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Cosmology | Lecture 4

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some reason I have a set of notes here from last week had to do begin notes from last week there are few warmer and then I made a mistake this week's notes I wrote them heads which are probably too long the scan 1 piece is pacific other kid I inherited a whole bunch Our of Yellow Pear Eds From the mathematics department it's mathematicians don't like riding on won't pay so I got about 25 of them the stupor what is it you have right that's hard to skin is butcher paper OK we got a couple minutes looked so let's begin with some questions just elects couple minutes yes but from the possible that lead you actually who thinks so as to what have 1 I fear was that it was more calm and that was kind of Torres was just really periodically periodically repeated itself so there a large part of it the rule looking at auditors repetition of saying things are another way of saying that we looking at ourselves the back door that has some consequences that has observable consequences he you might think it be really easy you just look out you see but now that doesn't work that way it's much harder and people have looked for a period the city and and her astronomical cosmological observations and there is no evidence some counter evidence are periodic or that there was nothing I don't see how it could be more observed were part of an interesting question I don't think so OK use of the universe they Spanish there's no so far I haven't done it on should eventually dark energy but so far we have not talked about fortunately our we will the Jr the uterus is definitely expanding by expansion Doole consequence of dark energy is not that the universe is expanding its but that's not a stuffy universes was expanding before anybody knew about dark energy dark energy just makes it expand faster buyers suppose that I expect yes works well no yacht North from ever we hear from you'll core holdings Euro your friends he and out space your distance between you and your friend would not expand review general expansion the general expansion off produces a kind of very very mild repulsive force between everything and everything else you would think of it that way the it's a way to think about it that everything has a little bit of repulsion of everything else our basically proportional to the Hubble constant but that would you are holding your friends here and that very very tiny repulsion between the the 2 of you is more than made up for a vastly more than made up for just attractive force of you holding on your friends hand so as long as you didn't like narrow or out of place mourners only didn withdrawal you would not participate general expansion you as a couple speak and factor is enough probably enough other kinds of forces between you beside told him that they would overwhelm this tiny tiny tendency for separate on that because you're closer together and the average the distance between your Adams your Adams in different atoms are closer together they have more traction in particular holding your hands will keep you from separating and you're not part of this general expansion the same thing tho the solar system the solar system was largely held together will 1st let's talk about how What about Adam do a wanted to come to salsa summer and Adam also is embedded in the face of that space is expanding why isn't Adams also expanding and the answers electrostatic forces more than overwhelmed by an enormous amount to more than overwhelmed the general tendency to expand now would you can say it is In an expanding universe slits even take the case of a our very accelerated expansion that this trainee little bit of our would force will pay to modify the Alamo little bit it will tend to make the Adam just a tiny the bigger Adam will have to
be slightly slightly expanded but other than that at the end of tiny tiny effect but will not close the atom per flight path not strong enough because why are saying fill the solar system solar system is held together by by gravity the gravitational pull of the sun is simply much much larger than this tendency to to expect so for that reason and With much stronger don't know last up to a small you need to do work through and I have to do a lot to do what really were thank you very much deputy especially now that is restored they entered Over From that was what energy conservation and expanding universe is different than energy conservation and static in Harlem energy conservation and we would do that this several ways to think about it but let me give you the simplest way to think about energy conservation is a consequence of time translation and invariant another words if everything is tying independent meaning the say space find and all experiments would reproduce exactly the same effect if they will later or earlier that the consequence of that is energy conservation on the other hand if the basic set up call the background spacetime itself if it's not static of its expanding or if it's changing with time then energy conservation doesn't apply there is no energy conservation in a world where that parameters of the world time depend in this case the radius of the universe as time-dependent energy conservation not quite what you thought and will not come work ability you lose energy conservation will use it in a certain form but doesn't say that the net amount of energy in the universe is fixed was not worth it basically says that changes of energy in the universe translate into kinetic energy of expansion so it was a back and forth between changes of energy and expansion plans that you don't have to ask where the energy came from it came from the expansion Oregon data from the scale no part on small scales sir falls apart on large jails as far as we can tell universe is homogeneous and isotropic the error in this room is largely homogeneous and isotropic every time somebody opened the door of course a draft but we kept the room closed and you can't get mouth shut and the air in the room would become highly homogeneous but not every scale on Kearney tiny scales were atoms molecules assault for a formal review over here about homogeneous the fact and even bigger scales fluctuations that take place density fluctuates sometimes moreover heroes last over here so on our whether something is homogeneous not is a scale dependent question the universe is not homogeneous from small-scale and small scales means you harm hundreds of millions of light-years on scales bigger than hundreds of millions of light years everything seems to be distributed uniformly unnatural wrenching you question but adventure some of the universe is expanding Carlos can fall that for now young known that's that's 1 we're working on it that is not a lot the resulting 3 yes yes with is it that matter talks of the 1 reason look at what the 3rd it's a statement about geometry Einstein's equations are the dynamics of how things change with time so for this is a pure a statement of geometry if you classify these geometries which are in some sense the same everywhere of love at every point His every of a point to what that means a means exact greed given the space honey describe a space you describe a space by Matt Uribe met triple-space some Jean Nuno of X . 2 units the only spaced about space and kind of ecology of X X now stands for all of the quarter that represents some geometry now you could make a coordinate transformation imagine you make it coordinate transformation the point x equals 0 which was the or origin of accord
next was equals 0 that now has a new value no coordinates could be called I what was originally x equals Iraq is no no longer the origin white equal 0 might be over here so this corresponds to some kind of transformation which replaces the origin with no origin now when you do that when you do that the metric when you do will coordinate transformation of the metric transforms transforms into an electrical cord Jeep prime an end of y and this is the metric and the Y coordinates typically the magic in the Y
coordinates will have a different form than the metric and the X coordinates that that could be for 2 reasons to resist 1st of all it could be because the space itself is different at this point in this point and so we transformer coordinates to this point we might discover that the curvature larger over here or something different over here in which case the metric will have a different look to to it when you expressed in terms of why didn't terms of X all nite b not because the point different but simply because you use screwy coordinates over here and some other screwy coordinates over here a homogeneous space means 1 that you could find a coordinate transformation which will replace any given pouring as origin Merck's early any given point by any other point in such a way that the form of the metric is identical both before and after the transformation for example for example let's just take flat space flight space has a rhetoric which is um DES squared equals DX square let's just take 2 dimensions DYC RDX squared the X 1 squared forced BX square X 1 squared plus 2 squared lets 1 next to where the coordinates black words that are supposed to make a translation of coordinates that means
let worry while the state effects of stands for a new set of quarters which is just equal are short X at Y 1 is X 1 plus a shift let's call it a 81 a does not stand for any other 80 that we use so far in this this course and white tool is Eagle Texas 0 plus 8 82 This is a shift this is a shift which show the X coordinates and the wire coordinates are shifted by a vector 80 what is metric In terms of white coordinates wealth we can do it by simply be expressing DX 1 in terms of D Y 1 but D while at the same as DX 1 if I differentiate of America little change in white it will be equal to little change in expert since this is a constant doesn't contribute and likewise for the year of the D. White to sold for this kind of transformation squared is equal also could be why 1 squared plus BY 2 square has exactly the same form as the metric cat in terms of Exeter's the implication of that is that the neighborhood of the origin exit x equals 0 has exactly the same properties as the neighborhood of the origin of white equals world II caught another words these 2 points have exactly the same property now since a could be anything there's a coordinate transformation which takes the origin over yet Hey other point to any other point whatever that preserve the form of the metric and that's what's called a homogeneous space whose properties are everywhere same so if they exist recorded transformation between Exxon worry that takes the origins any other porn such that the form of the metric in terms of y it is the same as in X then the space called homogeneous everywhere a seal biggest we Capote origin the South South poker the origin nor the former metric is we wrote it down last time arrive down again foryou br squared plus signed squared off our times let's say defy squared or deep please square forgotten what I called it later square that's the magic of the sphere now must make coordinate transformation by rotating the sphere rotating the sphere means choosing some other point-to-point over here for example instead of measuring are from the South Pole we measure aura from me from the from the east Paul is east of here we measure our from that point instead of measuring angles about the South Pole we measure angles about the East hold this issue coordinate transformations so little nasty in complicated to write down how does the distance from this point depend on the coordinates as measured our series will be complicated but when you figure it out your find meant when written in terms of the accord its relative to this point has exactly the same form as wrote for another hour bother writing down be exactly the same form our except that the meaning of a long there will be distances measured from here instead of Europe and angles measured about that point instead of about that point former emetic form of the because the same end since you can chose you coordinate transformation the take this point that 8 point it tells you that every point on the sphere is basically the same as every other . 3 1 test which works very well and could mentions off for a uniform geometry is that scalar curvature should be the same everywhere that's not good enough and of the dimensions of the basic idea that the geometry is the same everywhere is called a homogeneous geometry this is also true struggle flap playing its truly fear that homogeneous and it is also true of the hyperbolic twice a year negative curved analog of this year where you replaced signed by hyperbolic there are no others they have a good right there a bomb no 0 OK this yes it is true that the Taurus is armed translation invariant is the right word translation invariant means you could transform 80 . 20 other but as a matter of fact happens not to be isotropic wise and isotropic it's not isotropic because they're a definite remember Europe sources say a rectangle with Armed with periodic boundary conditions and the ax Our are preferred to go out here you come out here so this axis is preferred that accesses preferred and axis like that is basically different observations would be different below 45 degrees along a it yesterday true it's a homogeneous space but some isotropic space but they now 0 of course the idea that the space it is homogeneous and isotropic kind has a status somewheres between being a postulated therefore highly questionable and the 1 here and and somewhere in between that and observational fact storms certain scales it really does work homogeneous but on scales so big that we can't see them we don't not OK that 1 of the remark before we go on since this comes up over and over and over again I just want you to Cairo fearing but today that on this often the enormous amount of confusion when you say such things as for example our space as a spherical shape like people get into then heads the idea that it really is like a balloon and namely that hasn't inside and outside and they ask questions what happens if you move away from the or into the belong now you have to learn to think of geometries as In having their own intrinsic shape that has to do with what happens if you move around the geometry and not having to do with some imaginary possibly real or maybe imaginary additional directions that you could move all always from the from the space for example these were the most confusing kinds of space spaces is a one-dimensional space a one-dimensional space but they closed one-dimensional space close finite one-dimensional space was a close is a close adviser one-dimensional space is another 1 here's another 1 happens to be a square from the point of view of the the intrinsic geometry all that
counts is measuring distance along with the idea of along for the geometry no concept as far as the intrinsic geometry goes of moving away from so in fact how many different kinds of one-dimensional spaces are there from the intrinsic point of view the answer is 1 all of them are identical Ellen not quite an identical to each other this all 1 parameter family of them and the parameters is how far around you have to go a comeback place so if this circle here happens to have the same circumference as of this sum of peanut shaped ownership over here then intrinsically they are identical identical and in fact other them not curved and to see that the Mount curve think of them as strings superstring about a string theory strings just as strings without stretching the strings you can always deformed and make any piece of them straight this pulled straighten about our be want intrinsic fact about them is that they have closed the comeback In themselves but any piece of it could be straightened out without stretch without the forming the metric the distance between points one-dimensional spaces are all flat they have no curvature the fact that they draw curve that has to do with the way you drew them in two-dimensional sorrows you have to learn to think about the intrinsic geometrid hardened as a reason that hard allowing segment I segment it is also a one-dimensional space it has a different property in the closed one-dimensional space topologically different that has been points but the only thing they characterizes as it is the distance between me and points this mines segment is the same as that line segment that so they shapes it went space that that but only if it's a bit now I emphasize is over and over again when explaining some of these things to a general audience probably said to you before but Was it learns to live through special about 3 dimensions and so I had to pick he asked the question do you think that you can visualize a 5 dimensional space visual arts close rises take every race is not can't do that like Interleaf for not really know I complain I can now use a trick that hope visualize it but the direct visualization I cannot our cast now I say close your eyes and see if you could see a speedier about what's up because field circuit could you visualize a Q yet I could visualize the secure but conceded cubicle nature I concedes three-dimensionality and about the Obama mentioned let's go down to 2 dimensions I want you to do is to visualize an abstract two-dimensional space could you do that and everybody assured see it's a RICO surfaces and my response to that is yes you see a curved surface but the only way you could visualize is by visualizing it as In a bid in 3 dimensions unless you have some brain a different mine could you even visualize a one-dimensional space Sarah conceal I'm not what you see is that lying possibly embedded on a piece of paper to dimensions or possibly um embedded in 3 dimensions and even appoint an abstract point you cannot visualize what I've seen it suspended in 3 dimensions wasn't the special about 3 dimensions something really mathematically special not that you are architecture your brain architectures evolved for the purpose of navigating around the 3 dimensions and so it's not surprising that your ability and visualizing it is home up at 3 dimensions that doesn't mean that 3 dimensions aren't in east and anyway special course every dimensionality has its own special features but 3 dimensions is not special and feel as mathematically arms sophisticated people we just say if you wanna discussed true dimensions a mechanic so life if you are discussed 3 dimensions Jamaica next Hawaii if you are discussed for mentions you make an X a wife Theodor W. and you get that like to joke that to convey all the way to 26 that mentioned our 4 OK so I just want to remind you only over again that when you that I will do again I will not do it again but that that when we talk about the geometry of the space with talking about be intrinsic geometry and not the the way it in bid for purposes of visualization In some higher dimensional geometry it was yup but except except Thomas or more the mall would be very small yuck yuck yuck if I do this if I did that births but but but what 1st the the vet that's right they've discovered that's right but assumed that light rays only propagate along the surface of its a one-dimensional world that living in which knocked over a one-dimensional doesn't matter how we draw torn dimensional but assumed that has the properties optical fiber that light only propagates internally or longer and no such thing as messages are getting off the axis and back onto it that everything that takes place takes place on the line for ongoing segment with that stipulation there would be no difference between this and a straight line segment that would be be identical if it not for at the curvature they now than then they think no two-dimensional two-dimensional surfaces cannot be in general cannot be performed without stretching them to make them flat it cannot obtain arms that Kerviel lines segment here the right Turner is that the intrinsic curvature 0 extrinsic curvature is not 0 extrinsic curvature means what you naively think about when you say this is a curved line and has to do with the way the wine is placed in the higher dimension but from the point of view of little creatures that live on this line can't see anything off the line they have no experience of line no experience of anything until photos they're concerned there's only the line so that's the way you should think about it and not ask where the line is embedded in her now you know if these creatures could be fooled they might literally live on a optical fiber and think everything is moving along optical fiber but no doubt there would be some things that they could do for example they could invent a laser within their own fiber which high enough frequencies gamma rays and I assure you that gamma rays do not stay in the an
optical fibers they will jump across so it's good see that there that these creatures could discover that they really are living in a bigger space with more I don't think it's gonna happen with us None None the thing like that but 2 but the experimental back at the moment tell us that we're living in a 3 dimensional world and that three-dimensional world is as an intrinsic geometry be intrinsic geometry being go our way 1 of 3 kinds of yet while Wang your the wet fight what circuit yet right worse than that site the right that that is a good that's right that a good right that's exactly right there and so useful example a piece of paper that you take and curvet like this is no less flak from the intrinsic point of view than the flat piece of paper that you start with all relationships within the surface are the same as they were before you were a bent like so that as flat and but this is not true of the hand of a what's fear hemisphere again this year notice that this can easily be flattened without stretching it this cannot be flat churches so two-dimensional surfaces can be curved one-dimensional surfaces a one-dimensional lines curves cannot be Kerr that by that I think it my Our trials had difficulty with Congress's a similar program at gravitational field implies curvature for the yes yes yes yes as I mentioned due before curvature is a measure of tidal forces and tidal forces have to be due to some friends in general relativity they have to be due masses therefore they are title gravitational fields Due to some Mass yes OK goes along but very very known hopefully help to you longer questions OK let's move on now let's except that space is isotropic and homogeneous and therefore is 1 of the 3 stranded the tight spaces your name very colorful names the colorful manners Our K E equals 1 K equals 0 equals minus 1 Jerry the case stands the curvature curvature Riegler 0 that's flat space Katie Eagles 1 that positively curved space and Seattle ogress beer but of course we are not talking about a two-dimensional space is not two-dimensional sphere it could be be a three-dimensional sphere noggin keep in mind when I say that spaces in the flat for a sphere of hyperbolic plane and it's completely homogeneous that of course is not completely true it's definitely not completely true it could be true for the average properties of the space over distance scales are big enough that average things out at a statement comparable to saying the Earth sphere worried the surface of the Earth as a spirit of glassy surface of the earth as bumps on that has mountains and valleys has the cement of serving not a perfect sphere Our end however on how big a distance to you have the think about before mountains valleys elects overstuff average out 50 miles from Liberty Mount Everest is what said Miles case on a scale of a hundred miles by 100 miles years looked pretty flat off the scale of a thousand miles by a thousand miles of looks very flat so this idea of whether something is flat or not as a scale dependent idea not costs the earth did not have to be flat even on scales of a thousand miles a could've been shaped like a cigar and no scale would have been thought to be flat so it do some content and saying that the Earth big enough scales as homogeneous isotropic surface of it and that looks article same is true of cosmology OK so now but I think we discussed last time a little bit it we started discussed spacetime geometry were now moving from once we start dog rocker geometry of course we move away from the and we do we relativity general relativity general look at least starts with Mets at the start of the track and we are going to make an assumption about the space time that there were going to make the assumption that space and time dull are mixed with each other are mixed with each other in the metric another words symmetric has a former that looks like this minus squarish In relativity the time component of the metric always comes in with a minus sign and the space component of the record comes in with a plus sign some scale factor which depends on time in general has to do with the expansion of of the universe and so forth a metric of 1 of 3 types was types Katie equals II erased cables 1 cable 0 cable minus 1 but the 3 metrics which could be here are 1st assault cable 0 that's flat space just play OBX square plus DY squared plus disease glared that's Cagle 0 and the father concerns have just name cable 0 beers Kinko's plus 1 that means the positively curved space elegant he'll right that it is omega threes squared or are we doing right now In detail we can ride out that's equal to DR squared plus sign of OR squared they squared sorry where there were no mistake I would that the Omega who squared but that's that's um at Skate equals 1 Waco also written this warned the flat space in a similar former still going cable 0 we could have written it as DR squared plus square deal maker to square this is call three-dimensional polar coordinates same space but in three-dimensional polar coordinates where Oracle 0 is your location and made it 2 0 with just the angular world around 2 also just
cable 0 same thing change and finally came was minus 1 which is the hyperbolic as Shearer angels and devils the world except In 1 in and 3 dimensions and that's Dr scored a plus hyperbolic signed square of all are times as
saying deal make a toast to it they all look similar but qualitatively they part of special quantitatively but qualitatively very different this AFP squared is called the scale factor the distance between fixed points by fixed points I mean points with fixed coordinates In space the distance between 2 points will in general be proportional Teddy it times some characteristic difference for example on a three-dimensional sphere there might be some angular distance he ate some our daughters court dealt doctor of some angle but this doctor of Angola is just some fixed value which is the distance on the unit's sphere likewise the velocity between those same 2 points the same thing except time differentiated the ratio of the velocity of distance Hubble's law equals 8 . 4 times the distance barring used to the thing about 80 84 that's not a constant in general could be a constant by constant any independent of time could be a constant butt general it's not constant but it doesn't depend on position that the sense in which the Hubble constant they got over a is is a constant it doesn't depend on where you are although it may depend on when you are correct focus of their there we are with the basic geometry the basic set now will it over also set up last quarter we learned about Einstein's field equations we learned that their complicated is held that actually writing them down in detail is real mean In but we're not gonna write them down but our right down the general form of them but how do you do it you're right down the magnetic in this case space time climactic 8 of peace could the full baseline epic incidentally in case you were wondering this multiplies that's a clear now you're right down the space-time metric you calculate Einstein considered you said equal to what I'm supposed to be set equal to 0 and out of that operation columns equations firehouse 80 varies with time that's the Have you calculate that the Einstein field considered the left-hand side the energy momentum tensor on the right hand side Einstein's field equations you have to calculate Chris powerful symbols that nuisance have derivatives Walker kinds and so forth and there's a lot Christoval symbols in this case there aren't too many actually but a nuisance I'm not going to do on the blackboard we just outlined what the work the basic idea is and then upload are mentally plug in the Einstein field equations and spell out have not been what the geometry of the world or what a 0 8 is but what equations a satisfied OK so let's just to remind ourselves the Einstein field equations have a left-hand side of the right hand side like any equation the left side has to do with geometry if you remember if if you don't remember it's not terribly important but if you remember the left-hand side is called the Einstein tensor IDs bill out of the curvature tensor and I'm not gonna go into detail about the curvature tensor is it that the of the Ricci curvature it's got to 0 indices that can serve and I chose the indices to be upstairs and the seats that some curvature tensor minus 1 half the net no times times scalar curvature R Dr. Mark terribly important with the details of these this is the notice the left hand side has the curvature tensor and the right hand side has remember what somebody inside a booty the energy momentum tensor but it also has 8 5 Jean provided by 7 right now you know separates 3 the rate others remind you this was the same factors that appeared in the Newton's equations enter the energy momentum tensor both sides are a cancer care farfetched joint any frame true every frame is good cancer law laws of physics
1st question what With start with the right here but female Nova contains a complex of things which include the density of energy the flow of energy flux of energy the density of momentum and the flux of momentum different components of it in particular the kind kind component is the 1 that will get a fix on the time time component 8 GE all 3 times he timed time he Nought Nought the time time component is the energy density so let's call the energy density role would be calling a row previously was a scholar Rome stands for energy be ordinary energy in matter whatever whatever kind of material this energy momentum is described as way a ranking inside is completely sensitive to the kind of material that the universe is it particles of electromagnetic radiation something else it knows about the material nature of these of these ingredients that making up the universe the left-hand side has nothing to do with the left hand side is geometry on the right-hand side of the kind tying equation is the energy density and on the left side it is something that involves curvature nephew go back to the definition of the curvature of work at our you'll discover that there are 2 contributions to 1 of which has a 2nd derivatives with respect to the coordinates and the other has 1st derivatives square quadratic things in the 1st derivative fortunately for us this particular combination when you take the time time components only has 1 of the 2 of the things with 2nd derivatives adapted differentiate metric twice is made up of derivatives of the Met wrecked the things which involve differentiating metric twice actually cancel between these 2 for the prime-time component not for the space Space space components but for the time time so on the right-hand side here on vast inside Excuse me things strictly proportional 2 squares Of 1st quadratic things 1st derivatives are that's 1 thing the 2nd factor is that the energy momentum and that these are the Einstein tensor here has 2 contribution 1 comes from derivatives with respect to space the other comes from the rivers with respect to kind things which have to do with only derivatives with respect the space couldn't care less that this kind of problem where they have to do with they have to do with the curvature of space the curvature of space itself here it is this has positive curvature this is negative press about 0 curvature and this has negative curvature of the curvature of space is 1 contribution in here the other has to duel With the waste space changing with time but the only way in which space changes with time is so the factor of any of TT soldiers give 1 factor which will have to do with it derivatives of 80 square 1 terms in this equation here while have to do will will have a factor of a doctor some kind of factor With a dark squared your work it out just what you find a top scorer over a square yet nuisance but you could calculated a dark squared already square b other terms which has to do with armed the curvature of space itself comes with a plus sign he was and think about for a moment how curve of spaces as a function of its radius the earth where thousand times bigger than it is at Google agreed that it would be less curved at least locally would look flattered to rest their favorite for marble it would be small but it's curvature would be large the curvature of any surface scales in a certain way is radius and fact that 1 over the radius square the curvature of us sphere for example is 1 over the curvature of the radius squared and he comes in but proportional torn over a square yard and that I heard is positive faces sphere 0 space is flat minus 1 if space is a hyperboloid it means that this Acadia and Kate Plus minus 1 or 0 so just a placeholder here that tells you which of the 3 kinds of places you talking about this similar
Einstein's equations abound in fact let's just us switch this to the other side to switch to the other side of becomes Martinez Katie all over a square but this equation is absolutely identical to the Newtonian version if we think of role as the mass density Arabia equations a Verdi explored the only thing I know is that we now have an interpretation of this strategy remember what this term stood for a minute and the county an example it stood for the energy whether the energy was positive and negative 0 it had to do with whether you were above all below the escape velocity at the same exact Karen E. I here are but it has to do with something entirely different or something on the face of a very very different namely the curvature of space Jamie equation somewhat different physical interpretation United ask Winders this general theory of relativity which contains among other things the ingredients a special relativity what White isn't the same as course and the reason it is basically that if you look at suppose universes curved but we just look on a small little piece of it was more little piece of it we can't tell the occurred look at all the galaxies in there the way these galaxies move for a small enough PCs must be b curvature can be important for small once we realize that the curvature can't be smaller and that the equations are exactly the same for a small piece of a big piece we realized that we must somehow reproduced Newton's equations but but any case we do the new attorney in our version is correct but we still have to keeping remember that in no means physics this stood for the mass density and setting speed of light equal toward so messy equals M C squared just equals M energy momentum same units on this was the density of ordinary that ordinary rest it was assumed In Newton's equations that everything is moving much lower than the speed of light I have a collection of particles all moving much lower than the speed of light energy density is just the density of met these equations of more general follow from Einstein's equations Andrew example they do will apply the situations where particles may be moving fast relative to each other and fact they even apply here from the energy density here is due to photon restless radiation moving with the speed of light soros the Newtonian equations wouldn't know what to do with full time the Einstein equations know what to do with 4 times or radiation but otherwise the equations work very were picked yes the toe 2 questions In that better where we assumed that space are being set yet giving Eddie Lee set forth by the that that the lifting of die if you it's a consequence of the mice after P and 8 if isotropic stays isotropic and homogeneous and stays homogeneously really has no alternative but to prove that the rebel Pruitt tonight but our but that is a consequence of that 1 but but but and you could actually choose any of them for example you could choose a space-based components were to a get will again be like except instead of being This is the the energy equation from the busy ethical semi equation articles and equation does have a 2nd I'm there it has a double there are linear combination of the space space in the space time time time equations which instead of looking like the energy equation really do look like a son of the Newton sequels and equation but there will of course point is a role equivalent they better be equivalent Why is it that you don't need more than just 1 of them well for this simple example there's only 1 function work to to calculate a shift to a more general context where a geometry maybe wavy and away the weather things you may need all of the equations because just a lot of functions to compute here for this case the only unknown is a seats only 1 function as some really you only need 1 equation it could pick any 1 of them you could not picked the makes space time equation then you would just got 0 equals 0 but if you take a space-based components Fugate the same equation that OK let me just remind you 1 more set of facts and then I want to discuss with the guy will break all discussed the equation of state and how it determines information about this world what we my
information about World War I mean how role would itself varies as a function of the scale factor is not much you can do do with this equation defied unless you know something more about rogue we don't know more things about role for example the role just made of ordinary particles sitting and the universe expands it's quite clear that the density of energy decreases only the know-how already down the minute the radiation it decreases in another way so how rural depends on a E depends on the nature of the material it's making up the energy but it's clear In order to solve this equation for you to even think about us having any content we have to know something about how we rolled depends on a if we know how Arroyo depends on AD this just becomes an equation a different an ordinary differential equation for aII as a function of time so I'm just gonna write down now Michael break in them but before I wanna right down to 0 examples 2 examples of of ones adjustment mentioned here they are they have names where called matter dominated simply corresponds still ordinary particles moving slowly relative to the match particles which are not moving so fast that we have to worry about relativity or even a kinetic energy very much their energy standing next particles its energy relative to you is simply its mast our called matter dominated and it's a case where energy density role is equal to some constant role naught score divided by a incidentally what is any was the meaning of a let's have a case let's take the spherical case this and the struggle case the meaning of a is extremely clear it's the radius of the universe at any given instant the radius of the sphere but think that a has a definite meaning are Of course we have to provide some units we could measure meters of St. Peter's then being meters aII is simply the radius of the universe and was wrong what role not is just a density at that time when 80 was equal to 1 at the the uterus had a radius of 1 meter now we wanna go FOR really our maybe make a pass that could be a better idea but just conceptually conceptually the meaning of it meeting Rome nor is there is the density whatever that density was when the universe was 1 meter large that's a tall noble thing you could change it it's sort of initial conditions so role it can't stand but every time you double radius of the universe the order change the scale of the universe by every scaling AD this city changes by the cube power of a that straightforward question now is what what called the offer in the the big man meant that doesn't violate it really really bad for but less not In the latter case which is considered a itself has no invariant meaning nothing to compare it with a new around case you could compare with a radius of the whole universe also in the negatively curved space is a natural definition of the radius the hyperbolic geometry of that the other case we talked about was radiation dominated Raul is wrong not divided by a thought I would talk that 3rd at length about the difference between these 0 the fact that if you had a bit of both the very early on this would be dominant because 4 0 8 before what is likely to be bigger than 108 CAQ late times this is dominant we talked about that not change everything is exactly the the same as before but 1 new piece of information remember the sign came here determined whether the universe is gonna continued to to expand linearly or whether it's Canary collapse solely from now is a correlation incidentally this will change a little bit when we come to to talk about dark energy but up till now with these forms of energy if Katie is positive assess fear case that corresponds to a situation where the universe 3 collapse collapses it if it's flat but it's as if every galaxy was exactly at the escape pilosity so it continues to expand ever slowing down slowing down asymptotically coming to rest but it doesn't recall apps but also on knife-edge and the last case it is Katie negative in which case that corresponds to be above the escape velocity and that case at late times 80 just continues to increase linearly as a function of time but as I said we didn't waste our time by doing the Tony in case the Newtonian Cason and Kasten correlated let's take a break for a few minutes but we want
understand these kind of equations better we if if the only possibilities were matter dominated radiation dominated we might not care very much harder give a more general understanding of these course what's the point of a general understanding only learn cases but of course there are many cases many things in between matter dominated radiation dominated and now only in between the more extreme as a whole range of possible behaviors like theirs and their report airport to understand and so we want a deeper understanding of the connections for different kinds of material what's called material for different kinds of material of how the energy density would change as a function of scale what is it that you need to know a Aruna got to listen to steps where you got to it 2 steps I don't know will finish tonight on but could be important ingredient important ingredient in determining how we rode depends as a function of 80 is the equation state is was called the equation of state we'll have 2 steps the 1st will be to us assume an equation of state and I will tell you exactly where mean by an equation of state to assume an equation of state and from that derive equations this 2nd the 1st step will be to assume an equation of state and derive the appropriate formula of this type the 2nd part will be to derive a different kinds of materials the equation the state of get the both of the nite tonight angling to assume the equation of state for what different words collect and equations estate I'll tell you what they correspond to the next time we will derive the relationship between equation of state and the kind of material would talk about our e-mail equation state well and equation state it the basic with thermodynamic idea and it's the relationship between can basically the relationship between thermodynamic variables describing a system for romp purposes temperature will not play any big world temperature is not going to be important thank you for everything is going to be pressure and energy density and particularly the equation of state is a relationship between energy density and pressure now even for ordinary even for an ordinary gas is quite clear that there's a connection to an ordinary guests are moving go particles is quite clear that there's a connection in this room for example between the energy density and the pressure the higher the energy density in what energy density we're talking about basically the kinetic energy of molecule is the M C squared which is dominant by far much bigger than anything else but let's forget the MC Square from Omar just get the idea there's a cannot make energy emotion of modules and dome became Connecticut edgy is proportional to the square of the velocity and so forth those same molecules bounce off the walls of the room exert pressure pretty clear that the fastener move the bigger the pressure on walls there so it's clear there's a connection between energy density and pressure and fusilli the way it goes along with some some exceptions to it it is the higher the energy density the pressure these examples that I usually studied most frequently and in fact these coverup pretty much of the ground interest the cosmologists committee described by very simple equations of state squares and state is the equation which tells you what the pressure as P as a function of the energy density and the usual equation estate because knowledge a study of over an hour His equation state that says that the pressure is on constant called W
W was a constant times the energy density role now in many cases of course that's not really true ruled that feel that the pressure is strictly linear function of the energy density but as this said Ferrer just moralist by accident be interesting cases have this form of pressure is some number firms role on our tell you right now what W is where the 2 cases of interest here later we will come back and arrived for the man dominated case the matter dominated cases basically the case where the molecules in the room if we want to about our arrest them moving very slowly compared to anything else are moving very slowly compared to the speed of light energy is mostly just 3 M C squared energy so 1st approximation you could just the molecules are rest room molecules are rests in the room the pressure on walls 0 within creates pressure is collisions with over the wall and so for the matter dominated world the pressure is equal to 0 of very very very small compared with the energy density the big huge equals M C squared energy density and essentially negligible pressure because things are moving forward W was equal to 0 his matter dominated I deleted arrive at that that that was obvious that the musical's arose about the heart case which it says is that I don't think we'll get the tonight but the next time than and that's important support Finkel understand is radiation but Bollywood and that Chase is W. equals 1 3rd Way the 3rd come from the Nets what is number dimensions Commonwealth OK with will that out it is yes it is 3 dimensions pressure equals 1 3rd role for radiation or W. equals 1 3rd and w 0 for a matter dominated now is the equation of state have to do it and I think I'm going to do is use equation state to derived how the energy density changes as a function of scale factor deal we use the simplest kind of thermodynamic um identities supposedly have a box and get a box of toxic material could be gas Kabila quicker be whatever happens to be carbon molecules career radiation doesn't matter has some energy and the energy of course the total energy is equal to the energy density kinds of volume of Riyadh of a box of the boxer volume V the energy density is role and their are imagined changing the volume of the box a little bit it doesn't really matter whether you change the volume of the box firm sorrow isotropically in all directions or any other directions but let's imagine changing it equally in all directions but how much does the energy change what's the change in energy if you change the volume by amount DVD emotionally and not from this equation From the work done by the pressure on the the walls of the box P DVD pressure TV buzzer positive or negative if the volume expands our happens to the energy in a box Barry you do work it does work on the box and therefore the work on the box means the munitions the energy in a box of minors PDV that's all we need to use such an offer that sets the basic identity there's another churned out a big nor the other terms for a roof over the thermodynamics PDS cheered temperature the change in entropy when variations lower and the universe is expanding slowly by comparison with any of the current time scale and the aghast when changes take place slowly entropy doesn't change that's a rule court 80 about change for up purposes change in entropy is 0 and this is a formula D equals PDV OK now the energy is wrote times let's calculated D he E just had differential energy and that's equal to Rome times the change in volume plus the volume times a change in Rock also changing which AT the volume changing the volume with Cheney energy density in some way in some as yet unspecified way changing the volume will change wrote but we don't know how yet however however it however whatever the rumors D E equals wrote as row DVD was Vijay Rao that's the left-hand side the EU on the right-hand side we have P beauty well we have chewed show this year with DVD let's go the together with poem over the left hand side and keep them together because they both have the same differential DVD on the left-hand side we have VT role and the right hand side we have 2 terms for smaller minus sign spur parentheses around them and a DVD what goes inside the parentheses P federer Pepe plus a because the says tour to the other side he plus wrote but now if we didn't know any connection between PM Raul would become stock we would know what to do about her with us but let's assume that he is known in terms of rock and in particular let's take a very simple case where p His W times world Allocca writers as Mike this 1 plus W rome DVD want is just a row Devi the W is the PDT Valley are
something we can work with 3 have equation involving 2 2 variables at their differential equations volume and energy density let's rewrite it throughout board from that pumps V. B Rome physical minus 1 plus WLS is the No. 1 post W. just the number road beauty and that's really good the equation so that on 1 side I have everything involving rolled and the other side I have everything involving volume to do that I just divide by role to get all the road dependence on her left it Bureau all wrong and provide by volume was got we got rid of rural and right now I derived by volume and here's our equation Bureau overall is that the told indeed the differential of logarithm of rock Bureau overall is differential of lava rock where about BV over these differential of Lockerbie so we can integrate this equation and it just says that right 0 this proportional it says that while remember 0 it is equal till 1 minus 1 plus W logarithm of reach Ukraine a constant numerical constant to it but constant over here a solve this 4 0 in terms of it just exponentially it was a saver says that role is equal my assigned here is your won over a volume of power warrantless W With again a constant constant appear different cars if just take the logarithm are both sides logarithm of Rome on the left and the right minus 1 plus W. plans allowed by volume of volume of this box if we think of it as a box which is expanding with the general expansion of the universe the volume of the box is just basically a Q is proportional bottom a constant its proportional EQ so it this is telling us now is how row varies with the scale factor struck down the equation says that rolled his some constant which or about bank divided by AT the scale factor the 3 times 1 plus W Aceh thermodynamic so that sort termed the thermodynamics of variety of 80 a nice homogeneous material would tell us that's how roh varies with scale factor constant being some constant again you could say that the constant adjust the value of the energy density when is equal toward OK let's see what it says W was equal to 0 that's the case of matter Dominic dominance W is equal to 0 up case of there this just role is equaled or constant over a assess what ATF W was eagled one-third 1 one-third for 3rd for thirds times 3 of 4 so that this equation over here to see there is a general framework in which these things emerge and really what you need to know is doubly described because myology based on some sort of energy were due need to know is the equation of state and very little more very little other than that the equation state in the form of pressure is W arms energy journey question I think this is a natural place to stop because think of ago past this overload due next up of a state in the radiation and what is the radiation dose this is like that it's mostly background yes yes almost all of it our Wyoming near factor it's the microwave background other photons rose 0 rebounds sunlight and Starlight kind tiny fraction of the gap almost all the year CNB husband might and it's a very today at the present time it's a very very small fraction of the energy in ordinary Adams which in turn is a somewhat small fraction of the of the dark matter so most of it is thought that some fraction of that 20 per trip remember exactly some 20 % is protons neutrons Adams and so forth and that very very tiny fraction of that roughly a thousand army of thousands queue up to a billionth of a 1 thousand and 1 thousand billion it is radiation today but you backward in into the past crimes at some point this becomes bigger than theirs becomes radiation dominated the early universe with radiation dominated universe today if it were not for the dark energy not forever for cosmological constant would be matters dominated right and again if we didn't worry about the cosmological constant we would say that this parameter case is correlated with EU future history of the universe where it collapses of continues to expect not emphasized that was wrong because of 1 or other ingredients started searching armed well I could tell you are gone far enough that I can tell you a little about doctors if we don't care where the equations came from then cosmological constant CCD your dark energy or vacuum energy while the same thing cosmological constant dark energy back in energy or thing is W. let's see what we get W. equals minus 1 W
Eagles mines Alou God isn't that the pressure and be energy density have the opposite side how orders that four-year could you think of many of the situation in which pressure might have the opposite side of the energy yacht I can give you 1 just where these box but to replace the box just by line in the amount one-dimensional where the box box as the line integral parts of ornamental and the physics is that joins line the whole held together by a spring crazy it's reasonable description of a of energy density but does have the property that the pressure was negative wiser pressing negative because pulling together establishing a part of the pressure is negative we call attention when pressures negative called attention the tendency to pull together but the big AU stretch it the more the potential energy source has the property that event the increasing potential energy Tom makes pressure more and more negative vats there were no Rome are usually pressure or it can it can it can you got can under certain circumstances like in your direction Is it because course of absolute 0 you might think you might think Wow discussed on springs and yet but is also quantum uncertainty energy so we can go eat away but yes it is possible for pressure to be negative as possible for pressure To increase in the negative cents as the energy density goes out of the W it was minus 1 means wife TYA why such a pension should exist that's another question but let's just examined its consequences if W is equal to minus 1 saying here we are adhered W. equals minus 1 where we get role is constant role is constant that's the nature of dark energy it does not change when you expand the size of the box at the character of our energy density In a box doesn't depend on how big the boxers just changing the size of a boxer doesn't do anything to the energy density those James D. energy in the box because the density white because that energy density is a property of and these space an empty space doesn't buy Lutwin you stretch brokered by but so let's just let's just well I'll tell you what I think I won't don't go through this tonight next time we will examine more critically the reason Y W was have various values and then study the behavior of the universe under the various kinds of conditions in particular study what happens if we have energy that note no things that we haven't seen before they for tonight the Peckham a couple questions but can you get tired it I thought that yes yup anything which could be regarded as nonrelativistic particles but particle could be a black hole it could be a plant to be a star can be anything that's just jumped the material that there is moving slowly relative to world of the ambient the background and who was energy is mostly in the form of its next Yao Yao said we solved did we that optional Bridgeton equation not exactly equation know if it was that the interpretation of this type to wear of being above or below the the escape velocity that equation is exactly the same I remind you of our for the case with cable 0 I remember the incident very good Bakewell's love in that case for the matter dominated world 80 when Dennis Tito the two-thirds right to two-thirds for the radiation dominated world award the square root of tea that's still true here is the status 0 that reflects base case is not flat then you have to correct it but go back a previous lectures and or be equations were study them in order to have those holes were between all tests for that be Lord God is with the lawyer given the more glad I understand that there is acres word lot of it would be me and a recent polls the force that of armed you can't think of a you could think of that just as an expansion of the universe which we will work out but it is at least at the Newtonian level Ucon market up by just saying that gravitational force between every pair of particles has an additional terminal which is linear with distance a force proportional distance a very small coefficient and do go through that but you could market up to command got the effect of vacuum energy of the cosmological constant by saying that every particle has a force with every other particle proportional distant the cosmological constant is positive and repulsive yup and the cosmological constant is negative attractive needless to say if it attractive that will cause the university collapse even faster than a course about the cosmological constant was repulsive it gives a chance forever for continued expansion even the whole universe might be closed so world will work if there are more questions from formal our please visit us at
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Titel Cosmology | Lecture 4
Serientitel Lecture Collection | Cosmology
Teil 4
Anzahl der Teile 9
Autor Susskind, Leonard
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DOI 10.5446/15062
Herausgeber Stanford University
Erscheinungsjahr 2013
Sprache Englisch

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Fachgebiet Physik
Abstract (February 4, 2013) Leonard Susskind introduces the Einstein field equations of general relativity and thermodynamic equations of state to the analysis of the expanding universe.

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