Cosmology | Lecture 2

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Cosmology | Lecture 2
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(September 21, 2013) Leonard Susskind solves the expansion equation for universes with zero total energy, and then adds a non-zero total energy term, which leads to an exploration of matter versus radiation dominated universes.
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review of a bit and then I wanna move on to are generalizations of what we talked about so far I think we worked out we worked out the equations of an expanding universe it would know his equations that the promise of meals for the villagers equations really get it right yacht know equations does get it right for the most part need to explain why I signs equations have to do with Kurt space time now the universal or ultimately going to study it is it has curved spacetime all right and this fact some versions of it even have curved space as simply means that space itself forget things just basic stuff if you measure triangles on it if you do various kinds of geometric exercises on you discover perhaps that's it occurred at the moment it looks pretty flat but it's possible that it will turn out on that the average to be curved end if it is curved well maybe it looks like a three-dimensional version but said was 4 years old study later tonight that maybe maybe partly portion of a sphere that for some of fear where over here we worked out we can only see so much we can't even really see the tequila this year's curved but a large enough distance we may be able to see the spheres curve on really can't suppose we just decide to lock and very neighboring galaxies now very neighboring galaxies Komeito billion light years from Earth now very very neighboring galaxies much more then what we think the radius of curvature of this university with a looks flat and if it looks flat it should mean that at least for that portion if we're not in the best of things but we just interested in the local nearby behavior we should we should not have to worry about the fact that curve if that's correct that it means that the way these galaxies move relative to each other and how they move apart from each other at least in the small here To me studied using Newton's equation actually been doing we've been looking at the university in the small and studying how old are small little fraction of it is expanding or not expanding whatever doing perfectly legitimate and in fact entirely consistent with Einstein with with relativity except for 1 thing rivers except for 1 thing we would run into trouble if the galaxies all it ever is pressed galaxies particles present if they were really moving past each other With a significant fraction of the speed of light 1 only assumption is that the neighboring things are moving relatively slowly with respect to whichever something very very far away may be moving with a large velocity relative to you but as long as the things nearby are moving with nonrelativistic velocities you could study relative to you you can take a small patch now small could mean 10 billion light years but during take a small patch of it and study of without using any relativity really if we discover that there are particles moving with close to the speed of light past each other then of course we will have to modify the equation but on particles moving fast by comparison with the speed of light us where are they traders the trio's for 1 by photons not only photon from the sun I mean photons it would be the area of the was no son the universe filled in the same way that filled with galaxies it's also filled with radiation homogeneous radiation that homogeneous radiation does more with the speed of light that means that we have to modify our equations somehow to account but very very vastly moving rapidly moving material photons widow that tonight but I 104 about when review but we did last time quickly we 1st of all of us said suppose the base is homogeneous and those with galaxies marker Trevor drawer galaxies before making a gas if you like can fill the blackboard with a certain number of particles per room per cubic meter curative density density that we called Rome analysis the content of the universe In kilograms per cubic meter if your life you could use some other units but whatever you to write physical units kilograms per cubic meter recall the role we laid down a great on universe and light now the greeted it was clear ambiguity imagine that we lay down the grid at some specific time like today we lay down the grid and you can't ask What is the space in between with grade was a coordinate system but call accord X and the distance between X equals something Annex equal something plus 1 another words 1 greeted the 1 great separation here 1 lattice separation is a certain distance associated with that how big that center well with Colbert AD how is it that depends on the good that we laid down if we laid down every course screwed it would be 1 thing if we laid down a fine grid it would be another thing until they beat that our equations at least at the moment do not prefer any specific value of a we could lay down a different grade 4 different grades it could be twice his debts so here's the black forms 1 grade and the black and green together form of great if every looked at these more dense grid will also invented AD let's call it a prime is a prime is the distance between neighboring points on the day and the dense grid that would be one-half so
if you ask me what is a value of a Aguilar said think I can tell you until I know precisely what grizzly bear and so it a itself doesn't have a physical meaning at least at this stage later on we will discuss more with a means but the moment it on a flat blackboard doesn't mean anything by itself until you specify exactly what the size of the group OK so aid is just a bookkeeping device on the other hand really CEOs of a means something only give you an example supposedly I told you would is a function of time universe is expanding the distance between neighboring galaxies to galaxies are embedded in the Great War with agreed the actual physical distance between galaxies is growing I suppose in psychology over a period of time a doubled back has meaning that means the distance between every galaxy every pair of galaxies doubles so ratios of a particularly ratios of AD different time frames are recording a history of how the universe is expanding or contracting and that's why are equations and got only involve radios of things with a figure example let's take a dot the time derivative of a well that will defend on whether I use the black grid or the Black Queen great every 80 associated with black grid is twice as big as every aII associated with a black green grid of every 8 is twice as big a . time derivative will also be twice as big money other and if I take a dive abided by a and let's compare it with 8 trying . divided by a prime they will be the same they will be the same because the ambiguity in the scale of the grid will cancel out a prime will be a FAA 80 . crime will be a half 8 . 8 million and a factor of 2 will just cancel out so things involving ratios of Eddie those invariant physically meaningful things and if you remember a over a is just the Hubble constant at a given time so really does mean something but I itself ambiguity and how big it is but a once and for all ambiguity went to fix it acquitted OK another thing which is ambiguous was we introduced a constant no was known as the man's contained within a single pubic of the grid another words a grid element here 1 by 1 by 1 by 1 1 1 1 great unit the amount the Mets within that killed which recalled Noel is ambiguous because we haven't figured we haven't determined exactly what the greatest never alas once the greatest nevertheless the amount of mass with a single is called but it is also changes when you change the grid now let's
go to these equations that we arrived last time equations we derive last time remind you how we derive their week Tucker you Harris filled with galaxies called particles uniformly replaced ourselves right at the center of arbitrarily but still consistently replaced ourselves at the center looked at another galaxy which was at a specific location on the grid some X summer Exxon mcgrory end we studied look Who's equations for the motion of that Galaxy which will I which it would think would be horrendously complicated horrendously complicated because they're interacting with the loads and loads of galaxies but we used the famous noted about 1st of all imagine smoothing things How would so that we could think of the year of the distribution of these galaxies it's moved smoothly uniform center the center centered at us excuse me don't use know theorem that told us that relative dots In the pool would system will we ever at arrest the forests on that galaxy only depends on the man's within next year as if all of the for concentrated at the center we just said let's concentrate all the mass within next year at the center and then studied how under such circumstances the Galaxy X would move if it were under the influence of just that you're after were about masses on the outside you won't have to worry about a mess on the inside and we worked it out in a similar way that we worked about the simplest way to work it out was just the same if there was a particle over here moving under the influence a fictitious man at the center then energy of that article would be conserved energy consisted of 2 terms well I have to go out come back to the derivation again charges for review their visionary quickly but I'll come back to in a moment remind you what the equation was the equation there was derived was that a got over a square that from the kinetic energy was equal told the density and that are rolled measured In known physical units in other words measured in kilograms per cubic meter mothers no object over here but
rolled and I was a little more wasn't 8 party if energy G noted that was the equation that uh every derived by just looking at it at the conservation of energy Rollo hopes role raw sorry didn't 0 3 that 3 ultimately came from the volume of a sphere for 3rd part IRQ with 3 of the 4 8 2 times for applying pies pie 3 3 this is related to the volume of some fear OK that was the equation that we worked out that was in the case of Zero energy that was in the case where the potential energy of this point and the kinetic energy of it exactly canceled it could also be understood as the situation where every galaxy is exactly at the escape of city just a knife-edge between being able to escape and not escaped that Formula One we 1 more thing to it and the 1 more thing is we said that role Bravo which is the amount the Mets amount of per unit volume is related to know is the same as not use Newell is the amount of mass From grid box but a gray box your North volume is this is a mass within 1 Q if you want density and the density is related to Newell by dividing ruled by the volume of 1 of his boxers misses density for you because you're cubic meter this is for cubic box you divide that by the volume of box which is a Q and that's 1 more factor we put in that role is newly divided by a Q now is role constant with time no rule because any changes but no constant with time if no just represents ordinary particles sitting in the universe and never destroyed never created particles of photons let's forget the proton decays for the moment but times are forever and a galaxies of forever if for a moment last forever we all we won't believe that forever but we will believe for the moment the galaxies are forever get that or chaos galaxies were forever protons were forever the Knollwood because the number of protons with a box is the same at all times or happened the box grows so the proton spin out but the number of the box sex and so no it stays constant What is it its numerical value I can't tell you what the numerical value is because that has to do with the exact that I used a change great I changed up case once established all of the convention so I can replace role by Newell divided by 8 know it is a constant doesn't change with time 8 is a constant fight a constant 3 is a constant G is a constant all of this stuff here is just a numerical constant in fact by judiciously changing your definition of the grit since changing the great changes the magnitude of new I can't find life I can apply like choose 8 pie no over 3 could just be 1 it doesn't make any difference the point here is the numerical constants that appears here really doesn't make any big difference the way the university evolves and wait and see that is by changing definitions of the good you contains the constant so we could flee like just set a pie Jean who over 3 eagled 1 and then we would have a very nice differential equation a dot already squared is equal to 1 over killed now would you like to see us all that we saw that last part but we saw by
guessing and many people seen real Time and knowing you you'll see me solve their panties are hoping I'll make a mistake but yeah do we miss questions or cake while still bad the Solver would you do unmarked here sovereignty is only a sore always equation you 1st of all saw use 1st take the square root of both sides on the left side as easy it's just a dog already on the right hand side it's a little bit of a since it's 1 already aII kinds square related you see where that came from a cube is a squared times a square of regret a Times Square Root
OK but to multiply both equations a both sides by 8 to get rid of it in the Obama made at and actress tells us that 80 . is 1 divided by the square root of a rift that that's where would like to begin OK right this debate by DG which is what it is is equal to 1 over square today but now we can do something tricky the very tricky thing we're gonna do is instead of thinking of b as a function of time we think of kind as a function of age where take 8 to be the independent variables entered Hill that here is being that looks like a function of time and would differentiated in it but I turned upside down is deep meaning taking 1 over at its DTD aid and now it looks like it's the derivative of time with respect to and that's equal to the square root so how we saw that we look further function of aide whose school whose derivative is the square root of 8 was functions as is its derivative the square root of 8 803 3 right well two-thirds 803 3 hands so we find our team is equal to two-thirds of two-thirds is absorbent the constants that we've been numbed ignoring times 82 the the read head what we can solve that now we really wanted was a is function of time but that's easy our free just neglected to 3rd with adjust of like riding it to get the two-thirds this tells us that AD is equal to key through two-thirds a Eagles 238 Tito the two-thirds which is a function any key which looks like that and it gets flatter and flatter and flatter the flattening of it is deceleration universe is decelerating but it never comes the rest we could see that it never comes to rest because we look at this equation there was no point at which a diet of 80 becomes Iran I know it doesn't become 0 because the right hand side is never 0 Ricketts smaller and smaller they get bigger and bigger but never goes 0 and that's why the universe decelerate another way to see that it decelerated just a particle moving away from a fixed for sent there to decelerate because of the attraction by force and that was that was lesson number 1 I really did it because swirled where we think it's worthwhile or are we doing most important derivation but yes it's OK for amigo square wrote on was the logic the logic funny equation you cannot show with the universe is expanding contracting both are possible from the equation if you start it's exactly the same deal as saying you have a plan over here coffee is good planning rise is years apart as rock and you see the rock right over here the equations of Newton will not tell you the Caracas moving away or words moving toward there is 1 thing you can't be sure of let's unless let's ignore for the moment moving in orbit let's assume we know will that be the Caracas radially is 1 thing we know for sure that's not standing still may be momentarily at rest might have come up start but it's not going to stand still is accelerating its accelerating back toward the plan so it could be going out decelerating or it could be going in and you can't tell from the equation the same way you can't tell from the equation whether it's expanding contracting and that has to do with the 2 possible solutions arms whether DDT is positive contracting its negative it's expanding Foster OK the ball is always out movies radio all that all of us about whether this docket Dutch right that's right but still on the average came To move as if they were embedded in the script given that their embedded in the grid Ole the only option they have to move radially relative to each other now you're perfectly right up there Our motions on top of that average motion the emotions on Top of the average Martian don't of course satisfies the call by astronomers peculiar mounted a peculiar motion meaning however peculiar but our son moves around surrogates moves around the sun and is not a participating in this expansion wrote of son but there few go a few galaxies over the whole thing is Charles the real emotional real galaxies is a combination of the average floor figures of 4 River going downstream how fast as any given and molecule moving well I don't how players in a given was moving but I do not on average the clumps of molecules sufficiently average are moving with whatever the velocity of the rivers on top of that is the peculiar motion of the molecules relative to each other so that's the way you think about a flow and on top of the flow flocculation similar ignoring the fluctuation of the mob OK a quick how bestow directions I want to expand the number remarks Mr. directions I want to generalize it today the 1st well probably the 2nd the 2nd will have to do With what happens if you replace massive particles galaxies of protons and so forth by folk art by radiation how does the whole story look different if there was a universe of fictitious uterus which had nothing in it photons how different with this being a why my doing that but a silly thing to go Whitaker fictitious universe which is only made of photons well don't define is that earlier all In the history of the universe the most important thing numerous will time today the most and when I say most important I mean the largest concentrations of energy early on the most concentrated the most the biggest form of energy in the universe was radiation could AD the most dominant form of energy in the universe adjust the masses of particles equals MC Square kind of a kind of energy that this serious about its about the behavior of massive particles nonrelativistic slowly moving particles yacht that also presumably made of particles pretty much standing still talk along if if if 30
years ago we headed back by will you with still 30 is back with 25 years backers here you're right good good good tedious but documentary by contrast is part of the guests thank you I was helped of young or what I'm telling you now is an old story but it's a story was bases whose 6 yeah important to us OK 2 directions I wanna go 1st has to do with what happened at the universe is made up of the kind of stuff other than just a particles more arrest auto to whichever radiation particular and the other is the move beyond the assumption the energy is Iran was no reason for that with arbitrary we arbitrarily let's assume that every galaxy is moving away with exactly the escape velocity but back off that now and how we back off to go back to the energy equations and save energy of that particle moving in the fictitious field but the fictitious Gilberto field under fictitious concentrated Nancy the center is not a real galaxy now the combined man of Everything within the Greens fear it for particle out here molding outward and let's apply the conservation of energy to it but go back through the derivation apply the conservation of energy and then see how the equations change when we go away from the limit of 0 energy OK so what is the energy of the particle this whole glob here has met em for combat toward members moment but has missed him it's not a mass of the sun the massive any specific Galaxy it's a mess of everything in this sphere and the energy of this fellow over here it is 1 half M and is Mass Of this fellow marked a big mess in here the of this galaxy times velocity squared that's what it terms and the other terms is minus and GE over the distance between them but call distance D as we did last time I missed something good nite another factor right back product of the masses Newton's constant distance between them that seat at energy of the particle moving out what only know about it we know that it doesn't change with time it doesn't change with time because our model that that's a useful model is just the motion of a particle in a fix In a fixed background of mass energy is constant and so let's set the energy to be equal to some constant I was constant mean doesn't need a numerical constant that's independent of everything else know it doesn't it could depend on which particle with talking about court In fact if the homogeneous and isotropic the only real thing could depend on is the distance away so I will ask simplified the discussion from now by taking this particle could be at x equals 1 wet In other words will take a specific particle very definite 1 at equals 1 and focus on it if has some specific energy and that energy will never change so let's just call it he that is a constant what value the constant I don't know I can't tell you offhand no more than I can tell you if I know that this is not Over here a planet over here is no way I can tell you a priority with the energy of the particle is at the heart of molding so we have to study all possible cases so just put the energy and quality he can't take it to be a constant numerical constant let's right out now everything out provide His 1st of all divided by him malaria but survived by little and abide by little and analogy we get can now is a constant end a constant eel with him as a constant so I really have made any more complicated by dividing by and I'll leave it here is over area but the right side is just the numerical constant that's even do a little more multiply by 2 it's still just a constant and since I didn't tell you in the beginning what he was although I know was as is a constant source of much of its lead evaporate know about the velocity if this is a galaxy ad x equals 1 and then it's velocity but to distance the distance is any kind X the velocity it is 8 . 9 x and similar can just plug that India an X is what I chose an X to be 1 so it's very simple the dissidents Of the galaxy is just scale factor a called scale factor and velocity is just a dot so put it in here Mr. says a dark squared minus through a N G over D & D is it is equal the some constant thunderstruck Aussie constant on the right-hand side the says many of the elements of this equation here but this equation is nice equation we divided by a twice to get a of a square when I do that because I know in the back of my head that in itself is not a meaningful thing it's only ratios of ladies sold the back of my head I knew that what I really wanna get is an equation for ratios of Haiti's or 80 . divided by so I devised this equation by a square take noticed some things happened on the right-hand side here it's no longer constant is your constant but a square now has something new in it and finally we you is that the man divided by the volume Q now this is a sphere is a sphere of radius a X is equal to 1 a sphere of radius AT and the volume of a square of a sphere of radius is for a 3rd time a queue so I had a cube down here that's not quite the volume of the sphere let's fix it so that is the volume of the sphere that's multiplied much of this equation too much for free ride at a dart over a
squared minus twice L G all 8 Q is equal to or constant over a squared but now but to multiply and divide by for 3rd ii services pie over 3 and a half to multiply out here but for a 3rd fighting that becomes 8 thirds plight as were a 3rd pie came from OK for 3rd fight a tube of a phrase that is the volume of the old was the massive either by the volume density so by fiddling around I got the Freeman equation but right-hand side now a right-hand side that knows about the total energy if seized positive it meant the total energy was positive in kinetic energy outweigh the potential and dives means the galaxies and have continued to recede forever be beaten video escape velocity of season they did then energy is negative more potential energy more negative potential energy and a kinetic energy and that's a situation where you expect everything to go our way and then come back and crashed self we now have a new equitation the Friedman equation for non-zero energy energy could be positive or negative positive or negative both are would energy Pope the let's examine a little bit and see if we can see a little bit worried says it's not I want to solve for any given case but the votes it's not sovereign wealth 1 of that 1 of the stepped right over here that already squared and let's trends posed this factor to the other side equals a 3rd applying G times role but role is newly divided by a key member no no wasn't was mass per unit cell size is no over a cube row and G G is there plus see only a square This is a new thing COA squared is a new thing that was not here in our initial study of this equation was it do 5 so Houlihan let's look at it 1st Is it a real Friedman equations with this with this right him and it was derived from general relativity not from special relativity starting out from under but nevertheless we just arrived the from noted OK so here we are let's see what these things mean let's assume for a moment the universe continues to grow if he is positive if is positive and everything else on the right-hand side positive 8 by G newly a cube there'll positive than the right hand side is strictly positive an 8 . 0 a is positive under the words the uterus continues to grow 8 . is positive and a sponsor a doctor time to remove a positive universe continues to grow forever it may slow down slow down but it will continue to grow storm ripped open tractor will continue to grow so on if is positive universal continue expand it will eventually get arbitrarily Big might arbitrarily bigamy he will eventually exceed any bound and become very large so let's look at it in the limit that a becomes a much larger very very large where it they got Over right the square as part but the square and that has to be positive so let's assume that initially it is possible that uterus starts at expanding then the only way Connecticut to be negative it is of the goes through 0 Kinkel 2 0 of the right hand side pass to the point by the right hand side being permanently positive disallow was this from being 0 and therefore we continue to grow can't jump from positive to negative without going to 0 in the leg never 0 OK what's term he was bigger a school a square 1 of a like depends on whether it is legal small right but a particular let's go and think about 3 large L the small 1st but hey being small which is bigger 108 Cuba while a square 108 . if you take 8 could be small enough for this will always be with us and sofa sufficiently small 80 this is negligible compared the best we've already studied the case where are only know the answer when only answer long as a small under the words in the very early phases of the expansion when a is just starting out growing is just beginning to world this turn is not a point where we find we found out when the firm was absent that AT expands likely to the but to the other limits this a standard way of thinking about all kinds of things go to limit often an equation is this equation solver boards Bentley but nasty we don't have to solve we just have to know what it is doing it the tool In 2 wins means when a very small and when is very big OK so what I want is very big which terms this there this was the biggest Germany's big where a square is much much bigger than 1 of a few odd you may have to wait a while and how how long you have to the wait may depend on this constant here but eventually this firm would become much larger select study that equation but 70 equation 8 . all that a school is equal to some constant any constant already already squared when we 1st take the square root of it want a square constant another constants are won't change its name but it's really a is new see here the the square root of the Old City of course the reason I'm allowed to do that because it doesn't matter what the constant is here will get mail blue which is qualitatively the same for every state I'm a dollar a a knowledge multiply by a 8 . it is a constant 80 . is a constant a doctor's the velocity of the galaxy
at x equals 1 another words in a situation where the energy is positive when the Galaxy gets far enough away the effect of gravity becomes negligible and it just moves off the uniform velocity saying a is just constant and that tells us that aII is just proportional kind constancy and kind Galaxy just wolves off of uniform velocity so that tells us this is the kind a positive see a positive constant C is not the speed of light here years over what happens to be a velocity so it tells us that at late Kareem this does Moses a straight line With a constant slope 80 . is just a slope it was constant slope and the uterus just expands would lean years on accelerated none accelerated motion nothing is a deep is going on here my using nuts what I should be using Apple's woes saying his father's apart enough so the escapes from gravity with a good margin with a margin there with his mobile off uniformly with constant velocity so fact we find that we find exactly that what happens in between where's the crossover point leave that you you could figure out approximately where the crossover point is because as 8 of the Tito the two-thirds and the makes a transition to the key and fuzzy in between that's the matters dominated universe wiser called the matter dominated universe Our because these right here society Of this equation contains a turn which is just the density of ordinary matter ordinary nonrelativistic slowly moving matter that's OK so old that go to the other case what happens if he is negative that's the case of negative energy may should be able and gets a particle of the Apple moving away from the earth with firm West of escape velocity has negative energy in default back so that's actually not hard to work out understood tell you what happened what what happens 1st or happens and you have a negative sign here a negative sign here and the right he inside can become 0 initially this is the biggest at a later time this is the biggest hurdle but they have opposite signs that means somewhere in between this became 0 that is nothing but the point where a backed up going Apple simply is momentarily arrest when 8 . 0 8 0 0 4 the bonds and there opec so was see negative does a crossover and instead of moving off linearly like this back crashed Everything false together that's a matter dominated universe and it was a classic cosmology until a until other things with discovered OK any questions but a was my my assumption when I get no questions either of perfectly clear a perfectly it is what some draw judge yet were a word that said I made cells which goes right now I've become negative the I think that where your but I was that nite and right where they were wrote are you have to take a square root square root could have our party outside clients I know it's not only the right said so old without is a chance led by state Arizona no I made an assumption that that from the universe was expanding and that breaks down at a point where it comes to rest the mathematics is entirely identical terror 1st the yard Norwood a right arrive warm air every square is equal to GE M although the dissidents murmur they times arrest of the earth Our job minus this is equal to the energy of theology is negative votes court minus their Rice's plus Jew M M all basically or the distant descent equation lumber how does it happen if these guys he is negative he is negative and there can be a crossover point this is exactly the same exact with us you you know what happens in all the but these Apple just pulled back down but at the cost of taking the route of the of applause route and minus of a square root but if whatever over a few that if you otherwise south burgers negative that about we said that after reviewing for you Net hockey player at the point is Aitken never get big enough for lot turning asylum can never get big enough but a is bigger than this term was bigger I will go negative but never does get negative normal pocket but the right side never never gets you're right around inside and maybe it never get negative all that tells you is as an upper bound on how big a gets right a gets so big and stopped getting bigger and
comes back down 80 is always in a region where this is positive but goes to 0 going through zeros is coming to rest of backed down back and on the way back down you have to take the negative they got over 80 is negative square it's topic that that is not matter dominated numerous 3 possibilities posit managing negative energy 0 0 energy and 3 different behaviors decides whether the energy is positive or negative not only are find out that b connections on the positive list of negative news is connected with the geometry of the universe not tonight next time will talk about the connection with his constancy with a spatial geometry of the universe that's what that means thing at this stage the general relativity brought to bear on the equations are exactly the same for the significance of this constancy takes on a new dimension and has to do with geometry but not tonight 1 more thing limited tonight will take up the the question of what happens if instead of being made and material points slowly moving and what happens if the universe is made out of 4 parts made out of radiation I danced in that of course we really do have to think about relativity but really there's only 1 important idea it's just equals M C squared if we widowed done Einstein's equations the right-hand side of Einstein's equations right side left basically let partly with the connection of Einstein's equation on the left we have a already squared and that I'm going to a head plus always squared transport the left and the right hand side glow a pie over 3 G role nor obey orders role now is a mail density here but Einstein's equations are of the form that something on the left having to do with geometry is equal on the right side of things that have to do with the density of energy and momentum energy momentum tensor on the right-hand side geometry the left is clearly has to do with geometry the rate at which the universe is expanding our abroad tell you next time the sealed here has to do with the curvature of space of this idea here is Einstein's geometric side the right hand side the source namely the energy momentum whatever it is in universe what is the energy on the right-hand side of this equation it's just the Mass points and equals M C squared so all my goodness and calling the sea but please don't get the idea that the speed of light sometimes called Kappa miners cap but sometimes archery written minus Kappa unfortunately kaput is positive energy is negative and captors negative images positive throw from call kept OK after that the basic connection with general theory of relativity the only thing that we really need to do to this equation is to remember bad when you go from that when you go from Newton to relativity will was mass density becomes energy density all forms of energy density all forms of energy density always gone awry he inside well energy density times the speed of light squared is that right now and his iii oversee squared so its energy density divided by sea square which goes on here want sits equal to 1 when I worry about speed of light speed of light physical 1 with that idea in mind the right hand side instead of being the dense masters becomes the energy density total energy density some of which is just the Eagles in C squared energy of the particles arrest and some of Mike b Connecticut Energy of particles which we haven't really included at this point the kinetic energy of a relative of fast motion of them but if the universe is filled with times then it's radiation energy after what talk a radiation energy and how it is different from the Eagles MC Square energy always have to do is think about a box a box now the box box just corresponds to review it Q integrated here the Kwan unit Cuba the grid there's a volume of 8 Q but suppose it has a certain number of photon now photons have a wave so here's our box Delta x equals 1 each side as volume is a cube the not be O'Leary particles which are just sitting in that box and not moving they have an energy which is just a man doesn't change their energy doesn't change those particles adjusted ask nothing else but a photon behaves differently so I'll tell you how awful behaves the 1st thing cannot afford on electromagnetic radiation forcing them out of our time is that its energy is related to to its Waverly it's related to its wavelength the plants constant each roast lamb that a wavelength and the speed of light which are set equal to 1 important thing is that worn all over the wavelength now something which I am not going to prove but I'm going to tell you is that if you take a box with a photon in it ever given wavelength your photon given wavelength and you expand the box to expand the box fully for example use in the universe is expanding Presler takes 10 billion years for it could double in size British on you take a box and you expand the box slowly them what happens to the photons inside what happens is the photons inside Is there wavelength just stretches in proportion with a box now this phenomenon anybody who plays a guitar nor year the from armory well the box is replaced his string of the Katonah it's pinned down at 1 end of bridge over I at the other end its pinned down by your fingers at a threat and block it resource vibrate and then of slide if anger what
happens is is larger effectively changing the size of the box changing the size of the box became the size of a box the wavelength just changes and the cloned Serbia the note changes correspondence with the year with a change in the size of box the same thing happens to radiation a box radiation a box the wavelengths just stretch in proportion to the size of the box because the wavelengths change from afar that means energy of each photon changes as you change the size of a box you world was filled up with photons you could pretend that each of the 4 times the number of photons image box stays fixed but so but true statement number of photon stays fixed but then energy changes as you change the size of the box and particularly energy of each fault-tolerant will be proportional for 1 divided by the size of a box but we pick 1 box Jesus scale factor in the energy of each photon decreases same phenomena as he frequency going back down or up as you change the year of the lake the effect of lowering the video of a guitar string so well let me down traces of boss the what Birgitta saved by say they just free space did he said they what we're all of us get near the spot where you're you're up that 1 will think about 20 thick about it that better you about or and then they horrible I never used a word or a fast you did fans are depends on how would you could conclusion but but it but that that is willing work for the change yacht on we have to do a little more quantum mechanics of more or more classical electromagnetism in the presence of an expanding universe to justify let's take it let's take it for the moment a given elegant come back toward the answers it is a correct statement that that power and unsavory of think of an example Chad had offhand I can't think of a good practical example or where or something similar happens but nevertheless it is true that the radiation in the box the wavelength of the Ford Taurus readjust themselves to the sizes of bar of a box another words of photons stretch along with the users feel like you can just think of the space stretching and with it the photon wavelengths stretchers let's let's leave it that that the tonight and I'll leave it at that but just examine the consequences of a problem what's examine the consequences are the consequences are it is ahead that energy for photon decreases by 1 only 8 in contrast to the case with ordinary particles with their man stays the same doesn't vary sold by contrast with particles case ordinary particle case energy in the box the energy in the box the mess in the box effective energy on the not stay constant would decrease total energy in the box decreases like 1 over the scale factor every photo on its energy decreases wickets stretched so every single 1 of them and compared with the previous case is 1 more factor of a year in the denominator the energy in the box scores was like war already and the energy density was like 1 or later before instead of 1 108 killed previously br density here we said was a constant Newell divided by a Q now we get 1 more factor of 80 in the denominator and it's just the fact that every particle its energy decreases by 1 x power of 80 that means relative to the previous case is an extra 80 in the Promenade no could be taken now to to be the number of photons punitive are they in a box and energy would be 1 of before that's only new thing that happens this is matter dominated and this is radiation a . already squared and a by over 3 G some constant new way and an 8th of a now downstairs we can also put in this term he minus sealed a squared but let's study the case when the case correspond 0 energy just to see the different to see the difference with Stewart different behavior we get With this formula this formula says away again by appropriate choices Of the size of the grid you can make all the constants here a few light B 1 by appropriate choice of the size of the grade you can rearrange it so that it's just a . already school is 1 divided by heat before what was a before dilemma what are so we figure out what happens again with solve the equation now the steps of solving the equation for status where wrote 8 . 0 8 Eagles 1 of a squared talk that means that a doctor is just 1 more way a right could do right St ADT would do the same trick as before turn and upside down by D. any physical that so now if we think of a V
independent variable were looking for a function was derivative is just a what functioned as derivative Eddie a square right now a square for half a square to be exact so let's say is that he was like a squared maybe there's a one-half yet but that but that's not for thing is the key varies like a square alright but the scale factor varies like Quito the one-half square root of 8 well I'm not much artist and I can't really draw the difference between Keaton the two-thirds and keep 1 half their of the two-thirds Peter the 1 half is a little bit smaller grows fast data but look pretty similar and I was pretty much the same thing you have to be an astronomer a really interested in this up to care about the difference between the 1 by that I mean are the have sufficiently similar qualitatively similar but if you really wanna know what's going on the universe the difference between Tito the 1 Peter two-thirds to be very important OK so that the radiation dominated universe radiation dominated universe expands the square root of Katie that 3 2 Key to two-thirds Tito the two-thirds in Tetovo warned half Patricia what about the mixed case supposing universe has both ordinary particles and radiation like the real universe really got laughs worry about that but that's come about the mixed case either pure radiation not pure nonrelativistic particles and that Casey energy density is 2 components 1 for radiation icons like were made of fought and won for ordinary matter because like 1 of a Q so kind of equation that I have writer details on the right down the general form will be a doubt over a squared and that's going to have to to terms now 1 of which course constant No. 1 divided by Q. We ordinary particles constancy 1 is just some measure of harm a number of our number in each box and then another turn that both positive energy of particles and the energy of photons is positive and the other is some other accounts end of a for that's the equation of motion for a universe which contains like our real uterus does ordinary nonrelativistic matter plus radiation which 1 is more important 1 or a few the 108 0 4 those are they right when Renee's big which 1 is abroad buzz warned excuse me my mother always told me that the text would be by firmly after a small this is more important than a smaller a is the more more important areas when a is big this is more important than again swamps militarily when is really big so that tells us without that too much work it is that when a key is small and the beginning of the expansion and the only thing that's important is the radiation this was a radiation turn as dominant compared to a material protons neutrons galaxies normal galaxies the beginning and the radiation was the most important thing answer the universe started expanding like 1 half T but they eventually this term took over became the more important than animators premier again over color and switch has started to grow like Tito to 3rd but shares works yes yes that's correct that's correct and you might say well highly Noble Energy of matter doesn't get converted To energy radiation that's something we have to discuss the answer is they when he arm when things it cold enough the universe expands enough things cool down and there's not much exchange between radiation ordinary matter and they are pretty welcome each 1 separately they don't talk to each other too much in fact once things cooled down to a certain temperature at a at a high temperature of 1 thousand the Gripper Morga thousand degrees 10 thousand degrees but wants things cool down through certain temperature pretty much the photons are just free streaming in don't care much about America about the particles and the photons to wait too long wavelength Stevens scatter particles very much so that's right that's exactly right that we assume that the took constants here are pretty much time in fight that's that that's the nature of the universe built of 2 components that's only 2 components at only 2 components and there is a 3rd comply there is a 3rd component the discovered dark energy best organ talk about next time we have ordinary matter we have radiation those are commonplace Maine dark matter belongs to the matter category it's simply up part are very ordinary particle matter that indivisible simply because it doesn't have any charge was radiate much so we don't see a lot that going on that's part of see 1 this should be seek to hear rights to or maybe we should call C. N. from nadir and see radiation from radiation pass there's more than 1 component of the radiation for cartons is presumably Gravitt are and their neutrinos now neutrinos Have maps but the masses so tiny that even today he the neutrinos are moving with very close to the speed of light and simply mimic the same behavior as the year for servers radiation component the radiation component consists of all particles were just so light that that will remove close to the speed of light the 1 last consists of all particles which are heavy enough that European basically at rest relative passed nearby and the bats that's because Molidae it was known as it was known 30 years ago is the Cosmo due 30 years ago parts at the box that they wait wait wait stretching at the end the total I bought priest they said is the decreasing doing work on the walls to expand their wealth another laser says is going into the kinetic energy the kinetic energy of expansion busy quite is similarly was acquitted of conservation of energy fifa came from so what if you think about it that way and start tracing after this part of the energy decreases in Bend Ore let's let's put it this way let's write the equation different form from a minus sign over here
but the 1st separate times when miners put my and make it equals 0 I haven't changed anything now change sign of the whole thing might have plus plus this equation reads the following way because a certain amount of energy and medicine radiation changes with time that energy gets transferred for a negative Turner does what with a little bit bizarre is energy associated with the Connecticut Energy of expansion of the kinetic energy expansion this here is negative this can be read as a conservation of energy we read energy never changes because always 0 contains 3 terms matter radiation and turn that has to do with the rate of change of expansion set off normally in the ordinary world where the rare Werner rate which generate rate of expansion rate of expansion is very very small hanging number this year's kind numbered our world of the present part playing there's never changes very much and so will not be subzero that I think it is the premium a number but doesn't change much over the course of time the some of these 2 cases 0 0 and stay 0 that the conservation of energy by saying that right a mouse it right now energy what would you think about it that there is the fact that he is the key is the way to think about you could say the universe is a time-dependent background of everything moves and it cost time-dependent them Angel here equations have a time-dependent is no longer a time translations a retreat in energy is doesn't have to be so the other way to say it is it is a time translations symmetry did you start the universe at this time of that time at that time you get exactly the same response to stun the university equals minus 7 it would be exactly the same starting university was forced for and thinking about it that way you'll discover another attorney energy associated with the expansion rate us not gone toward Harrison was very resource for while I want spend at least 15 minutes on there were greater energy conservation but just to keep in mind that these equations their origin was energy conservation and so is no way that we could be violating energy now it should be what you could drive through bitterly mechanic store Carol that totally mechanics closes a black geometry and relativity they Kirkia trimmed geometry of space fears based on for Our geometry respects Einstein's equations permit flat face these are OK with that state's less but you can't drive from about face state that he had to clear where they are written say that figure races Yao but With if he 12 my Kapolei square this had to do with the public retreat the fixed energy if cap as positive in this equation energy is negative the captors negative energy positive sorry if it goes this way Cappaert is minor see energy century forever I know I'm wrong OK does 1 way or the other on this term over here has to do with the curvature of space wear study 3 cases there 3 interesting geometries would be interested in is a flat geometry where space itself was flat Gravano on endlessly homogeneously and triangles have the usual properties that's a case Cagle 0 is a case where the uterus is curved like a sphere of your around the corruptive assigned to him around the finite and compact that's a case Skakel's positive a positive number here corresponds to a radius of curvature of the year of the expanding universe expense likable in August the classic picture of the expanding balloons at Skakel's positive K equals negative negative weaker space have to talk about negatively Kurds spaces like less easy to visualize so the next time it's exactly what we are doing program the geometry of space and the 3 cases flat positively curved negatively curve they correspond to 0 positive or negative With that that 1 Cheney has well except didn't up the arm the case with take not being equal 0 if you want but they're too little bit of radio fake it now it is right a bottle of what they there's not going lot that but is both does work on Bybee read take a little box literal boxwood reflecting walls their boxer reflecting walls and ask what happens to the energy in the box or expand the year walls of for what does happen the pressure in a box this pressure exhibited by photons bouncing off a box the pressure when you expand the box does work on the walls of the box for a real box theology could go to a number of places they could go to stretching the distance between the molecules in the box it could go into making kinetic energy of the walls of the box a 1 way or another it doesn't work on the box and increases the energy of the box itself either by stretching area of books Lost Springs there inside the box the molecules together or by causing the box they have a kinetic energy the kinetic energy expansion or whatever else just heating the box just heeding the yes the bigger yarns and subtract your that will talk about that Auburn conservation more quickly wrapped but it's still it's to correct that analyze it it is not obvious but correct that's correct the analyze it as if they were stuff in a box and the stuff that gets reflected off here From real boxer reflecting walls is compensated for in the think box case by every particle which goes across here is approximately compensated for by a particle which comes in side on the average plus of the expanding space behaves like an expanding box particles instead of reflecting go from 1 box the other but on averages many particles go from the box box from that box for the Xbox and so it is stands we must stop by
saying that be particles reflect reflect both of box increasing the size of the box means a work is done against the walls and will have to talk about what that means in general relativity come to your house the 1st full day of work radiation would thoroughly and umber but 11 years and then sailed that she said the move was a little bit better get what that observed that he did 1 stench 7 because will that that's the expansion history of the universe and a member of the looking Out of the universe you effectively history looking to different distances using the university different time so by looking at the University different times assuming that homogeneous you can reconstruct the year of the history of expansion we'll talk about that but it's something that we can do basically by looking at a combination of density of galaxies is armed sky distance of galaxies would go spend a little bit of time going through that the combination on Keppel measurements Our distance density and so forth allows us to reconstruct the history and reconstruct the history because was seen at at different times like do he want with a new chapter Duff they will be off that yup yup and combining it with Percy and be endorsed things her dad was before galaxies formed the but were temperature history of universes there is plenty of evidence that the pictures right calf now key to the two-thirds is easy to see the key to the 1 half a deserters largely has to do with microwave background and things like that but get the equations that then will go and talk about how we know how much of his pictures right any answers North wrong the job OK now what I wanted to say is not all of it can beat to the dish yes said but not for what could you see like that bladder well you at their square really says or what their principal that things cannot can't exceed the speed of light yet what'd literally says as you can see anything going past you speed of light Our it doesn't say that 2 objects in an expanding universe can't be receding from each other and fast the speed of light that that as quite a and as you point out the error of the fuel have formula last equal distance by the Hubble constant and you make a distance big enough velocity will sure enough extreme dispute like mine it's connected with the idea of a horizon but but you're alright these equations say that things can move received from each other as fast as white and but for sure they don't allow our you to send signals in times faster than the speed of light get from 1 place to another and they don't allow you to witnessed something moving past you speed of light so far his cheeks so that the right before fisheries exceeds like richly disappeared since guest repressed Ferber horizon norms but that has to do with dark energy all that has be doctor and you that dark energy nothing would really move out of our our ability to see I will show you how the geometry work summer Orchard Norway's things a commune which real steps out of but yes is you it was worked out said that after 40 days exceeding its back would appear you got that it's more than that it's the accelerated expansion of the universe that that is true that is true but the universe will slow down and the sorry excuse me if this picture were right numerous with slowed down the and things which you are currently moving away from us faster than the speed of light would in the future be moving away from from us with less the speed of light and at that point we would be here we would that they would become visible so if this weather pattern eventually we could see everything no matter how far away this is not a part of the pattern that and if this pattern was correct then they will be in is an ultimate limitation that you cannot see past the point but what's what's fall back on track spend a lot of time on those particular thing I was ever really interesting things and will explore them the but for a while after Over but the where are your where is that the wavelength increases yet yes that that is correct the question is why do you have to suppose that as the universe expands that the way way of photon will match it why his wavering forked serve rigidly attached to the great if you like that's the question why is the wavelength of the photon somehow a rigidly attached to the these uh reveal degree anybody knows a little bit about the airwaves and Cobb wave propagation knows that there what Coughlin idiomatic abatic invariant BAT is the number of nodes the number of times a wave passes through 0 and as long as you change things slowly the number of nodes 6th this is called a Vatican invariant as they had 80 some sort of close circle that waves could propagate on this is something in principle we could build a have waves propagating on and somehow we yellow-striped think of an example I couldn't think of when I still can't think of 1 but imagine we have a dialog that allows us work to change the radius of this really here that things are propagating on another words change the circumference of changing the circumference of it will not change the number of Norwood's you might ask a self-help could it change a number of Nevada suddenly the day change the radius gradual or
continuously and what point would you expect Domino hear about the 10 or something like that you win on old means number of times at the Lavery 0 but now it has there has to be integer for 1 thing it has to be integer be waived going around a standing wave with circuit standing wave simplicity standing wave on here are Pasteur and be periodic so the number of nodes what has to be an integer neither stretching the space survey propagating on it can suddenly jumped at what point does a good jump from 7 0 state nor would it doesn't John what it does is the waves just get longer wavelength shorter wavelength to match the finger talk that sustained phenomenon except replaced this ringed by the University of number of nodes a number of zeros is invariant never changes and the only way that can be accommodated this is the year of the wavelength of the photon increases wavelength of the wage increases not any anyway even if it's not nice they like best supporters is a lob over here and nothing nothing over here by Fourier analysis it can still be expanded in waves that have the nice periodic smooth structure each 1 of them each 1 of them has a number of knows which is fixed and so that's good enough for us the individual waves of definite waved white can't jump the number of nodes and storms set instead the wavelength has to accommodate the size of the thing propagating on him heads of gallons University galaxies faster faster growth the media they met role of the Earth at theaters in the theaters because in the long way while UK away something is far enough apart when onwards aboard aboard a R you may enter my that rose continuously at some point exactly at speed life which means that must be met the All-Star get that people masses of thing Ameritrade less never change nest of electron as always there is no word yet by I think what the question itself that's what got your for so from mine shaft guest room really ought to juror for feet best thank you life but that he that it is not clear if they Brooker heard do broker will bosses made the IAEA With a a favor they didn't be it does rich worker we've got a of light that what you see is what you get both they led by With you when I my wait while we wait with talk a great deal about horizons and there are and geometry we can look at the point is we can analyze those kinds of questions by just thinking in terms of Newtonian or even special relativistic geometry where have to analyze them by saying his own rhetoric that there's a geometry and we have to analyze it geometry you do that you do that Rick industry can go only so far were were valid introducing real relativity so far we've been OK we didn't have to introduce relativity and the reason why is because we stopped tourists more of region always equations would derived by looking at galaxies which did not have to be very far away right we follow some galaxies richer reasonably close and we follow them if they're close enough dough moving with very tiny fraction of the speed of light so that's all we did we followed things which are close enough that they never got anywhere near the speed of light if we want to study the whole universe and out dissident out the distances where the Hubble constant times the distance is comparable to the speed of light if we want to study the universe on that scale then we can't do it without relativity we really can't go without general relativity so you're you're jumping ahead of the game and try to to get their thinking only about noon Newton and a special relativity there In the displaced setting what we do it single exploded view the dream but that there stressed we will use it could be the do you think of that that will be introduced although they energy itself decreases the easiest example is to think about the photon a box of violin strings the photon box exert pressure on the walls of the box because digs its pressure on walls of the box it does work on the boxes and expands the work is pressured by a change in volume and that's equal to the change in energy of the so photon is doing work on the box and in doing work on the box on energy use decreasing but this is also true and eventually Forget photon just take ordinary particles and gas in a box what happened to temperature in his box if you increase the volume down was at mean about the kinetic energy of each particle was due that few if you do it if a as look and if you infants as slowly if you do it into a test would slow it does or does not does if you do it very very sudden rate for happen was suddenly but it is a molecules of and you absolutely instantaneously suddenly increase the volume of a box what you find is a baby empty space the molecules a still in the original volume here and nothing has changed their energy the momentum or nothing else that's the 980 about case the adiabatic case means you increase of slowly and slowly means there's plenty of time for modules the bounce off the walls many times it will cool if you do things suddenly is likely to he even if you expand or contract to do things really suddenly is likely he armor or at least not cool but if you increase the volume of the box slowly it will call said get connected with disconnected but it's a phenomenal doesn't particularly have to do a photo arms is simply has to do with pressure against the wall and increasing the box means that work is done by the pressure on the walls of a box and to conserve energy it has to cool so this is this is the phenomena of the photons calling Brokers Group for more please
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