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Topics in String Theory  Lecture 8
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due to the fact that it that might both but she figured this electron on the violation actually actually occurring work but to correctly vacuum overturned profit from your death that the lecturer farther from virtual pair their production virtual parents who contribute to the vacuum doesn't mean that you should see electronpositron Paris popping out of nowhere does not mean there that means means that about you go back and a very short time subject to the uncertainty principle of quantum fluctuations and not real Virginia Lino particles produced as much like 0 . energy of a harmonic oscillator classically the harmonic oscillator just sit still at the bottom of its potential world Kwan mechanically it can not both sit still and be of the plan by the potential competition and that's resolved by radio population that Kwan fluctuation energy from which is at a half age borrow may get but always there King direct the least energy that they are so it can have the same as that of course that doesn't mean looking at the oscillator most choirmaster lives too small for column it is the ground state and as the it means the state in which nothing is happening to the extent that nothing can happen saying drew electronpositron pairs sameness true virtual photons harm benefactor it too closely related phenomenon B electromagnetic field for example in a cavity electromagnetic field the cavity is just a collection of harmonic oscillators each oscillator has a little bit of 0 . energy and the mall what you call out the back energy will lure I was saving that for a bit later that because a plays an important role in cosmology deals today a orders that you know I think I want them Leipheimer have mentioned but there are no car no I thought tonight in any case would mainly concentrate on cosmic horizon were horizon is wider horizons and there will be
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about her cosmology equations of cosmology how they lead to an accelerated expanding universe and why the accelerator expanding universe has event horizons I thought that would be a good thing to do marry since we spent a lot of time learning about where horizon when quite missed when her eyes and
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say you know that we have a lot of work to be getting that won't talk about what warrant that's a good neighbor will start regret maybe it's not really a murmur was of things to do tonight removed it with a run error thing is that it is also less that this off was trapped I just wanted to write they know that a lot of life like less on the vacuum energy and that's a good question to the question most energy that you think about it you have energy present here creates a situation where the world it is not Lorentz invariant I don't mean that the furious now for Epson and I mean that the configuration of the world is not there was a mean of what exactly does that mean 1st us something simple translation variants every places like every other place right now but Thatcher Stanford University's not like Berkeley our duty Earth is not like go the sun and that's not like interstellar space so what we mean when we say the world has translation invariant we don't need to know that the configuration of the world every place looks the same after we need if we took the whole thing in moved it would look so there would look the same but moving it means moving us with Berkeley moving everything else now in the same way are you talk about Lorenzen variants is the world Lorenzen variant now 31 minutes on the equations of physics Lorenzen variant yes cell was it mean for the world not the Buran well it means that for whatever reason the staff in the universe are does not look the same from every frame of reference and Violetta with by you a thousand miles a 2nd you would look different to me then I was standing still sold stuff in the world stuff and the world break the symmetry that we talk about create special locations they say they create special directions equations of physics was supposed to be rotationally invariant that means saying no matter how we rotate true enough the equations of physics are rotationally invariant but is wrong brother rotation certainly not if I look that way I see you a look that way I see something else sorrow the presence of staff breaks cemeteries and presence of stuff breaks Lorentz invariant The World Is Not Lorentz invariant by virtue of the fact that the stuff and now most of the time the staff energy and so forth as said is not Lorentz invariant and the result of that is In particular result unemotional light waves light waves moving through materials have different velocity than they have moving through empty space the rule about speed of light is that white always travels with a single varsity in empty space but that's another way of saying that in these babies is rents very empty space is Lorenza during empty spaces translation invariant rotation variant and these bases as has no preferred frames of reference ends the statement that life always travels With the a single are is a statement about a low rents invariant world where there is no preferred frame of reference work as a preferred frame of reference due to the presence of material light could travel at any velocity lower than the speed of light than usual speed right now vacuum energy is very special vacuum energy is the 1 exceptional situations where the presidents of that energy is rents variant was not pick out a reference if you weren't here the world wasn't here and I was absolutely empty space nevertheless there would be vacuum energy this rule a vacuum energy as a certain value where which medical value is certain number of Jules per hour per cubic meter very very tiny bomb but nevertheless finite Jim I think a lot sucker material being here that sort of stuff earlier and stuff being here would defying a frame of reference where the stuff would be at rest vacuum energy is very special at worried it does not define a frame of reference I could detect the vacuum energy with vacuum energy detected have built 1 belong while the rule will talk about how build a vacuum energy detected I could detect a vacuum energy as to how much is here we get some number I wouldn't go whizzing by as fast as I can go and do the same experiment it was ordinary material I would discover that the energy changes the energy that material changes were my velocity in particular Our if I passed against that of course the guests looks higher energy White all molecules appear to be moving past me vacuum energy is the 1 special case where my vacuum energy detector will give the same answer no matter how moving so vacuum energy is a special case where its president does not pick out a special friend reference nonviolent Lorenzen variants as a special case where light over the speed of light independently of the observer the woman with the speed of light means something it means it when light rail passes you president right in front of you know and you'll have your clocks and nearly detectors locally right near you you measure the same velocity of white be because the expansion of the universe and so forth becomes a more problematic question whether light moved relative us far far away when the light as far away from most doesn't move with the speed of light talk about that our theaters not general did was relative told local observers where that life is passing by them with the same speed of light but it doesn't necessarily move relative us with the speed of light bastards store is Russia back in yes absolutely right so let's begin their work would begin there if we take the universe and fill it with poor particles and witty ordinary particles protons fill it up I don't show up every protons against every other for propositus mean you create some uniform density of protons and dealers to something like that of different Zamora less uniform higher energy density rout of space that tends to be clustered in stars and stuff like that but scales bigger than super clusters of galaxies on its still homogeneously distributed rather the diffuse something if I remember something about 50 protons per cubic meter of money average Arthur
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Andersen energy density let's give that energy density in name a standard terminology for a density is Rome and I could stand for the mass density massive energy being the same thing but semester and consist of larger particles that what happens and sets the number of particles per unit volume or the number of particles kind is the massive each particle divided by a volume of space figure volume of space the density of energy is a massive each particle exposes only 1 kind of particle the particles a standing still enough photons suspect that normal photons right now just part standing still protons neutrons electrons Adams mass of an individual particle and the number of particles in a volume provided by the volume that the other no I had so that density as the universe begins to expand well number particles doesn't change at least not their ordinary particles the number particles doesn't change their man's doesn't change what does change is the volume of regional space occupied by particles so it take some particles that started out in a region of space a square on outside our 100 light years signed a 100 light years light years relate to it has a volume of a million cubic light me light years which is some number in cubic meters that's a volume goes in here number particle stays the same and now the universe expands everything expands with a sufficiently big scales the volume of space let's say over some period of time bubbles on the volume of space bubbles while I'm sorry and the radius of a mean the volume when we radius of this Q a blue linear dimension of the Cuban doubles what happens the density wow when the radius doubles the volume gets multiplied by 8 and the density becomes 1 8th of what started song ordinary material has the behavior and this is true for photons for anything this exact formulas of different for times the general rule is that as the universe expands the energy density by loops it decreased density of energy decreases just because everything separates again vacuum energy is the exception vacuum energy is the exception vacuum energy has the property back community density vacuum energy density has the property that as the universe expands on the volume of space expands amount of density of energy does not change the density the amount in the buyer in the box but the amount per cubic meter always stays the same if the universe what to expand by a factor of a hundred in every direction are the volume of regional space would go up by a million nevertheless the density of vacuum energy in this room would stay exactly the same sort of property of empty space which just doesn't change as time goes on it doesn't change as observe moved past it it doesn't change when the observer rotates if not clustered uniformly filling space picks out no direction picks out tolerance him and picks out no special kind in the center of the universe expands it doesn't change that's energy of that type is called back in energy if it exists at all it does seem to exist where we care why should we care viewers were filled with something that we call vacuum energy we do experiments and the only thing we're interested in this room before doing experiments in this room is differences of energy Zero Point the energy really usually doesn't matter for anything thing which usually count is differences of energy what's the energy difference between an excited adamant on excited about what's the difference of having on the energy with or without the average theology of anatomy counts but that's because we can compare it to the energy of a configuration without the Adam but just empty space whose carriers we could call could just put as much energy and their empty spaces we we like as long as that energy cannot turn any other kind of energy as long as it doesn't come into the questions of differences of energy why should we care if the something called vacuum energy in the room while the answer is West said we don't care about 0 energy on till gravity becomes report once gravity becomes important what's the source of the gravitational field the source of the gravitational field was messes energy vacuum energy Gravitt tapes if has a gravitational effects has an effect on space time and has an effect on spacetime which is different than if a warrant there so let's talk about there were 1st question is what's the origin of vacuum energy and as a sad or somebody is merely a vacuum energy is was quite asked me in these terms but vacuum energy is a feature of quantum field theory quantum field theory quantum electrodynamics quantum the standard model of particle physics all quantum field theories have vacuum energy it's just a 0 . energy of the oscillating field if you like as a contribution to it from electromagnetic field as a contribution to it from electronpositron field contribution to it from quirk fields even if no QuarkXPress 2 . is you don't need a had anything really present have that energy was due to withstood the virtual creation and annihilation virtual remember means that happened so fast that a little bit of energy that is created annihilated satisfies the uncertainty principle if you create some particles have nothing and they can only last for a very short time because of the energy time uncertainty principle of that obligor net result is that there is energy density Colorado fills up space everywhere Is Lorentz invariant a matter which showed how you moving until 4 back emerging what role vacuum no 1 orders it do it has an effect on the righthand side of Einstein's equations remind you were Einstein's equations are the details of them we don't need them I will write down as much detail as you need to know which is very little but just to remind you what Einstein's equations the left hand side arrived Einstein's equations cancer corgi Nuno GE stands for gravity off its attend it is made up of Wood of the metric components of space the metric of space made up Out of the metric tensor if the metric tensor is little genius of X it's our lefthand side it's got to do with curvature it's got to do with derivatives of the Met breaker Chris powerful symbols all sorts of awful staff and by the time you write it down takes up our June know is it picks up a piece of space but it's got to do with our life I still has to do with gravity and mature has to do with gravity I think maybe it has to do with geometry G is the
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geometric side of Einstein's equation says some property of the geometry of space time Mr. spacetime geometry of amateurs space along an already inside and lowered its gravitational field gravitational field geometric field however you wanna think about it and Einstein's field equations are that this summer lefthand side is equal to something when numerical constant but I'm not interested in America Merkel can't be piled with 3 but no report on the right hand side is the source of the gravitational field the source of the gravitational field energy and momentum energy also known as mass and the motion of masses which is also known as momentum sonar Ironside was another 10 it's called team you know I don't know why it's called off but the thing which is made up of particles of ordinary fields gravitational field electric field of magnetic field and if you know what this is you know what the edit would be ordinary energy density the density of particles himself or that Einstein's field equations and on the right hand side is all of various kinds of energy both those which particles ordinary stuff light the stuff which die loops when universe expands but also the stuff which doesn't die loot the vacuum energy soros that's the reason we have to know about effective energy no How geometry of spacetime behaves it was only vacuum energy here they were gonna write these equations right amount full but for 1 very very special case I told you that lefthand side is made up out of the metric considered and let's take a very very special case for geometry look forward spacetime geometry is like in some sense it's the simplest geometry that's not just static our space just sitting there think we actually rode out last time was expanding uniformly expanding space don't you think of infinite rubber sheet where the infant rubbish he is being stretched so that we coordinate marks on rates we pick rubber sheet with coordinate marks x equaled 1 Mexico's to lexical 3 articles for them we start to stretch the actual distance between neighboring point increases with time and that's represented by writing a metric DS glare space distance between Paris appoints neighboring pairs appoints just as it would be in ordinary Minkowski space we start with a minus Mrs. penned courtesy of a weak mind this BP squared as we would have an ordinary Minkowski space means flat spacetime and then lawyer have a Minkowski space would be plus DX squared RBX going stands for the X squared plus BY square past easy squared which I won't bother writing it stands for the ordinary you Clooney in metric of ordinary space it would be some speeds of light in here if I kept the speed of light around but I'm not but that's not be expanding space time expanding space time you introduce 1 more thing a thing called us scale factor 80 of T AZT is a function of time is not functional space and it tells you I'm sorry this should be a square of tea a square of what it says is that if you take 2 points but say exit was 0 then x equals 1 What's the distance between them but sworn you know it's not 1 unit because X doesn't actually measure physical distance it is this which measures physical distance the actual distance between them and any incident of where was a change with time distance between his mark points on the rubbish he world rubber world Mexico 0 articles won the mark points separated so a grows with time it doesn't matter if the separated by 1 unit 2 units if we want to separate them by radio services X equals a 7 x equals 0 13 then the distance between them is dubbed 13 miners 7 which is for secular 13 minus 7 times 8 50 4 15 minutes 7 visit for not fit the sex secular right OK that's the mean calm above his metric but have to be on the xaxis sickened me off the Xaxis but you get the idea OK this is a nontrivial geometry it has spacetime curvature on West AT is constant it has and will face constant and is flat space girl flat if a is varying with time doing things may be just expanding fast enough to make space turned up flat space may be flat but space time is a different story based on expanding space time is no kind of geometry that it's OK now let's right Einstein's equations what'll Einstein's equations have to do with they have to do with how a varies with time ADD is the only thing in this geometry is not known if they only a degree of freedom for this kind of geometry for geometry of this type all of the geometry is coded in the properties of AT of TV which means it is a function of time Einstein's field equations must be simply equations that how a varies with time they are here's what they say they say that be a bite DT square is equal hunger put all factors narrowed 8 piety all 3 and Newton's constant times energy density on the assumption here is the energy density of matter is completely uniform in space you could have such a thing of space was filled up with the uniform radiation that the same temperatures you have such a thing air from the uterus were filled up with Adams at a uniform density so we don't have the right role as a function of X because roads not function of X but of course we don't have the right role as a function of time but Rolison function of time but was Rose of France planned this if we all absurd is 1 of a squared here but on a square 8 . over a leastsquares dilemma from last time but a . over 8 years it's the Hubble constant but let's leave it there a . already square best OK now this equation also has another significant and in general apparently it's 1 which confuses people I get zillions of emails I get so many emails about this particular point that I have keep keep up with them and try this equation is also an equation for energy conservation expresses energy conservation in general
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relativity the righthand side is all of the energy of ordinary stuff but does not include the energy of the gravitational field the gravitational field because of varying with time if he also has energy and that energy is a kind of kinetic energy but to kinetic energy of motion are kinetic energy of the kind dependence of the video of the metric of space time what is that the energy that energy is simply energy of their varying gravitational varying arm scale factor energy of scaled back there it is call the scale factor 1 out odd thing about it is that it's negative in general reluctivity minus 8 . 80 square there's probably a factor of 3 over a pie G 3 all a piety G. I think it's not be sucked the number that's important it's important to have a formal structure mathematical structure of general a committee gravity has energy and some piece of the gravitational energy is negative and corresponds to a negative kind of kinetic energy of the gravitational field so what this equation could be ready as a community might as this plus this is equal to why is this past this is equal 0 was zeros conserved 0 doesn't change with time 0 0 doesn't change with time so among other things among the many things that says is says energy doesn't change a time also says the total energy of the world Zero but includes this piece coming from expanding space time which is not used to accounting we can't be users room we don't normally think about the fact that the universe is expanding we just ignore it we say the energy in the room was positive God but for evacuated it was this 8 cause theirs you must go further now let's ask what role of tea is like what the time dependence of energy density in the world is like so let's come back to the box as a real box I've taken a region of space was drawn a mathematical box the mathematical box as side built the X equals 1 on each side how much energy is in it rubs the number of particles inside it number of particles cider which is not going to change as the box expands the particles move with it sorrow that this is wrong this Rome was part of the expanding their universe you were a bunch of protons you would expand you would you would the protons would expand the distance between them were expand so as to keep the proton density uniform prices In protons and each 1 has and that much trouble masses and there was the volume of space the volume of space is dealt the extra dope X times Dr. X at 1 but turned 8 Q Y 8 killed because the actual here is acts times a suicide of a box chosen a box whose side at any given time as any of tea and defined the density I have provided by a of T Q them only important thing here is it time if I know the energy this damage and a friend or the energy at any given prior to that I know all alltime not because it stays the same but because it always scales white 1 always CQ so I say the role of clean and cell number which Colorado 0 throw some particularly since the time divided by a gift CCube finale income formula over here we're not going now have an equation for aII Einstein's equations for expanding universe nor goes over here some constant Rome all over 80 Q. When Gonsalves equation neighbor we should solve a neighbor which is sovereign some difficult equation it's much easier to solve if we take a look constants 8 pie G over 3 times Rome not and absorb them into a another constant was just call that other can't let's just call it Rome and thereby race that's OK Ali solve this equation here let's try that would a better consult my views saw the 1st or manipulator a little bit this is 8 . 0 Rice's GATT squared wasn't a square in the nominator but will multiplied through input on the other side so we multiplied by a square daily get Rome nor provided by AD is that right right no I take the square root of both sides know nor by A. I take the square root of both sides square root of Rome 8 of the 1 half multiply so that it reads born half firms D a Yea equals square root of Rome naught and DT 8 of the 1 half on this side Prime's DAT or just Rico caroticus worry new saw that have Saigon right do we were doing is a degrading both sides of the equation Michael suggested is to integrate both sides of this equation on right hand side we get square of raw not plans time interval of BT is just and this 1 is what 3 have 8 1 half of the integrated 3 halves now twothirds through their the 3 have cracked and now we know 80 as a function of time forgetting all the numerical constants through 3 role north where it says is that 8 grows like keyed the twothirds Powell the scale factor grows like keep the twothirds power because energy density has a particular behavior that's cosmetology that standard tha absolutely standard cosmological expansion a growing White Peter the twothirds what's is drawn a growing mistress draw a growing like tea as a growing like C a key feature the twothirds is smaller grows less rapidly then he itself this is large these more looks like that's not quite a parabola
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books that it slows down the you for online like this would be it AT which grows linearly with time who would be like a quarter of a particle moving off linear motion that's not what happens what happens is it turns over and appears divorce Lawrence Lawler the universe decelerate in this car cosmology offices of decelerating Cosmo what is it that causes the universe to decelerate grand ideas polling everything yacht there's another way think about this stuff started about moving away from our each other and gravity just grab the ahold slowed down and eventually gets Lawrence law OK now let's ask what happens if we have a different kind of energy on the righthand side we could put in a free 1 just for fun I am a door tonight we could put in radiation Energy Energy solely the radiation if no particles only photons and universe that would correspond to go work it out yourself that would correspond putting a for what he called article about analysis the fact you solve the equations and find out how the universe would expand it'll radiation dominated the particular version that I gave you a particle dominated most usually call matter dominated matters of postirradiation where the energy density decreases like 1 over a cube all cable y happens now if there's is a vacuum energy really is vacuum energy vacuum energy has the property that never changes under any circumstances the density itself the density itself does not change under any circumstances we need to give it a name let's put back for a moment to put back pie G's and so forth but he recovers the change quantum field theory of change that tells us that universal constant are that can be calculated now does that mean that the universe so we know with certainty that the universe is filled up With a kind of energy which doesn't buy loot theirs the universe expands fury doesn't tell us that work tells us that his experiment talk about experiment corollary observation theory tells us only want to put on the righthand side for various different hypotheses about what energy density of the energy density is vacuum energy in the sense of quantum field theory that really is just constant so that means what goes on the right hand side here does not vary with Eddie was not there is a universe expands and let's just call it roll back with scores of vacuum energy density roll vacuum is a pie G 3 8 a square a pie over 3 kinds constant value of our review energy density now 1st of if this is correct them 1 thing it says is that the Hubble constant a doubt over is the Hubble constant also H. square it tells us that each square H. also a pie G over 3 times vacuum energy that this doesn't change with time remember what the Hubble constant did the Hubble constant is the relationship between distance in velocity velocity is Hubble constant Prime's distance so it's as tho all time in this kind of world in this kind of willful all kinds of the relationship between velocity and distance is unchanged always the same numerical value that incident we would not be true in New year the other solution wrote about normally for other kinds of cosmology the Hubble constant decreases with time here just stays constant implication is that the dissidents out to which you have to go in order to get the speed of light stuff is moving away from you food with the speed of light that distance is always the same given by sea because each time they are cuomo a sea of age is equal to D. C is a constant H is a constant the distance Out of the place where things are receding with the speed of light from us 6 doesn't change with time so that's 1 implication of what this means the state is a constant value here let's say now we can know if we can now solve Einstein's equations for an expanding universe under the assumption that they only energy density now the only energy density is back in energy density don't have we have a dark already squared is a bull that 8 geometry Colorado but that's just the number 8 square was just call at age squared now I'm using age now not definition this is not definition now previously made over a was a definition of the Hubble constant no it's the numerical value of 8 pie G over 3 times roving number it's a number of that a 3rd way of who we don't know how calculated we don't know the theory of it but No. 8 squares no number perimeter understand the difference between saying squared is a battle that aid as a definition and H. Quinn is 8 a square as as no Maryland numerical relationship yet H it is not defined this way adjusters number is measured it is measured to raise number and it's not expected there was evidence that is not changing with part of the only saw this equation so we Daggett square root a . multiplied by a reads a dark Circle H. terms a I think some fraction of you equation recruits are cracked a buyer 8 as corporate he could be H T equation that tells us that the growth rate of 80 is proportional to itself exponential growth with kind thing which changes by an amount proportional to its own value exponentially grows and the growth factor the time factor time constant fervor his is age eagerly H he is the weight this universe is growing exponential Yucaipa but constant from a constant doesn't matter much so this is an exponentially growing universe that's the result of vacuum energy vacuum energy really is important and the way to detect it well give wasted detected 1 is astronomical measure the Hubble constant and measure the Hubble constant over a period of time and see whether it's staying constant although you could measure the scale factor over a period of time and see whether it's expanding exponentially as another way that you could measure in the laboratory on his no way you're at your do this but if you have a laboratory now means way out in outer space take 2 objects which are very light in there for their gravitational between them is not very important to stop them out
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arrest and daylight enough so the gravity between them as unimportant they will just expand the expansion and you could measure the rate at which the distance between them expands with time make sure there's nothing else present complete vacuum so the only possibility is vacuum energy distorting space and you will discover that these begin to accelerate and begin to expand separated from each other exponential rate so that's how you would build a vacuum energy not cost effect in the real world is so minuscule that you couldn't even begin the hope that detectives rise in the laboratory but you can detect uh cosmological idea detect because more logically a number of ways but they're not he said well we have to we have to check that h is constant with time that's go pick long long time but no not really we who we look out of the distant universe we see it at different times so now we can actually measure a Hubble constant at different times by looking at it at different points scales although we can look what's comes the the same thing is we can measures expansion regular can measure How a varies with time that's OK now how varies with time all wake shots what point something else out the real world doesn't just happen each squared it also has some other energy density wet sand and the other the energy density was called wrong not provided by a cue for example when a yes more universe a small scale factors small this Sharon as good a much bigger than that's why is that because aid is in the denominator that's why early on in the real world this is not very important and this may be very important when it is small in that case we worked it out a few minutes girl we found out that a goes like Quito the twothirds goes like this but now is 80 growers eventually this term will give small wears this was not getting smaller this was not looting this 1 is dialog so eventually they hate turned gifts smaller than the number and then after that the basically forget it once this stern gets smaller than this there you could forget it and have a behave aid behaves like eatery aged T so what to expect in a world that has both kinds of energy is that early on the scale factor for a radius of the universe that the radius of distance between distant galaxies grows this way but then at some time when the vacuum energy becomes more important than the other forms of energy it takes off shore exponentially exponentially something like that we a solid waste see if was our now doesn't agrees that I had a may great follow when measures measures you measure it and I said yes seat peers a course for new measure age you don't just measure 18 hours said you measure a by looking out deeper and deeper into distant galaxies you can measure the distance between different distant galaxies at earlier times and so it 1 time you could measure this whole curve up to some pour you can measure would be in the future but you could measure what it was in the past so if this is now we get the measure of all of this criminal look something like that would look something like that approximately something like this where there's a branch off here being fit rather accurately who are to an exponential I want I want tribe tell you how accurately our ships about 5 % of the proper figure of merit is about a 5 percent deal is an abbreviation exponential plus or minus 5 % OK so that books or most cosmologists looks fairly convincing that we live in a world of vacuum energy certainly we don't know anything else that would have much chance of doing precisely this and it's a it's not a factor in the same sense that we know that the electron exist the fine structure constant is 1 around 37 . Bob Obama but by now it's pretty much considered a fact of cosmology that we live the world were back in energy and that we live In exponentially expanding world that's a pretty remarkable right away that it will take our busy right now a lot slower down down down if that's slow the left of the order see that the Big Bang in the middle of the time you can see that the big Bang is an arm of another expanded waste for that I don't quite understand that workers were on mustard assure expanding by stressing that the Big Bang part of the world then this is is really a and cents above aboard a yacht right gobble it right in this exponentially increasing world here the Hubble skill was not related to the interview it's just a constant right but he that the Lura's Wilbur probe Roper roars we talk about event horizons a lot of different kinds of horizon on the market but the only warnings which deep and fundamental interest are event horizons so that other but event horizon selects but just but not back pocket as piece of knowledge of the world was exponentially expanding matter summarize everything I was on the blackboard but is an interesting fact the world continues to exponentially expand like this then all of the dive laudable matter the kind the energy by looks with expansion is just getting more and more and more diluted so in some number are over billions of years material in the universe will stretch out so much that it will become of negligible density and the only energy left over will be the backup energy become a very dull world Our bar so be it does that mean galaxies which are expanding today because we all their reward express our war will happen is the few galaxies which are close enough to to us which M which happened just by accident moral less to a B falling toward us there's work to galaxies trajectory for toward us our that's a service where we say that everything uniformly expands of course a
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uniformly expands plus a mind this little bits of fluctuation it happens that there's a little bit of fluctuation in our neighborhood where chapel the course the Andromeda galaxy to be somewhat moving toward us so Andromeda are not gonna separated from each other but but you have to go out a few more galaxies are certainly moving with the flow anything which is moving with the flow will eventually depart it would just grown and separated separated seperate and after a few hundred billion years all that will be left for us to see is hour galaxy and that's it we will be living in a world where be able Seattle anonymously large distances but you won't see anything astronomers born at that time and don't have a good record of history will come the amazing conclusion that they are alone in the universe now let's say you always exceed where additional pressure best organized about group can they had brought the group expands in the gravitationally bound that followed and the drummer With by that time probably will crest us in the form of a single structure each clusters their patient that can also earn it but no I will know depends on what you mean by cluster if you look at a few galaxies Our not very many just a few would work to more than moving away from even know they form what's called a cluster I don't I were Brown Northern not gravitationally gravitationally bound means that they will do you want to make ready down Yahoo but not very much but not all hardly media things if they're gravitationally bound just need weather Our just forget to get this expansion and just ask whether they're moving relative to each other with less than greater than the escape velocity at that would forever now it happens is or accidental factor drama that happens to be moving toward us a few galaxies out tho they're moving away from us and they will continue to move away from us moving away from us were great escape last no continue the new address wrap their classes but he out forced to do that is a function of Rose at present distance it grows linearly with this it this is not now what time it acceptance of forest things separate with time to object was separate with time the distance will become larger and for that reason we Nunavut repulsive force will grids but called for a more calls for is all up and over a period of billions of years with holes that's only because things get further away from each other but but what if the Galaxy 2 galaxies are the same distance of away current as they are now a component of their move repulsion from the vacuum energy will be the same for as that and that he might be finished if you can force a ban on with that said that she heard that same not the the distance from trimmed you have to object if you have 2 objects and distant are between them at any time there's a with direct took components for the Fort Worth November G times and and to a divided by a are squared and then there's another forests on each of them or is another forest which is proportional to it has a gene it but mainly it's proportional Toland repulsive force forces minus meaning attractive plus a repulsive force proportional starts the distant between them work coefficient of change what does change the distance between things is this is grows will become less important as will become more poor but ask where does this become important wrote that's the answers when the distance scales off pretty cosmological No 5 billion light years or something like that across over the coefficient of this Peace is so small that the crossover between a normal gravitational attraction and the bitter repulsion is on the billions of lightyears scale so that's where you can measure it by you in the laboratory so it's not just a function of the report of data space is representatives of both forced it back it is it is expanding space but you can you can't fake it by year but force which is pushing everything else apart from each other so a slight fake but above soda works right right armor OK we now that's that I think about my arrest bad they shot about an expansion is now where he back space really Illinois not gravity the gravitational factor wall relief when they get far enough apart In the repulsion becomes more important than the attraction of borough Jarvis was now imports nearly and that's another statement delay its Tuesday different statement to say that they gravitationally bound and that the important force was the ordinary levitation for us if we should a rocket off the surface of the 80 errors the 1 and a half times the escape velocity it is not gravitationally bound but still the most important force on busy ordinary gravitational forces of 2 different statements 1 statement gravitational bound his the comeback via the statement is this may be more important than the nearly universal was but it doesn't mean to grab a year galaxies were gravitationally bound it was still moving apart from each other with greater than escape tho most part pretty much Galotti shot out pretty much greater than the escape velocity between them book I'm roughly and at very close to the escape velocity very very close extremely close to this day possibly and then the either turn around and full backing for the accelerator with depending on whether there's this kind of new energy OK the figure will break and we come back so you more about the geometry of this kind of space and why it has horizons what it means for half horizons
1:07:21
summarily no that in a couple of hundred billion years the universe is vividly described as thought that's it's getting that way pretty quickly actually on the scale of tens of billions of years we're getting there pretty fast obligated to a conditional with the the geometry of spacetime is described by a metric and metric each of the 2 0 H that's a square times DX squares again X squared stands for the X square post to our square was busy square and this is a scale back on this space time to time has a geometric property it has a name and the maintenance procedures the it's the sitter space the recall de Sitter spacetime obviously booth called the sitter space and collect examines properties riverbed not because not because of we really sending a rocket ships out there to explore it that's could while but just for the intellectual value of understanding be space time that we live in a free weird it's pretty weird the castle looks weird that's weird the locks on its weird and never has Verizon's before we get into a horizons of de Sitter space let's just talk for a moment about what horizons or in general and I'll give it as the illustration the horizon of a black hole member we've worked out or a black hole formation of black coal which is created by in falling collection of material looks like only drew 8 can remember the Penrose diagram Penrose diagram McGregor before we do that we really really use Penrose diagrams so because the top 4 but its 1st rubber Penrose diagram of ordinary flat theirs ordinary flat basis points far away at spatial infinity time infinity light rays coming from here got from here a lot of stuff and that's just empty space a black hole spacetime looks like there's 1 of a point imagined an object a massive object not a photon a massive object In this space How does a massive object move through space time well Oh long list of objects windup at the same place they were Mansel subjects photons society moving with the speed of light massive objects they are all go to this point that that was not really a point is not really a point hold big space that been most do I want diagram to look small what obviously if we want to take all the space time and squeeze it onto the blackboard River have started pretty badly at 30 distorted appear as an infinite amount of space and time open this corner likewise here likewise here on all of the edges that all out of the massive objects eventually windup at this quarter that's not to say that they be close together is just to say that this is a very big place OK now let's think Of these massive objects in the black holes the black hole space you could have 1 of 2 things happen a massive object conformal into the horizon and hit the singularity let's avoid that but that let's talk about the objects which going forward of the singularity where do they go they go to hear they all do again that doesn't mean that they literally get close together it just means as all our space time out there all of these people Winder puts drove horizon the horizons earlier OK now let's asked what these people here currency as they look back into the past they kill see of only those things which are inside their actors like backward like all means take although light rays which can get to this point over here while libraries from Metromedia take that region that's the region that he will you Camille where looks back you can always see the things insiders like savor this guy over here saying this observer I could see things in their backward like now how much can they see give arbitrary amounts of time given arbitrary amount of time they can look back from later and later times and eventually they see all out his blue region here what they cannot see is what hear horizon is by definition a place which separates the region that these observers see from where the can't give it enough time you draw other backward like Kong of a very future point along that observers trajectory and that is the region you see everything on the other side is behind the horizon and the horizon is the separation between here does not arise observer away happier gets back looks back and sees it every everything day sees the entire space time not eyes and ears of horizon so I just remind you of that to tell you horizon we take this metric rebuild the mathematics not hard enough that it should be easy tricks it is not particularly abstract but it helped us visualize the spacetime wanna do with it His with we are not going to try and take all of space and squeeze onto a black leads space in Palermo the cake Louis take the time axis and when Irish Moshe it I'm Wannamaker coordinate transformation of time so that I could put the entire time from my innocence from remind us of the movie will be deep down in the floor but plus Infiniti will be of some finite place here because we don't want them to want to study what his world is like at late I would have liked to get the whole infinite future onto the blackboard but the hall and the future of space for the whole a future time so that may coordinate transformation would do this In a way that makes a light rays move on 45 degree angles in space always
1:15:26
good to do that always get to libraries moving a 45 degree angles and easy to see what can communicate with what I've said is track where change cordon really introduce a No . 9 Collett capital he and it's a function of the old please there's gonna have the property is a property it's going back from such that DTC squared where appears here is going to be equal to each of the 2 0 H T exactly what appears to be a kind of D capital P square This is the definition of capital at no time no time coordinate and its related to the alltime coordinated by a pretty simple equation with a solid and they would get a rewrite the metric with the new time and place of the old kind and metric will have a nice appearance that will allow us to explore very nicely OK how I saw a big key as a function of relocating the 1st thing I do is take the square root of both sides of the equation H T D capital and now divide by http were gonna lefthand side the divide by reached 8 minus a a minus HKT here and now I have will always have to do the relationship between Mbeki and little tea is to integrate the integral on right inside just gives us for about the last dance side anybody could do that integral vying for anybody for a H minus sign I need to minor stage ranked farcical buzzer constant but we can fix the constant bidder by just shifting key right is a constant but I could shift the cross border per chronic is a constant now it is redefined piece of the steep the admitted right OK now know is 1st of all capital T is negative assisting is think about what happens this former HUD buffs report let's see what happens to it what is it like and very remote past times but state wind a little see here is extremely negative what happens to minus it becomes very big if he is negative we have each other minus because because Ito a positive number a big positive number this gives very big but a negative sign out here so that means the remote passed with little is very very negative is also the remote passed from the point of view a big team when military gets very very negative sewers big T. right so in both variables the remote past his way down through the Florin basement and even deeper about the remote future what happened In the remote future to capital tree the remote future is with little P is very very big weapons so this will appears very big because 2 0 so big he goes to 0 in the remote future now we can draw the climax system which drug time big actors the big caxis goes into a very remote past negative region deep past but where is the infant a future in the future it is right at Capital she's equal 0 0 Cabral equals 0 also is little kid equals infinity big key runs from minus infinity 0 as 1 way of getting not the whole geometry on the blackboard getting remote future onto a blackboard girl OK let's uh let's see if we we can rewrite the metric to rewrite the the metric in terms of pink instead of solve he only have what's little Tsquare what's the relation between little bt here will be T. areas will be is equal he the big H T tribes big team right God was multiplied by EPA's BHP together or on the side are here and now I can't asthmatic over here we might is deeply square tho this is equal decays squared but that each of the 2 waged DBT square and there I have class each the 2 H T X squared another lawyers our aging period so that they eat the 2 HQ economy outside the equal eager to following news but commented Minkowski space this would be the metric of ordinary flat spacetime minus square post TX square post y script was busy squared but it has this factor in front of it about the fact from a very welcome we do with it I want to really express it in terms of capital city so it's do that 1st multiplied by each each other miners H T equals have been quite do was multiplied by Barry H. now let's square if I square at the minus sign goes away and I get eaten minus 2 H T decided age squared squared and finally I take 1 over survey each of the 2 H T is equal to 1 already each squared Tsquare got through it checked the arithmetic you find it works that worry and now we can write this metric just by substituting for each of the 2 1 over age square squares but about the virus right over here squared is equal to 1 owner H. squared he squared times minus T squared plus Square Pocket along starting nice simple result but the
1:23:33
mean point of the main point of it is that we've written the metric as an ordinary Minkowski space met times a function with the same function appearing multiplying all light move in this geometry what where we want to rule for life D D S 0 yes 0 right light rays MoveOn pants Our Zero property or that say that's says that you can forget this fact or altogether when you're thinking about a white race if you want to know how light removed you don't need to know about this factor will you need to use this fact see that your marlite nor was it was exactly as it would if this would just ordinary flat spacetime out libraries Moon flat based on straight lines at a 45 degree axis vertical so we now know it is geometry how light rays move with Stroh the geometry mirrored those starts out deep in the past can't go down that deep into the basement but it eighties missus based space novels this way and this worried it is at Capital Teakwood 0 How can the capital people 0 little tea when I go on forever well it's just that the metric were moved it would be key gives small metric trachea blows up gives big so that means there's all are trying here the metrical distance between neighboring points gets very large when goes 0 Source highly distorted minutes are highly distorted geometry the 1 advantage of it is that light breeze move on 45 deg so we now know how life is more and more like that them only have a blackboard they also more 45 degrees below protracted caterer now let's ask what observer considered Nagin an observer in this world make World their shares she standing still around reference frame trying is going on more and more and more time is unfolding and eventually she gets up know how long does it take according to her watch forever right takes forever for about half an hour but on this diagram on this formal mathematical abstract description of the sitter space with terminal pointers is over there at the end of her existence doesn't bother all because Clark there's gotta after let's ask what she can see what she concedes she looks back stroke array of issues rhinorrhea Alice and analysis looks back and sees Everything that she conceit within her like remembered the whole purpose of this coordinate transformation was to make it easy the light rays 45 degrees 2 or more dimensions will have their work extend South America cold out of it but you could see all the main features here she can see everything behind her but she waits a little while she sees more if you wait so long she sees more but no matter how long she waits for most the chicken embassy it is theirs what about things that he had not somebody out here she can never know about it in principle she has a arrives a red line out here it is event arrives used for private enterprise other people have different event horizon but this is hard private event horizon he cannot see anything out beyond print a matter of real principle she cannot the least thereby classical relativistic reasoning that classical like reasoning and so forth she cannot see anything out all kinds of things that be going on there a friend Bob might have originated over here and she might have been able to see bob the past from here she could even as Bob Alice bomb in the past Dallas was able to send messages Bob Bob was able to send messages to Alice and now Bob passes Alice's horizon from their Bob cannot send the message that our Bob may be able to see was conducted from this point on Bob has passed out of her horizons gone that's what's going happen unless of course if she's holding here is always a look like she's holding hands of volunteers or a mail yacht then there's now does this mean they really get closer and closer together no it doesn't it doesn't because the metric is getting bigger and bigger out here so the actual metrical distance between an actual proper distance between them could stay constant and it does stay constant there would appear to join up here but they are not really getting any closer together so that bound together for holding hands looks like in making continuously forever and ever send messages back and forth Our nor limits but if Bob lets go of ours and goes with the flow of his Let's go implies among other things that it's far enough away that the gravitational pull between them doesn't hold together let's go and passes outside horizon simply fall out of course all contact that expression a former cause or contact in this particular instance of Bob can send a message to Assad receive a message from hours but you can never send the message for our Alice up here cannot send a message to Bobby because Bob also has his own private event horizon but Bob's Event Horizon Green is Bob's Event Horizon Bob cannot see anything outside of this event horizon so he can see early on but we can see how soon after she passes out of her you won't be affected
1:31:41
if that had feared not only look at it as a Will you could never see anything at the same time period right you could never see your the only thing you could ever see is things which are back inside your past like never concede a buddy of St. Paul was barbers passed through here Alaskan never see if this really was flat space it really wasn't as peculiar the sitter space with the fact that you which terminates it there now keep going and backup Bob Bauers comes and the 4 World line over here simply cannot road back so this is the character of the sitters base has horizons in fact every single observer has private event horizon and this case Allison Barbara constitute 1 observer and they will talk a little bit more about these event horizon next next time on kind quit but we apparently live in such a world and in such a world not Allison Bob but galaxies Watson galaxies here we are now we could see about 10 that 22 galaxies around us as time goes on Paris through the horizon and passed out of were works but there were Alice bill passed out of Alice's horizon and Alice will not be able to see them and they will be our so there said Alice will be alone in the world for practical purposes you deserve their is for us we will not leave it he's got a threat we were or over a gal pals all but wrote you might see my OK with that let's say as good Our some very very late here conceived Bob back here are the very innocent no good looks like he can but since nor more bomb is Alicea Alice's trying steps look like best many many trim steps here now remains the Bob send terrorists signal and that's signal come up here are also gonna look back and look for Bob back here you say she can see him but in fact b costs her time clocks so bunched up over here it means the radiation that she sees from Bob incredibly rich if that Dominique beats of Bob's of Clark gets squeezed and Alice's prime scale here 1 beat clock says stretched out Over an enormous time scale from our world storm Alice looks back and sees Bob from way way up here he is enormously rich you can see very very rich things wavelength the away leg of his life and the energy of his light will be so stretched our soul and fact really Alice has no chance of really seen probably just the red shifted the health their arrived but it's exactly the same as the horizon like all so let's until very late time Alice will not be able to see any of the galaxies except them is exceedingly rich shifted stamps which is useless can't detect the lowenergy radiation and she will just find herself along the sole Bob all other galaxies having passed on the horizon is the same level it takes forever yeah but they OK and this is the same it is the same on well it is the same house looks back as Bob crossed problems Bob Bob over here as Bob across the horizon no Alice looks back as Bob across the horizon not our strokes back has Bob across the horizon now she never season across the horizon but to objects believed the black hole because it really rich what yes yes it's exactly the same thing it's exactly the same thing arm but the truth he is a black hole Harris who stays outside for hours outside young Alice outside keep the color coding blue blow for Alice I don't really know what was going From but but but if I had better is it is Alice ideally send Dallas into a black hole Castano skits to stay outside and he a clock clock is going faster and faster and faster faster that as not going faster and faster but I was diagram or time intervals a bunch that same as they are over here and Bob who is now Green slowing the horizon how's looks back but she ever skipped the sea Bob fall through now Bob's signals as light waves Ariz waves of his hand that he sends out appeared to get slower and slower to her because they drawn down a little longer and longer time scales which is another way of saying Alice becomes infinitely red away from Alice becomes infinitely red shifted from Bob's perspective so it's exactly the same thing now that I will chief she's going at a proper just but on the scale of its but if getting right where are the scale not 1st she's is taking along happily does but it it's not Our Bobby right of Bobby Bobby gave around us that fish that Chief Bob Dole that not ready but that was only season for managers Adams slowing down she sees the radiation coming how she whose death 3 cities not only Bob slowing down the GCT Adams slowing down she sees the emission of photons getting slower and slower before Khan's getting longer and longer wavelengths more and more frequency and eventually his photons becomes so monumentally long wavelength that they can be seen all the energy of the niche markets diminished to negligible well to do that she has to of from Mahmoud and make sure doesn't fall into the black hole to keep it from
1:39:47
falling through the black or to send them a from armory measure the temperature report back up the were recorded but that's not what you doing what Bob Bob is in the same incidentally same thing here enough it is analysts it would be the experiment Alice has a very long world long cable and the end of the cable is a thermometer sheet robbers about 40 knots Beyond the escape velocity from a local cluster of galaxies and wait a long time for it to migrate out toward the horizon arrives but she doesn't let it go through the horizon she then polls a back in after she pulled back in she sees what they know what the temperature of the recorded once she will also see a hightemperature would come at horizons are always hot but who come that this is so far the classical description of the shooters place full of arises as Horizons everywhere outerwear Alice it this somebody she's passing through somebody's horizon so wherever you are passing through somebody's arises say goodbye because you were to begin says there a slight the them the arrests saying this movie that I would that be of been exercise an exercise reply now that's a good exercise just over a month due under what I did that it my son is dead that were not only mobile home metric is blowing up including the ex parte so the distance between points if you keep the points at fixed cordoned separation here Bob in allies are next exit the distance between them is getting bigger and bigger but the attract distance between them blows up warming up room a similar fact of the sooner space none some legal work hours is you can't compare the distances between these 2 points with this distance between these 2 points from a distance between these 2 points between 2 distant comparing how far apart on how far part of these 2 how far primaries to how far part of East waiting yesterday the growing threat shrinking not staying the same things if you draw a horizontal and coworker is it for a horizontal line here and ask for the distance let's say from the center for of the horizon over here and compare it with distance from the center of the horizon over here compare with distance from the center of the horizon or you'll find all the same the coordinate distance is shrinking but the metric is increasing and the distance of the horizon is always the same that is connected with the fact that the Hubble constant is constant until the distance to where the speed of light is whether there will be things are moving away with the speed of light is constant sorrow I was doesn't see things changing with time everything stays the same she just sees a horizon enter horizon stays fixed distance from her village rejection of all things that have been rejected that arrived at the level of the Member go away but she sees it harder and harder to see them because they get richer now that doesn't take into account the Hawking radiation coming from the horizon but a but when you save right OK assuming that we don't pasta which others horizons will meet here next week for more please visit us
1:44:22
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Titel  Topics in String Theory  Lecture 8 
Serientitel  Lecture Collection  Topics in String Theory 
Teil  8 
Anzahl der Teile  9 
Autor 
Susskind, Leonard

Lizenz 
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DOI  10.5446/15109 
Herausgeber  Stanford University 
Erscheinungsjahr  2011 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Physik 
Abstract  (March 7, 2011) Leonard Susskind gives a lecture on string theory and particle physics that dives into the idea of cosmic horizons as well as the relationship between ultraviolet and infrared light. In the last of course of this series, Leonard Susskind continues his exploration of string theory that attempts to reconcile quantum mechanics and general relativity. In particular, the course focuses on string theory with regard to important issues in contemporary physics. 