discuss I'll do it as usual but let's get started okay let's begin just with the idea of space and the simplest possible space one dimensional space for simplicity then it'll immediately add two more dimensions space not spacetime and it's just good old X axis the x-axis imagine it goes on indefinitely both in the left direction in the right direction and imagine embedded in it are some objects which are equally spaced they're embedded in the space so I'm going to give them a name now they're called galaxies okay one dimensional galaxies but they're just particles or objects which are embedded in this line okay now the first question is let's let's label a line with a coordinate and let's label a line with a coordinate which in this particular case just measures off this is x equals zero and let's use the presence of these galaxies to define a coordinate so that by definition the next galaxy over is x equals one obviously this is an enormous idealization where I have the galaxies perfectly uniformly spaced this is of course not the case in the real world incidentally for those who may wonder the world is three-dimensional not one dimensional so it's a further idealization or a further fiction x equals two this is worse than flatland this is line land here's x equals minus one whoops minus one is x equals minus 2 and roughly speaking these coordinates simply label which galaxy were talking about we could think of them as the name of the galaxy this galaxy is called x equals two this one is called x equals one that just labels which move with the galaxies if the galaxies are moving if they're spreading apart or thick for if they're coming together this coordinate just as attached to the galaxy all right now how far is it from let's say x equals zero to x equal one well at some distance it's not one unit this is these are just the names of these points these don't say anything about distance I'm going to assume they're equally spaced but what is the spacing I haven't told you what the spacing between them is in meters or kilometers or lightyears pick your favorite unit for the time being the units are unimportant but what's that well they are one galaxies apart if by galaxies we mean the distance between galaxies hmm well we could use the size of each galaxy but we could use and whether well I don't want to use the size of the galaxies because it's going to raise a question I don't know but let's say meters meter sticks we measure them off and I'm going to call the distance between x equals 0 and x equals 1 in meters or whatever definite units we use the reason I don't want to use the size of galaxies is because the size of galaxies changes with time they tend to contract so I don't want to get into that but just good old meter sticks good solid meter sticks that don't change with time and let's say the distance between x equals 0 and x equals 1 let's just call it a e hey that's that's it what's the difference what's the distance between x equals 0 and x equals to twice a because I've assumed that they're really equally spaced in particular the distance between any two points is just let's call it Delta X that's the difference in x times a a times the difference in X if it's x equals 2 minus x equals 0 and 2 a this is the distance between any two galaxies Delta X means the difference of X this could be x equals let's see 1 2 3 4 5 the distance between the galaxies called x equals 5 and the galaxy called x equals 1 that's Delta X equals four times a is the distance eight times four is the same as 4 times a what about the distance between X equals 3 and x equals minus one that's also four times a and so that's the notation that's the notational thing we're going to do distance is the coordinate difference times a and a has a name it's called the scale factor it's called the scale factor its name it's name is the scale factor okay now the galaxies may be moving think of this line imagine this line for the moment as an elastic thread an infinite elastic thread but about infinite is too big that for right now a very large elastic thread and there are two giants at the end and the two giants are having fun with the people in this in this universe and they're pulling maybe they also even allow it to go the other way they change the total length of the of the world so to speak just by pulling on this thread and stretching it contracting it doing whatever it is they decide to do mostly stretch it but for the time being what does that mean about AE that simply means that a depends on time the distance between two neighboring galaxies will depend on time exactly as it would think of rubberband cut the rubberband open though I don't want to be double and think about stretching and instead of galaxies think about the rubber molecules you stretch that increases the distance between neighboring molecules you contract it that decreases the distance and so in that way in general a is a function of time in this world with these giants who are pulling on the end of the stretch this is not string theory incidentally nothing to do a string theory not yet anyway I've heard that in Berkeley there's a restaurant called noodle theory that can anybody confirm that this is true noodle theory okay okay so this is this is the universe and the universe is such that first of all it is homogeneous homogeneous means equal spacing that's all all right equal spacing it's everywhere is the same so to speak every one of these galaxies looks at its neighbor and says you are a certain number of meters away galaxy over here sees the same thing as a galaxy over here and so forth that's the term homogeneous everywhere is the same equal spacing but the spacing itself imagine that it's time-dependent okay let's now calculate the rate at which the distance between two galaxies grows or shrinks the rate at which the distance changes with time what's calculated there's another word for the rate at which distance changes what what velocity right actually speed but the velocity LA right velocity so when I say I'm calculating the rate at which the the galaxy at x equals four moves or increases relative to x equals one I could say it another way I'm calculating the velocity of x equals four relative to x equals one but I don't just want to call it the velocity I want to call it the velocity of one point relative to another all right we know what the distance is here's the distance it's a times Delta X now we're taking two fixed galaxies two fixed galaxies this one over here and this one over here we're not going to change those galaxies and that means that Delta X is not going to change with time we've just fixed the two galaxies we want to talk about but what does change with time a alright so what is the rate at which D increases with time that's just DDOT and for those who haven't been new to this class dot always means time derivative derivative with respect to time D by DT that's what about means derivative with respect to time I'll use that notation and it means the rate at which the thing is changing with time the rate of which D is changing with time Delta X is not increasing order or decreasing so d dot is Delta X that's unchanged but a is changing with time a dot the time rate of change of a product like this if one of the factors is not changing is just that factor times the time derivative of the other one so this is the rate at which D changes with time let's call it D of Delta X the distance between two points separated by Delta X that's just this now I'm going to do oh and what do we call this we call this the velocity the relative velocity between the two points it's the velocity that the the astronomer at x equals one would see x equals five moving away from him or moving toward him it's also the velocity that x equals five would see of x one either moving away or coming closer it depends on whether the derivative is positive or negative whether they're growing larger or distil s where they're approaching or receding all right so let's write this and now I'm going to write a totally unnecessary step I'm going to divide this and multiply it by a physicists and mathematicians are always dividing and multiplying by the same thing and you think why bother well let's look at it here's a times Delta X what is that that's D okay so I can rewrite this as a dot over a that's the ratio of the rate of change of a to a itself times the distance now I've discovered something interesting the velocity of recession well I haven't discovered anything interesting yet but I will have discovered something interesting in a moment does a dot and a depend on which galaxies I'm talking about no a and a dot are the same for every galaxy a in particular a is just the distance between neighboring galaxies but it doesn't depend on which galaxy we're talking about so a is the same for everybody the scale factor it's a universal idea which everybody agrees about what the distance to his neighbor is all right it will changes with time but a dot over a does not depend on X it doesn't depend on which galaxy we're talking about and what this equation then says is the velocity between two points is proportional to something which does not depend on which galaxy we're talking about a dot over AE we're going to give it another name in a moment times the distance between them anybody know what the name of the kind of a dot over a is the Hubble constant the Hubble constant velocity is proportional to distance is what this says with a number in there a dot over a now is this necessarily constant if the universe is expanding and contracting or doing whatever the devil these giants have decided to do is a dot over a a constant no it changes with time but it doesn't change with position it's a constant in the sense that it's the same everywhere but it is not a constant in the sense that it does not depend on time so the Hubble constant the thing which people I prefer to call it the Hubble parameter calling a constant makes you think it's like the electric charge of an electron or something else like that that it really doesn't depend on anything no the Hubble constant is a function of time but it's not a function of which galaxy we're talking about right so what does it say it simply says the further apart two galaxies are by that I mean further apart in real distance the faster they will be moving apart I say apart but again I could be more talking about coming together and the factor of proportionality is the Hubble constant well among other things it says that if you look out if an observer over here looks out to more distant galaxies the further away he looks the faster you will be seeing them move if he looks at his closest neighbor they'll see him moving away from him perhaps at a certain rate if he moves at the second if he looks at the second nearest neighbor he will see him moving a little bit faster if he will see the third nearest neighbor he will see him moving yet faster the further away he looks the faster that will be moving to the right the same is true there's nothing in this which has focused attention on x equals 0 or x equals 1 it's the relative separation of two things which are entirely symmetric and it's the same everywhere so there's no notion of a center to this line all points are the same as every other point but every pair of points are increasing their distance with a velocity that's proportional to H times D that's the hubble law now there's nothing especially one-dimensional about this let's do it in more dimensions are any questions about meaning of this formula the Gauss's final part in the we're going back to the state line right that will play a very important role in cosmology right you know what you're saying is correct and we will come back to that for sure it's one of the most important things okay one other thing if these again Alice and Bob this is Alice's Galaxy this is Bob's galaxy Alice and Bob are sending each other messages with light waves or whatever the FASTA Bob moves off to the right the more Doppler shift again they're sending their messages with standard message senders made in the same factory of exactly the same kind so they're sending light if you like of a very very definite frequency in their own reference frame but by the time it gets from Alice's signal or the time it gets to Bob the Doppler shift will have reddened that light or yes reddened it if I'm always going assume from now on that everything is being stretched and not contracted at least for the moment so Bob will detect Alice's message being slowed down but Alice will also detect Bob's message being slowed down everything is entirely symmetric between them ah and it's simply the problem of two people moving apart from each other sending pia sending light rays the faster they move apart though more redshifted the signal will be so ah that's a fact about an expanding universe the further all you are away the faster you move and the faster you move the more Doppler shifted will be messages now the kind of messages that we really have in mind of course is messages which are sent at frequencies which are determined by basic underlying physics the underlying physics might be transition of atoms from one energy level to another energy level spectral lines spectral lines are good standard frequency or wavelength what we call them they have very definite wavelengths their own frame of reference and so for example if Bob is centre of Alice ascending Bob a particular spectral line from the sodium from some sodium their transition that will be sent in his frame of reference at the usual frequency of a sodium line but by the time it gets the Bob over here it will be reddened or lowered in frequency one way that Alice can measure the distance to Bob is by monitoring the frequency of the light or the spectral lines that come from Bob's galaxy the faster is moving away the are the more written the frequency will be so that gives you a tool now we're not going to spend very much time talking about observation but that gives you a tool or gives Alice a tool for measuring the velocity of recession of any galaxy that's velocity how about the distance to any galaxy are the various ways other ways of measuring the actual distance to a galaxy well let me give you a very simple what it would more or less what Hubble actually did Hubble estimated in his first round of detections his method for estimating the distance to a galaxy was very simple if the galaxy was bright and big in the sky it was close if it was small and dim it was far the dimness of a given size object well from the size in the sky and from the brightness it could estimate the distance for reasons unknown to anybody Hubble decided to plot on a plot the distance of a given galaxy that he could see in the sky again pretty much just from the size of it there are much more sophisticated ways of measuring distance than just by size and and brightness but that's clearly one way to measure the distance and on the vertical axis he measured the velocity of the galaxies well first of all the first thing he discovered is that all of the distant galaxies now distant to Hubble doesn't mean what we would call distant galaxies today Hubble when was Hubble when when did Hubble do these wonderful things 20 sometime ah not 1820 1920 he used galaxies yeah oh it's an interesting historical fact that up until let's see when was it I'm not exactly sure get it was Hubble probably 1920 or something in 1917 1916 I'm not sure when exactly the idea of a galaxy didn't exist uh all we knew about was one bunch of stars the Milky Way across the sky and people spoke of an island universe that we lived in an island of stars with nothing out further than that they did discover that there were these blotchy things out there but it was not recognized that those blotchy things were other Island universes so to speak it wasn't really Sun till sometime in the in the middle of the early 2020s century that that Hubble realized that what he was seeing when he saw these smudges in the sky were collections of stars that there were collections of huge numbers of stars which rivaled in size and number of stars the Milky Way itself so that was the first discovery the universe is filled with galaxies and then he as I said measured the distance of galaxies and the velocity of galaxies and he discovered first thing that surprised them the galaxies are moving away from us apart from one or two galaxies which are very close to us our closest neighbor in fact just happens to be moving toward us it's a more or less accidental fact that it's so close to us that it's sort of trapped by the by our galaxy's gravity and it happens to be moving closer toward us what's the name of the closest galaxy Andromeda right so the Andromeda is moving toward us that is a more or less an exception if you look out beyond the Andromeda galaxy two galaxies three four five six times further away or a hundred times further away they're moving away so that was the first discovery almost all the galaxies and in particular all of the distant galaxies were moving away from us furthermore they were moving away from us by a velocity which depended on their distance and when he plotted it he got a perfect exact straight line nonsense of course he didn't get it he got a mishmash which was all over the place aerobars but somehow he looked at this and whether he was intuitive the right answer though he was a master of statistics which I don't think he was or whatever it was he drew a straight line through it and he said look velocity is proportional to distance he actually didn't know why he had no idea the simple argument did not exist it's been explained in an even simpler way than this but I know this picture here is correct and really at the heart of the matter but it was explained by other people just saying look why are you surprised that the fastest horse goes the farthest in a race that the if you look at a horse race expect that the fastest one has gone the furthest right the fastest galaxy is most likely to be the one that's gotten further away from you but that the that of course is just a um a that it's true but this is the right way to say it this is the right way to say it velocity is proportional the Hubble constant times D now Hubble was not very good at the time measuring the distances to the galaxies this wasn't his own fault he just thought crude crude tools at the time and he measured the galaxies as being maybe 10 times closer I think than they actually were so his Hubble constant was off by a bit like a factor of 10 that had implications let me explain why um the bigger the Hubble that of course meant that the Hubble constant was too big too big too small but he measured which way did he measure it he measured the Hubble constant being too too big as I remember yeah he measured the Hubble constant being too big by a factor of 10 now that meant that his estimate for example of this galaxies moving away was 10 times too big not that the velocity velocity was correct but but the estimate of the Hubble constant itself was 10 times too big it took a while to correct it we're going to interpret we're going to discover another interpretation of the Hubble constant it's basically one over the age of the universe I will tell you why later not right now so he had Mis estimated the age of the universe by a factor of 10 what's that it's really hard to tell her and the most sophisticated tools came into play but he miss estimated things by a factor of 10 we corrected them later he corrected them in any case that's the Hubble law it's very very central I said we can also do the same thing with more dimensions let's take two dimensions instead of one dimension just for simplicity x and y and again make a grid of course it's nonsense to say that the galaxies are at the corners of the grid but but that's an extreme idealization of let's imagine galaxies at the corners of the grid then every galaxy is labeled with two names the first name and the last name x and y and x and y stick with the galaxy x and y of a galaxy doesn't change with time it just is just a label that labels the galaxy but the distance between neighbors is again called a in both directions ah that brings up an interesting question why is it do we know do we know that the separation between galaxies in this grid model we could ask how do we know that the separation between galaxies along the x axis is the same as the separation between galaxies along the y axis do we know what do we know about it well supposing the galaxies were not equally spaced or not not that they were they're equally spaced but they were more crowded in the x direction than they were in the y direction at the corners like that when we look down into when we look out into the sky we would discover that things are different along the y axis than they are along the x axis we look out along the x axis and see very crowded galaxies we look along the y axis and discover they were very very separated we don't we see the same distribution of galaxies pretty much in every direction in the sky there's a principle involved that it's not a principle it's an it's actually just unnamed for that fact it is called the isotropy of universe two basic names the homogeneity of the universe that means everywhere is the same now this is everywhere is the same at least on a scale larger than the distance between these galaxies looked at our bigger scale we would say every place is the same as every other place everybody has a couple of neighbors around them and that's all and every place is the same as every other place this here is also homogeneous this place is the same as this place or has the same properties as this place over here this one is the same as this one there's no sense in which this one over here is special no different than that one no different than that one but there is something different about this than this it has to do with angles in the sky rather than positions it's not isotropic it isn't the same in every direction it's the same in every place but it isn't the same in every direction as a directionality to this a directional bias when we look in the sky we don't see different things in different directions of course that's not entirely true I mean when I look up I see something different than when I look down or when I look on a scale of our galaxy if I look into the plane of the galaxy I see the Milky Way if I look out of the plane of the galaxy I see a much darker sky ah but on a scale bigger than a few hundred galaxies scale bigger than a few hundred galaxies the sky looks the same in every direction so the principle is called homogeneity and isotropy now how well do we really know that the universe is or how well did Hubble or the people at that time really know that the universe was homogeneous and the answer is not very well at all they looked they were able to look through telescopes only to a minut fraction of what we can see today so really there was no good reason to believe that the universe was homogeneous on scales much bigger than they could see nevertheless all cosmologists assumed it and it turned out to be essentially correct that the universe is homogeneous and isotropic same in every direction same in every place that makes life simple much simpler than if it was just randomly different in different places okay we'll worry yeah let's come back now to two dimensions and the spacing of galaxies along the x axis and the y axis the grid size is the same what's the difference what's the distance between two galaxies separated by distance Delta X along the x axis and Delta Y along the y axis if anybody want to shoot that it what the distance square root of this is Pythagorean theorem well first of all what's the distance along the x axis it's a times Delta X the distance along the y axis in other words when we took two galaxies one over here and one over here this is the distant what's that yeah a times the delta X Y a yeah yeah that's right but let's say let's write it down anyway let's go through the steps the distance along the x axis is a times Delta X the distance along the y axis is a times Delta Y and then using the Pythagorean theorem the distance between them is just d is equal to the square root of Delta x squared times a squared plus Delta Y squared times a squared and now we can factor the a squared out of the square root and again as Michael says it's just proportional to a it's proportional to a the distance and now instead of just simply Delta X it's the square root of Delta x squared plus Delta Y squared what would happen if there was a third dimension plus Delta Z squared that's the three dimensional Pythagoras theorem which I don't know if Pythagoras knew or if he didn't know okay that's the distance what's the velocity between these two points one over here and one out here somewhere is it's just the time derivative of this alright so the velocity is given by the same square root which I won't bother to write times the time derivative of a a dot that's the velocity of recession and now we just multiply it and divide it by a a over a same trick as before and recognize that a time's the square root is just the distance so we get exactly the same formula as before distance times Hubble constant the Hubble constant doesn't depend on direction it doesn't depend on where you are but it does depend on time in general it may it may depend on time velocity equals distance times Hubble constant and it doesn't depend on the dimensionality of space okay that that's the hubble law and as I said it's a extremely simple idea now let's talk about the geometry of space just one question Ferdinand IV and you may get to this later but this is so it's bothering about the surface of it so the velocity of the Hubble constant on the distance this means the distance actually is increasing so the velocity is increasing so doc bob announced we're both being accelerated within our model but either one of them seems that the velocity is not well you don't know whether the velocity is increasing with time what you can say is the velocity increases with distance all right but here's a possibility the velocity of all of these young whether the velocity of a given galaxy is increasing or decreasing relative to us depends on how a is changing with time for example let's take a simple possibility supposing a was simply proportional to time the distance between two neighboring galaxies is just increasing with time with some velocity some particular velocity let's let's just call T velocity one you a was proportional to T that's just each link here in this lattice of galaxies is increasing linearly with time okay what how does the velocity what is the velocity between boys did TT TT delay the distance between them is a and the velocity is just the ad T which is one well I don't want it to be one because one stands for the speed of light and let's let's it 1/2 1/4 1/4 T supposing a was increasing like 1/4 T that's perfectly good law then the velocity between the two neighbors here is just 1/4 and it stays 1/4 for all time it doesn't change but what's the velocity between this point and this point twice the velocity between the neighboring points so it's 1/2 the velocity increases with respect to looking at more and more distant neighbors but it may or may not increase with time the velocity with time the velocity between any two galaxies here's another example supposing a increase like T squared the distance between this galaxies in this galaxy is increasing like T squared what is the velocity between them 2 T the velocity is just 2 T in this case the velocity between these two galaxies is increasing with time all right in any case it increases with distance depending on the details of how a depends on time the galaxies may be accelerating away from each other or even decelerating for example supposing a let's take another law well let's take the law a goes like e to the minus T is that a good law no it's not a good law okay let's take logged yeah let's take log T good thank you that was good log T what is the what is the velocity between these two neighboring points the velocity between them is the derivative of log T which is one over T in this case the velocity is decreasing with time as time gets bigger the velocity is decreasing that would be saying that they're decelerating relative to each other so you can have acceleration you can have deceleration but always the Hubble law is correct as a function of distance okay that's the basic Hubble law and is the basic fact that that all else is built on the you know it used to be thought that the universe was decelerating the reason was very simple well it was thought that gravity was pulling the universe together if gravity was pulling the universe together think about it starting expanding gravity would slow it down just like gravity slows down this pen as it's going up and will eventually turn it around it was thought that the universe was expanding but that the pull of gravity was pulling things back together and therefore decelerating it that's what was thought to be the case we now know that that's not the case and the universe is expanding at an accelerated rate okay everything is accelerating away from us rather than moving with a uniform velocity okay other questions again one yaki how do you set the initial value today since it's changing well your pixel yeah that's a good question but you pick some particular time at any time the initial value of a is somewhat arbitrary it's not really the initial value of a it's at which particular time that we set a equal to some potato do we do we measure a it doesn't matter you'll always get the same answer and the reason you might worry about when the particular details how they affect the Hubble constant I think you're asking what you're asking about is supposing you called this distance to a or three a would you get a different answer for the Hubble just by name do you just gave it a different name instead of the scale factor being the distance between these two galaxies let's take it to be the distance out to one-third of the way to the next galaxy we could do that all right so we would have another Hubble constant sorry another scale factor which was bigger or smaller it doesn't matter than our original Hubble original scale factor but the ratio of a dot to a would be the same for example supposing I defined another scale factor let's call it B B is three times a just another definition for the scale factor ah so that it's three times bigger than the original definition of it all right what about B dot over B you might get worried you might think you get a different answer for the Hubble constant but you don't you get three a dot over three a 3s cancel so what matters really is not the scale factor itself but how it's changing that's the important thing how it changes not its overall magnitude but we'll come back to that it's really a question of units the scale factor would be different numerically if we measured in meters or if we measured in kilometers and so forth and it would be different if we chose a different time at which we know it wouldn't be different than now what I said is right okay let's talk about geometry a little bit the geometry of space-time but first the geometry of space the geometry of space and space-time is determined by a metric the metric in exactly the same sense as we talked about last quarter I'll try to be self-contained for those who aren't here last quarter let's start with a very simple geometry again a one dimensional geometry but for simpler fun let's change it a little bit not the infinite straight-line but let's take all let's do let's first consider the infinite straight line yeah we have our coordinate X and we have that the distance between any two points is Delta x times AE let's think of that in terms of a metric remember what a metric is a metric is an expression for the distance between two very close points when the points are close we can replace this Delta X by the symbol DX same thing except when small magnitudes are small we replace them by calculus notation a times DX that's the distance between two neighboring points all right and what's the square of the distance between two neighboring points let's call it DS squared that's equal to a squared times the x squared well that tells us something about the metric the metric is just the coop definition of the metric the definition of the metric of a one-dimensional space this is a very simple case of a one-dimensional space the metric is just a squared and it's a thing that multiplies DX squared to find the distance squared S here stands for distance of course because wait a minute s isn't distance ssssss distance space spatial distance the S squared is a squared DX squared okay if a is changing with time then the metric is changing with time but still this is just a one-dimensional space with a metric and that metric might depend on time so you see where we're going we're starting to talk about geometry of spaces one dimensional spaces are too simple to have an interesting curvature or anything like that the metric of space is just a coefficient out here we could write that as G X X DX squared and what would G xx be it would just be a squared so eventually we're going to want to connect up the idea of the expanding universe with a metric of space-time but we'll come to that let's uh good now what about proper time remember what proper time is proper time D tau squared that is equal to DT squared this is the proper time along a trajectory let's now imagine what we're talking about is space time let's draw a spacetime your spacetime X and T our galaxies are not changing with X but wait a minute they are moving further and further apart so how can I possibly decide to draw them as if they were all parallel like this this looks like they're all at the same distance forever and ever that is where the scale the time-dependent scale factor comes in remember that the distance between these neighboring points is not just Delta X it's a times Delta X and that changes with time now what is the proper what is the proper time along the trajectory let's say of a particle between two neighboring points the proper time what did we call it D tau squared that's equal to DT squared minus the square of the spatial distance between them what's the spatial distance a squared the x squared right the square of the spatial distance between these two points is a squared DX squared not just the x squared because X isn't really spatial distance it's just a label which is labeling these points you could have given them different names Fred Harry Joe Nancy whatever you like but we gave them names which were x equals 1 x equals 2 x equals 3 x equals 4 the fact that X doesn't change for a particular galaxy is the same fact as your name doesn't change with time that doesn't mean that the distance between you between Fred and Harry doesn't change with time Fred and Harry are their names the distance between them is governed by both the difference of X and the scale factor a alright so here is our formula now for the proper time along a trajectory it's not just DT squared now T here is time measured by clocks moving along here ordinary clocks moving with these galaxies with this galaxy if X didn't change if we were talking about observations along a galaxy in other words moving with the galaxies we wouldn't have any DX squared and we would just say the proper time was the ordinary time but if a point is moving through the galaxies from one point to another then we would say that the proper time is the Tau squared minus a squared DX squared all right this becomes the spatial distance and a may depend on time so another way of thinking about the scale factor is that it's the space component of the metric let's just compare that with the Tau squared well I mean in compare it here's a geometry that's a space-time geometry and what is its metric it's metric again is a set of components the set of components represents the coefficients in this form here the DT squared has a 1 and the space - minus a squared 0 0 0 is n tacked that in this formula that could be terms but they aren't with DT times the X and the x times DT so there are no off diagonal terms and this is the geometry now this is the general relativity description of this expanding space the general relativity description of the expanding space going back to last quarter is that a certain component of the metric the space component of the metric is changing with time that's the way you describe a geometry which is either being stretched or contracted if the geometry is being stretched then this component of the metric call us G X X G X X is minus a squared it is the thing which changes with time as the universe changes its size or as the universe changes its scale factor scale factor and space space component of the metric are the same thing the same is true in three dimensions or two dimensions or however many dimensions the only difference being that the metric would contain more terms minus a squared dy squared minus a squared DZ squared the X square plus dy squared plus DZ squared this is just the Pythagorean theorem again but now multiplied by a squared so we will have a metric which will have a 1 in the time time component and a minus a squared minus a squared minus a squared 0s everywhere else that would be the metric of an ax form not a uniformly of an expanding space with a scale factor that was a function of time so another way to say expanding universe or contracting universe is that the metric of space is time dependent in other words it's a special case of general relativity metric of space time where the components of the metric change with time a growing with time expanding a shrinking with time contracting all right so that connects then this idea of the time dependence of the separation between galaxies with the basics of einsteinium geometry requires in all three a's and that matrix are the same function because of homogeneity and isotropy yeah let me let me make a slightly finer distinction they're not dependent on position because of homogeneity they're not dependent on direction all right the fact that they don't depend on position moving around in X as opposed to T they could depend on T but they don't depend on X that's homogeneity the fact that all three are the same that's isotropy ok at the different three directions all look the same that's what says that these three are the same the fact that everything is the same from place to place says that a depends possibly depends on time does depend on power what does not depend on position in space so it's an extremely simple metrical geometry where all of the coefficients are independent the position in space and isotropic meaning the thing the three are the same but time dependent time dependent but not space dependent and not angle dependent or not access dependent all right now that the incidentally not supposed to make a DX as a distance yes the X is not the distance a times the X is the distance it's just a label it labels the galaxies if you like here's the way to think about it there's a bunch of galaxies out there let's let's and at some particular time it doesn't really matter when but at some particular time somebody drew a coordinate grid through them and now the rule is the coordinate grid follows the galaxies as the galaxies expand there's a galaxy over here as a galaxy over here there's a galaxy over here the coordinate mesh follows the galaxies all right that means the distance between here is not Delta X Delta X is always the same between these two galaxies by definition because this coordinate grid follows the galaxies okay the distance is a times Delta X that would be the answer readers or the kilometers or so which one okay between two galaxies also the distance and white years of the a times that's right as opposed to death - that's right that's right Delta X is dividing my C square yeah units of distance into DX so you could you could put the units of distance into X but I choose not to I put the jovi the units of distance into a yeah that's that's arbitrary that's an arbitrary decision on mice apart whether I want to put the units of distance into X into a a product communities which we think it is a DX squared yes that now media square as soon as I put this work yes yeah nitric coefficient for time is equal to mine is that just by convention also that is a convention not a there's no there's no physics in that arm supposing I changed it let's change it now what is not a convention is that the coefficient doesn't depend on position but it could depend on time so let's suppose that it was T squared DT squared just for fun okay just for fun let's suppose that we have T squared DT squared this can be written another way T DT squared right I'm going to put a 4 here just because you'll see why in a moment that's twice T DT squared what's twice DT DT it's also the same as D T squared squared the differential change in T squared is DT squared in other words whatever is here it can be written how to say this right you can always change variables given a thing like this you can always change variables to a new definition of time where this becomes just DT squared if it's not DT squared you can find a new change of variables of time variable where it becomes DT squared yeah this is another possible question about whether there should be C squared DT squared oh yeah you're using C well I would put the C square over here or prison but you can also absorb it into the scale factor you can absorb it into the scale factor if you like so that's a but my rules are going to be C equals 1 1 light year per year okay that's the speed of light one unit in light years per year light years per year is not such a bad unit for time you can put this in die size general theory yes fuel equations will you get for the energy momentum vector well so far that we're really careful alright a couple of Corrections energy momentum tensor yeah no that's a good question ah and this is something we're going to spend a lot of time with so let's not do it now that's right right this is a metric you could say you can put it into Einstein's equations remember that Einstein's equations have on the left-hand side something that involves just the metric and on the right-hand side has the energy momentum density or the mass density or whatever energy momentum mass all the same stuff on the right-hand side energy and momentum so if we put in this geometry we could calculate what kind of energy momentum distribution we would have to have in order to sign order that solves Einstein's equations so we're going to come to that that's a very important part of the whole thing but for the moment for now we're just doing geometry and not dynamics dynamics is what determines how the metric changes with time geometry is just the abstract description of a space or space-time ok so an expanding universe is one where the space components of the metric change with time and given the homogeneity and isotropy of the universe a is not a function of position and it doesn't depend on which direction we're talking about it's basic rules of cosmology and they're not absolute rules are in the sense that one could discover it's inconceivable that one will discover that the universe is not isotropic in fact it's not exactly isotropic it's certainly not exactly isotropic in different directions you actually do see slight differences it's certainly not homogeneous it's not the same here as it is a light year away from here so it's certainly not exactly homogeneous it's not exactly isotropic and we could discover in principle that the universe is fairly far from isotropic and homogeneous thus far we've made no such discovery and I think it's unlikely that we will because I think we've seen about as much as the universe as we're ever going to see the universe is bigger than the portion we can see and it's bigger than the portion we can ever see so we don't know that on distance scales bigger than we can see we obviously don't know very much we don't know if the universe is a homogeneous isotropic on those very large scales but it has turned out that over the size and scales of the universe that we can measure astronomically or by whatever means the universe is isotropic and homogeneous apart from local variations on small scales you know it's kind of like a farmer's field if the farmer has planted his corn well and accurately then on a scale of a hundred feet it looks pretty home pretty homogeneous equally spaced rows and so forth on the scale of two inches it's very non-homogeneous homogeneous and on a big enough scale namely a scale bigger than the whole farm or the whole field it's again not homogeneous well we know about the universe is that on scales between distances between galaxies 100 galaxies apart or so forth that out the scales as far as we can see the universe is homogeneous we don't know whether it's homogeneous on bigger scales and we probably will never directly be able to find out okay let's let's come now to close the origins with it never directly find out does that imply you thinking wow I think most thinking these days is that it's not homogeneous on big enough scales but that's certainly something we'll talk about before we're finished my bed very strongly is the universe is not homogeneous on large scales and I will never win like that we're also gonna lose my dad the only one well I can only write a book if I win my bet or lose my bet I mean you know I'd be willing to write a book on losing a bet that book won't be written because for physic for basic fundamental reasons we won't be able to see much further than we can see now that's called the existence of a horizon it's a problem but couldn't you derive the idea from gravity to look at an asset beyond what you can see no way tell what's up to beyond of your so you negotiation for to deliver my team no no if we're sitting here in this room we are in the field of the Sun and we're circulating around the Sun but there's no easy way there's no way for us in this room on this earth without being able to see out beyond the room to tell that the Sun is there now we can measure tiny tiny um tidal forces but the tidal forces in this room I don't feel stretched do you feel stretched I'm not talking about the tidal forces due to the earth the certainly tidal forces due to the earth I feel I actually can't feel those either if I jump off a springboard a diving board or something while I'm in freefall I don't feel squeezed and pulled and I can't tell that that the earth is there I closed my eyes I could be in freefall a zillion miles from the earth so the answer is that you can't tell that there's a large mass out there except by its tidal forces it only has tidal forces if it's close enough to you and you are big enough to feel the variations of the gravitational field from place to place so if there's a large lump of mass very very far out there we're just freely falling in the presence of it and we don't feel it as I accept I say through possible tidal forces which are too small to feel if things are far enough away right so the horizon which we haven't even done to discuss yet is an ultimate barrier to and then the frightening thing frightening I know is frightening frustrating frightening fascinating thing is we know with virtual certainty that the universe is at least a thousand times bigger in volume than the horizon then we can ever ever ever in principle ever see so we know this stuff out there that we will never be able to detect and we can't tell what it is we can't tell if it's similar to us we can't tell that's why you took this course to find out microwave anisotropy measurement of the curvature but if I don't answer that remind me to answer it but not now okay let's talk about the idea of a closed universe now a closed universe well it's best described by example the infinite line is an open universe closed and bounded I'm not so closed and bounded closed and bounded the infinite line is certainly not bounded a line segment like that is bounded only as a certain size bounded means it has boundaries it has it comes to an end well that's not what bounded means excuse me doesn't mean that has boundaries it just means in some sense it's a finite extent now the universe is not as far as we know anything like this line it doesn't have to be a line we could be talking about a square or a cube a cube with walls the universe could all be inside a cube with walls what does this violate it violates homogeneity every place is not the same as and I saw trippy for the case of a cube yeah and I saw trippy somebody's standing right at the center see something quite different than somebody standing up here near the corner number one number two even at the center here even if you are at the center you look in different directions you see different things in one direction you see a corner in another direction you see a smooth face I see I mean detect by whatever means you can detect the shape of the universe so a bounded universe seems like it's out of bounds in the sense that if the universe is truly isotropic and homogeneous then let's concentrate on homogeneous then it can't be a cubed with walls good okay so spheres in in a very interesting case here's a sphere the circle is good enough I mean the interior of the circle we have this region here somebody at the center looks out in different directions and sees the same thing in every direction so from his perspective or her perspective over here the universe appears isotropic all right but it's certainly not homogeneous somebody over here does not see the same thing as somebody over here this person over here sees a wall close to him on the right or upper right here and far from here in the other direction in fact this observer over here on this galaxy does not see an isotropic universe it doesn't see an isotropic universe because this the direction is different than that direction it's neither it happens to be isotropic for one special person the person at the center but for everybody else it looks non isotropic and it's certainly not homogeneous because the properties over here are different than the properties over here all right so that boundary is a is a we don't know as I said that the universe is truly on very very large scales isotropic and homogeneous but let's take it as a principle in fact it is called the cosmological principle that the universe is isotropic and homogeneous now I remember as a young as a young physicist being absolutely convinced that it must be true because it's a principle and then when you ask why does this principle come from well we assume it's true so it is the cosmological principle but it could any day of the week we could make some radical discovery that it is not true I don't think it's going to happen for reasons that you'll understand yeah does I I stopped with me everywhere Michael J yes that's the point I sokrati everywhere is implies for better if I said I saw RPS homogeneous hmm in other words it sighs doctor fee is no genius the universe is not homogeneous somebody super genius it does mean something no no that's right that's right for example here is a universe this is a universe of arrows it's a universe made of arrows and they're all pointing in the same direction they densely fill space let's say that's very homogeneous this goes on non non it's homogeneous in that it's every place the same but it's not isotropic it has a direction associated with it right and I saw trippy doesn't imply homogeneity this fellow at the center here sees I saw trippy when he looks out but this is not homogeneous but if everybody sees I saw trippy everybody sees things being the same in every direction that happens yes to imply homogeneity so they're not quite independent the two different concepts they're related though in a particular way yeah ah the surface of the sphere that's that's that's another thing on a very bald sphere bald meaning featureless every place is the same as every other place every direction that you look in is the same as every other direction that you look in on the surface of a sphere and it's bounded remember bounded does not mean that it has a boundary it's now we're talking about little creatures let's imagine little two-dimensional creatures flatland creatures except instead of living on a plane they live on the surface of a perfect sphere and they can't look away from the surface of the sphere they can't look into the sphere and reason is because light only travels on the sphere itself on the surface of the sphere in this theory or in this world if light only travels on the surface of the sphere all they can do is look around within the surface of the sphere look that way that way that way that way and on the surface of the sphere all directions are the same and all places are the same so that's basically the key to combining I satrapy homogeneity and boundedness again boundedness does not mean a boundary these creatures do not fall off the edge of the earth when they get to a certain point they go around so boundedness does not mean boundary it means boundedness finiteness in some sense right okay all right the simplest example would be life on a closed curve for example life on a circle okay the circle has a metric let's talk about a life on a circle a nice parameter a nice parameter which could be a coordinate again there might be galaxies sprinkled around on this circle every point on the circle is the same as every other point not quite true of course the point points between the galaxies are different than the points on the galaxies but on scales bigger than the size of the galaxy every point is the same as every incidentally what does AI satrapy mean in one dimension I saw trippy is clear in two dimensions you look in different directions what left and right being the same that's right so here for example is a one-dimensional world which is homogeneous but not isotropic it consists of little arrows pointing in one direction all along it left is different than right all right left is different than right but every place is the same as every other place so in a one-dimensional world you can also have an isotropy non isotropy what it means is simply left and right along the line are different ok so if I just have a population of galaxies like this on the average left and right are the same every point is the same as every other point and I would say this was homogeneous isotropic and bounded let's talk about the metric let's talk about a geometry how do we what might we characterize the position by well we could use X but X is an inconvenient variable on a circle a convenient that a variable on a circle is angle so we could pick one galaxy and call it theta equals 0 theta is my notation for angle or at least one possible notation for angle and we could characterize every other galaxy this is theta equals 0 by some theta and what's this galaxy over here well we could call it minus a negative angle or we could call it 364 degrees it's almost all the ways around so what did I say to her blaster 364 oh that's the number of days in a year dated 10 and 65 355 sorry I got the comic days in the year confused with the number of degrees on a circle Wow okay right 359° not quite all the ways around not quite 360 degrees okay but we could use an angle to characterize the position again let's let's suppose for the moment that the radius of this circle or the circumference of it the little bugs who live in this world have no sense of radius because they have no sense of where the center of it is it's not in their world but they do have a sense of circumference they can measure the circumference by marching along it till they come back to the same place they can measure the circumference the circumference mean something and of course they could call the circumference 2 pi times something if they wanted and that's something they might call the radius but they would have no particular reason to but they could so they have a sense of circumference and let's now talk about the distance between the distance around the circle the size of the circle oh we can call let's call let's see I let's call the total distance around the circle do I want to call it a you'll do I want to call it 2pi times 80 I think I'll just call it a scale factor scale factor is the total distance around now supposing we have let's go why am i calling that scale factor we'll see in a moment but let's take the distance between two points theta equals zero and theta equals theta just theta what's the distance between those two points a times theta let's define it to be a times Delta Theta our a is not the let's see we've got to do this right let's get the numbers in there correctly we're going to measure angles in radians all right so the total distance around in radians is 2pi total distance around is 2pi so this point is 0 and if I go all the ways around then I come to the point just below it that's two pi radians around okay the total distance around is 2pi let's call it 2pi times a i'm going to call it 2pi times a where a would be the radius if I decided to plot it as a circle and I decided to plot and I remember these people who live on here can't tell whether it's a circle or whether it's a curve like that they have no way of telling because they can't see into the interior all they know is the distance around but let's pretend that we drew it as a circle then the total distance around that we call 2pi times a but what's the distance between two neighboring points let's suppose the points well separated by a distance Delta Theta then the distance between them would be a times Delta Theta the radius of the sphere the radius of the circle times the change in angle the distance between two points on a circle a circle of radius a is a times the angle the angular distance well this is very much like a times Delta X it's just I've used theta and theta happens to have the property that when you go around the circle it comes back to itself that you come back to yourself but otherwise this is very much like 8 times Delta X in fact we have a metric we can say the distance between two points squared between two neighboring points squared is a squared d theta squared so in fact we have a metric on a circle very very similar to the case of the infinite straight line except now the space is bounded it's bounded because when theta comes all the ways around by 2 pi you come back to the same place this is a closed and bounded universe and here is its metric now this is not a space-time metric this is just a space metric all right so you could imagine the creatures who live on this thing measure this distance around here and then they know what a is they know what a is ok but you can also imagine a changing with time again a changing with time would correspond to the radius of this circle changing with time in that case we would write the s squared distance between two points is a of T squared times D theta squared and we might even write a spacetime metric we might write D tau square the space-time metric is DT squared minus a squared of T squared eh pretty squared in the first a lot of second or not here about that a squared of these a square a of T squared right yeah D theta squared good yeah yeah right all right what does this geometry look like let's try to think about it as a spacetime as a spacetime will start the vertical axis is going to be time let's start at one instant when a has some particular value here's a let's say a equals 1 or 7 or something else so it looks like a circle that's at one particular time now the vertical axis is the time axis as time goes on a changes that means let's say gets bigger that means that the space-time geometry grows with time let's suppose at some time it stopped growing and turned around and shrunk start at the shrink and it's not the shrink maybe it does something crazy oscillates starts to grow again the surface the space-time looks like a surface it looks like a surface of a look like a Vaz or vase whatever you call it in fact we might even imagine a space-time which starts what would the scale factor be at this point if the universe started infinitely shrunken here it is it starts infinitely shrunken like a circle of infinitely small size and it starts to expand expands out to some maximum size I'm not thinking this as seriously that it expands out the maximum size expands out starts to shrink again starts to grow again does whatever it does what would be the scale factor at this point the a equals equals zero at that point so in this closed and bounded universe a equals zero corresponds the thing shrunk to zero size right and then it starts to grow grows from zero and very quickly become something not zero and then changes its size maybe in and out and in and out whatever it does whatever it does okay so that's a geometry it's quite clear that this geometry is curved quite curved it's curved the sense of Einstein that's a closed and bounded this is the simplest example of a closed and bounded universe which also happens to be time-dependent in this universe hubble's law is still correct for exactly the same reasons as before the distance between two points again involves a times Delta Theta distance is a Delta Theta the velocity is a dot times Delta Theta and that's the same as a dot over a times a delta theta which is Hubble constant data over a times distance so Hubble's law is still true this the velocity between these points is proportional to the distance between them you get into a little bit of confusion when a distance is so large that it comes around the other side let's forget that for a moment but an astronomer over here looking not too far away would see Hubble's law and that's the idea of a closed and bounded universe notice that the closed and bounded universe is not an explosion from a point in three-dimensional space where stuff gets ejected out it's not like the explosion of a hydrogen bomb where something is concentrated in space and then goes flying out into space it's the same Everywhere's everywhere is it's the same and yet if we imagine galaxies let's imagine galaxies or particles or whatever here they are spread out on here they separate because the distance between them gets larger they start to come together again they stop it oh yeah what happens to them when we look way back at very early times they're very close to each other now let's say there are 360 galaxies those 360 galaxies are well spread out here but they're very very close together here here they're even closer together they get too close together they're not separate galaxies anymore get them even closer than not even separate atoms anymore but notice there's no special point in space space now does not mean this three-dimensional room it means this one-dimensional closed and bounded universe the little bugs live on no special place no sense of being ejected out of some particular place like like the bomb explosion but instead just a uniform rapid expansion taking place everywhere is at the same time satisfying the hubble law okay but things get pretty crowded down here infinitely crowded if the universe began as a as a point so when somebody comes and asks you well where did the Big Bang take place where did this Big Bang take place where were all these things ejected out of the answer is everywhere yeah times Curtis if that was assumed was in again you say the space-time is curved would be curved that were cylinder no if it were cylinder it would not be curved if it was a cylinder in other words if a were constant with time if a were constant with time that would be like this that would be this world ah oh good a cone isn't even it's a good good cone all right all right first of all cylinder a cylinder is the case where a doesn't depend on time at all right good that's just a constant day up and down that is not curved that's not correct that's obviously not curved no no the cylinder the space-time is not curved the it's it's it looks like it's curved because I try to draw it in three dimensions but it's not curved remember why you could take a scissors and cut it open I think a scissors cut it open without stretching it without stretching it you could lay it out as a on the plane and it would be you don't have to stretch it to lay it out on the plane okay so that's not curved if you laid it out on the plane so there it is I've cut it open laid it out on the plane just mathematically cut it open in my imagination then the cylinder is in here but this point and that point at the same point in the sense that when I cut it here this point which half of which went with this way and half of which went that way become these two points which are really the same point all right so no the cylinder is flat okay or equivalently in this particular case for a one-dimensional universe not the same three-dimensional is different but for a one-dimensional universe the time in the pen the universe is flat okay flats in the space-time since now what about the cone first of all what is the cone correspond to it corresponds to where's the cone the space-time cone how does the scale factor grow with time in the space-time cone linear linear the size of the cone is linear in a time and you double you double the time that means double the distance along here you double the size of the alright so that means a is proportional to time again 1/4 1/5 1/10 whatever the scale factor is proportional to time is this or is this not flat as a space-time it's not flat at the bottom that's for sure it's got a conical singularity there but cones if you slice them and open them up a flat alright so a cone is flat you make a cone at a flat sheet of paper by I mean the same deal as we studied last time just by sewing the edges together you make a cone for the only place that the cone is curved is right at the tip of the cone here that's the only place where there's any real curvature so yeah this is also flat except for the tip the tip is the origin of prime here the big bang or whatever we haven't gotten the big bangs yet all right so I think I think it's time to quit now next time we will discuss not the circular universe but the spherical universe or the universe which is a three dimensional sphere R and the geometry of it and then we'll go on to discuss what is the thing which the Germans how the universe grows what is the dynamics which causes it to grow or shrink or whatever it does and that will bring us to the end of next week the preceding program is copyrighted by Stanford University please visit us at stanford.edu

Channel: Stanford

Views: 478,295

Rating: 4.8914027 out of 5

Keywords: science, physics, Big, Bang, theory, space-time, geometry, inflation, dark, matter, energy, cosmological, constant, expansion, galaxy, doppler, shift, relativity, alice, bob, frequency, Hubble, homogeneity, isotropy, velocity, of, recession

Id: 32wIKaLkvc4

Length: 103min 2sec (6182 seconds)

Published: Fri Jun 12 2009