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Cosmology | Lecture 3

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Stecher University we haven't
talked much about geometry we've supposed that the basis black space that we have even talk about space time geometry of called scored do start as a matter of observation space is very flat but it's not a principle it's important to understand very important understand what cosmology would be like if space for not flat and as a emphasized over over it's not that we know the basis flat which is not very big orders flat positively curve negatively curved we really don't know them so what's important to investigate the various possibilities now we don't know all lot on scales of space much larger than 10 billion light years 20 billion light years but we will make some assumptions the assumption I will not be the basis flat but at least for the time being but continual to assume that homogeneous homogeneous and isotropic is that true maybe it's not but will never know until we learned the find out its consequences and the only way to find out of consequences is to assume it and see what it said that we should assume the opposite and see what it says but because almost all cosmology is based on that assumption it's a good thing to our who explore that space is not flat but homogeneous there was need space not being flat means its curved what kind of spaces curved 1 assumes curve AT paraboloid his career but saw his curve and side with a bomb planted His curve so all kinds of curved surfaces that you could think I'm not thinking about two-dimensional surfaces will come the three-dimensional space is soon enough also lots of spaces you could imagine are curve but only a very very small number of possibilities are consistent with home maybe MADE means the space is everywhere the same somebody located at a particular place in the space looks around and sees everything around them and see exactly the same thing that somebody else these and any other position that's called homogeneous 8 ellipsoid overlooks the paraboloid paraboloid is most certainly not homogeneous it's more curve near the tip of the paraboloid it's less occurred far-right ellipsoid our along pointy ellipsoid is different nearly polls of the ellipsoid is a waste of your where the equator of and if you walking around on it you would notice a difference you would notice a difference seriously you would a not however not be fooled unless of course it was huge in you couldn't sample the curvature so ellipsoid spaces bumps on the surface of the earth if we take into account the mountains in the valleys on about current services but they are not homogeneous what kind of spaces can be curbed and homogeneous well basically there are only 2 kinds an error 2 classes that apart from the size overall size of them they're only 3 kinds Excuse me 3 kinds of 1st flat space let's the Webster begin by thinking about on the metric for space not space-time magistrates the metric of space flight space is as good or flat rates very fast bases just go out there and if we were talking about two-dimensional flat space mean plane and the plane distance Salomon the distance between 2 neighboring points with lawyers Mr. D X squared plus BY square the everyone I add a 3rd dimension or we have to do is add a 3rd dimension straightforwardly easy square that's the way we describe spaces but by giving them at are giving their that break the distance between any 2 neighboring point but we know that we know everything about space this is ordinary flap space you correct we don't but Euclid OK let's think about the two-dimensional example and instead of working in Cartesian coordinates with work and polar coordinates poll of coordinates have some nice feature when you think about cosmology they have a center and if you think of the center as you walk they have a natural place to put you you look around you look around the sky hook up into sky you look around you looking around at angles your visual field the field of angles if the world was only two-dimensional than literally looking around you would be looking around the angles and what you would see in front of you would be laid out in the angular space it's useful to think in polar coordinates in America and Cosmo orgy in the two-dimensional example we introduce an angle theta and a radial variable are the usual thing we've done this over and over and I think everybody knows the metric metric in polar
coordinates St. metric fiasco is equal to DR squared the radially plus not be square but what are Square squared off this is just a statement that the distance Intel from given indeed as a given a small angle gets bigger and bigger and bigger as you move away and it gets bigger linearly with distance of the square of the distance grows quadratic we would our square so that's see him that's the metric of the ordinary plane and the will be given another name give Hogan and that the new term for these data squared was is status quo deep this quick itself is a metric it's a metric for a particular space it's a metric for a search for a circle if I had a circle and was circle it is a kind of sphere it's a one-dimensional sphere it's one-dimensional removal or get somebody living on it we just think living on a one-dimensional space and sometimes a circle is called a warns one-dimensional but it has apology that the closes back on itself sometimes circle is called Omega 1 Omega doesn't where omega came from but 1 is justified its one-dimensional always sometimes refer to they the square as just deal Omega 1 score they just stay areas for the metric only unit circle these is metric of the eulogy circle of unit radius unit circle I mean radius equal to 1 the metric would just be deal may go 1 square where about that notation cover over Norma we're always wore I have the right to details of a metric so I would mean for a deal maker 1 square picked would say that these days this that only saying that measures the square distant of us young young gap measures songs conference right now it wouldn't make any difference if we took the circle figure was a piece of string going around a circle and the former work like this tho it's still be trouble that you could still label the circle by the same angular coordinates still label it by the same angular coordinate so that equal intervals along equal angular separation and you would say that the metric of the Our performed circles would still be exactly the same thing became square so it doesn't matter he draws a circle the circle is just a space along with if you go a certain distance you come back the thing place after a distant and this case the distance would be part a division leaders and you live them a circle and the circle happens to be a circumference of 2 party leaders you would call it a unit circle in the metric system will just think about the abstract unit circle which has only been rounded off to party but they when it goes into the polar coordinate thermometric each circle as storm radius of the radio might increase and so we put our square in front of the Omega 1 square and that's the metric of video of RV flat plane at flat based in this orders it looks like there's a special place but didn't a special place every place is the same as every other place is just do we use Our place as the center of a drug that age a series of family of circles now I can't think of the flat space as a kind of nested sequence of circles of increasing size think of it that way they got a flat space as being composed of nested series of circles each 1 having a radius of the circle as a radius equal to its distance from the origin here our next Let's go to the sphere the sphere is also a homogeneous surface every place on the surface of a sphere is exactly the same as every other places so it's not the point but it is homogeneous much discussed it's that begins Nov aren't talking about the two-seater we talked about the warns spirit that's Omega 1 now we're going to talk about 2 spirit the surface of the earth as to speed its two-dimensional fear a sphere of course has a particular meaning but for us it may mean that has is that it has uniform curvature it everywhere is the same as a property that if you walk around it to come back to the same place in any depth and direction that's the main properties of it that we care about so what to do about it is ordinary spheres and let's also think of it as a sequence of circles this case rooms start at this point the collapse he is where we are I'm right at this point on this year now I look out at different distances ROSE but I see a series of Shiraz nested spheres I look
out nearly I see the 1st 4 years I say Look out there Muammar struck looking out away from my own position at the two-dimensional astronomers that 3 dimensions then at 1 light
year I see everything arranged a light years I see everything arranged on another fear another sphere others but something different about this series of nested spheres and that as I move instead of growing the spheres stop growing they grow more slowly as they move away from the center here and eventually point to grow anymore and then contract services not with time is districts the ones years circles Out of which the 2 spheres of former instead of Rawlings linearly with radius role for a while and then they shrink and in fact we know how big each 1 of these fears is that the spheres a characterized by an angle let's call that angle all are distance from this point as measured let's say in a angle so are 0 over here Arias Hi over here as just a waiter labeled as free store us the quart mixture describes we're right where we are that's Oracle 0 the furthest vacancy and those few closes up on itself at the back end will call at Oracle's pie and what is the the metric Of the sphere in terms of it looks a lot like this but slightly different Rice's flat now sphere s squared equals DR square what's the radius of each 1 of these fears which will be circles sign are trying hard this for a equal to 0 sigh Nora's equal to 0 what signed a pipe also 0 sign of pilot tool his pirate to prior to pile on the equator equator are the radius of each 1 of these 4 years instead of growing we're distance like this stop growing comes back the answer is signed square of all signed square means square was signed the metric over Circle unit circle are squared free signs quits are we can write status square here we like to think it would be the angle around here but we can also call would deal may go 1 square James said of the deal maker warned scored fight so we built and now now war is a symmetric other it symmetrical the two-seater France give to Sephira new name the new name of the 2 sphere is D foam may go back to square on ghetto it to see this as a writer Herb a mail 1 is a circle well missed pattern just continues but I want to make a three-dimensional sphere a three-dimensional spheres of three-dimensional space everywhere is saying If you go out in directions you come back to yourself think about looking out the sky again to look at the sky you see things at a certain distance former aide to steer around circle a former to around you you look a little further another tombs fearing bigger you look a little further and you see our very big sphere but then what happens if you look a little further the 2 spheres to get smaller and smaller and smaller and smaller until you see around the other side of it wearer a where's shrinks again . 3 spheres are just as good as bases to a harder to visualize euro a visual cortex doesn't have these machinery to be able to visualize directly 3 spheres but they just as good and what they are is instead of a series of nested ones for years which grow and come back and collapses you go further and further away there a series of nested 2 spheres once feared North inside a bigger 1 that collapsed so you could think of it as as as you move further and further away in all are at 1st you see a small still around you a small to when you see a bigger Iran Telogia 2 spheres conceal and they say that they that further distance and then I started its more Oregon where strength as 0 an Oracle 0 and they gained at Oracle's pot not now observer is on all none not the observer is at saying is is city of this series of fears here refers to the few you see around you see around your the closest 1 my head this mostly around me right is pretty small as was formerly here I looked out at a distance of meter and I see biggest fear I look out distance 10 meters I see a biggest fear around me if I were living on a recent years but I would look at our To a distance which would be a biggest to steer and that they would start a shrink again that's exactly the same thing as here you see earned your small sphere you see a bigger 1 or circle a bigger Circle of 1 and then they started its small over here so we could draw on by saying it is saying we have a point a little bit further away we have a circle and they have a bigger circle and we have a bigger circle and circles shrank again now the case of the tourists here I can draw as a fear that a recognizable sphere but just think of it as a series Of circles which grow and collapsed growing collapsed as a function of art from Oracle 0 particles pie if you are go unnoticed debt the three-dimensional spheres you think of them as a nested series Of condescend trick to Sears around your OK and that you should be able to guess what the metric of a free spirit is this is a record of 3 sphere steelmaker twos glared equals again as a square there's always DR square and distance away from you and that is the angular part and the angular part now would not involve circles but the angular part will involve 2 spheres a series of 2 spheres of renewal and that will be signed square are the to square not deal maker 1 squared the deal maker to square such yes flat two-dimensional space was about flat three-dimensional space hero these 2 spaces are Spears to spheres and recent years should be Odie omega-3 square the omega-3 square which is 3 4 we may be living in a 3 sphere possibly we do live our 3 space be a three-dimensional thing like this and this was reached easier to a magic analog his flat two-dimensional space where about flat three-dimensional space you guessed to the guest reflect three-dimensional space in this former yes to have angle and that other angles forms in Omega Tour do justice flat three-dimensional space in polar coordinates looks like this where this is a toast it is corresponds again to the 2 spheres that orbits around
you a lukewarm as is stated in a patient standard notation to call a the metric of a sphere deal maker squared approval index indicate coming dimensions if there's another way to view serious of 21 Eds in some ways a little more intuitive well on own much room intuitive but it requires some extra baggage Hays Circle Our circle if your little including on the circle and you could look off Circle you could look into the circle all you could do was receive light from longer circle to communicate with neighbors but you would have no way of telling whether it was truly a circle or whether it was her thing like this war even if there was no other dimension perhaps all there is is year space along the line with no sense of moving perpendicular to line as a creature living on his line who couldn't see off the line in there somebody who lived on optical fiber maybe could receive no light except from along the fiber would have no way of telling whether that fiber but in 3 dimensions was truly circular has some of the shape and you wouldn't care ending more might actually just be living on the one-dimensional space with no sense of a year of perpendicular directions but still nevertheless we can if we light described circle by and betting it in 2 dimensions only one-dimensional but we can and that it into dimensions and how we do that we're right Mr. Cohen X X square foot Y squared equals 1 Vassar Circle right common distance every point saying distance from the origin namely in this case the distance 1 of unit circle the yield it to
smear we introduced a 3rd direction noticed that describe it to In this way we have to what we have no choice but to introduce a fake a 3rd dimension now 3rd dimension the case of the surface of the earth is real you can move in the perpendicular direction but again you thought about
a world flat plan if you thought of flat land where creatures can only receive light from within the surface itself and the extra dimension would just be a year a trick for describing the circle sorry describing sphere who described his ex grep was why square forces eastward equals 1 that's a surface in 3 dimensions but that in its own right as a space and it does have exact has exactly this if we measure distance away from 1 of the polls in the same way the metric of the 2 spheres This is the true spirit is exactly what I wrote while could go another step in island construct a free-spirited constructive 3 syrup in this way you have to embedded in the four-dimensional space again now the four-dimensional space may really be a fake may be only a video of the three-dimensional surface makes any sense but you would add 1 more ladder but but and this surfaced the three-dimensional surface in the four-dimensional space the 3 sphere again you coordinate dies it like distance from some point this is a metric of the 3 sphere and then it high dimensional space may or may not need not make real cents armories really have physical significance as the surface of the earth it is embedded in three-dimensional space if we live on a 3 spheres chances are it is not embedded in the same way in a four-dimensional space River we don't have the best answer that question OK what's the difference In terms of what you perceive if you live on a sphere versus if you live our infant flat Plane how could you tell will miss telescopes in those telescopes allow you to work hard to determine the distance to distant objects just a distant moderate determined arm of the galaxies which are embedded in your space telescope itself of course doesn't allow you to tell us the distance but let's suppose you had some various tracts are to be able withheld How far things were away standard trick cost spectroscopy but turned using go the hobbled off a combination of things but this particular case what truly uses trick tell how how far the Galaxy was away look at luminosity have bride so it could add to its lookout bright amateur luminosity we're getting from from Bull Bull a light bulb faraway looks less luminous Obama like Bobola close up so but assume we can go on and let's look at end object unknown object to Galaxy let's assume all galaxies of strain is of course not true but on the average the average populations of different kinds of galaxies we can click to look at a distant galaxy and we can from the type of galaxy galaxy alike are wrong with roughly the same number of stars we can really see the individual stars but nevertheless to Galaxy like own will assume that has about the same size is a 100 thousand light years across a wherever and so we look at his galaxies we begin tell how far they are and the simplicity let's just say all the same size and alike and ask how much angle the lady the sub Clinton than the sky we see a galaxy an obviously the further away the galaxy is small angle with some tens of the sky up hands me how big it the from big Englewood looks the sky so let's take let's begin with a flat space and they ask galaxies all galaxies have a diameter of these offices the diameter of of the standard spiral galaxy our standard tariff by galaxy has diameter were looking at it from different distances or it's different distances from last public anger was observed by so we look at the matter is that correct let's still flat faces flat space then the simplicity just take the two-dimensional case the matter was logo just look look on radio circle answers galaxies out here galaxies year galaxies caused by galaxies very far away each galaxy has diameter D and let's think about angle that some things in the sky as the angle that tens in sky has signed D D E is it's true sigh so that means that cornered from here to here is just these square on the other hand it's equal to deal squared well don't care about oranges looking between here and he was too points are the same radial distance so if I look at this the whine of the surface across here DR is due 0 0 this is a line at a fixed aura of Aziz's Iran is equal Forest squared Decatur scored just as part well is ah Square D favor square but for the ordinary flat space two-dimensional space to limit be squared the actual size of the galaxy is D squared sequel Dora square dictator scored nothing new so what he favor it is the size of the galaxy divided by all are just the to solve this by writing to Data squared Tuesday squared of our square taken square root OK so the angular size and this is obvious the move saying anything you don't the angular size in the sky is proportional to the actual size of the galaxy divided by distant from meal so we could check this week and measure the distance the galaxies we can see how big a look at the sky if we lived in flat space we would discover the angular size of them decreased like 1 divided by dissidents now still exactly the same thing on the 2 skiers incidentally this fact it is true truant 3 dimensions of straw and any number of dimensions now let's do it on the sphere and the simplicity let's just imagine toast as a hilly art world he was looking out at the galaxies which are all about the same size they fill space pretty much homogeneously we can tell how far they are from us in the same way that we call before we can measure angle but see what do we get began the size of the galaxy is D squared and now instead of being Square they squared Sonny Square Pa d status squared work out here would reverse galaxy overhear angle some attended satisfies D squared his sign squared are favor script these data is
equal to the size of the galaxy not the letter but all are but by divided by sign of our but does doesn't mean which is bigger at a given distance was signed is smaller than are are increases linearly sigh are turns over so signed it is smaller than are that means deflated is bigger than it would have been in the flat case dissidents would gather sorry angle by a galaxy of let's say a thousand a couple of million light years away from us but to make it more than that a few billion light years away from us if we lived on the sphere that Galaxy would look bigger it would look bigger because signed our decreases but doesn't increase as fast as are in fact when you get around to the other side of the sphere with a sign of our starts to decrease the galaxies out here look bigger than the ones in clubs because galaxy over here and compared to 1 almost at me and the other end of the universe together under the uterus the is almost 0 again so the Galaxy over here it looks about as big as the Galaxy over here so you could tell you could tell it would look think there because it's far away but the door a have river properties a distant galaxies post but as a sick if you have a trick to tell how far away it was you would find something different about the spherical geometry mainly up to pouring up to a point we are a distant galaxies would looks more and more men will kill you got halfway around and then they would start increasing in size again in fact if there was a galaxy right at the end the Interpublic here you would see it in every direction that you look so you would see it look that way you see if you look that we would see the it that way it would fill the sky that would be an extreme case of rests that that the further you go the larger things look now it is good is analogous to determining the curvature of space looking my microbe or things which who societies you're able to decipher the cosmic microwave background that is the size of certain acoustic go lumps of those who can't lives not galaxies we look galaxies Michael Ray rework at the oscillating lumps of staff but basically is same OK so there we have the 3 sphere and we discover what it means to an observer at the center of the universe was the center of the universe there is no sense but being very self-centered with ourselves of the center we look at different distances Ellicott can also look at another thing we can count the number of galaxies and different distances but on some of the year if you look at sorry you see fewer galaxy but if you look the same distance on the plane the plane saw opens up the Quran contract and fact way out at the maximum distance you basically have a chance of seeing 1 Galaxy whereas the same distance in flat space you would see a lot of galaxies counting galaxies is is another way go camping galaxies in you configure at yourself comedy galaxies you see at each other radial distance on the sphere and harmony galaxies you each radial distance on Bonnie infinite flat Iran I think there's a lot of it was hot in the S if that lets us see you at a steel at the moment with thinking at the moment I'm just telling you about the geometry figures at the moment we not asking about increasing or decreasing were what made this fear that size were just thinking about units years mightily thinking about units Gaza thought of sphere twice as big as a unit to the same things would be trouble but now we just thinking about a world which is likely Earth except its three-dimensional and it has a fixed loans side is our doesn't change with time that's or talking about now of course that's not really the case really the case of the world does change with time grows but there before we talk about the time-dependent we talk now about tool kind of geometries which a homogeneous meeting to say they're everywhere the same as a 3rd 3rd has various names on and just going to court a hyperbolic space it's not as easy to amend as a sphere the full I'll tell you what before we do that let's talk about another way of representing the sphere this will be candy when we go to think about the hyperbolic space as a way of thinking about spheres which is get useful moved summer some ideas from right but more important is the that useful way of thinking about a hyperbolic geometry artistry this called stereo graphic projection lot of you know it is you take this fear on ordinary space and you rested on an infinite plane mathematically you just passing the changing planes of the South local point on the fear this could be our point the point where we are now every point on the sphere can be labeled on map to a unique point on the plane is a little bit of trouble about the North Pole but the North Pole was very far away without worry about every point atmosphere can be identified with the point this plane by so-called stereo graphic projection you got to the North Pole and you take the point that you're interested always said we are at the South Pole where at a southpaw but that's our us we're over here were at but visited due to stereo graphically projector a point on sphere to draw a line in a straight line through the point comes out and hits the plane somewhere it another point it's a plane here is a poor way up near the North Pole that map to some vary very distant point where about the point that the North Pole itself was echoed coastal cities it doesn't matter what direction every point that the North Paul it is out
circle a huge circle and it's even nicer not think about points but the figure will circle little circles or let's say of the same size they could stand for galaxies little circles on the same side by this projection the circles on the sphere mapped some kind of closed curves on the plane it's a little bit of magic but circles map circles that something Yukon proved to have the patience for those who take a little circle on the sphere and that by mapping every point on the you'll find circles on the plane that's obviously 1 of the South Pole what about the size of these circles however the size of the circles depend on how far off near the North Pole you are only answer is that circles of given size near the south pole look small and not only small they look about the actual size of the year of the circle that as you move further and further as points more further and further the circles appear were bigger and bigger In fact the circle up near the North Pole he love would be art except for the extreme Martock circle that looks like a giant circle very very far away so that's a way to think about the Sphere mapping the plane and mapped onto the plane has the bizarre properties that further toward the North Pole you are the bigger things look on the plane but does that mean that the sphere is not homogeneous know this is just a way of drawing it is just a way of representing at every 1 of these circles is on the scene intrinsic side on so I'm not people playing like this given drummer plane you distorting things you could do the same thing with a 3 3 year you could also map to an infinite flat three-dimensional space similar things happen 2 years which are near you map small spheres To use far away now legal so this is called stereo graphic projection number tell you about the hyperbolic space behind them but space was made in exactly the same worry I'll take the case Our VER three-dimensional hyperbolic space organ about 3 and 2 dimensions yachts out to dimensions instead of calling on get Cora H. Court D. H. squared H stands for hyperbolic H. squared and now I'm talking about the two-dimensional hyperbolic space exactly the same br squared plus something square turned Omega 1 squared Fletcher a chief flight feet to the Omega talk for you in here we put hyperbolic sign all are squared may remind you of the hyperbolic sine up for 1st let me remind you about ordinary sign sign is equal to eat the IR minus Ito the minus are divided by 2 white another formula formula for signed in terms of complex exponential hyperbolic signed is even easier it's just eat are minors each of minus all are all too In particular hyperbolic signed for a very large on is nominated by Ito the authority to the miners are a very small number of each the past is a very big number sometimes I just call exchange Of all are common turned hyperbolic sine synch of art is a function which grows very quickly it grows exponentially signed as far away shrinks to go far away from all sides shrink Specter 0 hyperbolic sine just blows up and get bigger and bigger rapidly exponentially again this is based that when you look around you you still see circles we're still in lawyer damage still two-dimensional you a new family of circles but the size of the circles goes very rapidly laws are very very rapidly the to the omega-3 so to go to a V H 3 squared which is the candidate geometry that we live in a candidate saying thank squared plus cinch squares R D or maker to square now we have the situation with surrounded by clues 2 spheres but the size of those whose views gets really big really fast as we move away want you guessed happens be angles attended by a galaxy as you go further and further away with work it out with work out on the hyperbolic geometry remember the signs here we are were at the center looking at a certain distance will looking an object which has size is indeed intrinsic size that occupies an angle on the sky data but she wanted it we can replace the Omega 1 square just data or data squared with half the size the actual intrinsic sizes D squared and that should equal hyperbolic sine Ross inch square inch of ah square DD let's squared favor squares saying the steelmaker 1 square what are just these data the size in the sky is equal to D divided race in of aura OK now that's plug-in Watson Gerard is a large our when you look far away we don't have to worry about this for long are it's just eatery are divided by 2 of 12 defined these data is approximately equal D times to win over the now eagerly arm goes very quickly as you go away so what happens to this angle associated with a galaxy of size as you go further and further away it shrinks really fat furthermore the number of galaxies grows especially fast number of galaxies grows very fast and the size of their shrinks to match so if you live in the hyperbolic world and you look about you would notice that distant galaxies look too small you don't come you don't have this phenomenon where after a point is nothing left budgetary back 0 kids growing but you would discover many more galaxies at a given distance long 4 wickets for worried you find a
huge increase in the number of them would find each 1 of them anomalously small compared to flat space so the geometries really have meaning OK this is this it is hyperbolic spends the purposes but tell you how you get this picture of gas summits Excuse me you're too but we although was who he is a case excuse now so is that all of them were still goes down based a case of beer and on record said that whatever but are there are 5 not a case that I've thought about it How is it that you had a and when we starion graphically projected this the years it was on to the infinite plane now because I was at the cost of enormous distortion where things far away just look much too big to the opposite takes place on the hyperbolic plane show your way through wistaria graphically draw the of the hyperbolic plane if we ever to lose here let's take the tombs your case I can draw a three-dimensional figures especially when they occurred to kill so I'll stick with two-dimensional case on yacht but but so dimensional of two-dimensional severe X square plus Y squared Posay squared equals 1 are now going to draw a two-dimensional hyperbolic space the two-dimensional hyperbolic space again you start with 3 coordinates you could call an ex-wife disease but I'm gonna call ex-wife tea capital P capital this is not it's not really kind this is just a trick withdrawing space it's not time nevertheless summer call it sheet cake X Y Z not XYP space and construct not just here but instead he squared minus X squared Money is why scored Eagles 1 instead of a T square plastic squared plus Y squared equals 1 which would be a sphere take he squared minus X square minus Y squared equals 1 of you know what kind of a surface this it's a plus sign sir spheres plus sign of the summit circle but a hyperboloid you only have 1 of these it would be a hyperbola with 2 of them becomes a hyperboloid hears about from Lord looks like the plot he upward X and Y horizontally earlier and now drawn calling we're not really doing relativity but it looks an awful lot like relativity but warned that this is just a trick for drier surface drawer or call and it will upset some very good commented dead harsh right Oklahoma and it corresponds to the surface he squared minus X swimmers Y squared equals what but Cone 0 hour these squared equals X squared plus Y squared coal man ahead now but 1 well let's see 1st of all said X and Y equals 0 let's go on key access use the vertical axis that corresponds Thaksin white equals 0 Where is the point he squared equals 1 Watts is the point equals 1 2 with them on top and the bottom was just concentrate on part of people's 1 what happens is you move away from that point to former personal a hyperboloid of revolution high around hyperboloid of revolution whats ought to bulb it does not look like every point on this hyperboloid is the same as every other point it really looks like this 1 is special but not not if you use the metric not be Quinn plus DX plus BY squared but you measure distances on here as deep he squared might be square might be are squares chart the PX square but you want squared minus 2 key squares you measure distances on here using the relativistic metric pretending this was high it's not just the embedding space to draw the Gerard herbal or if you do things that way than this hyperboloid is absolutely uniform howling know because the go from 1 point to another point hyperboloid is equivalent to Lorentz transformation which was the climax of various ways but if you're not comfortable with that we just take approved as a true fact that hyperboloid when distances are measured with the metric BX square plus 2 BY squared minus D squared in this hyperboloid really is completely uniforms and thus a pulsar I want the stereo graphically projected here is a waiter stereo graphically projected that's useful but gay draw a change plane tangent plane now is through the bottom pouring think of us as living at the bottom pouring here we live at the bottom pouring we look out around us we look out in distances distance out here this year but now take every point on the green hyperboloid but and get to a point on the blow plane how we do it we just draw a line from the center of a year for the particular point on the hyperboloid and it intersects the planes mourners point on the hyperboloid % don't just is at the center the point on hyperboloid over here intersects mourners over here but over here wherever how bounced How far note on this please do you go as you move out my hyperboloid you get closer and closer to the Aston prompts about herbal idea but they have sunk survived Ferber like intersect the plane on a struggle here so no we no point who's ever mapped out on the circle even the asymptotically far away points very very far away simply map points near the boundary here they circle source donors mapping again every point wines up in a particular place forge good but there's a lot of distortion a lot of distortion and in particular things very very far out there squashed very very close to the edge of the circle things near the center described pretty faithfully things out near the boundary are much too small quite the opposite exactly the opposite from of what happens when you took the sphere the plane is
freely spread out on a plane everything far away was much too big a hyperbolic geometry things faraway look much too small for you look at this picture is a opposed thinking a mind every Angel and Devil is exactly the same size as every other ones in fact there there are motions that you couldn't do coordinate transformations that allow you to move the Devils and the Angels around analogous to a rotation of a sphere to take sphere and you rotate the mirror the galaxies along around if you projected onto the plane appeal from all over the place but really billing is rotating sphere same kinds of things here they can all be thought of as the same size everyone sees exactly the same thing if you look carefully you'll see that every Devil arranger essentially sees exactly the same features around him and if you take into account that size scales have been performed letters may happen so when you look at things don't pay a lot of attention to size but don't pay attention for example harmony angels and devils each 1 season the neighboring to arm the see that this really is homogeneous every point is exactly the same as every other point this is sort of like a the sense that everything it is like that however this is not a space time Purup space Mrs. purely space notice as you look very very far away that things look very small angles attended by a very distant Angel looks anomalously small compared to what would happen if they were spread out playing a uniform white notice also that the number of them increases vary dramatically as you move far away increases exponentially so that's the that's a ruse is geometry various names Cloncurry gets two-dimensional version of it it's the Lobachevsky plane hyperbolic geometry shirt has other names it's also uniformly negatively curb space the sphere uniformly positively curb space and that actually mean something tactical was a certain components of the curvature which are really positive and this geometry has the opposite sign for its curvature sorrow mathematically it is deep space of uniform negative curvature and as I said it's the same everywhere so we have 3 kinds of bases there and three-dimensional version of and the three-dimensional version instead of having it did you have a ball notable always the ball is a three-dimensional solids staff contained atmosphere sphere always means the surface terminology sphere means the bounding surface Sony say the Earth is here are you talking about the surface of the year when you see the Earth as a ball you talking about a two-year over everything contained within the bounding surface OK so arm for that I mentioned that her case a remember every 3 kinds of spaces but that's right down all incidentally just as Syria has a natural radius the hyperbolic geometry also has a radio the radius is the thing which appears on the right-hand side 4 square that's square of a radius This is the unit hyperboloid it's the Una hyperboloid in that send them a point over here is 1 unit away from the origin of course we just as you could also draw a sphere of radius is a serious 1 he is a sphere radius do you could also draw hyperboloid of radius to what you do is just go up till units a network that it would be somewhat flattered at the center Arabia draw very big hyperboloid big hyperboloid now would be much flatter but it would be the same shape except expand about uniformly bath and you would indicate that by putting a radius of curvature in he has scored some square radius of curvature right so the point is that each 1 of these geometries in particular CEOs and hyperbolic plane hyperbolic disks or hyperbolic bowls however radius associated with them and if you include the radius and then that that if you want to think of not about a unit geometry leaders multiply the metric by the square of the radius so let's would do that now but think about last fear ordinary sphere the metric of an ordinary sphere of radius a real ale the sphere radius a fear has DS squared is equal to the aura squares plus sigh square are these data square or Omega squares and settling case I didn't mention it there are forced to use 5 to 6 to 7 years in each case if you want the 100 dimensional speedier right exactly the same formula except you put the nite in the 1990 them into he always 1 less than mentioned around your area are dimensions altogether OK that's opposing I want the sphere whose radius is not 1 but whose radius is real a warily I just multiply sizes by and sizes squared get multiplied by a square square will a square of the radius of my sphere fixed radius of the spheres AT & the metric is exactly the same except when multiplied by a square unit sphere and he is a sphere of radius any pointers multiply why a square why not 8 will because this is a square of reversed where about the hyperbolic geometry saying things yes squared is equaled a squared times br squared plus hyper box school and are the Omega square and that's a hyperbolic geometry of radius so we cannot only accommodate units use in Ulan hyperboloid but arbitrary hyperboloid of arbitrary dimension be bigger collateral hyperboloid issues but they also similar to each other similar in the technical sense they're just um expansions and contractions of the same thing it is assumed in most of Cosmo orgy that the space that we live is 1 of these 3 spaces they go the break to
double the size of a fear what happens to it curvature he said it is set us not the Waco Richelieu defined forum head bump compare basketball with the surface of Europe which 1 would you say fewer explorer landmark looks more 1st of all a small the geometry the hierarch curvature now he said they said they are out of
order what but they with what is that they are of qualities here's what laughter what or something else I and I know on scales Out as can as far as completely Ted it looks flat our feel so large guess how about a 3rd of areas that talks should ended as much as I see it is thought that that far duly explore new bike nearby means distances small Mike comparison with a radius of curvature with a radius of the year then you can't distinguish 3 of them they all looked flat India Young said he is not this was a disaster right and the answers we can see literally microwave infrared to water a 20 billion light years or something like that summer like that because it looks so flat out to that distant we know players at least at the absolute minimum our 10 times larger and that means a thousand times larger volume that is a really just can't very very unlikely residual comp on almost certainly much more than they need what flat corresponds to limit of infinite being if that we beg forces fled could there be toroidal and maybe will talk about that but there's no reason to believe it now got that or other kinds of geometries is not that this is the only kind of geometry in fact there even under ideal I said something not quite true I said these are the only homogeneous geometries but with just remind you just now but that's not quite true there are other geometries namely the surface of the Taurus the service of the now service a rebound that does not work flat OK but that's only because you took the surface of the doughnut tried to put it in the three-dimensional space I'll tell you what a shares surface of the Bar Rod mathematically where mathematician means by porch you start what a rectangle obstruct its 1st drug forest the bagel they as a property that surfaces two-dimensional in Ucon in 2 directions and there are 2 8 independent ways that you can go around the torus income back to the same point you could go around it by starting out horizontally and you come back to the same place or you can grow vertically and come back to restrain price there are also other ways where you saw spiral around and you go you spiral 1 as you go the other but these are the 2 primary independent cycles of Torres now you could represent the Salazar topology across from a mathematician speaks of a Taurus usually speaking topologically take a rectangle and start walking on a rectangle horizontally so in color coding has come to the energy but when you get to the edge make an identification of every point on energy is identified as being identical with point on the opposite edge over here so that when you over here you come here you'll your every Yukon combat Missouri barge that's the analog or not be analog mathematically saying is going on the horizontal cycle here you could also go on the vertical cycle if you remember identify points on the upper boundary with points on law boundary to come around here this point with 2 cycles adjusters point over here this is a Taurus a rectangle with opposite sides identified topologically Torres mom it does have the property 1st black life is a flat but his drew it flat on the blackboard it's obviously flat just video thing which earlier drawn on the blackboard but for but periodic periodic in 2 dimensions of bubbly periodic a mathematician so it's topologically a chorus it is flat and every point is the same as every other points it doesn't really look like that looks like this point being closer to the
boundary is different than the point the center but there is no boundary there really is no memory walk out here to come over here every point is exactly the same as every other poor it so it is also a homogeneous so when I said that it was only whoever with only 3 possible geometries Dyer was was being a little bit looks there are others pluralist Torres it is a particular case our but in many ways you just consider this flat consider a flat with periodic boundary conditions so just just clarified now could we live on a torus yes yes we could look artists our voters oppose was big enough we them a small chorus we would not look out CEO of families from the direction look at 2 different ways of Celia periodically update but down but as long as it Torres was big enough we couldn't tell we couldn't tell the difference between it and the a flat base was possible it's not a popular right idea sorry Torres tosses of two-dimensional space the stores you could do the same exact thing in any number of dimensions you make instead of a rectangle you make a rectangular parallelopiped Perry it's called and you do the same thing when you come out this edge over here you reappear here and now you come out this end he reappeared the other and so forth up and down so that a three-dimensional Taurus Nucor Torres of any number of dimensions what's one-dimensional chorus this would start over two-dimensional Torres adieu three-dimensional circle circuit right may go warned a shrewd is anything you have they face not a forest where are we are could you yes it is possible or go there would be yesterday said you could look out that their actions to to your tail if you could do that a different will be different be different distortions would be different here you would be essentially no yellow be different you if half through directions in which you would see yourself another directions would see something much more complicated that you would you would but they would you would think a few of them it would be equivalent that's completely equivalent largest saying you live in flat space bullet everything in the universe being repeated periodically here is who you hear you saying areas you Perez hero it's entirely because if you look at the size of use the Taurus you would look out and see yourself from the other side here look out and you see is somebody who does looks exactly like so it's completely equivalent to warm to saying that the first-place is periodic and that he would just see replicas of the same thing in each rectangular self would that be visible well it depends on how big these cells are yourself here you see yourself here see here to yourself up Pierre Crystal array of yourselves US a different that's pretty different by the what it would you do it what else could endure you could In another sphere on like that day but I want to have 2 Spears how we gotta go together there's a sphere is different of course it's not possible in a nice way to warm to continue this fear pest Pest Pi no reason is because it shrinks because the circles shrink 0 at that point the Taurus when you God doesn't shrink 0 for the shrink away any near this thing shrinks away 2 0 stars as you move over him so it's not on my server prospect passed through the point of 0 Sunday's bite but let's school was on the space and harder rumor but just review for a minute the metric of space time the matter of the space time in special relativity has to pieces a kind peace the space priests Moses remind you of looks like setting speed of light equal to 1 ordinary Minkowski speaks space such just flat space-time special relativity DES squared is equal a minor speedy squared plus whatever DX square to why square plus DG squared notice that they had square post BY square was busy squared is just the metric of flat space so if taken time and space and put them together and as always in special relativity tying
component of the electric is negative space components of positive if you want they ask how like Ramos has a removed what's the year was the trajectory of white it's trajectory in which DES is equal to 0 already was let's see I just have move along the x-axis for forget y and z it would say that he squared plus DX course along the trajectories equal 0 all just that would be X His eagle of plus or minus T VX Eagles plus a minus DTD is a light removing the left with unit velocity or lightly will in right you know what you're velocity so no like rays on meaning not year standing for yes equals 0 5 a more general kind of space time is always pioneer in fact was always base but we're going to form or steamed should the kind of space we're talking about keeping time just as it is but substituting for 3 dimensional flat plains here 1 out of the 3 kinds of geometry that we studied 1 possibility is the plane 1 possibility is a sphere of a possibility is a hyperbolic John Bertrand but include 1 other thing scale factor let's take the case of the truce but case of the of the you must look to here to see adjust their visualization squared plus the Omega to squared why this is a world with time but in which space is just to steer his space the usual thing and that is fine but talk space and time but we're going to allow for the possibility that the radius of the tooth few changes with time and that easily done with just remember the change the radius of the sphere we include an 80 square where is radius of the sphere but now we allow a square to depend on time clean squares did took that's because myology our new world which spaces two-dimensional and in which sighs of the universe the radius of the universe is time and again we can write down a free like how light rays more column light rays and didn't move along in all directions work out some examples let's not do that right now this is space kind geometry world two-dimensional world class time were speaks of 2 plus one-dimensional world to plus 1 means dimensions of space one-dimensional time with the kind dependent radius where spaces to smear Wigand the same thing for 3 demand and there was a look like me who looks like is a basketball 2 suicides changes with time but that's all right you wanna three-dimensional world areas say world what about think a pair of points take any pair of points on the sphere separated by a given angular distance Swiss call rectangular dissident leader what the actual distance between the 2 pouring whose angular distances they now or later 8 is a radius of the sphere Decatur right evaporite abuse go wrong the distance between those 2 . 80 s a radius of the sphere times the angular distance What's see velocity Ali relative velocity of the 2 points be close a dark times freighter were keeping the status of the 2 . 6 for keeping the 2 points fixed on sphere but is letting its radius change with time what about the velocity as a function of distance velocity as a function of distance is 8 . 4 with a distance we've seen this formula before the Hubble with 8 . 4 with a being the Hubble constant tho points on this expanding local tracking spirit doesn't matter points the velocity and distance between their actual distance between their satisfy hub along with as the previous examples the Hubble constant H is equal pay 8 ready so we're talking about something quite similar to what we talked about before notice that this fact doesn't depend very much on whether this was to smear it would also be true for a hyperboloid we took 2 points fixed on hyperboloid allow the hyperboloid radius to grow old exactly the same thing would be true in fact if we took a the flat plain and took 2 points at fixed coordinate their stands allowed the coefficient here vary with time we would still have the hobbled off so let's right down the method now of the 3 cases of the space-time metric of 3 cases 1st of all flat DES squared is equal to 0 minus squared but this is not an ordinary flat space-time Mrs. flat face when they time but with are growing geometry or shrinking geometry that we put here plus aII of T squared SA DX square post BY square post easy square prospects for this corresponds to is a flat world would remain at Granada the BXY easy grid but exactly as we studied in the county and physics we can that the distance between neighboring points in the grid changing is a function of time like any of t I so this into geometry spacetime geometry This is the metric is metric of the flat spatial universe with a scale factor which depends on time same thing is true of took 2 galaxies fixed and the grid the distance between them would increase with a velocity would depend on a . ennoble law would still be at the flat geometry nor say flattening flat space How about the spherical geometry squared saying things minus D squared plus a 50 square announced writing a big things from just gonna write Dr Omega 3 square this deal maker 3
square stands for the metric 3 sphere of Munich Re sphere but the actual radius of the universe changes if he squared Excuse me more about the hyperbolic case hyperbolic casing squared equals minus T squares plus a few Square D H 3 square Wallace means is any fixed time the geometry His Haider's Freedom hyperboloid or a flat space but the scale of the space and the distance between galaxies embedded in the space the galaxies embedded In the space that will change according to AT and T . OK best-seller that Our cosmetology what what do we need in order to make this cosmology have sung by exam next to where we need equations house AT changes with time and we're going to want to know how does the universe doesn't expand would continued to expand at what rate does it explained experiment was Tito the two-thirds of have for exponential what does it do to that we have to have equations for it changes with time now we can't really use new his equations although we will find the equations are no Fuller can start with the equations with talking about curved spacetime really not talking about anything new would have were written down we're talking about geometries which curved in both space sometimes in space but definitely in space times since changing with time and what are the rules for such geometries the way the physics works general relativity so we have to do is we have the right down the equations of general relativity for the special case of geometries which have this form and they will translate it will become equations of motion for safety will find 3 cases 3 cases depending on the world was flat positively curve negatively curve and most 3 cases were already seen those 3 cases of art have the same equation as the Newtonian equations for positive energy 0 energy and negative energy uterus riches beyond the Escape Velocity FT escape velocity below the escape velocity so far the Sivakumar Oregon ago next we will not spend a lot of time going through calculating or release symbols and Einstein's general rapidity was a sketch out for you are the equations and what they lead to walk in terms of equations for a requires a simple With before other Friedman equations once we have a more we know what they mean we can we can explore the Cosmo OK now is off to is his work is a good day I say is absolutely Idaho you have certainly a complicated because argues arm and you might ask why don't you include them a lot of them will simply evolve into this is a natural sorrow of placed you evolve but there's no question you can have more complicated cosmetology reasonably easy 1 these are the easy ones and the ones that core popular not a if but the should picked up that they have is best With don't talk about freedom as about no we're talking about a a freeze for is a three-dimensional space that's not boundary anything it is based do got word the stop visualizing the year as a theme which is and baby in higher dimensions you gotta think only of the spherical surface or 3 dimensions 1 more dimension is is hard to do your mind does not want to visualize the 3 spheres of serve the only way to visualize this through the equations for it but it's not in bed in our higher dimensional space that you could move off the surface of onto the surface it is just it is Stetzel effect not is not not they were not if peak he known as a whole lot of there's a whole range of different our 1 way is to use the Hubble locked through a distance of velocity and then measure the velocity by the red shift by the Doppler shift from spectral lines other ways in which a useful it is honored knowledge on the role as Islamic varsity luminosity a particular kinds of galaxies supernova right supernova tomorrow Harvard definite their absolute luminosities from natural rubber tappers using will luminosity but that he did not it that some space but well ahead that's the easiest case from you do your combination of measuring luminosity measuring red redshift measuring spectral lines taking into account the possible a different possibilities for expansion of the universe were positively occurs negatively liquor think all until account measure all of them and you get a START calm positive but not to say it simply as long as spaces approximately flap over bigger of regional news luminosity cool worth to determine distances is the easiest way I think about it but there are other ways that must be it they that I will not be there would not be where does not the other sectors but that's the way I bought it but that is not what it but that started all of this gets taken into account new expert to do it they keep track of their reading big computer car roads and there were 2 saves right over there is a lot if they lost only if the 5th till about 1 % 1 % means that the curvature is 0 2 with a more but 1 that translates into the statement the uterus released 10 times bigger than the visible visible region formal or
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Metadaten

Formale Metadaten

Titel Cosmology | Lecture 3
Serientitel Lecture Collection | Cosmology
Teil 3
Anzahl der Teile 9
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/15063
Herausgeber Stanford University
Erscheinungsjahr 2013
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (January 28, 2013) Leonard Susskind presents three possible geometries of homogeneous space: flat, spherical, and hyperbolic, and develops the metric for these spatial geometries in spherical coordinates.

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