Cosmology Lecture 2

[Music] stanford university let's review a little bit and then i want to move on to um generalizations of what we've talked about so far i think we worked out we worked out the equations of an expanding universe they were newton's equations let's talk about something else first does newton's equations really get it right yeah newton's equations does get it right for the most part and let me let me let me explain why einstein's equations have to do with curved space-time now the universe that we're ultimately going to study is has curved space-time all right and in fact some versions of it even have curved space that simply means that space itself forget space-time just space itself if you measure triangles on it if you do various kinds of geometric exercises on it you'll discover perhaps that space is curved at the moment it looks pretty flat but it's possible that it will turn out on the on the average to be curved and if it is curved well maybe it looks like a three-dimensional version let's say of a sphere well we're going to study later not tonight and maybe maybe partly tonight it's a portion of a sphere that's a portion of a sphere we're over here we look out we can only see so much we can't even really see that the cure the sphere is curved okay but at a large enough distance we may be able to see that the sphere is curved on the other hand supposing we just decide to look at very neighboring galaxies now very neighboring galaxies can mean a billion light years from us now very very neighboring galaxies much smaller than what we think the radius of curvature of this universe is well then it looks flat and if it looks flat it should mean that at least for that portion if we're not interested in the whole thing but we're just interested in the local nearby behavior we should we should not have to worry about the fact that it's curved 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 can be studied using newton's equations that's what we've been doing we've been looking at the universe in the small and studying how a small little fraction of it is expanding or not expanding whatever it's doing and it's perfectly legitimate and in fact entirely consistent with einstein with with relativity except for one thing except for one thing we would run into trouble if the galaxies or whatever is present galaxies particles whatever is present if they were really moving past each other with a significant fraction of the speed of light one of the assumptions is that the neighboring things are moving relatively slowly with respect to each other 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 non-relativistic velocities you can study relative to you you can take a small patch of it now small could mean 10 billion light years okay but you can take a small patch of it and study it 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 equations but there are particles moving fast by comparison with the speed of light past us what are they well neutrinos for one but photons not only photons from the sun i mean photons that would be there even if there was no sun the universe is filled in the same way that it's filled with galaxies it's also filled with radiation homogeneous radiation and that homogeneous radiation does move with the speed of light that means that we have to modify our equations somehow to account for this very very fastly moving rapidly moving material photons we're going to do that tonight but i want to uh you want to just review what we did last time quickly we first of all said suppose that space is homogeneous and filled with galaxies and i'm not going to try to draw all the galaxies they form a a gas if you like that kind of fill the blackboard with a certain number of particles per uh per cubic meter in other words a density a density that we called rho and that was the content of the universe in kilograms per cubic meter if you like you could use some other units but whatever units you like physical units kilograms per cubic meter and we called it rho we laid down a grid on this universe and laying down the grid there was clear ambiguity imagine that we laid down the grid at some specific time like today we laid down the grid and you could ask what is the spacing the grid was a coordinate system let's call the coordinates x and the distance between x equals something and x equals something plus one in other words one grid one grid separation here one ladder separation there's a certain distance associated with it how big is that distance well we called it a but how big is a that depends on the grid that we laid down if we laid down a very coarse grid it would be one thing if we laid down a fine grid it would be another thing and so it had better be that our equations at least at the moment do not prefer any specific value of a we could lay down a different grid the different grid could be twice as so here's the black forms one grid and the black and green together form another grid if we looked at the more dense grid we would also invent an a let's call it a prime a prime is the distance between neighboring points on the density on the dense grid that would be one-half a so if you ask me what is the value of a i'm going to say i can't tell you until i know precisely what grid is laid down and so a itself doesn't have a physical meaning at least at this stage later on we'll we'll discuss more what a means but at the moment on a flat blackboard doesn't mean anything by itself until you specify exactly what the size of the grid is okay so a is just a sort of bookkeeping device on the other hand ratios of a may mean something let me give you an example supposing i told you now a is a function of time universe is expanding the distance between neighboring galaxies the galaxies are embedded in the grid they move with a grid the actual physical distance between galaxies is growing okay supposing i told you that over a period of time a doubled that has meaning that means the distance between every galaxy every pair of galaxies doubles so ratios of a particularly ratios of a at different times are recording a history of how the universe is expanding or contracting and that's why our equations tend to only involve ratios of things with a let me give you an example let's take a dot the time derivative of a well that will depend on whether i use the black grid or the black green grid every a associated with the black grid is twice as big as every a associated with a black green grid and if every a is twice as big a dot the time derivative will also be twice as big on the other hand if i take a dot divided by a and let's compare it with a prime dot 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 half a a dot prime will be a half a dot a you know what i mean and the factor of 2 will just cancel out so things involving ratios of a those are invariant physically meaningful things and if you remember a dot over a is just the hubble constant at a given time so it really does mean something but a itself an ambiguity in how big it is but it's a once and for all ambiguity once you fix it then you stick with it okay another thing which is ambiguous was we introduced a constant nu remember what nu was nu was the mass contained within a single cubic cell of the grid in other words a grid element here one by one by one by one what one what one grid unit the amount of mass within that cube which we called new is ambiguous because we haven't figured that we haven't determined exactly what the grid is nevertheless once the grid is fixed nevertheless the amount of mass within a single cube is called new but it also changes when you change the grid now let's go to the equations that we derived last time the equations we derived last time i'll just remind you how we derived them we took the universe filled with galaxies we call them particles uniformly and we placed ourselves right at the center arbitrarily but still consistently we placed ourselves at the center looked at another galaxy which was at a specific location on the grid some x some x on the grid and we studied newton's equations for the motion of that galaxy which which you would think would be horrendously complicated horrendously complicated because they're interacting with uh with uh loads and loads of galaxies but we used the famous theorem newton well first of all we imagine smoothing things out so that we could think of the uh of the distribution of these galaxies as smooth smooth and uniform we took a sphere centered at the center centered at us excuse me and we used newton's theorem that told us that relative to us in the coordinate system where we are at rest the force on that galaxy only depends on the mass within that sphere as if all of the mass were concentrated at the center so we just said let's concentrate all the mass within that sphere at the center and then study how under such circumstances the galaxy at x would move if it were under the influence of just that mass you don't have to worry about the masses on the outside you only have to worry about the masses on the inside and we worked it out and the simple way that we worked it out the simplest way we worked it out was just to say if there was a particle over here moving under the influence of a fictitious mass at the center then the energy of that particle would be conserved the energy consisted of two terms well i think i'll yeah let me come back to the derivation again i just want to review the derivation very quickly but i'll come back to it in a moment let me remind you what the equation was the equation that was derived was that a dot over a squared that came from the kinetic energy was equal to the density of matter rho measured in uh in physical units in other words measured in kilograms per cubic meter not this new object over here but rho and uh there was a little more there was an eight a pi and a g g newton that was the equation that uh that we derived by just looking at the at the conservation of energy row oops yeah row r oh sorry you didn't catch me over three that three ultimately came from the volume of a sphere four thirds pi r cubed okay that's where the three that's at the the four the eight is two times four uh pi pi is pi and three is three this is related to the volume of some sphere okay 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 this escape velocity just in the knife edge between being able to escape and not escape that's what that formula was we added one more thing to it and the one more thing is we said that rho rho which is the amount of mass it's the amount of mass per unit volume is related to nu is it the same as new no because nu is the amount of mass per grid box but the grid box you don't know what its volume is this is the mass within one cube if you want the density then the density is related to nu by dividing new by the volume of one of these boxes this is density per ubi cubic meter this is per cubic box you divide that by the volume of a box which is a cubed and that's one more fact that we put in that rho is new divided by a cubed now is rho constant with time no because a changes but is new constant with time if new just represents ordinary particles sitting in the universe which are never destroyed never created particles of uh protons let's forget whether the proton decays for the moment protons are forever and uh galaxies are forever for a moment not forever we won't we won't believe that forever but we will believe for the moment that galaxies are forever did you get that yeah okay if galaxies were forever or protons were forever then nu would be constant the number of protons within a box is the same at all times all that happens is the box grows so the protons thin out but the number of them in a box stays fixed and so nu stays constant what is it what is its numerical value i can't tell you what the numerical value is because that has to do with the exact grid that i use if i change grids i change new okay so but in any case once i've established all of the conventions i can replace row by nu divided by a cubed nu is a constant it doesn't change with time 8 is a constant pi is 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 grid since changing the grid changes the magnitude of new i can if i like i can if i like choose eight pi g nu over three to just be one it doesn't make any difference the point here is that the numerical constant that appears here really doesn't make any big difference to the way the universe evolves and the way you can see that is by changing definitions of the grid you can change the constant so we could if we like just set 8 pi g nu over 3 equal to 1 and then we would have the very nice differential equation a dot over a squared is equal to 1 over a cubed now would you like to see how to solve that we solved it last time but we solved it by guessing how many people want to see me solve it in real time i know when you want to see me solve it in real time because you're hoping i'll make a mistake yeah you're gonna ask question okay well it's too bad i'm gonna solve it whether you want me to or not here's the way you're solving here's the way you solve all these equations you first of all solve you first of all take the square root of both sides on the left side that's easy it's just a dot over a on the right hand side it's a little bit of a nuisance it's 1 over a times square root of a do you see where that came from a cubed is a squared times a the square root of it is a times the square root of a okay let's multiply both equations both sides by a to get rid of it in the denominator and that just tells us that a dot is 1 divided by the square root of a that's where we would like to begin okay now we're going to write this d a by dt which is what it is is equal to 1 over square root of a but now we're going to do something tricky the very tricky thing we're going to do is instead of thinking of a as a function of time we're going to think of time as a function of a we're going to take a to be the independent variable and to do that here's the adt that looks like a is a function of time and we're differentiating it but if i turn it upside down it's d meaning taking one over it it's d t d a and now it looks like it's the derivative of time with respect to a and that's equal to the square root of a so how do we solve it we look for a function of a whose square whose derivative is the square root of a what function has is its derivative the square root of a eight to the three halves right well two thirds eight to the three halves so we find out that t is equal to two-thirds the two-thirds is absorbed into other constants that we've been um uh ignoring times a to the three-halves all right we can solve that now what we really wanted was a is a function of time but that's easy uh if we just neglect the two thirds i just don't feel like writing it to get the two thirds this tells us that a is equal to t to the two thirds a is equal to t to the two thirds which is a function a t which looks like that and it gets flatter and flatter and flatter the flattening of it is deceleration the universe is decelerating but it never comes to rest we could see that it never comes to rest because when we look at this equation there is no point at which a dot over a becomes zero how do i know it doesn't become zero because the right hand side is never zero it gets smaller and smaller as a gets bigger and bigger but it never goes to zero and that's why the universe decelerates another way to see that it decelerates it's just a particle moving away from a fixed force center it will decelerate because of the attraction by the force center and that was that was lesson number one all right i redid it because well i always think it's worthwhile uh redoing the most important derivations yes it's okay to throw away the negative square root um what's the logic the logic from the equation you cannot tell whether the universe is expanding or contracting both are possible from the equation if you start it's exactly the same deal as saying you have a planet over here yeah coffee is a good planet well i and the almond here is a part as a rock and you see the rock is right over here the equations of newton will not tell you whether the rock is moving away or whether it's moving toward there's one thing you can be sure of let's let's ignore for the moment moving in orbit let's assume we know that the uh that the rock is moving radially there's one thing we know for sure that it's not standing still it may be momentarily at rest it might have come up and stopped but it's not going to stand still it's accelerating it's accelerating back toward the planet okay so it could be going out and decelerating or it could be going in you can't tell from the equation the same way you can't tell from the equation whether it's expanding or contracting and that has to do with the two possible solutions um whether the adt is positive or negative if it's contracting it's negative it's expanding it's positive okay excuse me sort of all of this all these objects are kind of moving radially with respect to each other but of course i think of all these objects out there they're all you know kind of spinning around each other well okay so that's right that's right but still on the average they tend to move as if they were embedded in this grid given that they are embedded in the grid all the only option they have is to move radially relative to each other now you're perfectly right there are motions on top of that average motion the motions on top of the average motion don't of course satisfy this they're called by astronomers peculiar the peculiar motion meaning i don't know are they peculiar well i don't know uh the sun moves around sorry the earth moves around the sun and the earth is not participating in this expansion relative to the sun but you know if you go a few galaxies over the whole thing is so the real motion of real galaxies is a combination of this average flow think of it as a flow a river going downstream how fast is any given molecule moving well i don't know how fast any given molecule is moving but i do know on the average that clumps of molecules sufficiently averaged are moving with whatever the velocity of the river is on top of that there's the peculiar motion of the molecules relative to each other so that's the way to think about it a flow and on top of the flow fluctuation and we're ignoring the fluctuation at the moment okay there's two directions i want to expand it i will not explain there's two directions i want to generalize in today the first well probably the second the second will have to do with what happens if you replace massive particles galaxies or protons and so forth by photons by radiation how does this whole story look different if there was a universe a fictitious universe which had nothing in it but photons how different would this be now why am i doing that what a silly thing to do why take a fictitious universe which is only made of photons well what we're going to find is that early on in the history of the universe the most important thing in the universe was photons 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 today the more dominant form of energy in the universe is just the masses of particles e equals m c squared kind of uh kind of energy that's what this theory is about it's about the behavior of massive particles non-relativistic slowly moving particles that's also presumably made of particles that are pretty much standing still oh 30 years ago we would have said that right we're still 30 years back or 25 years back or something you're right good good good yes but dark matter by contrast is a part of this story so yes thank you that was helpful yeah well what i'm telling you now is an old story but it's a story whose base whose basics are important to us okay two directions i want to go the first has to do with what happens if the universe is made out of other kinds of stuff other than just um particles more or less relative to each other radiation in particular and the other is to move beyond the assumption that the energy is zero there was no reason for that it was arbitrary we said arbitrarily let's assume that every galaxy is moving away with exactly the escape velocity let's back off that now and how do we back off it we go back to the energy equations and say the energy of that particle moving in the fictitious field not the fictitious field but the field of the fictitious concentrated mass at the center this is not a real galaxy now this is the combined mass of everything within the green sphere particle out here moving outward and let's apply the conservation of energy to it let's 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 zero energy okay so what is the energy of this particle this whole glob here has mass m we'll come back to what m is in a moment but it has mass m it's not the mass of the sun and it's not the mass of any specific galaxy it's the mass of everything in the sphere okay and the energy of this fellow over here is one half m m is the mass of this fellow not the big mass in here the mass of this galaxy times its velocity squared that's one term and then the other term is minus uh m g over the distance between them let's call the distance d as we did last time oops i missed something didn't i another factor of little m right right product of the masses newton's constant distance between them that's the that's the energy of this particle moving out and what do we know about it we know that it doesn't change with time it doesn't change with time because our model that's that's a useful model is just the motion of a particle in a fixed uh in a fixed background of mass the energy is constant and so let's set the energy to be equal to some constant now what does constant mean does it mean a numerical constant that's independent of everything else no it doesn't it could depend on which particle we're talking about could in fact if the universe is homogeneous and isotropic the only real thing it could depend on is the distance away so i'll i'll simplify the discussion for now by taking this particle to b at x equals one let's in other words we're going to take a specific particle a very definite one at x equals one and focus on it it has some specific energy and that energy will never change so let's just call it e and that's a constant what value of the constant i don't know i can't tell you offhand no more than i can tell you if i know that there's a a nut over here and there's a planet over here there's no way i can tell you a priori what the energy of that particle is i don't know how fast it's moving right so we have to study all possible cases all right so we just put the energy and call it e and take it to be a constant a numerical constant let's write out now everything let's divide this let's first of all divide it by m little m let's divide by little m divide by little m and then over here we get e over m now e is a constant m is a constant e over m is a constant so i really haven't made it any more complicated by dividing by m i'll leave it here z over m but the right hand side is just a numerical constant let's even do a little more let's multiply by two it's still just a constant and since i didn't tell you in the beginning what e was all i know is that this is a constant so let's well let's let's leave it that way now what about the velocity if this is the galaxy at x equals one then its velocity well its distance the distance is a times x the velocity is a dot times x and so now we can just plug that in here and x is one i've chosen x to be one so it's very simple the distance of the galaxy is just the scale factor a it's called the scale factor and its velocity is just a dot so let's plug it in here this just says a dot squared minus 2 m g over d and d is a is equal to some constant now i'm just going to call it c constant on the right hand side this has many of the elements of this equation here but this equation this nice equation we divided by a twice to get a dot over a squared why did i do that because i know in the back of my head that a itself is not a meaningful thing it's only ratios of a's so in the back of my head i knew that what i really want to get is an equation for ratios of a's or a dot divided by a so i divided this equation by a squared a cubed notice something's happened on the right hand side here it's no longer a constant c is still a constant but a squared now has something new in it and finally we used that the mass divided by the volume cubed now this is a sphere this is a sphere of radius a x is equal to one a sphere of radius a and the volume of a squish of a sphere of radius a is four thirds pi a cubed so i have a cubed down here that's not quite the volume of the sphere let's fix it so that it is the volume of the sphere let's multiply well i don't want to i don't want to botch up this equation too much let's rewrite it a dot over a squared minus twice m g over a cubed is equal to a constant over a squared but now let's multiply and divide by four thirds pi so this is four pi over three and we have to multiply out here by four thirds pi that becomes eight thirds pi that's where our eighth thirds pi came from okay four thirds pi eight cubed over three that is the volume of the sphere and what is the mass divided by the volume the density so by fiddling around i got the friedman equation but with a right hand side now a right hand side that knows about the total energy if c is positive it meant the total energy was positive it meant the kinetic energy outweigh the potential sounds like and it does means that the galaxies are going to continue to recede forever you've beaten the um escape velocity if c is negative then the energy is negative more potential energy more negative potential energy than uh than kinetic energy and that's the situation where you expect everything to go out of ways and then come back and crash so we now have a new equation the friedman equation for non-zero energy the energy can be positive or negative positive or negative both are allowed total energy and let's examine it a little bit and see if we can see a little bit what it says it's not too hard to solve for any given case but let's not solve it oh one other step one other step let's write it over here a dot over a squared and let's transpose this factor to the other side equals eight thirds pi g times rho but rho is nu divided by a cubed remember nu nu was in was the mass per unit cell size nu over a cube that's rho and there's g oh g is there plus c over a squared this is the new thing c over a squared is a new thing that was not here in our initial study of this equation what does it do all right so let's look at it first this is the real friedman equation with this uh with this other term on the right hand side and it was derived from general relativity not from special relativity sorry not from newton but nevertheless we just derived it from newton good okay so here we are let's see what these things mean let's assume for the moment that the universe continues to grow if c is positive if c is positive and everything else on the right hand side is positive 8 pi g nu a cubed they're all positive then the right hand side is strictly positive and a dot over a is positive in other words the universe continues to grow if a dot is positive and a is positive if a dot the time derivative of a is positive the universe continues to grow forever it may slow down and slow down but it will continue to grow or at least it won't it won't contract and it will continue to grow so if c is positive the universe will continue to expand it will eventually get arbitrarily big a by arbitrarily big i mean that a will eventually exceed any bound and become very large so let's look at it in the limit that a becomes much larger very very large why does um a dot over a have to be positive i mean the square root obviously has to be positive right the square of it is part that's right good the square of it has to be positive so let's assume that initially it is positive that the universe starts out expanding okay then the only way it can ever get to be negative is if it goes through zero and it can't go through zero if the right hand side is positive good so that's a good point the right hand side being permanently positive disallows this from being zero and therefore it can only continue to grow can't jump from positive to negative without going to zero and this is never zero okay now which term here is bigger one over a square one over a squared or one over a cubed well that depends on whether a is big or small all right but in particular let's go and think about very large ah let's start with small a first for a being small which is bigger one over a cube the one over a squared one over a cubed uh if you take a to be small enough this will always beat this and so for a sufficiently small a this is negligible compared to this we've already studied the case with out this and we know the answer we know the answer as long as a is small in other words in the very early phases of the expansion when a is just starting out growing it's just beginning to grow this term is not important and what do we find we found out when the term was absent that a expands like t to the two thirds but now let's go to the other limit this is a standard way of thinking about all kinds of things go to limits often an equation this is this equation is solvable incidentally but it's nasty we don't have to solve it we just have to know what it's doing at the two ends the two ends mean when a is very small and when a is very big okay so what about when a is very big which term is big this term is the biggest term when a is big one over a squared is much much bigger than one over a cubed you may have to wait a while and how long you have to wait may depend on this constant here but eventually this term will become much larger so let's study that equation let's study the equation a dot over a squared is equal to some constant any constant over a over a squared we we first take the square root of it and what's the square root of a constant another constant so i won't change its name but it's really uh this new c here is the square root of the old c and of course the reason i'm allowed to do that is because it doesn't matter what the constant is here we'll get an answer which uh which is qualitatively the same for every c all right a dot over a is c over a now let's multiply by a a dot is a constant a dot is a constant a dot is the velocity of the of the galaxy at x equals one in other words in the situation where the energy is positive when the galaxy gets far enough away the effect of gravity becomes negligible and it just moves off with uniform velocity that's what this is saying a dot is just constant and that tells us that a is just proportional to time the constant c times time the galaxy just moves off with uniform velocity and so that tells us this is for const for positive c for positive constant c is not the speed of light here incidentally although it happens to be a velocity all right so it tells us that at late time this just moves as a straight line with constant slope a dot is just a slope moves with constant slope and the universe just expands with linear and unaccelerated non-accelerated motion nothing as deep is going on here why am i using nuts when i should be using apples all this is saying is if you throw this up hard enough so that it escapes from gravity with a good margin with a margin then it will just move off uniformly with constant velocity so in fact we find that we find exactly that what happens in between where's the crossover point well i'll leave that to you you can figure out approximately where the crossover point is it goes as a to the t to the two thirds and then makes a transition to just t and sort of fuzzy in between that's the matter dominated universe why is it called the matter dominated universe because the right hand side of this equation contains a term which is just the density of ordinary matter ordinary non-relativistic slowly moving matter that's it okay so oh let's go to the other case what happens if c is negative that's the case of negative energy now you should be able to guess a particle or the apple moving away from the earth with uh with um less than escape velocity has negative energy and it just falls back down so that's actually not hard to work out i'm just going to tell you what happens what happens first of all what happens then you have a negative sign here you have a negative sign here and now the right hand side can become zero initially this is the biggest term at late time this is the biggest term but they have opposite sign that means somewhere in between this became zero that is nothing but the point where the up going apple simply is momentarily at rest when a dot over a is zero [Music] okay so with c negative there's a crossover and instead of moving off linearly like this bang crash everything falls together that's the matter dominated universe and it was a classic cosmology until until other things were discovered okay any questions i must be there no my my assumption when i get no questions is either i was perfectly clear or perfectly yeah i'm trying to understand it so at some point those two terms cancel each other you get eight prime over a squared equals zero but the other side always has to be non-negative so which does the right hand side does have to be non-negative no this can become negative it's not going to be negative over a squared try it again your left hand side is always got to be non-negative hence your right hand side i always have to do that we have to take a square root you have to take a square root square root can have a pi right just that equation there says that that right hand side since the left hand side is always not negative then the right-hand side is always not negative yeah so but that's just wondering what happens when it becomes zero then it must become positive again or stay zero or something no i made an assumption that that um that the universe was expanding and that breaks down at the point where it comes to rest the mathematics is entirely identical to to this yeah the uh what would you write you would write one half mv squared is equal to g m over the distance am i missing anything times the mass of the earth uh no minus this is equal to the energy if the energy is negative let's call it minus yeah then this is plus g m m over r basically or over distance the same equation how does it happen if the if e is negative if e is negative then there can be a crossover point it's the same exactly the same thing exactly the same thing and you know what happens you know that the uh that the uh apple just falls back down to earth it's just a question of taking the root the v there's a plus root and a minus root of a square root yeah whatever over a cubed has to be greater than c over a squared otherwise something squared goes negative which is mathematically impossible you say this has to be bigger than this more equal than that yeah i think it's what you mean the point is a can never get big enough to turn the sign of this a can never get big enough i'd say if a is big then this term is bigger and it will go negative but it never does get negative no no okay good the right hand side never never gets negative you're right the right hand side can negative never get negative all that tells you is there's an upper bound on how big a gets right a gets so big and then stops getting bigger and comes back down a is always in a region where this is positive but it goes through zero and going through zero is just coming to rest and falling back down right and on the way back down you have to take the negative root a dot over a is negative square root good all right yes okay so that's that is the matter dominated universe three possibilities positive energy negative energy or zero energy and three different behaviors who decides whether the energy is positive or negative who knows but we're going to find out that the connection of that positiveness or negativeness is connected with the geometry of the universe not tonight next time we'll talk about the connection with this constant c with the spatial geometry of the universe that's what that's the main thing at this stage the general relativity brought to bear on this the equations are exactly the same but the significance of this constant c takes on a new dimension and has to do with geometry but not tonight one more thing we're going to do tonight we're going to take up the question of what happens if instead of being made out of material points slowly moving what happens if the universe is made out of photons made out of radiation all right to understand that of course we really do have to think about relativity but really there's only one important idea and it's just e equals m c squared if we were to have done einstein's equations the right hand side of einstein's equations here the right hand side the left basically all right let's let's i'll tell you what the connection with einstein's equations is on the left we have a dot over a squared and then i'm going to add plus c over a squared transpose transpose it to the left and on the right hand side good old 8 pi over 3 g rho nu over h cubed or just rho now this is mass 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 to things that have to do with the density of energy and momentum energy momentum tensor on the right hand side geometry on the left this clearly has to do with geometry the rate at which the universe is expanding uh i'm going to tell you next time that c over here has to do with the curvature of space so this side here is einstein's geometric side the right hand side is the source namely the energy momentum whatever it is in the universe what is the energy on the right hand side of this equation it's just the mass points and e equals m c squared so oh my goodness i'm calling this c but please don't get the idea that it's the speed of light it's sometimes called kappa minus kappa it's sometimes actually written minus kappa unfortunately kappa is positive when the energy is negative and kappa is negative when the energy is positive but it's often called kappa okay uh that's the basic connection with the general theory of relativity but the only thing that we really need to do to this equation is to remember that when you go from when you go from newton to relativity what was mass density becomes energy density all forms of energy density all forms of energy density are what go on the right hand side here well it's energy density times the speed of light squared is that right m no m is e over c squared so it's energy density divided by c squared which goes on here we'll just set c equal to one we're not going to worry about the speed of light speed of light c is equal to one with that idea in mind the right hand side instead of being the density of mass density becomes the energy density total energy density some of which is just the e equals m c squared energy of the particles at rest and some of it might be kinetic energy of particles which we haven't really included at this point the kinetic energy of a relative fast motion of them but if the universe is filled with photons then it's radiation energy all right so let's talk about radiation energy and how it's different from the e equals m c squared energy all we have to do is think about a box a box now the box the box just corresponds to a unit cube in the grid here take one unit cube of the grid it has a volume of a cubed let's suppose it has a certain number of photons in it now photons have a wavelength so here's our box it's delta x equals one on each side its volume is a cubed the not the ordinary particles which are just sitting in that box and not moving they have an energy which is just their mass it doesn't change right the energy doesn't change those particles they're just the mass and nothing else but a photon behaves differently so i'll tell you how a photon behaves the first thing to know about a photon or electromagnetic radiation first thing to know about a photon is that it's energy is related to its wavelength it's related to its wavelength through planck's constant h what else lambda the wavelength and the speed of light which i'll set equal to one important thing is that it's one 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 of a given wavelength let's say you have a photon of a given wavelength and you expand the box you expand the box slowly for example you slow then the universe is expanding pretty slowly takes 10 billion years for it to double in size it's pretty slow you take a box and you expand the box slowly then what happens to the photons inside it what happens is the photons inside is their wavelength just stretches in proportion with the box now this phenomenon anybody who uh plays a guitar knows the uh the phenomena very well the box is replaced here's the string of the guitar it's pinned down at one end at the bridge over there at the other end it's pinned down by your finger at the fret and you pluck it starts to vibrate and then if you slide your finger what happens is you slide your finger you're effectively changing the size of the box this is changing the size of the box you change the size of the box the wavelength just changes and the tone uh the the note changes in correspondence with the uh with a change in the size of the box the same thing happens to radiation in a box radiation in a box the wavelengths just stretch in proportion to the size of the box because the wavelengths change for the photons that means the energy of each photon changes as you change the size of the box the world is filled up with photons you can pretend that each of the photons the number of photons in each box stays fixed that's that's a true statement number of photons stays fixed but their energy changes as you change the size of the box in particular the energy of each photon will be proportional to one divided by the size of the box so we take one box change the scale factor then the energy of each photon decreases same phenomena as the frequency going down down or up as you change the uh the length the effective length of the of the guitar string so what does that mean yeah that makes sense as the box is conducting and the electric field has to go zero times then that makes sense but you don't have connecting boxes in space just get the safe same thing if just you have free space expanding get the same result one way i've thought about it is like if you had mirrors this box is made out of mirrors the beer is moving away you get a doppler shift that's one way to think about it it's one way to think about it last week you said that the the expanding universe is sort of a mathematical artifact and then you can just think of particles i never used the word artifact you did but you agree with me it depends on it depends on how what you what your conclusion is okay well that's what i'm trying to understand is that clearly there's something more going on or the way it wouldn't change yeah that's right we'd have to do a little more quantum mechanics or a little more or a little more classical electromagnetism in the presence of an expanding universe to justify what i said okay let's take it let's take it for a moment as a given and we can come back to it the answer is it is a correct statement that that [Music] and let me see if i can think of an example uh offhand i can't think of a good practical example or where something similar happens but nevertheless it is true that uh that the radiation in the box the wavelengths of the photons readjust themselves to the sizes of the box of the box in other words the photons stretch along with the universe if you like you can just think of the space stretching and with it the photon wavelength stretches let's let's leave it at that for tonight for tonight let's leave it at that let's just examine the consequences of it um let's examine the consequences of it the consequences of it is that the energy per photon decreases like one over a in contrast to the case with ordinary particles where their mass stays the same and doesn't vary all right so by contrast with the particle case ordinary particle case the energy in the box the energy in the box the mass in the box effective energy or mass does not stay constant but it decreases total energy in the box decreases like one over the scale factor every photon its energy decreases when it gets stretched so every single one of them and so compared with the previous case there's one more factor of a in the denominator the energy in the box goes like one over a and the energy density goes like one over eight to the fourth instead of one over a cubed remember previously the the density here we said was some constant nu divided by a cubed now we get one more factor of a in the denominator and it's just the fact that every particle its energy decreases by one extra power of a that means relative to the previous case there's an extra a in the denominator could be taken now to be the number of photons per unit in a box and the energy would be one over eight to the fourth that's the only new thing that happens this is matter dominated and this is radiation a dot over a squared again 8 pi over 3 g some constant nu and then 8 to the fourth downstairs we can also put in this term here minus c over a squared but let's study the case when the case that corresponded to zero energy just to see the difference to see the difference let's see what different behavior we get with this formula this formula says again by appropriate choice of the size of the grid you can make all the constants here if you like be one by appropriate choice of the size of the grid you can rearrange it so that it's just a dot over a squared is one divided by a to the fourth what was it before do you remember 1 over a cubed okay so let's see if we can figure out what happens again we're going to solve the equation now i'm going to go through the steps of solving the equation first take the square root a dot over a equals 1 over a squared that means that a dot is just 1 over a right did i do that right that's the adt all right we'll do the same trick as before turn it upside down dt by da is equal to a so now if we think of a as the independent variable we're looking for a function whose derivative is just a what function has its derivative a a squared right yeah a squared one half a squared to be exact so that says the t is like a squared maybe there's a one half here but that that's not important the thing is that t varies like a squared or that the scale factor varies like t to the one half square root of a well i'm not much of an artist and i can't really draw the difference between t to the two-thirds and t to the one-half there's t to the two-thirds t to the one-half is a little bit smaller huh doesn't grow as fast so tita but it looks pretty similar and does pretty much the same thing you'd have to be an astronomer who were really interested in this to care about the difference between t to the one half by by that i mean uh you know they're sufficiently similar qualitatively similar but if you really want to know what's going on in the universe the difference between t to the one half and t to can be very important okay so that's the radiation dominated universe the radiation dominated universe expands like the square root of t not t to the three halves t to the two thirds t to the two thirds t to the two thirds and t to the one half what about the mixed case supposing a universe has both ordinary particles and radiation like the real universe really does let's worry about that let's uh let's come to that the mixed case neither pure radiation nor pure non-relativistic particles in that case the energy density is two components one for radiation that goes like one over eight to the fourth and one for ordinary matter that goes like one over a cubed so the kind of equation that we're going to have i'm not going to write the details i'm going to write down the general form of it will be a dot over a squared and that's going to have two terms now one of which call it constant number one divided by a cubed that's ordinary particles constant c1 is just some measure of the number of uh of the number in each box and then another term and they're both positive the energy of particles and the energy of photons is positive and the other is some other constant over a to the fourth that's the equation of motion for a universe which contains like a real universe does ordinary non-relativistic matter plus radiation which one is more important one over a cube the one over eight to the fourth depends on a right when a is big which one is more important this one excuse me my mother always told me not to take such big bites from the apple for small a this is the more important and the smaller the a is the more more important it is when a is big this is the more important and again swamps the other term when a is really big so what that tells us without too much work is that when a is small in the beginning of the expansion the only thing that's important is the radiation this one the radiation term is dominant compared to the material protons neutrons galaxies there were no galaxies at the beginning but uh and the radiation was the most important thing and so the universe started expanding like t to the one half but then eventually this term took over became the more important term and it made a switch let me get another color it made a switch and started to grow like t to the two thirds yes yes that's correct that's correct and you might say well how do you know that energy of matter doesn't get converted to energy of radiation that's something we're going to have to discuss the answer is when the when the um when when things get cold enough when the universe expands enough things cool down and there's not much exchange between radiation and ordinary matter and they are pretty well conserved each one separately they don't talk to each other too much in fact once things cool down to a certain temperature it's a rather high temperature one a thousand degrees or more than a thousand degrees 10 000 degrees but once things cool down to a certain temperature pretty much the photons are just free screaming and don't care much about them about the particles and the photons are two way too long wavelength to even scatter the particles very much so so that's right that's exactly right that uh we've assumed that the two constants here are pretty much time independent all right that's that's the nature of a universe built of two components now that's only two components that's only two components and there is a third component there is a third component it's the discovered dark energy that's what we're going to talk about next time we have ordinary matter we have radiation those are commonplace things dark matter belongs to the matter category it's simply a part of the ordinary particle matter that's invisible simply because it doesn't have any charge it doesn't radiate much so we don't see it optically that's part of c1 this should be c2 here i meant to write c2 or maybe we should call it cm for matter and see radiation for radiation there's more than one component to the radiation there's photons there's presumably gravitons and there are neutrinos now neutrinos have mass okay but the mass is so tiny that even today the neutrinos are moving with very close to the speed of light and simply mimic the same behavior as the uh as the photons so there's a radiation component the radiation component consists of all particles which are so light that they're moving with close to the speed of light the c1 or the c mass here consists of all particles which are heavy enough that uh that they're basically at rest relative to us nearby and uh that's uh that's cosmology sort of as it was known as it was known 30 years ago this is the cosmology of 30 years ago yeah all right so in that box that's expanding and the wavelength of the wave is stretching out the energy the total energy inside the box is decreasing where's that energy going the energy is doing what decreasing decreasing yeah doing work on the walls to expand it well another way to say it is it's going into the kinetic energy the kinetic energy of expansion this equation is remember what it was it was the equation of conservation of energy that's where it came from so if you think about it that way then start tracing uh if this part of the energy decreases then or let's let's put it this way let's write the equation in a different form let's put a minus sign over here well let's let's first one step at a time put minus put minus and make it equal to zero i haven't changed anything now change the sign of the whole thing minus plus plus this equation reads the following way that there's a certain amount of energy and mass and radiation if it changes with time that energy gets transferred to a negative term this what's what's a little bit bizarre is that the energy associated with the kinetic energy of expansion the kinetic energy of expansion this term here is negative this can be read as a conservation of energy it can be read energy never changes because it's always zero contains three terms matter radiation and a term that has to do with the rate of change of expansion normally in the ordinary world where the when the rate of rate of expansion the rate of expansion is very very small it's a tiny tiny number this here is a tiny tiny number in our world at the present time tiny number this never changes very much and so uh the it's not zero i i take it back this is not a tiny number it's a number but it doesn't change much over a course of time the sum of these two is zero and it stays zero that's the conservation of energy am i saying that right i'm not saying it right yeah conservation of energy is a symmetry with time when you think about it in terms of the northern charge would the fact that a is time dependent sort of uh excuse you from conservation of energy there's two ways to think about it you could say the universe is a time dependent background that everything moves in and because the universe is time dependent that means all of your equations have a time dependence there is no longer a time translation symmetry and energies doesn't have to be conserved the other way to say it is that there's a time translation symmetry if you start the universe at this time or that time or that time or that time you get exactly the same response if you started the universe at t equals minus seven it would be exactly the same as starting the universe at t equals plus four and thinking about it that way you'll discover there's another term in the energy associated with the expansion rate let's not go into it now this is this is this is very very subtle and i want to i want to spend at least 15 minutes on it uh later energy conservation but just to keep in mind that these equations their origin was energy conservation so there's no way that we can be violating energy conservation yeah dad you you said that treatment equations are interesting because you can derive them through newtonian mechanics or or general relativity newtonian mechanics presupposes a flat geometry and relativity assumes a curved geometry the geometry of space forget space time for a minute okay right geometry of space einstein's equations permit flat space these are okay would that mean that space is flat because you can derive it from either not space time into space would that be a conclusion where's our equation i think you erased it yeah okay minus some kappa over a squared okay this had to do with the total energy the fixed total energy if kappa is positive in this equation the energy is negative if the kappa is negative the energy's positive sorry it goes this way kappa is minus the energy essentially do i have it right no i have it wrong i have it wrong okay doesn't matter one way or the other um this term over here has to do with the curvature of space so we're going to study three cases there are three interesting geometries that we'll be interested in there's a flat geometry where space itself is flat this goes on and on endlessly homogeneously and triangles have the usual properties that's the case k equals zero there's the case where the universe is curved like a sphere so if you go around it you come out the other side you you go around it finite and uh compact we say that's the case k equals positive any positive number here corresponds to a radius of curvature of the year of the expanding universe expands like a balloon you know the classic picture of the expanding balloon that's k equals positive k equals negative is a negatively curved space and we're going to have to talk about what a negatively curved space is like less easy to visualize than the sphere so the next time it's exactly what we're going to do we're going to talk about the geometry of space and the three cases flat positively curved and negatively curved they correspond to k being zero positive or negative so so would that mean then that in the newtonian case k has to be zero well except that you can sort of mimic up the um the case with k not being equal to zero if you want but it's a little bit of a fake yeah okay right a box full of photons where the boxes expanded and the number of hosts on stays the same the energy should be going down that box what happens where does that energy kill you may answer that question but i don't understand where did that energy go [Music] it does work on the body take a literal box a literal box with reflecting walls and a box with reflecting walls and ask what happens to the energy in the box when you expand the walls okay well what does happen this pressure in the box this pressure exerted by the photons bouncing off the box the pressure when you expand the box does work on the walls of the box uh for a real box the energy 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 but one way or another it does work on the box and increases the energy of the box itself either by stretching the you know the little hooks law springs that are inside the box that hold the molecules together or uh causing the box to have a kinetic energy a little kinetic energy of expansion or whatever else just heating the box just heating the box you said the grid is actually absorbing the energy yeah in a sense that's right yeah right we'll talk about that we'll talk about energy conservation more clearly right but it's still it's a correct thing to analyze it this is not obvious but it's correct it's correct to analyze it as if it were stuff in a box and the stuff that gets reflected off here from a real box with reflecting walls is compensated for in the fake box case by every particle which goes across here is approximately compensated for by a particle which comes in the other side so on the average uh the uh the expanding space behaves like an expanding box particles instead of reflecting go from one box to the other but on the average as many particles go from this box to this box as from that box to this box and so in a sense it can be mocked up by saying that the particles reflect off the off the box increasing the size of the box means that work is done against the walls and uh we'll have to talk about what that means in general relativity when we come to it yeah a couple of questions one how soon after the big bang did the universe cool down to where the matter and radiation components were not really interacting or even changing um about 100 000 years and then um the second one was what astronomical evidence do we have to see the t to one half behavior is that what what do we have to say it again what evidence do we have to observe the t to the one half expansion behavior is that in the cosmic microwave background uh that's that's the expansion history of the universe and remember that looking out at the universe you effectively see a history looking to different distances you're seeing the universe at different times so by looking at the universe at different times assuming that it's homogeneous you can reconstruct the uh the history of expansion we'll talk about that that's uh that's something that we can do basically by looking at the combination of density of galaxies on the sky distance of galaxies and we'll spend a little bit of time going through that the combination of careful measurements of distance density and so forth allows us to reconstruct the history and we can reconstruct the history because we're seeing it at different times and you know combining it with cmb and all of those things uh yeah yeah it was before galaxies formed but uh but uh uh in the temperature history of the universe there's there's plenty of uh there's plenty of evidence that uh that this picture is right no you're right the t to the two thirds is easy to see the t to the one half uh is uh is largely has to do with microwave background and things like that so get the equations down and then we'll go and talk about [Music] how we know how much of this picture is right and the answer is we know it's wrong okay yeah so i think that this isn't a problem i can't hear you i said i i think that this is not a problem but i don't understand why okay the apparent speed of an object relative to any observer is a function of how far away it is so conceptually it seems like an object far enough away could appear to be exceeding the speed of light but i i don't think that doesn't violate this idea of nothing can exceed the speed of life but i don't understand why well you have to ask what it really says what that principle that things can or can't exceed the speed of light what it literally says is you can't see anything going past you faster than the speed of light um [Music] it doesn't say that two objects in an expanding universe can't be receding from each other at faster than the speed of light that that is quite allowed and as you point out that if you have a formula which says velocity is equal to distance times the hubble constant then you make the distance big enough the velocity will sure enough exceed the speed of light it's connected with the idea of a horizon but but you're right these equations say that things can move recede from each other at faster than the speed of light and uh but for sure they don't allow you to send signals in time faster than the speed of light can get from one place to another and they don't allow you to witness something moving past you faster than the speed of light so that's right does this mean that objects sort of at the limit right before where the expansion rate exceeds the speed of light eventually disappear in a sense yes they'll pass through the horizon you won't see them right but that has to do with dark energy oh that has to do with dark energy without dark energy nothing would really move out of our uh ability to see right and i will show you how the geometry works and i will show you how all these things can be understood but little steps good okay yes we'll never see more of the universe no matter how long we wait is that related to the fact that objects farther away are receding at a speed faster than it would appear to be traveling fast 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 no sorry excuse me if this picture were right and the universe were to slow down then things which are currently moving away from us faster than the speed of light would in the future be moving away from them from us with less than the speed of light and at that point we would be we would they would become visible so if this was a pattern we could see everything no matter how far away okay this is not the pattern the pattern is that and if this pattern is correct then there will be then there is an ultimate limitation that you cannot see past a certain point uh but let's let's hold back on that we're gonna spend a lot of time on those particular things those are the really interesting things and uh and um we'll explore them the way i've always thought about the question of why the uh when the box expands the radiation the photons lose energy is that there's a wave phenomena going on the wavelength's increasing reasonable way to think about it that that the wavelength increases yes yes is decreasing because that's that is correct the question is why do you have to suppose that as the universe expands that the wavelength of a photon will match it why is the wavelength of the photon sort of rigidly attached to the grid if you like that's the question why is the wavelength of the photon somehow rigidly attached to the uh to the uh to the grid now anybody who knows a little bit about the waves and prop wave propagation knows that there's what's called an adiabatic invariant the adiabatic invariant is the number of nodes the number of times the wave passes through zero and as long as you change things slowly the number of nodes stays fixed this is called an adiabatic invariant and if we had a some sort of closed circle that waves could propagate on this is something in principle we could build and we have waves propagating on it and somehow we well i was trying to think of an example i couldn't think of one i still can't think of one but imagine we have a dial that allows us to uh to change the radius of this ring here that things are propagating on in other words change the circumference of it changing the circumference of it will not change the number of nodes you might ask yourself how could it change the number of nodes suddenly i mean we're going to expand we're going to change the radius um gradually or continuously at what point would you expect here how many nodes were here about 10 or something like that you know everybody know what a node means number of times that the wave goes to zero right it has to it has to be integer for one thing it has to be integer the wave in going around a standing wave let's take a standing wave for simplicity a standing wave on here has to go around and be periodic so the number of nodes has to be an integer now you start stretching the space that it's that it's propagating on it can't suddenly jump at what point is it going to jump from seven nodes to eight nodes it doesn't jump what it does is the waves just get longer wavelength and shorter wavelength to match the thing that it's on it's the same phenomenon except replace this ring by the universe itself number of nodes number of zeros is invariant never changes and the only way that can be accommodated is as if the if the wavelength of the photon increases wavelength of the wave increases now any any wave even if it's not a nice thing like this let's suppose it's just a lump over here and then nothing and nothing over here by fourier analysis it can still be expanded in waves that have this nice periodic smooth structure each one of them each one of them has a number of nodes which is fixed and so that's good enough for us the individual waves of definite wavelength can't jump the number of nodes and so instead instead the wavelength has to accommodate the size of the thing that it's propagating on yeah as you look out in the universe and galaxy move faster and faster relativity says the the uh the mass of them relative to us would grow as things faster and faster correct the energy the energy the energy yeah okay the kinetic energy and if we were to weigh it it would weigh more well you can't weigh something that's far enough or far away no no it's important um it's important uh but go ahead yeah i i think you may have answered my question but uh assuming you can it grows continuously at some point it goes exactly at the speed of light which means the mass must be zero intimate mass no oh sorry yeah that's right the the momentum mass is a thing that never changes mass never changes mass of the electron is always um uh whatever it is yeah i i think what it means that the question itself is silly because you can't talk about the mass you have to talk about the energy you're photographing energy so from a naive perspective mass goes from really really a lot to zero since part things with mass can't go at the speed of light but as you just said that is not their way of thinking all right well look to do yeah okay it takes more faster than light wouldn't the time and space what's that time and uh space actually it gets into change right i mean how does everything go that was which thing worked out i mean [Music] why don't we wait why don't we wait we're going to talk a great deal about horizons and uh and the geometry we cannot the point is we cannot analyze those kinds of questions by just thinking in terms of newtonian or even special relativistic geometry we have to analyze them by saying there's a metric that there's a geometry and we have to analyze that geometry we'll do that we'll do that we can just we can go only so far without uh without introducing real relativity so far we've been okay we didn't have to introduce relativity and the reason why is because we stuck to a small enough region all these equations were derived by looking at galaxies which did not have to be very far away all right we follow some galaxies which are reasonably close and we follow them if they're close enough they'll be moving with a very tiny fraction of the speed of light so that's what we did we followed things which were close enough that they never got anywhere near the speed of light if we want to study the whole universe and out to distances 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 and we really can't do it without general relativity right so you're you're jumping ahead of the game and trying to get there thinking only about newton or newton and special relativity just by stretching the wavelength how does it decrease the energy of a single photon it's more the density that has decreased right what's that say it again if you stretch the wavelength you said that the energy of an individual photon decreased all that happened was that the energy has spread over a longer wavelength no no no the energy itself decreases and the easiest example is to think about the photon in the box or the violin string the photon in the box exerts pressure on the walls of the box because it exerts pressure on the walls of the box it does work on the box as it expands the work is the pressure times the change in volume and that's equal to the change in energy of the photon so the photon is doing work on the box and in doing work on the box its own energy is decreasing look this is also true incidentally forget photons just take ordinary particles a gas in a box what happens to the temperature in this box if you increase the volume down what does that mean about the kinetic energy of each particle it goes down if you do it adiabatically if you do it uh if you do it infinitesimally slow it does or does not it does yeah if you do it very very suddenly what will happen if you do it suddenly the here's the here's the molecules okay and you absolutely instantaneously suddenly increase the volume of the box what you find is a big empty space the molecules are still in the original volume here and nothing has changed their energy their momentum or nothing else that's the non-adiabatic case the adiabatic case means you increase it slowly and slowly means there's plenty of time for molecules to bounce off the walls many times then it uh it will um cool if you do things suddenly it's likely to heat even if you expand or contract if you do things really suddenly it's likely to heat or at least not cool but if you increase the volume of the box slowly then it will cool the same as this wavelength it's connected yeah they're disconnected but it's a phenomenon that doesn't particularly have to do with photons it 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 the box and to conserve energy it has to cool so this is this is the phenomena of the photons cooling okay good for more please visit us at stanford.edu

Channel: Stanford

Views: 249,309

Keywords: modern physics, mathematics, universe, cosmology

Id: 938_TNP4aUs

Length: 106min 7sec (6367 seconds)

Published: Tue Jan 29 2013