Quantum Entanglements, Part 3 | Lecture 9

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Quantum Entanglements, Part 3 | Lecture 9
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Weapon Saw April Hot working Speed of light Scale (map) Measurement Summer (George Winston album) Cosmic distance ladder Ring strain Separation process Turning Packaging and labeling Astronomer Spare part Star Eraser Sky Duty cycle Kitchen stove Metre Excited state Angle of attack Universe Measurement Astronomisches Fenster Flatcar Sheet metal Ground (electricity)
Rear-view mirror Spannvorrichtung April Geokorona Scale (map) Order and disorder (physics) Cosmic distance ladder Particle Bird vocalization Separation process Turning Tool Line segment Steering Device driver Goldrubinglas Electronics FACTS (newspaper) Focus (optics) Direct current Belt (mechanical) Cartridge (firearms) Fahrgeschwindigkeit Plane (tool) Morning Flatcar Near field communication Storm Ruler Hubble's law Trajectory VW-Jetta Year Differential (mechanical device) Measurement Green politics Galaxy Summer (George Winston album) Basis (linear algebra) Field-effect transistor Roots-type supercharger Lecture/Conference Automobile Spare part Musical ensemble Rail transport operations Station wagon Defecation Big Bang Array data structure Extraction of petroleum Ballpoint pen Transmission tower Supercluster Bulk modulus White Map Finger protocol Universe Expansionsturbine Ground (electricity) Movement (clockwork)
Magnetic core Scale (map) Bubble chamber Amplitude Order and disorder (physics) Interstellar medium Electrostatics Negativer Widerstand Cosmic distance ladder Light Particle Phase (matter) Bird vocalization Roll forming Packaging and labeling Ship naming and launching Limiter Cardinal direction Merinowolle Charge density Electric power distribution Hall effect Electronics FACTS (newspaper) Excited state Belt (mechanical) Series and parallel circuits Fahrgeschwindigkeit Binoculars Mass Measurement Book design Saddle blanket Throttle Orbit Ruler Hot working Trajectory Speed of light Lawn mower Measurement Year Summer (George Winston album) Printing press Force Galaxy Atomhülle Gemstone Cluster (physics) Lecture/Conference Alcohol proof Ship class Automobile Spare part Musical ensemble Hose coupling Electric charge Chronologie Material Juli Fernández Typesetting Surfing Key (engineering) White Kopfstütze Classical mechanics Spant Metre Finger protocol Angle of attack Ship Standard cell Schwache Lokalisation Moving walkway Hour Electric current Universe Remotely operated underwater vehicle Restkern Last
Girl (band) Weapon Capital ship Cosmic ray Geokorona Scale (map) Dark matter Cosmic distance ladder Particle Roll forming Turning Projectile Atmosphere of Earth Water vapor Progressive lens Uninterruptible power supply Source (album) FACTS (newspaper) Excited state Density Direct current Weight Fahrgeschwindigkeit Cartridge (firearms) Mass VEB Mechanik Vorm. Seidel & Naumann Measurement Book design Hubble's law Speed of light Year Force Atomhülle Galaxy Reichen & Robert Automobile Steckverbinder Spare part Material Mast (sailing) Star Ballpoint pen Data conversion White Kopfstütze Metre Finger protocol Sizing Game Standard cell Hour Universe Week Last
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your all give acts of expanding universe made 1 for some which is motivated mostly body observation that the universe spatially flat spatially flat means space at a given instant of tired there's just ordinary ARU Clooney and three-dimensional space but the space time itself is not ordinary you clearly Emison say clarity and it's not ordinary Minkowski space Kowsky space which is compared to Minkowski space just to remind you viewers a space of a special relativity face time of special relativity and that has 80 met that our profit time element that Turin by the storm of 4th fans different of terms he scored a minus X square our purpose bracket minus DX square BY square the useful and now this is the year of the term a bracket was they giveaway of course the space flat just Golkar and ordinary are you could space where the size of a white element BXU just right the viruses Forney three-dimensional Frederick before we know America is cosmologic and now when I say the new element property 1st of war said nor element the general theory of relativity Is that the coefficients our which we face 1 minus 1 might 4 minus 1 1st although more aware there could be things like the XTY what but the coefficients depend on position In space foreign that means forever b coefficient off-field and this clear part generalization of new members graphic or no terms of fields but we configure from the fields that generalization of the colony heels of gravity coefficient of which go now are observation from Marjorie appears to be universal telescopes appears to be on all Alcoa feature arrest of Gaza has features features a galaxies clusters clusters baht that's all small potatoes also more about on the scale of billions of light years the U.S. certainly your billions of white leaders on a scale of us several hundred million light years universes very feature found the same everywhere homogeneous the word homogeneous isotropic who looked Stringer every direction on such a large scale then this is somewhat of our slightly larger surprised to grow Rogers the isotropy and on the has been with us since the very beginning of Cosmo orgy the very beginning of cosmology post Hubble cosmetology 19 thirties or forties or whatever our cause Margie modern cross-border started it was always assumed that space for 4 years based on the basis of homogeneous and isotropic and I was given these shows called the cosmological principle once it's a principle that has to be right and so rare for voluntary principle but we proved it the broader by calling a principle of course crosses mathematically proved never argued for many basic theoretical underlying principle there is by remarkably more that we look out the space the more received the principal looks to be very very accurate on large enough scales sorrow ah Souter Majamay and 1 additional fact space looks flak space looks flat which means that we will not so far by writing Square was BY square was easy square but while looks flat who also is expanding so the that she should have roughly in the back of your here are yeah a flat sheet of Robert and
the flat sheet of rubber it is being stretched a certain rate but it it that that's flat what minimum viewers really and how well the how well the windows the universe it is our duty of flat but with flat course separately the goes on and on forever but we I'm lesser some saw a walls are there's nobody walls are there arms it will gone forever and ever or a flat universe is also likely to be in the universe but of course we don't really know the universe as I or perfectly flat we ought to the same degree that if we go out in the backyard of a football field and measures survey geometry of a football field Ali could drive stakes and the football field stretch strains between the banks by stretching the space between the states the states as tightly as possible we construct straight-line lose part of our star state and really triangles and measure the angles of triangles and what we form of course were very high precision is the triangles behave just like triangles flat namely the summer angle that upon the agreed where we learned from that we learned that if you're curved the curvature small which is another way of saying that if the earth the finite here it's a big studio much much bigger than the field not a use were feel on of gravitational field about the feel that we surveyed off that's precisely the same cause Margie astronomers who were kind of cosmic survey measuring triangles thing in the sky over there in the sky over there the jury angle you may have some reason the believe that you know distances involved both the objects and also between the objects assuming you have you do have good reason reasonably to believe those things you are measuring the properties of triangles end remarkably growth triangles I of course and simplifying the story what you really measure was a little bit less record bar there are about will be effectively comes wrap would measure triangles and reform triangles behave like Euclidean are flat triangles ordinary equally inspects saw apparently high-precision space itself was flat but it's expanded make expanding have have to do is put in reverse equation he square the also my yes a scale factor have squared at squares because everything in the equation squared the EU limits our work has always had the speed of light and you want units prior to the universe face are we knew its Ms. equations arbor square was and so if we take be all lengths very worrying the distance between troops particular points as points separated part we should also think of axes as not having any damage this to waste of think about you community that they have no editions of the Texas have dimensions of lying are the more conventional way to think about it is axes here have not dimensions simply labels of points when you say that that distance between April points or the separation between the points is dealt X equals over 35 that's not a statement about how far part they are that's just a state of the are our work is always think about it again think about the expanding rubber sheet we draw coordinator in the expanding knows coordinates stretch and expand our with the expansion so that going from sequels of Iran backs equaled 1 over here corresponds to different actual physical separation was a different time at an early time the actual physical separation and meters between 0 and 1 might being some kind kind of will wake late the universe have numerous expanded the distance between 0 1 may be very large that this is characterized by of the actual distance between Corn Belt backs for far would be a stove all foreign aid and a visit which varies with hard it must have some buying them there must be some equations governing there must be some
requirements governing how if expands with time so that all this morning plans are both he and Mr. worked out some of the equations Barbara fate now 1st value right now the general theory of relativity allows curb space prices based I don't based on the basis that allows space to be curve and as 1 of its homogeneous which means the same everywhere there are 3 kinds of spaces which are homogeneous any 1 of which could be the right thing right down here as I said observation we were lock it looks like faces flat so we authorize making their complicated the other 2 possibilities are based around like the surface of urine sphere but of course it's not a two-dimensional sphere it's a three-dimensional steer feel ordinary years ago it's a mirror terminology when talks about we both from horror from Bobby of Europe bulk of Europe rocks in the inside the mathematical terms of Wald bowl leaves a solid 1st terms here refers to the boundary refers to researcher for Europe SOA here is a two-dimensional surface let's let's just talk about a little bit was fears 1st circle was subtle you just think of the locus of points X squared plus Y squared equals sunk our squared the farmers want Europe's circle service focus of points which satisfy square crosswise squared equals ask where squared is just a circle and a struggle as a one-dimensional space and mathematicians would call it 81 here for once a year because as one-dimensional what you call the interior Tom for orders a special case of regions has a special meeting of the song called this but we want to use the generalized mathematical terminology we recall at a ball through a two-dimensional war but the boundary of the war is of course a circle and 1 Spears Price offering Hey square at best we get a deuce near surface of the earth or something like that that pretty easy visual artist I concede refer resource for your right here I cannot see the freed myself on the the 3rd dimension I cannot see the of see that's the euro without also seeing what outside this year and what's inside here but I know the mathematically the surface of this year without regard to what's inside or outside is a mathematical space fair and does not have to the be embedded in 3 dimensions of the reason we have such a ban hard times seems two-dimensional spheres without being inside and outside it is evolutionary we have built constructed a recognized three-dimensional space and we have not been built the destructive of the proceed directly two-dimensional space without being imbedded in the 3rd dimension but they encounter something even worse we come this to off the our these surfaces a two-seater and the interior is a three-dimensional space we need the the ball or the three-dimensional ball but they get worse we can corner now I gave brought for your again for you but I can tell you that were as this is the locusts on all points which owns it good then middle distance from the origins fixture are word of our draw but I can't because I have to be able to picture for baggage and just in order to be able to cede the three-dimensional sphere This is a three-dimensional sphere just as when I just had 3 coordinates I was thinking about Green that elections in which you would ever better the two-dimensional speedier through originals you want lower dimension and the number of coordinates if I am not caught then I construct a recent the interior of which is of for the ball but get the interior forget the exterior 1 possible client geometry that we could live is the three-dimensional sphere the Frisbee now you could see from the coast near meant every point on those fears estate every Of course in real both the North Pole is different than the the equator but just as a geometry as a geometric construction ASEAN is really quite homogeneous every point is like every other points a so I had originated not force space to be flat it allows for example a of if your Apr . com feared you look in different directions everything looks the same those fears about a homogeneous and isotropic a person on the street and we could invent a Cosmo orgy in fact are whose wars always the most popular cosmology and TO EXPAND and Sommer said well maybe it's not right which was replaced before I 20 year bars are more right down the rhetoric of this year but just think about this year was a space multiplied by they have attained as he grows or shrinks because France were shrinking or expanding sphere and it does not make since they ask well to rethink about the big bang at a time when the year was very small remember we're not talking about the rituals you're talking about freedom rituals nevertheless the best I could do in from the drawing is to draw two-dimensional sphere of which solid very very small the all the particles are are squeezed together obviously there is no sense at a special placed on the rear it's not that at all a particle from wastes that shot out of that place I want space they want us Europe this year was very small and other particles whatever they wore a galaxy of electrons they were all very very close together which is a very small space how small we don't know how far we can trace back but we can probably trace things back the point where we observe universe boras no bigger than our Claire Mario so we can trace it back a long way is and and that's for gold were operation so we start with a space which might be as big because as a cranberry filled with some larger larger number of particles and then started expanding particle spread out further apart they are fast RIP Castillo Maltese rules segments in between from our
fact they won't moving relative to each other faster they could further apart further apart they are best removed that's closed finite our circle universe that was originally the most popular and I was maybe like this it's conceivable that is like that but if it is this serious albeit from like Earth we can't the those curvature bodies small-scale surveying but Rice a small-scale 10 billion light years apparently small-scale because we don't see curvature salt we might as well they and replace replaces year by flat cleaner three-dimensional plane which is three-dimensional space but the picture was a saying we have our said we have not playing off articles ahead of you like you could think of was part of Richard fixed but being stretched apart being spread apart it's off and as it spreads apart from points which are then there are also spreading apart from each other and the spreading apart at a rate which is proportional to the distance between us a hobbled off now we care to take a space Alec enjoy coordinates are call us that's bull Cazares basic spare parts removed for a meter say that as the space expands recorded separations between neighboring point expand the with corn like that which expanded would be expanding space called Kohl moving car having coordinates are cordoned whose face which explains with the most coordinates every galaxy is fixed place it's a fixed place because of bad of course an approximation that's a approximation of idealization but on a big enough scale of pretty much draws that's not of individual galaxies is true super clusters laughter and as the universe expands the locations in the accord Messier year of any given Galaxy thinks but the separation between them increases costs are the scale so if you like the scale factor when you will always coordinates and you call equals 0 x equaled 1 than the scale factor is just a physical distance between equals 0 electrical 1 the visitors between Exor Guan Mexico's tool is twice as big solo maps are we to write back Paris distance distance between points is equal what's called 6 now I showed you last time I wanna go through it again because it what we off of deriving novel off this is a physical distance between 2 points and this stop X could be measure axis of course the fire happened measurable Longley extracts of sorts for long extracts of don't barracks here reduces forced Europe's keep along the x-axis while the galaxies don't move in accord with X better means if I teII differentiate by galaxies and Delta taxes is now a separation between the 2 galaxies belt X was marked change would part surprise is what party differentiated from like questioned by differentiated left him aside because the loss of between your storm laughed Pine movement reserves unless we have the velocity on right where have the time remove on the a scale of 5 years apart the root of being represented by a bot funds those facts such Ross however device driver and move blinders Marie why would I do that because I want to get rid adult the X by itself and replace it or something more physically measurable Michael real this summer at the wide and divided by a year or dealt the X here could be replaced by a more I get I get velocity paid Eddie Harding's distance between 4 there are a lot already AD may depend on time the scale factor depends on time so as part of it but it doesn't depend on where you are solve this equation which tells of velocity In terms of this end in terms of a quantity which doesn't depend on where you are is Hubble's possibly is equal the Hubble constant which is really a constant distance for now we know With the meaning of the Hubble constant fears the Hubble constant is Horatio 8 . 4 8 what we what we measure when we measure the Hubble constant look out measures for Milosevic of galaxies abided by the distance to a galaxy the same numbers apparently overlook out galaxies in that direction prayer 100 light years are aware of where to look up a 100 billion light years we were person galaxy bad we compute are Hubble constant by dividing its velocity but this we look our motherland that election witches 600 million light years away we the velocity but distance we get the same the Hubble constant sorrow the cover but if you will hold no right afterward are bar but only we look at exceedingly large this was the reason is the Hubble constant itself about over a varies on time-scale of roughly universe slug of we looking up distances which a small by comparison were you do universe poll as if they were to take taken account to take a variation in what can we have been or are with them how any and they got very bad reasons we have a heart not only for a reason for many reasons we would like to know the equations which determined now you you varies with heart Soweto will work out tonight all there was 1 more geometry 1 more mathematical G. do it is still alive so bad With that the now I'm so let as while the rules for white properties also I propagation is that all-white pray he towers equal to 0 Justin are more white stressed draw Rose Russell Road and why comment with Georgetta a wider array Carmack trajectory he
acts is equal to detail 1st Weiss a 45 degrees course at way for the speed of light as USAir equaled war and they're always be a while before along he ate BX physical combined this bt Boeing both branches deeply is equal to the X square he square minus X squared is 0 but another way of saying that a proper time library is always 0 1 way of thinking about white Ready Reserve trajectory of 0 proper time high that were extends into the general theory of relativity peak hours ago 0 little Europe's mad or lightweight move along the x y and z not changing its is along X we forget y z was says is that is equal that any of he and VX work done this just written I want to make this whole thing 0 or so are you why and easier equal to 0 by Gregorio along the x-axis and that means D. C squared must equal a squares yet squared promotional positive axis that we just say DT is a Quebec safety VX now we could work out a kind of velocity of light not the ordinary velocity of life we ask Friday launch a distance built X belt the actors not showed this remember Mr. tho label for we could ask his library what is the amount of Delta XOB X in a given time that's easy we just provide rugby I would find that the X by city is not cast their votes 1 already have be solved in these coordinates light seems to slow down as the universe expanded is not really slowing Baron it it's just that giving the acts Carlos forest bigger and bigger distance as time goes on so in order for like that have a uniform velocity and VAX gets to be distance through physical distance our for a given EX grow they misses the Justice statement wrapped each rule ordinarily conventional velocity of white uniform Rock if expressed in terms of VX and we go white bracing to slow down where are you quote and quote what they say baby the couple solidly like a sorry just like those lowered its EU that viruses standing you could probably find a way to armed twisted In the a statement of polite peers launch mowers you will further away but you have to do some of all kinds of things going there probably not the best way to think about but quote is area the sector Obama isn't so that it is smaller lighter going at Rikers he might yet well we are a regular speed of light healthy could liver what would what would regular speed of light light isn't going as fast as light said Bridget say buyers often get rid of us 2 well I've ever met a friend mentioned was constant nature of the same way Syria also Jarrah was half what Moller bar the violation was speed of white drama let's try think of summer are of stones physical you tired Mike Timex watches now a 1 or you lose the time it takes to electrons in all of it the 1st excited orbit where to go that's a that's a that's a good humor of swept that in my use of time said I want you lose base Falwell Lewis phase could be the start of matter but work How long does it take for a white How many revolution from your firm was paid in order for white cross our across the Arab back appears to be the same everywhere the 1st time that would take light to travel Crossen Adam with kind is measured in some physical you we'll have to use could use some next units for example instead of our instead of using B or recover electron we could use some vibrational frequency of nucleus take some particularly localism with some particular vibrational frequency Maryland pine for 1 oscillation we could use that as a unit of time and then we could ask how long does it take imaginative time for a light across an act breath a strong relationship between skills and look Williams Arab Mr. nontrivial fact that seems to be the same and replaced him you're Accorsi argument is indirect that seems to be a good growth and that's what needs to say that there were quite travels In speed everywhere it's hard to imagine what would mean for like cannot travel with the speed of light what God want fire press while we were Safra wool but but they warn y queer Mary normal when it was quite letter quit her support supply the review of cod core why Crebert erect but that's a basic set not as said all say there's 1 or other mathematical geometry that has the property that the same everywhere into limitations it's the negatively weaker of the is a positive weaker geometry was Anita said positive weaker of the 2 states statement are trying their hero take a trying Austria as pretty apparent the angles or a bowl out and so some of the angles on as always greater than 180 degrees the Vatican maker triangle with me with 390 recites the 390 degree angle easily Claire so some of the angles or greater the July that's called a positively varies and its full both in the two-dimensional sphere and three-dimensional four-dimensional sphere reviewed make a few make triangles some materials are by our greater 180 degrees in that call positive curvature who's also the yuan and negatively geometry as part of the draw to saddle surface going to work out a mathematical properties today
but it can be it it stands on its own terms as geometry without having to embedded they provide embedded in space start you could see a knowing look like a saddle and then I will look to you like every point as a safe as every other pouring but In mathematics of the saddle surfers is such that it has uniform curvature everywhere was negatively curve medium backers of huge right triangles triangles Osama bin and they have some angles West this is also uniform isotropic homogeneous and they never at any of these 3 geometries is a candidate for an isotropic homogeneous cosmology as I said experimentally observation only at least on the scales that were familiar with it looks quite OK now lives let's cast truly be equations governing the expansion of universe we start last Armbrust Baccarat let's call this point of India past that's analysts Ross brass that's a big fear is the little things may be many galaxies and clusters of course 4 I must look at the all year this more away from us not bad must write them off offers 1st act 1st fat and this move fact really if space is flat fee was off while a course From a factor loophole EU gravitational series of false from Newton's gravitational walk from moves equations the miracle was what made it possible for Newton to compute things about the about gravity is that if you have a sphere like and this Everything is distributing spherically symmetric weighed in other words if everything is I still topic relative to appoint arts but much last year was here's here's here's years but everything has nice article symmetry then it's a about fears about the inverse square war class or particle over here is some general use there were working with corn in which we are at rest room we will just arbitrarily music when the system frame of reference where we are at rest Vasily's relative twice accelerations main roads are velocities accelerations mentioned melamine Israel arts then it's a fact of Newtonian physics that they always gravity all forms on this point here only comes from things inside this year and things outside this year before us is just cancel our this is maybe well-known you in electrostatics you have 80 Hall region and charged some charge distribution on outside as long as as it's Veracruz there was no forests particle on the inside it's all charges on interior with and which exert force use of force on particle all summer particle columns from messages the interior a former the next statement is that the gravitational force our spherically symmetric distribution of America it is very simple I'm but we symmetric their distribution of matter on the inside is very simple and it's just take manner means I and think of it as being concentrated those point so the motion of this galaxy relative to us is exactly the same as if all of the rest the northern SME outside of the show and they called on the inside of the and placed at right at that point not that makes it very easy to compute still see what's going on with this article with particle over here it behaves exactly as a particle in a gravitational field where all the masses of the sector end after that concentrated this is a key thing that made it possible for me do a lot brings with gravity of pay which allows us Our to compute the properties of chronology OK let's put us at x equals 0 and last call this point just X Riverkeeper the actors course it doesn't have to be on the x-axis 1st thought equation what is the distance out back let's call but are now forest stands a radius of actually means physical distance What's a measure meters art to work Corey well the answer was its ex parte Zobel cars not ex parte scale factor of bubble at times Guelph deliver relationship between actual physical distances and scale factor is that you multiply the War distance by the scale factor get the actual physical districts of actual distance rule the subject of India are is equal to current scale factor and scale factor which changes with time move because the scale back move away from us because the scale factor changes with time was velocity velocity of moving away from us it X turns 80 . now let's remember there are that we can't pretend we can pretend that this is an equivalent problems are of a key project bio moving away from us under the influence Botha gravitating point at the center was energy a rare point moving away what will be energy be if all messed concentrated there and we could ignore all rest on outside while there are 2 terms in the energy potential and kinetic the potential energy or let's go what's called a massive particle here Lula and massive everything inside the ship I worry about why our back scored big year began the massive everything inside the shell will land is masses of this galaxy that will follow I was saying was the energy of a distant galaxy its its comedic energy 1 half Reese cornered class the gravitational potential energy so want thing you have to know is the potential energy of an object of a gravitational field but not knowing some people G. M. Mormons will the problem that so solos potential and you what sorry that it may give because of attraction Seoul where farmers from the book also poster poster Of course it gets more energetic but the energy goes back about because things tend to be pulled the direction of
decreasing energy source was a mother turned me on my best known is kind a product of the masses which masters mess of a projectile man concentrated at the center of In our imagine area Our Majorie simplification of all of inside this year provided by emerging reveals why not are you over that energy is constant doesn't vary with time that's why I because there were it could be part possibly was a positive kinetic energy it could be negative of GMO laws is bigger man and square 1 square of the whatever tho it is not allowed change were parked constant that would be the property of the projectile moving away from I will study the 1 special cases which corresponds 0 energy over here Red are comeback from your connections between the energy underwriting insiders and the geometry of space but just simplicity let's take the total energy gravitational plus can ever be 0 is it makes sense that the energy is 0 the officer positive and negative you make 0 was only gave zero-mean for project throughout so it means it'll be escape velocity that means Jack exactly what if is positive that we on newspaper last negative all right so I'll come back 3 cases the energy could be positive be 0 it to be negative unwilling to take the simplest case set now we've come back in the other cases the simplest case is the reduce 0 cast bronze things being shot How exactly these newspaper last week it's simplest cosmology got on our normal capital and does not vary returned it's all particles year but remember that they are pulled together so as that here explained the same number part of that extremely good points are extremely question was a lousy statement was of request Bob the same number of particles as always in that sphere so it doesn't but now let's write it in terms of the density of matter in the universe the density of matter in the universe mass per unit volume that does change with time as sealed 1st of all the material was means for real value real genuine volume measured cubic meters Serbian arsenic cubic meter cubic meters Our cubic meters just cubic meter the material bows out of it and asked me in this city does go down the problem space fakes density decreases the density decreases the volume increases our service is a mess as a young man per unit volume what is really problem mast inside that shower Don inside their child is density Rosa standard there standard patient bears density and volume volume was volume sphere for a 3rd but you'll water birds high raw that's fair keen mind that role it is depends on time will discuss later how it depends on pies but basically it depends on time just because the volume of sphere increases and the mess doesn't agree so volume increases in that doesn't increase wrote depends on that's a problem Master so now we can write down or but slipped through his steps 1st war this is a formula for mass inside ball in war if I had 0 on the right inside and this is 1 of the reasons are taking a case zoo 0 on the right side makes a little bit easier weakened the volume by the mass of the projectile so let's do that supervisor our from make any difference it doesn't matter if that is a galaxy pebble we just a bite out its mast that's gotten out of the equation setting a difference of 2 things to 0 . 0 same thing is setting equal to each other so much of the 1st wary do Matt plug-ins for the man's let's plug-in were we have here so I don't think I want through a . mess up equation progress collects leave up there were prepared here there's just do it now equation marks for capital where says that 1 the square velocity a Of the outgoing project our G are narrowest put it in the form of high over Veriee but are killed constant came from the right hand side of the energy equation all also from the energy and then the rest the mast inside the shell of reviews although now let's use Homles war all reserves but also 30 is a quote from a hobble concept cars investors it can also be written and another way to do it right velocity is equal they not only 80 hours a distance because the Hubble constant is just the race EUR 0 . over a we're sure we get our idea 2nd continued court at the Hubble constant why half a hobble constant squared times the distance squared on the left hand side is proportional from a distance of about going down 3 squares White because of law or age of course depends on time but we can use it in this equation and that's equal G Force Aurora state Afrikaans 1st for Peru's Piney density of matter books for Peru's GE density of matter the urged party GE sensitive 5 Suu all wrong facts are squared all of us are nice where Arafat are squared on both sides of the equation that means we can cancel our but
further enters egg because word tells us behind Cancellara you worry because the Hummel constant if we had some which have some way of calculating it which we do now the Hummel constant pulled the plan on how far away the galaxy that was studying the supposed to be the same forward works sort red or out of the equation news if they're going to tell us what bobble constant in terms of other things it had better not depend on how far away the galaxy that was studied and doesn't we could have like we could have asked we could this galaxy we could use this galaxy we could this galaxy all give us the same formula for libel cars for the study it says that the harbor constant is in some way some way the square of the Hubble constant writer traditional rate was around the Hubble constant is parlay over 3 Bruins constant the density of matter universe and the density of massive universe to boot also so now on what enough scales the density of material universes can't book 0 5 Inter a function of hard turns over the Hubble constant now young right you go like this another way writers not square is any fighting over 3 times to you you we were just a loss president or the energy was there about . 4 on a floor for 1 simple priest rose we can measure bobble confident we can measure the density of lab a muralist 2 of them but don't tell you what was said not but provide told direct rock the movie yet but Boca West but were all forms nets are attractive for a number of possible you hear the term dark energy or energy amassed a sense equals and square see is just the conversion factor from energy from s saw what I speak massacre also be speaking of energy in units in which the speed of light as 1 road could taken the energy density energy density density saying there no variety of different kinds of energy in the universe different kinds of massive requires a energy is the masses of stars whereas intergalactic boss is something called dark matter Her about everything else was accurately with all of it is editor got the answer is yes these 2 equations court found who are warned forward see what the we can't keep up the other can away which is just constant we could keep prowl tell you eventually what was poker so young new 1st play the game of finding all the energy in the universe everything we know about putting on the right hand side measuring Hubble's constant learned after many decades and finding this kind of energy and that kind of energy of this equation looks however does not mean he does not mean that the university escape velocity because I have made would wear makes remain girls from West area some assumptions that came in not how role Paramount but Sabri war across Brooker murmur part of raw we take some box in the universe with a certain volume it's the lesson that box was quite M arrived by allowing their of various kinds of situations Let's take the 1st Georgia simple situation of the boxes filled with particles the particles are moving very fast more or less arrests that what we call galaxies or else the arrest Robert Court Rejects Naumann seem Quark 0 fund so we are given box now X on this side of the wine on the side opposite the other direction were following the boxes expected it has a certain number particles let's say that No. particles face bricks each particle has amassed and it's not more than not moving relative throughout the coordinates them what's the energy in the air Mary just hope won't pass which number particles which they my brother by volume was a volume the value is the size of the box X kind aII Q Ottawa whatever is in that box the density gets looted by the fact that grows over time this is just the statement a box if they don't bigger and given lot mass in a box gets bigger and bigger the density will get smaller and smaller and it wants more according toward divided by a few so it is say that the prevents density of the universe is some constant provided by a Q what is the constant well as 1 simple way the specifier cars through a simple way to specify what constitutes is to say was a density of mess today from the left hand side I could put the density of rest today she former right-hand side backup the scale factor a very good thing now I can multiply as they were arms are that's a correct equation now how mother acres but tried the same equation except that any kind of role of is able to see provided by cube the cheese and that's the right these 2 equations this 1 by 1 that tells us Graf role at any kind city the riven by wrote vary this provided by
various is the relay hero became receives cancel out and it's just race a Q ready provided by a queue of Ross the writer by wrote today he is 8th today he arrived by a few 50 arrest because role in each case is proportional Gore rates your adjust the biting these 2 equations and that we can write 8 car and the density of matter wrestling Raul has just whatever it is steady missed the race I was gay factor today and the scale factor and arbitrary part of these get factor today is drop back for we equations but basically a finger tells us Is that the destiny of material decreases like 1 already Q. We cannot preventing here we have on the left-hand side something involving the Prime derivative of aII itself and I not only rose tea but now we know something about with what's right again dark provided by a square it is equal GE referee and now wrote today which will measure by not density material on the density of mass of the universe is roughly that just the idea starts negligible neutrino Burns said and when I wanted to be there but let's not all of it because out thought Dr. lock matter and dark energy Her include although it's about 50 Adams bar it's a matter of right through and it's something that principle we could and possible we could measure it directly and principle direct measure of war fully energy firm Saul Floyd or whereas today where her AQ theory the Nevada by 8 few of laugh a prodigy Saudi relativity of HU theory are can't constants that is to say they don't vary with time ran out the wrath of robberies forever today used as wrote that a number of measures about a Asia's whether is solve it's just frozen doesn't change with time the only things which change in time 80 . 8 2 over here so I'll together we haven't equation which looks something like this that already Square is a focus on big constant was just call constancy contained a prodigy 3 wrote a cube today courses which don't change with time provided by Hugh this is a differential equation that in principle I will tell us how a varies with time from a solid for Saleh priorities you solve bomb the 1st thing we're scared worked let's just multiply and they all sides Murray a squares to get a square of the thermometer on both sides Seoul what's to that we have not square per year got squared His report a seat that is survived by a of multiply our most size by a square but we can now square root square constant is just the mullah constant are however says is the time to remove 80 is proportional 1 over the the square root of whether strife rather than the those solid are doing the world what let's just drives already but they're aII this summer however all the kind with some power some some co-authors showed some coefficient constant may remain for concert k k plans Jeter P old was a . 8 . 2 cost K. P. T. Philippine miners want end on the right-hand side of the square root of this constant and now we have 1 by the square root of 8 which is 80 someone use one-half With there are 2 things are going well right on the constant Katie and I don't know how or when I look at this equation here then all really solid this is a mistake in not Our restorative services stress was must put them it is 8 1 out but a minus 1 have escaped minus 1 have Miners fuel to 801 F Katie Kumar here F P 3rd Rosser Tito appeal so just for 8 minus Warner half Kato minus 1 him and teach America's for this equation Mike things have the chorus of team have marriage that would just tell us that the left side of the right hand side vary the same way with time then the coefficients after be that only an to determine both the power and the coefficient so other the power of Paris says he might 1 equals might disappear over to war I'm just equating P minus 1 from right field to make sure that the left hand side and the right here sigh varied same where we are and so I was like give vectors he is eager 1 of get your or the west side of rest creepy over 2 0 0 1 warrant P equals 2 years that tell us that tells us that the universe experiments like Peter the two-thirds detent of their data there is not as fast just T t to two-thirds increases more slowly than he itself and they see square root of p whether there's just there aII glows like to offer voters get bigger and bigger and bigger cause in this matter here OK now I know what he is pleased to parents we can look at the constant constant coefficient of acts as Perahia KP used Rivera two-thirds K D if it is equal square put city of records to use by now terms takeover one-half so now
we have provided by Kate warned him have caper 1 have America we have k equals seed parsing Iowa for Busch Series but remember what sees but this time yeah after all a ghostwriter what says but from over here and I don't always says burned-out our words work with Katie equals mind for what's city equals 4 which is not so interesting is not forgotten somebody reminding so what where is it against Was it by how jail or 3 said a few hours but but who causes go and were Cape Times 2 burgers but he feared two-thirds equals hate some draw this equation to the Tuesday and also something wrong ballroom file but right and what's at the main the main thing here is we determined how the universe Express expands on the scale factor increases as two-thirds power of pine 8 2 Mary Rose see where some northern coefficient on led something or was there some of the White Book of this equation we should have the same power of AD on both sides of this equation still it too we are wrong it has do that boys well let's let's do it a different stores deferring we have 8 . they already was proportional some constant divided by a right there clinic I swear Revay we're so was multiplied by IBM multiply by 8 the 1 half Of point made the one-half year would give 8 . much of that would give us None off some progress it's a one-half roses Our large track I'm lost track all those about the idea about all already square it is not always square remote but was 1st multiplied by a squared review squares garment that gave us a squared foreigners right now is sealed over a darker arrived prior Mrs. D a nearby square was multiplied by a and take the square root of both squared of our size gives a square root of a GATT equals archives Toyota cars was just Ricardo Mello multiply by DT an acre great Causey the role of constant firms what Carson 1st Street terms to Constable what and the role of a toward him it would so 803 shares is proportional to me and he is for a is proportional to their so deserters right so this is an example of Our Howell nothing more than Newton's equations was not tell you are the expansion properties of the universe then it just continues to expand whether I would suggest that's what happens to the Hubble constant what will happen the allowable cards what part but we know that a song can't different constant of different going on a constant feed the where about 80 . 8 dock is 2 or 3 city clean the media two-thirds of minus 1 just differentiating people two-thirds you get a minus 1 or 2 thirds times miners would not take a doubt over that's the Hubble constant the Hubble constant is 8 . 0 aII about two-thirds seat Tito a minus 1 3rd provided by C the two-thirds Stacy careful you you find the Hubble constant is two-thirds already that tells us how the Hubble constant will change with time so this he answers the question a relief well with the assumption that we may need this the question of how the Hubble constant varies with pride yes this is the way things really work or we will look back into the past and we should account for the fact that the Hubble constant increases by quality service would give us we now have detailed predictions for we should see very far away we should see things moving away from us for the Hubble's law changes was kind of a further away we are the bigger the Hubble constant should look Doris according to this formula and that but that's a pretty good there it's a pretty good representation were actually happened to lobby on this we have to know something about general fear and that's beyond or for what would doors from back we discuss the fact that there are 3 geometries but described homogeneous isotropic positively curved 0 curve was case doing and negative occur there are also 3 possible values for the total energy positive 0 and negative he's gotta get the 3 possible values of the energy of 3 possible signs for the energy plus your mind on one-to-one correspondence and this only falls from Einstein's equation is you cannot arrive just using your Bull but positive total energy 1st of all means that the university on the escape velocity and will continue the escape will continue curb Gore out positive focal energy means negative curvature such a cosmetology would not only be
eternal into the future by will also be eternally inspects negative curved space Provence forever but it wouldn't be flat space negative 0 total energy that's the case we did that Carlos farms 0 curvature fly Megan a energy forever across forms positive euros workers fear for positive energy source maneuver to they have energy also means that you belong obviously philosophy Megan you go the implication of that is the uterus out expand and then turn around and 3 collapse all I have to do is focus on that 1 particle out there other 1 Dow artsy that we focus on his or there your velocity it is about the escape velocity escape velocity for all the Macedon here concentrated the point of the escape velocity of Rabat only escape velocity easier used positive earnings a uterus will continue to expand forever therefore both while the escape velocity merits a caseworker land use negative then every single Galaxy will turn around for him at least according to the assumptions we made now the number of things all wronged Will we get the most important is that we have assumed a arm a model for the energy density and the way of varies with said in a given volume was a fixed amount of mass and that s will simply stay the same as volume increases and that was the origin of the statement that Riyadh the density of the best vessel because like 1 already killed volume increases Malcolm S. inside raises that necessary well there are lots of examples where natural but said the following case supposing that of galaxies or badly balls even protons and neutrons supposing what's in his box this vote columns photons of all different kinds of this is energy sold were interested in is the amount of energy in this box as it expands what's a connection between the energy of a photon whose wavelength 1st of all he glitch Norway so the energy of each individual faux pas His times its frequency our we can write that in terms of a wavelength frequency or way life frequency or way rights of the bigger the wavelength swallow the energy of a photon now what happens to if we have a box we expand the box what happens to a folk column June wave well the answer is it just stretches out match a box so the Waverly the the photon will get bigger as the box increases the number of photons in the box will stay the same off the average number of photons in a box was saying the box gets bigger the In the wavelength will get bigger energy will decree it's so we have a box and not bunch of particles were fixed energy but a bunch of particles was individual energy decrease said his boss
saying phenomenon that happens to have a forestry a guitar string between compose you put a thing Obama over here so that effective deposit friend there's only this 1 you plop In the new movie finger while you hear the call girl from the associated with this wave of string Foote Cone associated with this library of string what happens it is about the hour the way you started with just gets stretched out so car are you
could series yawner guitar if you play the guitar you very well so we don't if they contain about a bunch of photons Iran's increases Assad's van to your energy reach for Congress back vote on those found the means of energy density in there decreases even tho passed through their war if each vote stayed with the same amount of energy which say the same amount of energy that's a situation where the density is see already Q. What happens a box of font times as it grows COA former power decreases more rapidly as the euro that's easy to derive you just say that each for foreign instead of having a fixed energy has manager goes like 1 0 a key as a universe expands you get the same formula extra power over Obama made your right back over here instead of saying their logos or 1 already Q. Will you say that were logos like 1 already From afar and the analysis in the end you get a different answer you don't get a two-thirds you get a 2 1 here nor is a universe expands a little more slowly room or swallowing award later two-thirds expands a little fast ever made to two-thirds right that Tito their the 1 was a little slower than Europe your research now Missouri focused on me answer depends on the nature of the energy very very early in the universe numerous was filled with half 4 times most of the energy was in the form of 4 most are areas Of course if it's true that you expand the box the energy of each photon decreases it's also true that you squeeze of Box Theatre du Pont increases who take a box with a bunch of all kinds low-energy photons very logical very much approach Collins the protons and neutrons and electrons assault for Adams Romy Adams would ordinarily be most of a master the of squeezing RBOCs the Addams each and stays the same but each fault aren't you increase the energy of it so eventually when you squeeze enough me photons will become the dominant form of energy the universe is going back in time squeezing numerous means going back time some sometime in the past when the euro's was about a thousand times more radius and is now scale factor was about thousand times more the dominant form of energy war not particles ordinary particles were at that time the universe was expanding white Peterborough 1 half 3 years incidentally by now we have good evidence of good evidence we all of us before the time was a times crossover time where numerous became dominated by ordinary matter and before that was dominated by radiation this is called the radiation dominated universe Tito two-thirds of its called America dominated universe was what other example which is extremely important and it's the energy of the vacuum are not going to tell you much about you would you back your work for your only 1 thing about the which is true of vacuum energy and there is vacuum energy or your theory it's a component of the energy which was not when you expand wherever the vacuum energy a property of empty space and no matter how big the universe gets the vacuum energy density very far purposes that the findings vacuum energy a component of the energy that just doesn't go Baron as the universe Express let's just see what I would do surpassing those a component of the energy which just didnt refused to decrease just call I can't say it's a vacuum energy and fact it's about 70 per cent of all the energy in the universe so really is dominant prayers today it was good parent is also true visual backward in part because go off if you take all the other forms of energy squeeze in what's more argue the energy bills up stormed the class vacuum was quarters in the future of all the other terms the energy gets more and more smaller 1 over a few won over 84 off the vacuum energy will become the most important component is already the most what upon by a factor of 2 more 70 % vacuum let's solve this equation was it say that says 1890 while already taking the square root of the square root of 8 parties GE over the radio constant wrote one-half that's also a constant everything on the right side grants a multiply equation by says that the time derivative aII was proportional constant what's a solution of exponential time derivative is a constant firms offering itself OK the solution of this is exponential are square root of about prime space most of the stuff so this is an exponentially expanding universe the universe which is dominated by vacuum energy expands exponentially are all wrong was paid for it Babylon well it looks like with the present value of the vacuum and it takes about 10 billion years double so it looks like we're heading off to a world where the uterus will simply be doubling a roughly every chemical billion years are white true we not purple fact it appears that appears to be a purple fact is a component of the vacuum energy which is constant which doesn't lose you expand which causes the universe exponentially agree on and when we look at our into university does appear to be exponentially increase if work far enough away very very large distances to Oxford universes exponentially and sewer that's have told you a small faction of everything there is to know about cosmology but I've told you really mean that pools of Maine none the main facts about it was not her that that is a part of a gap in absolute said a word about where we are today that I think the best way to think about his terms
of sheet picture are space is like a rubber sheet General Workers End on it the pounds of course don't nor whether we defined in but they can't tell regional spaces doubled size they can't tell America 1 Waverly our Warner region of size 1 wavering and a certain amount of time a bubble size or from whatever prose that's really all I can tell tell stretching Del what I don't know anything about the galaxies that's basis by the the army could you could another those photons another like think about this the ship farms but shaky if yard where script whether photons come from universe became star let's drop picture of universe from wrong perspective where sitting at the center what they are around us shallows of different a radius said there was strong like back down to you around us which like that the distances out a Commerce class which is that it if we got far enough we call a place called surface of last Saturday now is lot deeper and deeper into the universe the material was more and more squeeze because it was squeezed was higher so as look out for you not only see earlier times because you were far enough material out here were based on offer and hot enough but it was opaque was as opaque as the star somewheres out here was as high as a side and you simply can't see the sum was OK opaque because it's full hour on matched electrons and protons free electron plasma like can move plasma well far enough in their early few earlier universe uterus was a plasma for when we look back we look back in time and eventually see back there rules plasma bats call the surface of West scattered we were back so some steel around us rats bats looks as houses are but is also moving away from us will also the speed of light why because of Hubbell's lawyer so today that they would log numbers or we were just a look back use video views or we here we would discover that have surfaced last gathering it is moving away from Mosul was moving away from us of course looking back in time but receiving away from us were almost the speed of light that means thought on the comes from there is topless bar a lot of stop which is a lot because the surface is moving away very very rapid broker now what about want to know topless you normal something about the way the wavelengths of the radiation coming from the top starters are about the same as the wavelengths of light coming from the start the Doppler shifted north regardless shifted where you want we see them being a father roughly a thousand times wave then the photons of the start of a thousand times less energetic so we see Sula unwavering photons here because the which not go back into the past and this with the same picture would look like much much earlier much much earlier the surface of West began moving closer us if we could go back and coroner has our ancient ancestors are 4 or 5 billion years ago what the universe look like they would sit still picture except that they would say the surface unless was closed and moving more slowly boring more slowly so they would see less of a Ranger less of a rich right there was U.S. Sheriff because they receive a service Darren Morgan away from them somewhat more slowly and the fact that your way way back the surface scaring was very near hardly moving at all are moving your 1st time are receded and sped up about this program so the felt tons if we went back in several billion years before time that we see him coming from the surface of West gathering where shorter wavelength source tangles armed photons that we see that make up the microwave background for example are as time goes on the photon so we actually see right here have longer away because they originated from a place which were moving aggressively away from threats another way to think about also wiser corrected could you think of the universe was expanding each wave the valve so that's why we see longer worries with kind or you can imagine the same kind of waves were from surface of less Darren but by the time they reach us more Richard the leader we do not know how good the sources of the class vision of a full of which submit their lawyers votes this is about us but was that is do slow so S the basic equation we wrote for Cosmo was nothing much energy conservation with this equation you can violate energy conservation drive to remember there are 2 kinds of ah by the time we go for it this term over here has more than 80 a square the expansion of the the universe the other term here is justly energy density of the universe this his energy conservation and assesses the kinds of energy in the world was the energy of expansion the kind of kinetic energy expansion and the other is just the usual things a quantities I have never installs is concerned with that did not answer serve for world where normal remorse for them after Serb survivors Scholes although the combination of the Earth domestic worries as we use showing Prime as the energy let's say was bad poem and you can't their kids transferred into Eddie are be comedic image of expansion soon yes yes we were you could think of kind of pressure on me last energy conservation supposing you really did have a box end let's imagine this is a box with movable walls the walls to expand if you make it moralist it's our box a cylinder with Kyrstin bucket there the and was thinking in it in or out said We have also voter cards are here for a while after slept box that part expands because of the pressure of photon on piston or habits each photon photon decreases its energy because of wavering where is our energy go because of the kinetic energy of the distance as with those from Greek mail or by Gold fortunes but that's the course idealized it no friction
because of a kinetic energy of course going on here we the energy the decrease of the energy the folk artist is being transferred to kinetic energy of expenditure for VeriSign are off energy is funny in general fear brokers choric reserve but also exactly 0 hour in this case here it is our 1st made exactly the rules for energy general theory of relativity Our peculiar I hear you take into account all the energies of around including nearly Gere expansion here which case conserve above 0 for forget this expanding energy and just energy Serer Our him a general theory of relativity any World War II draft a where are all Sunday aside with a 4 0 in a row of rose because seeks the universe because that's when Holly Energy come from our vacuum energy in the universe expands constant energy density and the universe expands rapidly like the energy of the vacuum is increasing and that component is increasing where is it going it's going into the expansion drive the expansion even more rapidly and the other forms of energy and thinking about energy conservation general you must worry about this turn this is very important that for the expansion of the universe and simply cancels changes in this canceled part of that I thought that was a factor in what was viewed as it all are all crew have multiply for would make a difference jury on your but stuff emissions OK let's so let's suppose it also multiply that are part of a empty pions us just rights simply did he squared minus VX square paying Wang this is a square this is just a redefinition of called cotton works invent and no quantity no Torre which is such that he he knew Torah which is equal that any of its he DG regained the grave all sides of this equation where we can find were big he isn't turns over there and a square finds little Square is the same as just a big square you can always get rid of a square in front of T square by a change of time that's done that want get rid of its multiplying VX scored stores still harbor Martinez aII of little people also of incurred a big score solve the issue of whether they have tea multiplies squares questions definition bomb mathematically there's no difference between bomb ripped through forms although you do have to change the definition of time to record for now you must but following orders but both are that depends on whether that depends on the magnitude G but suppose it was Back in the universe will positively curve the question is analogous to a farm question supposing we should a rocket out with less energy for less velocity last we look at what telescope Hoosier moving out of the moving in could be moving out of the 1 thing we know for sure is we wait long enough it will be coming in we could've clot allows going out about their and we kept the universe is expanding was no about that the issue is not whether it's expanding now or not what we see the issue was if we wait long enough of the future will everything around crashed back from Apple us army if there was no vacuum energy it's or molecular and so would be the universal positively would collapse back on itself the vacuum energy its large enough will prevent where certainly does look larger so it looks like our future independently of curved or were positively curve negatively flat looks like our future is 1 in which the universe will simply exponentially that's not such a good thing you exponentially it's not a that sign of future where the exponentially expands eventually All-Stars will burn our car everything will cool everything will buy removal ordinary forms of energy will exponentially by the universe is exponentially expanding its volume was exponentially expand all the energy densities of all the other kinds of the vacuum energy density will also exponentially and will just be a universe with me forward heard hindered their energy density sold me there is no good in the story of pose a greater sense of today's we grew out about a photo of a man who had both of us that they had better that way as you look back at the Paris race you arm folk horns you ordinary particles Bond where ought to have I I would say warrants the number of 4 doesn't get larger than the number of that's already that I universe the number of 4 times as much much larger number of Adam kantor 10 times larger so we are dominated by 4 times I don't mean the photons light all that only was signed I don't mean the photons starters in the microwave background we are so we dominated by your vote as you will back in time in the race's yellow photons of protons that is about the same but the photons yet more and more energetic the proposed so as far as energy goes as you go back into the past energy will be dominated by the time the number of photons will stay the same but there dual increases you go back the number broke palms stay saying but there energy won't increase fast but OK I'm tired what's gone wrong was last quest but Her
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