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Topics in String Theory  Lecture 6
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Erkannte Entitäten
Sprachtranskript
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E. Steiger University where I had prepared
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for tonight was a lesson are Harold string theory gave it a resolution questions of what is it that carries the entropy of a black hole from a throw it because later ready but what started the same this system has a name b a large entropy in particular is another way of saying that there's a large number of microscopic degrees of freedom which are too small and too numerous to you to keep track of who was the backed up for hot water has a entropy of Cantor 30th or whatever happens to be there implicitly making a statement that the microscopic structure there which you can't see Olivia can't see it but er bird you choose not to see it you will too many particles show numerous and there that they are carrying 10th of of 30 of bits of information which for practical purposes Is he admitted so it becomes a question that occurred an alright you deal the thermodynamics you study water you'll he did a little bit to to measure its energy you measure it temperature use the laws of thermodynamics and you discover that it has a certain entropy insult it then becomes a question what Is it what are those microscopic degrees of freedom that carrying a year earlier given that hasn't entropy you know they're there but that doesn't tell you what those degrees of freedom are the value of objects are find the same is drove a black all the same is true general reluctivity in particular but in particular a black hole where Becker starring put forward the idea that black holes have interpreted the natural question was What are these tiny microscopic things which apparently in some way on the surface of a black hole because said the entropy was proportional to the area but there was nothing in the general theory of relativity which gave any clue as to what they might be 5 the reason is not so different than the statement of use study fluid dynamics fluid dynamics makes no real reference to the microscopic nature of fluids or it Dougherty course green description of the macroscopic flow of fluids Bossidy every point and the fluid density of the fluid a few things like right and makes no reference to a microscopic theory but when you change the temperature of the fluid you change the energy of the fluid you discover the entropy and tells user microscopic structure just like fluid dynamics doesn't tell you what that the fluid dynamics together with some thermodynamics does not tell you what that fundamental microscopic structure if it kills you with their our what and you all invites you cool start to consider the question of what Is there beyond the simple fluid dynamics questioned you could have the same way the entropy of black holes invited us to stop asking Is there a microscopic structure the black hole of the gravitational fields or to space or whatever that were reduced explained entropy now string theory did provide an explanation Of the interplay of black holes but may not be the only explanation string theory might not be the right here we Blackall might be the right period nature we don't know but in the context of string theory it provided a picture and explanation of what the microscopic objects are that carry the the entropy of the black or so I thought I would take you to the very very simplest way of us and it's not so simple but the simplest example but can't think the entropy of a black hole using string theory what's early on at the Noto much about string theory very little in fact the argument is basically qualitative unfortunately it's a little bit too crude to see absolute exactness but got enough to get a forget the order of magnitude and right field will go through From the 1st thing you need to know is some facts about the constants of string theory and gravity 1st thing is both in gravitation and in string theory a unit of length EU underling that's important for the structure of strings is different unit of Lane that's important her structure gravity me gravitational length scale is called the plant Corey but Cordell P some combinations of may remember it are contains GE each Barron's see a workers square G H bar oversee Q possess ring a bell but right that's punk Glinka tantalum minus 33 centimeters very very small whether work with units in which edged boggle equals 1 so when those units Jedi square root of G is just a plant point no constant and units like this the square root of it is plant quite or right away but plant squared is Newton's constant no no constant has another meaning of course knows constant is the constant the gravitation those are Forest Law between between massive objects will come back so maybe point now is known as constant as units of area that's why we say that we that the entropy of a black hole is its area measured in plant so it's it's area measured in units of G OK that's wants to know about string theory string theory also has a unit of length it's not a plant point a pay what goes on if you are to take a typical string in string theory heated up you would find out that it would lose all replace but you find out if you look at the microscope that the size of the wiggle R's it would be of a particular characteristics size scale but you find out that Brown director wiggle Zoran never on smallest smaller scale a certain scale and that's scales called
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the strangling scaled it's not terribly important exactly how Woods defined it is the roughly speaking it's size and oscillating strain that has won him of oscillation certainly scale goes industry fearing that has to do we would leaders of strings and late scale along which it whatever it is it's called El so summon called El strained a recalled string squared the thermometers L. strained and finally there's another constant in string theory and it's the strain coupling constant minimum order string coupling constant is word or offers 4th World War media flood the probability that restraint crosses itself it breaks so it's a probability but if you have a strain and it pinches off like this it's the probability that a breaks as called GET for 2 amplitude for string break it plays the same role in string theory as electric charge doesn't electrodynamics of electrodynamics the electric charge is amplitude for charged particle to ease made a fulltime the string coupling constant is the amplitude for an oscillating strings to we made a small straight the small splinters of course gravity Song is seeking amplitude or oscillating string lids doing perimeter grab finally let's talk about gravitational forces Adelaide think 1st electromagnetic forces how we think about electromagnetic forces withdraw Fineman diagrams and photons exchanged now lists assembly taken with a little bit of rain of salt that's not exactly really of being shot From when charged particle toe another but the diagram of a has some real meaning there is In mission similar photon over here the amplitude for that is E electric charge and an absorption he adds another factor electric charge 2 factors of electric charge the meaning of In calmer language it is that the between 2 charged particles caused by the folk art is proportional to the product of the 2 charges 1 for ignition 1 for absorption Well what about gravity of gravity is controlled by string theory then we analogous then would be to objects both of which are made out of string everything is made of string in string theory 1 a middle ground time in those areas where around Mr. Gravitt the amplitude for that G Robert kind was this idea and reconnect with that industry theory that is the stores Of the gravitational interaction between tow object it's analogous to fine and diagram of the electrodynamics and they've point Is that a force between 2 massive objects is proportional to the school area on the string coupling constant 1 for absorption 1 free mission it also contains incidentally the product of the masses of the object and the distance between them square but that's not song what the important thing is that is G squared appears In the force well there's another object that some forms appears for so long as the G squared as things the usual maternity in forests you put big Jeannette Big G square Luigi Sala it sounds very much like little square must be big G but that's not quite true because they have different dimensions mangy has units of linked squared l'il G that's just a probability it's a probability that 2 strings will seperate probabilities on dimensionless no dimensions at all so it must be that little squared must be collected the big G by something which carries dimensions what's what's the units a big G again Wang square what's the problem Rudy Rudy what goes here is the stringing Lane squared it's dimensionless dimensionally consistent announced dimensionally consistent area on 1 side area on the other but it's also true that the plant point squared Is a Clark square so just arrived the relationship between the plant the string and the coupling constant removing the square root coupling constant times string Lane is equal to apply quite that typically we imagined that GE is a small number the probability for strings breaking up is a small number if it's a big number we don't know how it started doing string theory weak typically think of as a small number so the plant calling this typically of small fraction of the string but that we keep their money this is 1 of our basic equations L talks squared is the newly constant and plant is equal to the string coupling constant and strangling scale visit to important equations right man if we don't get to his burial fully may get through it and I would make life that's 1 set of facts or track choose to facts a G a G is just 1 of them is usually less young so that means the the string scale is typically bigger than the plant scale In general the plant scale the smallest gal you won the thinking about things get alchemy bigger showcase next entropy stowed Enderby a black holes was a symbol for Inter peak that's right into priority areas brief what's therefore jurors in
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beggars by talking try day s black hole is equal to now when I killed worry very much about the numerical constants 1st of all it's equal to the area of the horizon divided by GE makes sense is dimensionally consistent serve earlier as units of the area and jeans units of area is a foreign member worry about before I we could write it is to remember that the area is proportional to the square of the Schwartz the radius provided by GE and this what silver radius Mr. Wednesday so this becomes proportional that and squared G squared Ng it the radius and Square D squared the area and now we can factor our G so the entropy of a black hole is a man's squared times Newton's constant but it's also man's squared finds a plant squared followed is the same saying man's squared times plant links squared comes from here so as the black oughta peak now let's think about a string amended that we have a stranger who rule 1 time but now we hit it hard we had a heart and we do a lot of energy we heated up I'll heated up my bombarding it with lot of particles which is putting a frying pan boiling it whatever we do with it we come on a lot of energy into 1 happens with pump a lot of energy that thoughts of vibrators thought the hostility and typically From West we do it especially careful wait it will form a big triangle of string Mary all over place 3 heated up that's about will get as I have been wallboard the big blow with your eyes you go fishing you remember what you real looks like after road the panel met on big tangle messes have entropy them entropy just because there are many many triangles which are hard to tell apart what dropping enter PRB a thing is the number of configurations which are hard to tell apart which to want to tell apart because the pieces are too small and numerous solely strained vibrating string like this also has interpreted let's try to get so let's try to guests what are they entropy vibrating strings to do that make a model very very simple marble the strain it's an always simplification it gives a right answer it said certainly an oversimplification but qualitatively it's the right picture and his idea instead of thinking of string theory and then ordinary space let's I'm Maginot space is replaced by a lattice cubic lattice that no other a string it is a collection our links lines of the lattice just terminology lines of the lattice accord links the nodes are called society site links the square was build a fancy name for the squares of artists are the plaque kept the kept but we're not that when I can do you do like gets was the links the lattice stray would be a connected set of links words are lying curve through space is replaced by a sequence of the late King go back over themselves no rule that says only can't go back over itself but we follow it let's say we followed from beginning to end I'm not in the a big John stress a random walking string how many configurations of are there I start at 1 and we'll ignore the question of whether to close open string it doesn't matter his purposes To give the same answer pretty much support only a start at 1 AM How many possibilities are there I joined twodimensional world indoor threedimensional world we would have some lattice coming out a black political fourdimensional world but the basic picture would be the argument for the twodimensional world simplicity how many ways are most starting out but born do OK we get through this already here could grant any former what's the next a number of ways to continue what were allowed to were allowed to cut back on ourselves plans for suppository altogether a restrained as the little in links incidentally is marble I wanted to emerge in the signs of a link is a string Elaine L. string sighs were doing strength wire so what else could possibly be doing string theory each 1 of these size L straight so positive show tropical lady of the strain was an units states are rare what Enright right for tapis to the waited a surprise for me but suppose it was a threedimensional lattice what would be 6 to the end a dozen whatever it is it's some ordinary number to me and power suppose was a lattice which wasn't a cubic lattice but was a hexagonal closepacked lattice or some other ridiculous like the number would still be some number be in power now was a definition of the entropy that's how many random states but in fact it's an interesting fact of course some of them are very special the 1 state which is just a straight line that's a pretty rare possibility the chances that you would get a straight line pretty remote 1 metaphor in March if you look around this computer and just randomly and get a random number generator and just start generating string configurations and is relatively large with say a hundred built all look about as saying
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that although formal ball random walking stuff all look about the same hour the chances that you get a you under typical looking string of very very small 100 links boy every day they really look very similar I saw that means that the number of states which are all similar to each of hard to tell apart is for it to be and what's the entropy of that In love for right the entropy is by definition the logarithm of the number of let's Colon macroscopically indistinguishable states so knowing all the entropy of a random string out over here s straightened it is equal to the number of links times logarithm of forest when hearing number let's not the 1st I worry about logarithm of 4 is proportional to the the number of links on strain noted proposal to link the strain but end is dimensionless is just an integer Lane is dimensional as units wherever units are so how would make this well what to wear a phone or another message that this is correct this is out of this is correct this is correct but is another way that we can write the link of the strength and going further strengthening following around is aIl L is the weight of the strain How many newly links does it have lawyer and I was questions were there hello over strain right as the tell the length of the string provided up into his unit a streamlined that in so we can also write that these entropy of a string is its lady divided by the Times provided by L. strength let me rewrite this formula over here it's a over G but G Is L. plunk school With this something both similar and totally different about these formulas area is replaced by Lane L. plant squared is replaced by Hill string they are different they're not the same thing contained different powers this is Olivia knowing this is quadratic in the size of the of main contains L. string downstairs swim contains plant found that don't some some similarity but they are different now let's rewrite of way instead theory each 1 of these little weeks has amassed romance on 1 leg works but consider it called will Imre amassed a mass of 1 William L Katie was at equal to locate in alloy Unix he knew where our units units arrived here see equals age bar equals 1 the what's the real what's the relationship between units of Masson units Elaine in reps masses in and restoring Southern mess on 1 of these links in string theory since we doing string theory was only 1 thing it could be 1 all over L strength that's the massive later OK what about the mass of the whole strain at stake the massive string now we arrange this what's a mess of the whole long string home all over it is all wail but worthy and I think I lost am I was L Olvera strain in itself was string so romance string no link of the string divided by the square of that's the length of the string divided by a string lengths glared that makes sense this is a constant of nature are reviewed the sizzling following the strength following a string around the curve of us I'm sure mallets eliminate failed L is a Quebec and string found L string square and what plant the formula for entropy Roy debt NO it is the Mets a restrained fund or string right the radio arrived or the rest of the elements that called the mess of the string of massively of the whole thing whatever it is constrained theory strings can be anything you are a stranger in a strange that star is a string or collection of strings so I did we see something else that sort of Samoa the entropy of the strains of into it entropy of the black hole it is the square the mass times of plant went where the entropy strain is a mess of the strain and the strangling is it can a loose cannon here as Sue patterns on apparently 1 because after all would really like to say is big ball of tangled straight is a black hole while not turning Yuri picked a strategic alliance together with terrific terrific for you make a big jumble of strained like could possibly be there must be a black hole but is not working as a working and something's wrong working but don't work almost working it's working some pattern is right but stick break I need a break now I'll get it tonight a musher bird were bird armed U.S. too many questions and intend leave your suspends there right here are the basic formulas you need follow what I'd may not her 1st tonight Our this simple formulas rider man will get an airing it we don't finish tonight remember the only constant
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that's nothing but they plunk area constant Park area the relation between the plant going and the string going through the string coupling constant goes this way of
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way black all entropy is area divided by Plunkett area and that's also mass squared cars plant blamed square free of the G here we like and repeal strained is the length of the string instead of the area abided by want power of strain it's also happens to be 1 power of mass not plans the plunk linked but by the string about Bozo the basic working equation Mr. till you now it is strength theory is something you could change you can imagine are the coupling constant g is something that somebody could have I'm not that Cheney let's not SK build such an arm built into string theory that Our things like coupling constants of things that could vary so let's a magic that they could vary from imagined were pecan white hair barriers and that's thought let's start game with the coupling constant g being very small very small negligible maybe even 0 0 0 strings blue all across each other strings don't Split and jump across of strings there is no gravity gravity is infinitely week their followers string coupling constant 0 2 start point and knowledge With a big tangled string so much energy has been put into it that it's massively big as big as a star of bigger they can be a black or why can be a black hole because the relationships between the entropy of a string entropy or by call amount right although it is a strange attribute proportional Woodslane army now let's imagine increasing G G. what starts happen what stance that happened it is to where we but what they're stuck at as weak increases let's say well our let's hold Els string fixed the convenient convenient to work in units in which L. strings health effects so GDX bigger tho plant get bigger but that's another way of saying that the gravitational coupling constant but that's not surprising if we increase G it means little strings to break off Wigan break off a good jump across gravitation becomes port so I asked with starts to increase gravity starts to become port platters this thing is gravity starts to become important grab stuff to put together oppose it together shrinks the size of it what happens to to be energy evidence apparently it was an increase of decrease it decreases decreases that's because gravitational attraction and the potential energy is negative and when it gets pulled together lowers the energy when object it pulled together by gravity typically its energy decreases for of mass decreases but becomes but at some point just like any other object you squeeze it more and more at some point it structure will passage Schwartz a radius of becomes smaller misfortune radius of what happened it becomes a black hole so that's imaginative is very slowly we slowly slowly slowly change G begins shrinking shrink and some restraint can mean to say that over an hour of the whole ball of playing begins the contract and at some point it turns into a black hole I'll tell you all about Trevor tell you when it turns into a black hole or having his idea but what the crossover is between string of black hole pop Joel rescinded his G equals 0 0 0 gravity gravity has not turned on the 4 4 now we saw increasing G the strength of the shrink and eventually when GE is large enough becomes a black or iron ore this is some value of G the value of Jean depends on the mass of the object more massive it is the smaller GE has to be a worker make a black hole that's pretty clear the more massive is more easily become Michael Saleh at some point depending on what we started with it will become a black hole flat it is G 0 0 1 half of that state's between here and warmer because convenient is useful to think of as being between 0 and 1 on the map the story is not pretty when G gets bigger than 1 we don't go so yeah our fans that would continue think of G a small 50 is small it may take a very big object Europe could turn into a black hole there will be sold to make a rather large turned the radical but whatever G is whether G is basically was an object which is big enough that when you buy time to turn gee up command value it will turn into a black hole OK so that's what happens if you slowly is another word for slowly the idiomatic idiomatic as a physicist pants seat turn changing control parameter very slowly turning a bioterrorist Saunders 80 abatic changes G this turns into a black hole body has happens if you event take the black In turn G back in you turn it to be so small that none of gravity the holder black hole together was most Wells could begin what was good and it is
42:04
more string of gravity sorrow also rose this right now is actually a technical point here it's a tactical point about a relative entropy or a single strained and many strings of the same total amount romance you might think well when you will back again perhaps it doesn't turn into a single string perhaps it turns into a disconnected pieces of string 400 thousand disconnected pieces of string know there was a technical point about counting the number of states of strings and the answer is there are many many more states of a single string bean aura of multiple strings that the counterintuitive demand from France to string Law of the Sea met Thursday another man's divided up depends Springs many configurations are there relative to the whole mess made 1 big string out of it they may need more configurations 1 string surprising but it's a fact so the most likely thing you get when your back is a mother Sagle string of Passaic unimagined somebody with this Dial turning a oneway street becomes black hole turning it back again black hole becomes strain so for now what happens to the stringy staff but when you turn little black or a string disappeared as it get busy a get sucked into the black hole watching this missing thing from the outside but last but before watching from the outside nothing of across the horizon so the answer is not inside the black hole its on the horizon or near the horizon of the black hole here it is Of course it's only of only a good macroscopic size black hole when the black holes much bigger than the string length scale and so when the black hole was really much bigger than the string scale the string string it is just a little bit of fuzziness on the horizon and that's it you would see from the outside as you turned up gravitation the object would collapse would you would see the strings collect on the horizon of the black hole what did let's think about the black or woes of different societies relative to the string length scale the strangling scale is basically size Of these loops of string here let's
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imagine a smaller black hole it is a smaller by call and here's elope so you think it makes since the think about a black hole which is itself a much smaller than they late so the loops of strings which make up our which make up a black hole but there's no way you can answer that earned them a little bit more about string theory we talked about but the answer is not the point at which the Schwartz shield radio gets as small as loops of string that's the transition point so if you magic and if you imagine the varying the constants so that the relative size of V Schwartz chilled radius and the strangling scale vary somehow you do then at the point where the string length scale is comparable to the Schwartz field radios that's the point at which shall be black hole becomes a string string becomes a black holes on way to say it it's also the point at which the black hole does not have a to create a gravitational field goal hold the string down Montuori the fluctuations of the string will simply be a cause it to float up off the horizon so long 1 more element in isn't going back and forth between strings and black calls the point of point of separation between strains black holes was called the transition point the transition pouring transition is Wernisch what children ATS His reddish watch over radius far short show is equal approximately equal the string weighing scale the wristwatch still like all radius is smaller than the loops spring it doesn't mean anything anymore the it's a black hole it just becomes a strict this Jameson the transition pouring with 3 riders August watch over radius is white and G mood toes don't care where they will occur toes tho Oracle 1 physical tell Plant square actually leaking go 1 step further can't wait we can relate L plant
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tail string To which way show we do this but get rid of old plant in this formula here Our so this would be rewrite this and found plonk square G squared Bell string squared His L string let's go back over what I did was go do it again for you of the steps again Troy children equals Australia's Schwartz silver radius is Ng equals strengthen J Mass pond hello plant squared equals L. strained and now we use that plant is Jeep plant cells verses and plant G squared L. string squared equals 0 strain do yacht green cancels something here caring it will award but what Brown in a riot in writer WarnerLambert is let's see what that means for a given maps as you start to decrease the coupling constant g this quantity you eventually go down to about 1 and then that's the point at which the transition between strings and black holes takes place so we need to know this equation here it is out of bed or if they see LSC was up bargaining broke Berger Pirie a rim actor a source where about that for a fish like a now and only down or will the equations will you were not an FileShare tonight and I want to go and study those equations remind yourself and then put together we put together and out of them we're going to see a derivation of the entropy black hole Obama coming from the entropy of strings were basically going to derive this formula From this string theory the rides by OK is 1 more element 1 more moving parts of China the layout all and Sequoia it in a sentence a simple argument we don't even an integral neither a derivative of along an integral over the other hand some logical pieces daily intricate so I'll try the go slowly and spell them out 1 more so fight that would you change a system 80 abatic now control parameter coupling constant magnetic field electric field and then have some Sister Mary could be armed bomb box over a container of fluid gas and all is a huge magnetic field and use slowly slowly slowly vary the magnetic field what happened to interpret it's a constant it doesn't change paramagnetic field up all sorts of things happening energy changes volume might change biggest against Mike press against them all kinds of stuff happened is the 1 thing that doesn't happen His entered because changed it is partly a consequence something which have emphasized Over and over a new structures the conservation of information entropy is hidden information when you change when you change the environment not be environment in terms atmosphere and things like that but when you take control parameter and use slowly change is the amount of patent information doesn't change at the same saying interpret doesn't change so in a AT batik process but we're talking about here we use slowly change the coupling constant the entropy doesn't change us 1 more thing write it down others writer idiomatic Cheney Mr. are anything but a particular Jerry does not change a a tow if lot is contingent on abroad to a from version phase transitions are characteristic of infinite systems place transitions to really strict phase transitions only happen when the system hasn't volume systems of finite volume never have strictly speaking a phase transition if you go slow enough euros alone for the change the because changed part of the city a lost for us all said parts of bargain Nevada picture volume you put of spring in the air still volume and now you think about the possibility of 1 long strained the several small now work it would seem so it doesn't rank happens not to be true explicit calculation on which there is a reason that our perpetuity but will join reminding that there is a reason that factories simple a surprising but simple French OK so what's a what's what's a strategy Ghanem be the strategy is going to be start with a black hole start with a black hole with a given mass start wearing black hole where they give a mass at coupling constant let's go In order and Gene this is not target object this is what we're interested in a black hole of men's and not when the coupling constant is Jean Marc and now we're start slowly changing Jean genotype as we still only to Jean the character of this object Mansour tall structures gonna change and it's going to us more and eventually went G is small enough namely that this transition point the gun at turn into a string once transit but your stance is a black or start is a black hole and now we start decreasing G until it turns into a string once it turns into a strange we use of string theory could tell what the entropy years but the Inter because the changed during that process we call Anteby just doesn't change during 80 back price once it's turned into a string we continue string theory docility entropy is that we figure out what the original black hole entropy words as simple as that said they were always shell is when delay this formula for the strained becomes this formula for a black hole so that's a remarkable fact were there Mr. check may need many many different kinds of black holes rotating by called charged by calls hired them into black holes it's not an accident are removed the bird accidental agreement something which has been checked for a wide wide variety of different kinds of black holes rotating that rotating charge from charge magnetically charged higher dimensional you name it so in that sense there is a good candidate for the understanding of the into pure black hole with you
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light some rough and ready it's just me stringy stuff that's track near the horizon Wendy's her when a black hole for a 4th roof it creates a flawless string of a given only given Machamer masses proportional would very given the Our strain we've turned off gravity returned in the process of
58:48
turning off gravity became a string want to become a string a but straighten there were fill a certain volume of pulp mess M V the entered it greatest the jobless that define the made a to a finite probably made the being strings and find the answer would not change very much to straight once trained and the string you get a similar and it doesn't break up until last broke up and allows him to get a very different the statistical camping and is fancy can the statistical mechanics of camping configurations and saying when you go back and watch the black hole which is in a sense from maximum entropy that you could squeeze in a region of space morphs into a maximum entropy that you contained a strain both very interesting launcher we can carry it out for the simplest case beyond the simplest case OK will finish sort for
1:00:18
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Titel  Topics in String Theory  Lecture 6 
Serientitel  Lecture Collection  Topics in String Theory 
Teil  6 
Anzahl der Teile  9 
Autor 
Susskind, Leonard

Lizenz 
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DOI  10.5446/15120 
Herausgeber  Stanford University 
Erscheinungsjahr  2011 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Physik 
Abstract  (February 14, 2011) Leonard Susskind gives a lecture on string theory and particle physics that focuses on how string theory gives a resolution to the question regarding the entropy in 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. 