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Topics in String Theory | Lecture 7

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back and think we were in the midst of trying to explain to you how string theory gives a rough explanations of the entropy of black holes but doesn't pass if you remember evolves remind you if you did remember on the a slight some equations on the blackboard here which will be useful but I don't know that that things a he is a key hair care was false Richard them as indistinguishable will take them where we treat them indistinguishable if they are identical ahead you can a counter positions within the box yah yah putting them into a box of a certain size namely the argues that soon will be extreme would pick up if we woke there that of the on OK I want to write some equations on the blackboard remind you that simple equations and then I want set target goal for simple little derivation of the entropy of a black hole it's kind indirect minutes very subtle but very easy doesn't doesn't want anything is complicated I don't think it involves even the derivative let alone in the world certainly no infinite some odd it's pretty easy but still it is subtle sometimes Osub listings of the easiest ones in the sense of easy mathematically and this is 1 of the pilots so 1st 1st equation GDR he is Newton's constant we're going to work in units in which C equals age bar equals 1 and that that Newton's constant equal to 1 we believe that flooring so as 1 because for quite see because we measure space and time in the same units the speed of light is 1 constant be equal to 1 that's something go that permissible to do and it's just units which the uncertainty principle says that the product of the uncertainty of position on the uncertainty of momentum is equal to 1 unit 1 unit of something they Nunavut options ROC people 1 but we won't fit along with all traversable Newton's crop comes from more than you at all the results Eagles age follicle toward the loop but you have to specify you can either take it to be all or mass and I'm not cast is in leg man isn't in for does remind you why that is the fact that energy which is the same as last year you will need to do write equations find some equations after energy and look at the various dimensions and see what happens energy for example quantum mechanics energy is given by plants can't stand are divided by the wavelength times well
and that is the way photon for example this would be a connection between the energy of a photon adds wavering using it for anything except to to say what relationship between energy Our between these units In units are now this is not the romance To make a massive think we have abided by squares right ER overseas squared is equal and lots of people who know that you overseas Quaid was a Quebec em but it took a genius of Einstein to realize it and people to see squared price would Chloromycetin Yancey downstairs not only a cad novices I don't they can wear overseas our because we're a cervical but so the only point of this was to say the U.S. ask what we said Siegel born on the same as a unit of upon constant over land but also said plants constant equaled 1 now a unit of mass on 1 the key never mind mass is 1 Lowery Europe's Unit 1 day now what about a unit of GE not said called 1 gene has units set to hide the figure would be units of a G R we just take again equation that has G in it and those look at who what's assemblers equation contains G which he gets acceleration equals GE and also our square now bereft inside what's the units of the left-hand side of is overtime squares and time at the same units whimsy is equal more this is just following squared which is 108 right what a square of course it is laying square costly squared and now we can multiply both sides of the equation by laying squared his becomes line upstairs where we know that before it in ways that sphere radius ish children 80 years or proportional radius of black hole but now on is late in the 1st length and on we can ride a mass is 1 of our main transferred to the other side we discover that in these units GE has units of an area what area His only area around a bit could be the plant area I too GE is in fact not only an area that's also the a plug area square it's important equation and the fact that helps us understand why it is that the entropy of a black hole is an earlier divided by 4 Heinz G. forget before a seat it a divided by GE at sea area in plant so whose we're OK so that's 1 inquiries and now there's is another factor and that's that jail of course also has not just owing squared it does have to do with the strength of gravity so let's think about it in the context of string theory as having to do with the force of gravity What does the force of gravity come from the force of gravity and no explanation of this tonight this is not for tonight we talked about before this is justified cartoon The 1st of gravity between 2 objects this could be the a communal not he is due to the exchange a rabbit out as happens string well everything is made up of strings of string breaks off funds of often takes off when the stream courses itself there is a probability that won a string of the CB strain wears core boy no
turned this there and keep that he and stick out when the stream classes itself that can be arranged the string closes it's also could rearrange that probability of the square root of that probably would restrict speaking square root of it is cold we amplitude for rearrangement Austrian breaking or whatever but some breaking its rearrangement of Squire little Judy with talk about before Fuji remained about it is it a problem at square the probability a probability has no Unix dimensionless His G which is the probability that we're straying crosses itself it breaks off a little piece that OPEC jumps across 4 other things really Rangers again the shooting jump from 1 side to the other that in string theory is the diagram of the analog of fighting diagram which has to do with the forces between the 2 sides and but the thing is proportional to powers of this Jew you want to limit total for those to absorb liberal Stern that tells us the force of gravity is proportional Judy square so what must tell them is that Lucy squared must be proportional the notice CoreStates known as constant as the strength of gravity liturgy is the probability for break a string breaking but this can't be right laughter insiders dimensionless has no dimensions is just a probability the right-hand side has units of so other Huston something Ellison hero for make up the units give you and is on the left side was that that has to do with the fundamental link scale of theories it's not a punk like it's the fundamentally length scale of string theory is basically if you look at a lot of illegally staying you'll discover inevitably consists of little wiggle at random walks but as with and is basically the size of Lulu with the basic size of Art and it's cold strength with talk about before founders reminding you it's all other variable around and the problem is his plunked squared beside strength a beside his gravity gravity tells us that this side is our plant squared off only sensible equation like right but only contains things that have to do with string theory on side is to say that this is Jae squared times l string square and that's the collect equation physical like equation that the plant closed squared which is the same as big G is little D squared times L. string scrimmages thing about them and that means that if we G well over 20 20 number gravity will be very weak and tell us is that was much much more restrained for further fact is school G which appears he could also be thought of as having could do more With the relative size Of the plant length scale length scale it has to do do with gravity and streamlined scaled back we can even take the square root of this equation do so it says that plant is G Times strength cassettes another equation right Darren ya plant is equal to GE Times l strength over equations that we need cut next equation just make a list of them before which I use them and before we tightened arriving next equation and they are they strengthen the entropy of a string X entropy basically counts the number of decisions yes No War up up then owned or left to right decisions that you have to make the describe the system let's just leave it at that that's the number of could finally was a matter of binary it could be trying very hard quaternary some small number number of decisions that you have to make the specified configuration I get a little example all a model for string theory and is it it's a reasonably good model for string theory is well that statistical mechanics people use all the time laughter string theory but other purposes where Place restrained by a series
of Lake ladders and so forth wake no backer of self could do it wants that depending on the number of dimensions of space let's just be simple and don't think about 2 dimensions as basis for city then the on gets attacks as show us that can make a moving further NATO for directions items not that's not binary of each decision requires binary decision our the biennially decisions could would you do if you have but possibilities at equivalent to 2 to 2 heads tails kind of decisions but that distinguish weight of difference between 4 and to the number of elementary decisions to describe the configuration of the strain is clearly proportional to to Woods Lane so the entropy of Austria and is a number of decisions is going to be proportional to the length of the string not right that rate arrived now held by my pursuing leads but the length of the string was just caught end our but that's not a consistent equation why not and B is dimensionless entropy has no units at least in modern statistical mechanics were were familiar with the idea that entropy is really energy laughter and now it appears that temperatures energy on entropy has no units little has units so this equation can't be right the problem of the equation it is it's not telling us How long each 1 of these Lu excursions areas we're talking about when the walk and not to talk about when the block you have to talk about peace the size scale of the individual steps of the random walk these of the steps of the random walk and guess what the size of the steps of a random walk for a string is hailed strength the natural things to count up half a wheels or steps are that's gonna be the length of the strain provided by L. strength that's the yard that's the entropy follow a strict the notice it divided by sales strain will add another equation for in a minute what about them and knows I assumed be convention that entropy Tyler right Our editors termite entropy OK what about the romance of the Schroeder strings also clearly proportional which length it's just a matter of the sum of the little bit don't on oversimplification but not much indeed the let's look at the Connecticut energy of motion of the strains Well it's it's not too hard to prove and statistical mechanics that the kinetic energy plus potential essentially equal to the potential energy our it is correct statement in the area that been massive the strain is just the sound of the masses or for each 1 of the group's segments but the lesson ascendant that leaves us with the question of what is a mail service said was a mess of the most clue of basic rupees which they scored In I dealt with his bitter and ambiguity I'm going to write and strain but don't rule the mess of the whole big strain I mean romance Of the Luo bait of elementary stress that feeling of submitted link L. A. bits of land and have to remember is that Ms. units on the singers In here mess units are the same as in wrestler length units the although possible equation that we can write down for this is that it's equal pulled 1 divided by the spill Australia's Daily associated with it this is the bigger the mascot so their 1st the basic unit of for a string it's kind of like a 6 0 lead and centimeters and the Mets over the strings 1st every 10 overpass exilic something or others smoke this the bigger scale and now could write down here the mass of the strength of probable massive restraint is equal to the number of those segments on the other segments the number the number 0 segments is divided by illustrating finds romance each little segment and that gives us another L. string downstairs we can minutes this is a mess Of the big thing with scoreless the rest begin stands for the Hall Fame mountain manipulate these equations in various ways we can also like this as the length of the string kind will and substring because sitting amid the dominated the same as string of data and so forth and so on what we could do a bit more what we can do is replacing it with em but replaced by M was just right equations going nothing here and so is strings squared times capital and so was iconic give us that's going to give us our L strings squared terms capital area that is columns a new strain but Asia those Wilbur L. straying right so it is a good result I think I'll just condenses into 2 equations because parts of them are not interested in S is divided by the string but it also happens to be the Mets times a string of coal Gomez of problems and current string OK now I want to compare some of these equations with black hole for 6 I don't think I need this right down and they need to begin by maybe I will know what this is string theory if you like I sho headed the column here string theory string theory OK let's put over here black freely which also called gravitational theory of general relativity but a little more general relativity general relativity plus talking and back China post the entropy of black holes Iittle here we put Blackwell physics all of this of course is probably better and interested in me entropy of a black hole the entropy of a black hole according to enshrined is equal to the area Of the horizon AP divided by either plant squared writers Clark squared all capital G is 1 and the same thing so I'm not a cheating your anything end this the the new local coefficient in more than a radical coefficient was they are for where the unity of the calculate that Fausto hard but this is the if this is the scaling misses the parametric behavior of the earlier divided by a L plant squared come paint that With this equation over here the entropy is light In school it's by another system later provided by streamlining for streams and divided by plunk area for black holes now I'm not saying these are the same the Upper St. McQuillan up the same unjust pointing out be
similarities and differences we can also write another way be earlier it is equal to the square which watch over ideas and this watch over a radius it is ng now have Toulouse inflows and things like that and usually is D. Schwartz radius we want the school that and squared GE square feet area and wanted the violated by ideology or plant square divided by duty plant squared just aims squared trying Judy and squared times G & E is wrong squared so we rewrite mistrust and below plant squares let's the entropy religiosa no barriers rich products or deal was a similarity and the difference the similarity is messed up some color of some time some power of being different and Samoa to some power mass comes from power of streamlined 0 over here equals M squared L Fox quipped what if anything is the connection between any connection me in way the same claim saying the restricted was chosen like all the black holes a string also could not Kobe in string theory but restrained bill is nothing in string theory except string of black colors strained would stand to lose but should satisfy some wheels of strength theory and you might think this is the right formula books this is missing something important in this city v strongly interactions between the parts of the strain mainly the gravitational interaction between which has an effect of squeezing the string together we haven't accounted for we haven't take care of our own announced like attending the mistake of making something like this if you want to calculate the properties of a star given mass me giving temperature we might take that amount demands and put it in a box about as big as a starter the particles no payment boxed and rebuffs because but a box bigger than the size of the start and teach the box up till we can't Butcher of the stars would that be be a good description of the stock now it would be a good description of the start with that 1 important thing the gravitational forces between the pipes of the start which put together make the size of the box smaller than it would be well that would be the same gas at the same temperature as saying entropy and after that and the fact chill the relationship between entropy an energy it changes the relationship between entropy an energy Bacall changes the energy of the star of factor the energy of the star is less than it would be if it wasn't for gravity is less than a would-be because forces pulling it together are trapped there and the polluter together lowers the energy so you have to account for that we have to account for complicated physics of what pulls a string together and transit in the tiny little nugget of a black hole and of course actually following that is difficult no 5 4 a straight out house that's my late Tuesday it is a little cruises raced off that less of victory would be less what I did say that but there are still has this great had created now if you broke up peace news they're in the same volume the interview would be left but a little child rest as lower the entropy riverbed this is a a radically process No no no and actually does not the was a massive this is the particles which jump back and forth between the year of a virtual particles which jumped back and forth between the sun and the earth From a kind of equilibrium and the equilibrium by the increases nor decreases the mass of the sun in itself then the No. 1 jumps the other way let's just leave it at that equilibrium set up when I get the quantum field theory command said the picture of things jumping back and forth cut is too simple our were real discussion with the use of what was to say that capital GE was proportional collude G squared turned that I should do it were true OK now would to make a hypothesis stories guests and the glass is going to be a first-ever Rigases Marguez if you take a strict and in particular you take us going under the circumstances that GE is so small Maginot liberal GE starts Our scheduled Kazuo probability of all workers Split enjoying that's saying is truly off the strength of gravity letting GBR arbitrarily smaller euro is saying turning off gravity altogether under those circumstances in tell you want year turning off other things but turning off will G sending it 0 but a strain which has a bunch of and it's is fitting for which you don't have to worry about what happens when a stream courses itself orders right through when nothing happened so fast of the here and now you imagine starting to increase GE will GE Your Majesty starting to increase Little G gravelly starts turning on but keeping it going fix a video tribute yours ask what it was what was keeping fixed and keeping illustrate affects the size of these and now I start on Lou G. looking constant starts getting bigger gravity starts becoming put for this object and eventually does the same thing with the gravity would do it the start make gravity strong enough and the style will turn into a black hole but now interesting thing about string theory is G is a quantity which could change in strength theory you cannot match Argyle which terms it is a variable in theory would not get people interact with his team from string theory change little G this happens to a big blob of highly excited strength as increased duties was a string together eventually turns a simple black hole what happens if you take the black all Meryl and you start decreasing injury eventually Yukon the point where there's not enough
gravity the holder going to call the black altogether but eventually unimagined wearing little genius so I don't know all cities you owe them is wary left what can the black altering only 1 thing around for it to turn into an that limit strings at is industry and that's all others it has to turn back into rates well that turned on 1 going concluded into strings the 3 strings a good turn of 4 strings maybe a zillion strings that might be best guess I unless he move an overwhelming majority of configurations was a single string the basic guessed it was that kind of 80 reversible game take a highly excited strain on gravity it turns into a black hole cut off gravity it turns back in string don't slowly coal adiabatic string terms the black hole backing in the string back-and-forth again the song about moving slowly is it doesn't change be interpreted as a property of entropy in the 19th century it was discovered that entropy is an 80 abatic in various the meaning of that and aiding the Vatican there was a quantity which doesn't change when you turn when you change things slowly throws emotional maybe better career was not our in various contexts that entropy was an 80 Vatican very few things well so here's a game of them a attractive place with fire say Look let's start with a black hole but said this is not a target black hole over here it has May good capital M for big mess because molest it let's call them not to just indicated it's not just indicating that would interest end there is a coupling constant g not so that gravity is strong enough to hold it together wrinkle Blackall and energy such that it is a black or and not somebody comes along and start turning the dial G GDR decreasing Judy decreases June I watched gravity effectively turns off what happens to the black hole instantly when you start turning off gravity when he sought decreasing strength of gravity quotas a shrink because they got at the love it too badly on it to fix that starts going and at some point it turns back into strained with the guests at what point that happens will make told guesses is that not really guesses there are some things we could Wilbur mathematics confirmed treat them as intuitive guesses La Mer 1st thing it has to do with the entropy of a black or I'm not going to use this formula be assessed for younger tried approved this is the target is to prove this formula that the entropy of a black hole is M Square Park Square using the fact that the entropy of Australian it is and Times l strength as tried to use this fact the pool of this fact now these equations can't possibly UK in you can't take these equations Stop Loving me go from here here it's the wrong idea I R will see White OK so that's about all of this using this and using this idea that we can go back and forth from strings the black holes by turning on and off for the coupling constant let's start with entropy of a black hole now he is a formula and squared plug squared entropy use them ensures the entropy of a large black coal what would depend on question is what can depend on then of course it can depend on the arrests a mess not mention worse is nothing will come right down that involves only the morass that could give you can grab man's arrest squared rescue mayors 15 power most of the work what must you put here in order to get a dimension was thing you must put a plot mess times L plant Mets climbs widow the length scale associated with gravity no wouldn't black hole physics over he doesn't know yet about string theory we haven't even will end on days when talking 1st hockey beckons signed 1st discovered the fact that the entropy black holes have entered and you could ask at that time actually entropy depend on Hocking in baggage and everybody else who would have said the only thing that I could depend on is the man's kind of plot but that doesn't tell you that may ask where times or plant squared it could be Maersk you times plant to mass times of plant this doesn't say any such thing was discussed at all it says is that this is a dimensionless combination of variables that Escon depend on OK don't let can't reported fewer job diagram draw graphs often the deaths were a plot to things with plant Mississippi William a plot the massive amount Jack Morgan start here with a relatively large pretty large and I'm and the mess is not really gonna be small it's just I need a lot of space he allowed vertical space so put the Mets down here rather businesses big businesses pretty pretty big could be seen as show arrests pay whatever Mestre like for Bacal and GTE is also pretty big messes big G was big it's a black hole was no question for black coal black coal and let's give it a mess and and coupling gene or This is a target object that we want to understand we want to calculate it entropy but we I don't really get on magician who turn the knob of who stops decreasing at Assad's decreasing Ed Whitacre subject here it is only decreasing Judy what happened the masses we decrease G as we decreased GE negative gravitational energy disappears so the mess increases the best increases because what where do we think they do gravitational are binding energy and Sotomayor starts the inquiries and doesn't increase well if you don't slowly the entropy doesn't change by end of his only of functions on mass times of plant terms of website at a motorway I tell you that this is really easy and it's also very subtle it's difficult because there's a lot of pieces every single piece is no more I don't think there's anything is complicated as multiplication and addition in the same formula the worst is multiplication and division and we have used additions are not be integral derivative but they'll are pieces don't talk was I don't want is G Times Els strings so let's play that India is beginning to getting stuff for free and we are on is
Judy columns I woke the whole illustrating the sex would belong string theory would not hold the strangling scale effects and it tells us that entropy is a function Of that M & G but if it's still strain on either string fixes a choice of Unix if we hotel string fixed call the EU the basic unit of mass and the entropy is a function of the product of an injury another words this allows here must be aligned along which in Times G His 6 and times G fixed why is when I go back again we their G and so we don't change the entropy but they entropy is not going to be a function olive and columns G so must be that ingenious fixed along here engineers flexed here that it has the equal and not at this point because I can't tell you immediately with scourges can anybody tell me back but it was hyperbole and G is equal and Jane or its I Bobalong product of energy is sex and it's fixed so that at this point that it has been equal and noise he this is what we found we signed with what would be called statistical mechanics we found isentropic aligning the line in the parameter space with entropy is fixed or the idiot or the idiot battered sometimes called AT about a 118 who along with entropy is fixed at 1st Barbara cabdrivers met the 2nd part is among only go from black coltish where does this black coal some Egypt Qatar and turned into Eddie although work that we can analyze using string and not have to worry about gravity and I'll tell you what happens happens a magic it happened at the point where the Schwartz shield radius of the object gets smaller instruments curtailed his idea as you start decreasing the coupling constant contact a flight a string of this small black holes that 6 bearing its thoughts to expand the Sasser expand the expand and eventually it gets to a point where these could limp so strange are not the English watch showed radius of the black hole that the black hole has no meaning with just a ball of string to so square children radius is smaller than the string scale would mean anything it's also the point at which the yuan the which glad is not strong the hold the string down on the horizon but happens at the point where the rigors of the about as big as Schwartz shell radius won't be be enough it could remain a black coal remembers across all that happens the transition between the 2 of them happens all right over here that happens were much watch old radios MG becomes equal bail string at that point a black who was sure what should radius is smaller than the strangling scale is not a proper black hole has turned into a strict once that happens we use string theory OK let's see where this is what severe figure out where this is harmless diagram were looks like it's also a curve it's also wiser curve because it's a curve and now big GE is little Luigi school said Els spilling squared equals strength again with keeping fixed just think of it as a number of becomes a curve and G squared equals 1 of all Els string Francis size the right hand side is just a number of its whatever we choose the you for rail strike the left-hand side is and G AMD squared with Jedi it's not enough that Ng Square is 1 of a string all of a sudden it turns into a string of black coal turns into a string of curve on here His other energy squared equals 1 of stressing it looks like but ever right yacht you have a go at it this way when Hu Jia is a large this falls off like 1 or a G-string square this falls off more rapidly slimmer Monday with these 2 curves are 1 of them it is just a curve of of constant entropy for a black hole B other is transition point of view transition could will you go from right side or left sorry you go from black or restraint so for coupling constant is was Monday the depends on us but a left of this road wife was lump of stuff behaves like a string right of behaves like a black hole on all along worry about the way that's where the entropy constant OK so now let's for all of this black hole has followed it won't the linked with turns and Australia let's see if we can find out what math the romance and couple in particular with the Mets and the coupling constant offer at that point were turned bullish once it doesn't Australia we can struggle theory to computer center so I saw all those that entropy is a function of energy but we don't know what function of energy right so let's follow because here that took them to function at 2 things 1 of them what's right damn both equations forthright family equations for the took those which intersect at that point 1 of them it was a joke but it rose and lodging off just don't mind again and the gene is the target object that we're trying to understand were playing a little game which is changing it but 0 2 0 5 that's what equation for the black line the equation for the red Line erased very well was it was and for all of us squared I think is equal to 1 or sprang right let's asylum solve these simultaneously from solemn simultaneously by squaring this 1 and now the violent body that I get rid of G point at which they crossed maybe I should've labeled this Torre and called the started pouring in maybe we should call and start and star and start Angie stuff namely the values of MG act the crossover pouring here OK Sinologist provide 1 by the other half the ratio of this 1 he will be him start
right frenetic and that will equal I'm not school said Jean Todt square 3rd divided by going right models have not realize this woman who has to be our summer here you'd better be because yeah life please please let me say yes vote less drama OK 1 here Tuesday at the time of her death of but now I think we're right down OK but multiply both sides by no stretch and then we get a lot Jean square hello straying square it is equal to em scholar L string well as Jean Monnet squirt Pennzoil quartered capital Jedi at this pointing at the target point its capital GE at the point where we were trying to investigate the object so we can like this equation and not squared Connors was just Koji not whatever the constant was at this point over here Digital room will be moved here was just a truck about where we thought it was that and stop point over here crimes Telstra what solid year I'm not think pungent spot Citicorp which boast change services where's where's the equation this equation over here that's entropy which doesn't change its the entropy at this point but it's also the entropy over here some particular it's see entropy of interest sorrow a float is the connection between the 2 sets of ideas it's not the subject he thought of as a string and use them use a lot change it until it becomes a strain and off the entropy from the properties Of the string of point and that relates and not squared Jean which is why clear formula all rear someplace relates the bloc also Mueller at that point from stream formula at that point that was the theology of the connection between our understanding the entry fees of black holes and relating it the properties of stray this much more sophisticated versions of its version of it was not powerful enough to determine the numerical constant medical constant one-quarter this was much too crude for that hour why was it too crude among among other reasons as always was a bit of sloppiness in exactly where this transition happened the transition between string end of black hole is not a shot sudden transition and even if just trust transition will not have the power to predict exactly where it is all we can do is Paris bureau of so this
technique was not powerful enough to them the factor were quarter but where was powerful enough for for hours to get the same kinds of formulas for White a wide variety of different kinds of black coal objects different dimensions of space different things called Black things called black a black holes in all different dimensions charged black holes all the whole works the same technique work but not good enough to get a numerical factor the factor was provided by mathematically sophisticated version of which had a great deal of symmetry supersymmetry things when I can do I didn't it did in anything conceptually Avery knew it entered the color of symmetry and make very precise complications and the end result was a factor but was correctly identified OK that's so much slower strings and entropy but I just some it was deemed it doesn't string theory is correct it means that is mathematically consistent description of gravity and quantum mechanics which does contain black holes has within it only framework an explanation of the entropy of black holes YTD is an explanation of entropy black all's well armed entropy as opposed to climb again is an indication of the existence of microscopic substructure which equations are you a rough coarse-grained description may not see so the fact that black clothes hadn't would be woes evidence that there was some underlying small step Tuesday describing whenever a black aware of the horizon of a black hole is an To do that string theory was 1 1 and 1 other thing which is destined to is at the Apollo came out right here in my have come out Kobe
Cuba 0 plant Q. What be interpretation of out of there None this is something like this this this of course was the point of this computation that gave me in there and squared plant quarters it did not give in a plant the interpretation of plant earlier about it ability entropy was proportional to the volume of the black coal that is a actually thank you have a cup of coffee the Entebbe is proportional go about your entropy is proportional about you that off the entropy is proportional to the area and the extremely puzzling that still is puzzling this doesn't In August his relations was not for the purpose of making you feel better lose her but if a had a degree of historical of value in convincing people that the entropy of black holes really did mean but there was a young microscopic substructure that General Relativity just missed I think I probably discussed with you on the field peas in which which macroscopic theories which missed the substructure or microscopic structures that fluid dynamics fluid dynamics makes use of a set of variables which is which are averaged over Mini many many particles of fluid flow of fluid density and things like that the temperature of the fluid locally in them and the fact that the fluid has entropy was any indication that it had some kind of additional small distance structure that was too small to see the equation update I want 1 from that of a holographic principal O'Neill Williams who is very quickly there's an enormous amount that came out of the old bastard principal unexpectedly to me under the principle itself is simple visible unexpected but but the again simple those new requiring the girls and it goes like this the question now is not what is entropy of assist the question is what is the maximum entropy the system can hack the maximum notion of maximum entropy now systems maximum entropy is infinite from many mathematical systems that we know that we think about mathematical physics you could have arbitrary for big but seduced by the concentrate trade on a class of systems their classes systems which can be described our I suppose you would describe them as systems that have built that killed it secure bit busy yes no description us a bit it's a bit the sense of the bits new computer except you stand for Quantum quite debates about that this is the key that the important thing here on wicket takers and how we could subdivided into cells make a cubic lattice ever that lets simplify the discussions oppose everything in the room was made out of 1 kind of Adams with glasses a false statement on what kinds of Adams but let's simplify and say world all made up fabric of the 2 of you need only meet on a neat O'Neill Adams of it's too hard to say when the league minimum Adams and thus has made beyond me don't annoy me and but there is subdivided into a large number of cells the aged being about as big as a matter of the only question but but that we can ask about cell is Is there Avenue in it or not this is a matter of minutes we compared to black was not a matter and a white we could make up a lot of different kinds of configurations Abbas our we can make up configurations that look like a bunch of people sitting here we need to extend that so that we have a few more elements of course to do that but With this very simple marble we can have a room which is complicated enough to have us sitting here and in which the only question to ask about cell is filled or is it not for the number of configurations that you could make is clearly to a number of cells and the number of cells is the volume of violated by the volume of a ourselves in this case the value of the sale would be the smallest volume that would be appropriate to physics problem in this case the physics problems being putting Adams in not putting Adams is so we could put in the volume of an and total volume over the volume of atom that's the number of different ways of populating this rule and with batons and entropy is just water versus the entropy B is a lot of reverse of the logarithm of the natural logarithm was the usual definition of allow are missing
he is vii provided by VAT allowed remember tool now I take it back but it is not me it could be of ritual that's maximum entropy there are no more states and that now the entropy of this rule and is certainly nothing like a maximum of this rule the maximum Anteby of this room would be a configuration of the room which was completely chaotic and if we put that much entropy and would be very hot with equality figuration relatively low entropy This is the maximum entropy that could be in this room and the situation became anymore and we're like that has a maximum time things the notice bar 1 of them is proportional to the volume that may or may not be the deep fat on the thing is proportional to remember of basic decisions let's call it the number of degrees of freedom number degree of freedom not Naomi thing like a coordinator but a yes no discuss a binary question is proportional to the number of degrees of freedom and of course is wrong the number of degrees of freedom be proportional to the volume also proportional from warlord who want to ask whether visitors really very general feature of nature that the maximum entropy is proportional to the volume I can't get that the maximum entropy is always proportional to the number of degrees of freedom that's not that's too deep to fool around with the reclusive that but that's a very very deep idea that the entropy counts the number of yes-no questions after really get whether it's always the case that proportional volume and a detector In a case that includes gravity so there busy argument that was put forward 15 years ago after what a maximum entropy of region bases is original space but could to be a sphere for simplicity a candy populated with all sorts of stuff but add-ons are our photons elements may have enough cookies cookies that but to tell the difference between cookie my cookies plant but um course busses highly configurations are there was the maximum number of distinct states is is larger and the was really really simple suppose there's an entropy at a certain amount of energy altogether but scored M this is the in their southern maximum entropy attributes in the air and amassed now I think we're entitled to assume that the Mets end is she but the the mess a black hole whose horizons would be just that big if the movement the Verizon would be bigger than this and would who fled the region so asking how much entropy can be is staff which doesn't flow the boundary of this region and title was them but the Mets must be less than the mass of black coal close hope lies in which a still the region of Pfizer's I said his mess bigger overflowing region have to for the region now imagined that somebody created a shell with doesn't matter what the show was made out of it could call it from the city it could be made of light could be made times could be made over a Swiss cheese was matter and that's comes in and it happens the have precisely enough mail precisely enough that it together with the memory happens to be the mass of a black hole which just goes out region assuming the mess is less than the that of that black hole but take the red Shell have just the man necessary To create so that there some together with former Oracle and legend by on light heavy enough to fall in all we give another pushed to make a full and fair for but therefore was so bad it crosses this region Everybody does general relativity agree at the point where it crosses the region such that this is just big enough of all things a black hole that will form a black hole person from a black hole harbored black hole the black hole will be exactly as the major ordered that worry about only what
you just put enough Masonite read shell so that together with the lesson here is just enough to make a black hole that big What is the entropy of the final configuration the entropy of the final configurations just entropy of the black all them so we can write a book of course as initial tests a final it is the area the abided by forging the area of this region Taylor weakened state my by virtue of the 2nd law of thermodynamics is the final entropy is bigger than initially be so trimmed the theorem with almost no work we proved the theorem and the CO says the entropy in a region of space can not be bigger than the entropy of a black hole which would just barely filled it cannot because you always create a black hole and thereby creating an entropy of final entropy which were the area although for G we could that way with the maximum entropy S Max the same object talking about over here is equal to the area provided by forging it's another way of saying that the number of degrees of freedom of religion space that everything they can be described in a region of space is no bigger than Of the bounding surface of that region measured and plant he has called holographic principle it said does sometimes Co was very very good reasons the state Wilbert differently it stated saying that the region space can be described by a collection of degrees of freedom many regions based we describe collection of degrees of freedom which are non put plant area on the boundary of the region called a holographic principle it was a complete surprise our butts Valerie official part of standard of physics are not going to do much with it the most going to show you how it's used words or serve applications of I would just take us too far afield has had had has had very interesting applications but the what's most interesting about to learn how different it is than people imagine say it again and you'd think you if you convey something of this various something else over here you could something else over here that there because of locality that you could make any combination of variations and you would think a number of variations of Ukiah makers proportional volume the area or region measurement plant units is vastly vastly smaller This is not swell difference the rest between after that we have a number of stocks sales on the boundary number plug cells in the EU and the volume of the room a huge ratio Our and Docket were abuse cancer that is if you have a choice it is part the year last is had half a million women we're are not really do and wearable bucket but yell prom but let's say let's talk about such relationship part of while talking about our planet it I think I'll come back to it and they'll come back to its collector was something called the ultraviolet infrared connection which I can talk about what a return from the moment that's a holographic principle and that all in the same about for the moment will come back to it but I want to tell you about horizons in cosmology that's a different kinds of things of people call horizons and cosmetology them really only interested in ones that are analogous throughout the horizons of
black holes if they don't horizons of enterprises are by definition those places which separated them from where you can or cannot get a signal from the other side will you never can get a signal quite some events organizers dozens of could in certain kinds of cosmology certain kinds of cars bodies of certain kinds of solutions of Einstein's field equations applied the cosmology when that could as equations who I'm going to just what solutions limit talk about solutions of Einstein's equations but when I travel regularly like equations down with like the solutions down let's begin with mouse the basic concept of modern cosmology mainly the Met Beckham and expand the universe now various religions of expanding universes particularly characterized by the topology of space not space time but that space could be closed and mountain bike a year ago Cercocebus fear not the surface of a two-dimensional sphere service three-dimensional sphere it could be hyperbolic which is a kind of space which is said that and negatively could quite the opposite of the spirit of the eve of hours tonight with only to assumed that space it is just nucleons space not spaces just fallen nearly could space while we talking about does it cosmology while we just talking about ordinary Ukrainians face will won't be talking about space what are talking about space time and space time of the talk about what what what we are this kind analog of Ukrainians based notes called think House space right we're not going to be talking about Minkowski space will be talking about a space which is more complicated and expanding space of few think about it you know the usual picture for thinking about expanding close space is the ban one-dimensional going down mentioned view spaces two-dimensional and then think of it as a boon blowing up as blowing up with on the moon drawings in those little drawings are galaxies and a expands the Galaxy separated from each other I did once tried to show that a class by getting a bit below the big big things and blow up the blow up the before the market with this stuff and both reports that was mistake each robber plume of Bangalore on now that's not such a good idea for a further describing cosmology for 1 simple reason carbon molecules give cleanup and other properties of space for local patches space look like a getting thinner and thinner every 1 of the consequences of that is a separation between molecules eventually exceed the radius of a molecular forces and that of course is when the blue Blake's not because I paint their with that stuff but because the separation between molecules exceed these forces attractive forces so you that nothing they had a little bit you have to imagine you have a reservoir of other molecules as you blow up the blow to increase size every time a space where the distance between molecules begins to exceed audit closed the distant characteristic go were the year the molecular forces and another molecule in and of pocket keep doing it and then of course you could make the keep growing growing growing so all of cosmetology has that aspect to it that bad not that spaces just expanding but the sent being replenished the center which is being replenished is that the properties of a local regions based on change as the space get bigger and bigger space doesn't give him now whatever double that would mean muscle world our on but well not going to be doing close the space if you wanted to do the kind of space for talking about unimagined isn't infinite flat sheet of rubber while an infinite flat she might be of a little bit too extreme let's just make it a very big flat sheets because imagine people at the corners of the sheet and they're calling out causing it to uniformly expand with Prime Service the flat she
glows acetylation between the molecule was increases with time and will replenish the molecules as these galaxies which are drawn on the service sector late because of the geometry is growing and it's kind of say their distances between things as time-dependent because they space itself is a stretcher but now those there was 1 of most loads of the have mathematical meaning if they do a mathematical meaning mathematical meetings through the metric through the the magic of space biometric spends time so I would like down a metric for you my USA bt squared off he squared equals might be peaceful enough I would just writing Minkowski space I would write plus B X school square years since DX Webb was wise square was busy squared which I'm back on the right down earn our general will be minus squared plus DX square if you want the speed of light in Europe but the speed of light in the seas Square studio squared plus squared parameter premier speed of light this would be the thing you would normally put and that this space while expanding would cut multiplying the space component of the fact that cause DX EX by the stands for DX equipped with BY square Christie's square throughput what's called a scale factor AT T square but you put century of right now what's the distance between a pair points separated by acts separated by a distance Delta X a distance between not separated by a distance separated by court will admit interval Delta X the distance between 2 such point is not just for square of the distance short-circuit square of the distance between the 2 points it's not that OPEC's squatted its Delta x squared times a square of the distance between neighboring points is a squared times DX squared if that happens to be a fairly large interval will be dealt the X X but the actual distance between 2 points at call D between . 1 and . 2 0 2 . 1 and point to each were characterized by a definite location on the pouring Uzi X Y whatever you go points and this separated by a court Worldnet distance from along the x-axis dealt X how far they really improper distance that's what the man tells you matrix of fraud and the answer is the D is going to be dealt the X times an 80 of tea but for real Isidro are fixed in the plane anemia in the quarter new plane and the plume has ballooned blown up by this process approval molecules that 2 points was separated in the distance between them will be whether the death tax was times any of tea that much is fears that that means so as not going Minkowski space with the distance between 2 fixed points included space fix that some kind of expanding version of let's talk about the velocity between those 2 points velocity between those 2 points while I only the Exs 6th Cir points in the plane effects they don't change what's the velocity between the 2 points V 1 to while that's just the continuity of the distance between them right a time derivative what's kind of Of the distance between them while the exodus a being kept fixed so don't X difference of the extras are fixed not time-dependent time-dependent is an 80 case I haven't said it is called the scale factors scale factor which is time-dependent that's food differentiate and velocity is dealt X times of scored a . upstanding for time derivative now let's consider the relationship between the distance between the points and 4 velocity between their velocity between the 2 points is given by the distance between the 2 points but not quite velocity has a not the distance has a E and so there must be an 80 . all over a a . of over and has made not distances and a further rise in the same so velocity is distance times . 0 8 are a dime a dozen it's called the Hubble constant calls each velocity equals dissidents and Hubble constant Why don't call constant the robbery of a city is is a function of time and a dark can be a function of time in general the issue could be a function of time while the caller mean when we say that the Hubble constant constant we don't need the constant with respect to prawns we need it a constant that don't doesn't depend on the pair of points that he chose a pair of points and again at a given instant the price of relation between velocity in distance between them is common to all pairs of point and she between velocity and distance is given by a got over a witch doesn't depend on position but does depend on time so the Hummel constant is kind of a little bit of a misnomer it's certainly not constant with respect to Prague in general but it is independent of the pair points talking about the cause is harbors no others were a lot of expansion velocity it's amazing that was discovered earlier by fearless particularly as just thinking about the being discovered by moving through telescopes Her by was discovered by somebody else somebody make correct me and me warned of being corrected that it was discovered by somebody else but OK had such Hubble's last he got senior the best behind his differentiated 1st since we JRC cities yes see well like it was all right that In particular if you're looking at very very distant galaxies distant galaxies when you look back at me looking back at time which is much earlier than your local time and the question is then should you use the Hubble constant then or now well the answer is a combination of both that's but I this equation is at the instant of prime it does not correspond to what you see when you look back at work but it would talk about Webster space time diagram his time his
space now it was at the point and then maintained the same quarter positions nevertheless spaces expanding between fixed isn't the time this separating with a velocity which is related to the dissidents by the Hubble constant at that instant of time but that's not what they see when they look at each other what they see when they look at each other as they look back and they see 1 another and earlier part so this fellow over here sees this 1 moving away from him not the velocity that it has the same incident of time velocity that he had earlier and so you have to use a more complicated formula if you want to answer the question what somebody sees but this is what it is true at a distance of foreign elements of the plan that affected show was mayor of hers may not be as interesting as what you could actually be attacked by looking back but the rush is quick OK now 10 the Hubble constant beat constant is something that is capable of really just be a number and not changing with time what was quoted as all before we do what sort of reels but some if this is true a given instant of Prime indeed given instant of time that the velocity is the distance the Hubble constant and spaces Is it delivered the elections us and is a distant scale which is large enough that things are moving away from you with the speed of light and factors a distance which is large enough of them moving away with from you with 10 times the speed of light it is was not a mistake this is a true statement that because biology in an expanding universe like this could to the points can be be moving away from each other with greater than the speed of light the kid has each other with greater than the speed of light some those not you can't could buy you as the speed of light but 2 things sufficient would fall apart can and will be the year of the relationship between the Hubble constant than distance scale in order that you are at the speed of light you would just get pregnancy here see equals distance times age says you see meaning the speed of light the some arbitrary constant let's at the speed of light as we usually do and they say at a distance equal all over the Hubble constant that that instant of time things and not the moving away from you at a speed of light million most instances not because most allergies Ivins scores Montes that we use today means most Cosmo that was that the hairs on the Hubble constant decreased with time the Hubble constant decreased with time as time went on the Hubble constant of smaller and smaller which in the distance outerwear things are moving with the speed of light relative to you get a larger and larger with time you might think if his of our galaxy out there which is only from you with the speed of light on the that you could never see but the Hubble constant is decreasing you wait a little while and you find out that the distance to which you concede is a lodger Hoover was distant store that things are moving with the speed of light is larger and so announced because not allergies only because my orgies it was a case that there that there was no bound them when you could eventually see Bible gets smaller and smaller and smaller some distant outerwear things are moving with the speed of light we get larger and larger and you eventually see their mother because Macias uppercase modern Cosmo the Hubble constant or the Hubble constant thought open the Hubble constant meaning opened in the future appears as if it's becoming comes truly constant from some with foreign our Epsom observational factor I can tell you how we're not getting the observational cosmology here but there is good evidence that we live very strong evidence that we live in a universe where the Hubble constant is pending to a constant true-crime so let's investigate the consequences of the universe In which the Hubble is can't and they what's like if that is the case the bubble constant is a constant vandalism universal distance out which will you look at that distant things will be moving relative to you with the speed of light and that distance won't change with time but so let's investigate that possibility what does it said was a mathematically just spend a couple of minutes ever to might continue next time was says the Hubble constant is 8 . 0 8 already achieved it was age and ask constant now now age is just and constant has some units were due what time right right so only when you specify the Hubble constant Europe specifying applying scale innocuous the relevant time scale is about wild about 15 billion years so other kinds of the very small number but nevertheless appears to be constantly figure the become spent as says tree writers equation as DAY beauty physical the age relative signed that Is that the rate of change of something is proportional proving itself was a solution of equation like that exponential exponential 80 equals a at some fixed tying the matter what kind we don't matter important crimes eagerly H the scale factor increases exponentially with distance between pairs appoints a good increases exponentially Coleman and inflating universe that's called space for this property space with this property with a being before he put 2 H that put to death and see where it is square hazy BHP a square to late in this space has it called the de Sitter space base through beyond our mathematical astronomer physicist 1st discovered it the sooner what will and Willem de Sitter Dutchman but 0 4 4 4 or Swope smoky space sitter exponentially experimental armed I think we are I think we're stop now what the next time it is that the Soto space spaces with this property have event horizon event dozens which are in many respects similar a black hole horizons Our but they're almost like inside out horizon when you think about a black hole outside the black coal and looking at and you see a region where things on the inside are moving away from you faster than the speed of light roughly that's why the kick out our allies of de Sitter space is quite the opposite exactly the opposite of religion out there was large enough distance with things are moving away from you with the speed of light you kind of on the inside of it and things are moving away from you with the speed of light for whatever them and you
will never catch up very passing through that relies and are there were things are moving with the speed of light that as the those system with the passing the horizon of a black-owned the only differences as far as we know this space does not evaporate them with a black hole of her so talk about the city's face the problem at its horizons and will talk about why it that creates a lot of confusion and interest in modern pathology for more please
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Titel Topics in String Theory | Lecture 7
Serientitel Lecture Collection | Topics in String Theory
Teil 7
Anzahl der Teile 9
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/15119
Herausgeber Stanford University
Erscheinungsjahr 2011
Sprache Englisch

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Fachgebiet Physik
Abstract (February 28, 2011) Leonard Susskind gives a lecture on string theory and particle physics that continues the theory behind calculating the entropy of a black hole. 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.

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