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Topics in String Theory | Lecture 2

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Stecher University we'll move
on and talk about them because myology and that string
theory has to offer our cosmology Esparza could spend 2 minutes just telling you what I think the accomplishments of the failures of string theory bayonet Arthur I will do that another time it's getting late and we want to get on with some substance but let's say did that through the special theory of relativity from moments and just remember what's all about it's about thinking of space and time as a single entity that we call space-time the space-time as geometry geometry described by that if heavily we aren't and describes biometric metric cancer special relativity special relativity the metric is especially sample the metric it is just that we take 2 points in space and time from neighboring points could you see them now but as a matter of point in space and time a needlepoint space there are separated the call this the X-axis Corbis he access and tomorrow but where these tool arbitrary but neighboring points and has separated by a bomb D X DTE E wiII be separated here and
we defined a new concept the new concept is the relativistic distance between them the proper time and the proper time defined To be deeply squared minus 1 over the speed of light squared square plus Y squared was busy square different than you might have written down if 1st ordinary you could be in four-dimensional space before you clearly in four-dimensional space you probably would oppose a plus sign for a plus sign that had left out the 1 the squared or lower overseas squared is just a arbitrary factor you could always get rid of it and we scaling the space dimensions of the condiments the proper another words by choosing units for space and time in which the velocity of light is 1 of hours do that but slated for a moment this is the a proper time between 2 points along the trajectory Simitis concept as long as deep sea is bigger than the X square lies with squared you the proper time for school I have the proper time particular ought to be a positive thing but not as that if the yet square Post e squid was busy square is bigger than he squared something bad happens this expression namely deep Square was negative but never mind nevertheless this it is the definition of proper time between any 2 points on the the most important thing for us tonight is how light breeze moved I I raise more let's suppose a light gray passes those 2 points and the role the
real a lightwave but forget the XBY square and busy square moment was just take the X squared here so rating is moving along the x-axis who infinitesimal separations along like already have only a DX and D see the rule about light rays is that a along the trajectory of white the proper time 0 somebody travel in the library if you could which you can't would discover that a o'clock standstill which is another way of saying that a proper time along the trajectory of a light waves 0 if you want to know what kind of trajectory a light gray consists of of you simply set after the right-hand side of the sequel to get our this is equal to 0 FOR a the trajectory of alike say the same thing is that BP square miles at sea Square D squared is equal to the X squared just multiplied by 2 squares and a closed 1 side of right off taking the square root CDT is equal to DX what was still a possibility for our CGT is acts as other saying the light waves moves with the speed of light that DX by that's the velocity of light ready is equal to see all RDX by detainees minors 1 of the light raised with a plus sign was right the other light ready with a minus sign here most but the main point here is just that the way you diagnose a geometry geometry like this to find out how likely is moved is you solve the equation that says um the proper time along more like trajectories 0 I now come to more complicated by geometries of general relativity general activity geometric is more complicated and varies from place to place the geometry varies from place to place and that's described by a more complicated object netted Kansas which itself can vary from place to place only done these things before so I assuming a little bit about it you have a metric tensor GIC article Jeannine among patient would know of variables which run all over the floor dimensions of space time usually we take them go from 0 to 3 0 0 being kind an X 1 X 2 X 3 being the ordinary component of of space squares genial times the X new acts no Exmoor next new are the court before coordinative time and as always In an expression like this it is automatic that we some Over the repeated index is repeated in the city's so that stands for things like Jean ought not be glared text last Jedi not 1 D C D X blah blah blah blah blah all possible combinations with metric again if we want to know how light we simply words of want defined the little infinitesimal gaps that correspond to a light from 1 to the other we set the sequel 0 that's the rule sabbatical to 0 and that gives us constraints on army of allowable trajectories of light that the content incidentally forcing the light rays moves with the speed of light weather the speed of light is 1 the Aussie or to use some of the unit uh something else D. that doesn't matter be invariant statement is light rays move on trajectories up 0 0 this to be halved trajectories which correspond Kazuo public policy but that will only do with his role in to examine a particular matter it is not of a black hole the assigned equations which is to write down the answer and little inspected for a while and learn about it tonight and we're only spend a few minutes with metric showing you basically 1 property about the way life is moving sick but whether get right to the point tonight about the quantum mechanics of the quantum
mechanics of black holes very strange in some ways it very strange in some ways it extremely or very effective very strange how or after giving a bizarre black holes are as classical and quantum mechanical entities OK so that's right now but just explicitly right down the metric that's like the emblematic of flats based on the set from here on in I'm going to set the speed of light equal to 1 In that case what I would like Barry for metrical flats place would be the key areas might miss the Expo amp plus BY square plus might get square mile as to why square feet square minus you wise when miners disease or this way but know this he there's nothing but ordinary method of ordinary flat Euclidean space here would come out of the way we come polar coordinates of sing about Pollock wardens for moment for local admits are these corn admits that I would use I'm standing in the center of which I remote resource and the animal was instead of using coal admits X Y and Z I'm going to use coordinates which are distance from me off the court on this for me and I was playing the represent a clear direction of might be examined for hard and 2 angles defected toward angles all our often some used by the capital May got that's actually stands for a unit the unit sphere wiII units fear figure 2nd figure myself surrounded by a unit sphere point on the two-year turns of director of the similar Omega often stands for a unit sphere and this will be Omega school Leonard stands for them that the community fear they lose write them it just be really car concrete to represent the units need to angles but I call them as a new phone call a polar angles a longitudinal added to angles they didn't fight and the matter of the unit sphere is the light element Don units beer it is just later square is the polar angle past trying squared but data we fight that's said that gun units fired her by the angle around the Paul and his distance along an angular distance from work From North Pole all babies are not is actually Our distance from the equator and this the food usually called just just abbreviated is just an abbreviation it's called or make a squared and meters the mythic public pool the individual ordinary two-dimensional sphere of unit later that the CIA had some other a radius of the sphere at some other radius then that that that their covered with simply Have square of the radius most plying the whole thing will be joining you have got similar let's but put what happens if the radius of the sphere is not known if the radius of the sphere is not 1 that we were right squared Ramos the radius of the sphere times geometry thing square assigned square data if I squared off just ask squared deal maker square OK square deal makers glitches the mecca Of the sphere of radius off last in the center of local army I think this separated before November separated by an ad the exit BY Missouri but now I think members separated by a D or a day later and flight he or different different distance from a different angle so what what's the distance between those 2 neighboring points the distance between those names points in ordering lot conventional space was the thing which really replaces this thing this piece of it and polar coordinates it br square plaster that's the difference in radial distance along over a longer at the rate of beans and then contribution coming from the a distance Our aren't angular spaces and that's just this past our squared the Omega square he but
France show as a child tests must you're not here is not to be dropped if they're not but Oh no Olympics or Suarez shield bonds you'll Ariz is OK Ireland How Moses get modified in the presence of a black hole is a black hole warned that I would describe the again in polar coordinates nice around the object nice spherically symmetric object vocal admits over other convene coordinates arise in terms of the fact generally true we have something which has nice Oracle's symmetry then polar coordinates are usually the best course of what's best over go very far from the black hole the influence of the black hole should be the hearse should go should they should fade as you move away 1st that that far from the black hole this should be pretty good right then the internal just check that it the enters 1 minus now black all has amassed religious right out of your variables are interesting is the mass of a black hole in this I think that will call this schwa chilled radius or the radius of the horizon core semester that's for sure chilled votes were chilled and that happens to be related to him twice M G just once I will put the speed of light after that will set a speed Article 1 well I money here and there the site but Back in just 4 illustrative purposes of profile but there are short chose to injury you know that I'm saying speed away with 1 on 2 0 and G squared little Clark will on the radial coordinate Dolores radio Cohen we and brackets with that was also like this as 1 minus all our charge to all our ratio the distance the point with or without here is the point of interest Polish what Cheer the radius of the horizon were interested in point far from the black hole for moment of reach out beyond the horizon of the black hole so little or is bigger and bigger are that gets multiplied by D. C square replacing this says I was dead when and we go far from the black all this becomes this year becomes we just get back on 1 key square and mind has the same thing except in the denominator 1 minus 2 energy overseas Square Pa times they are square guinea worm really ordered to launch the anomaly a becomes warned this just or square and the best India that's easy just past are squared makers for this is the metric orange what chill black hole analyzes of great thing forever for cocktail parties yes thank you could feel at end-to-end GOC squares just are short draw the mutt called and but but OK right selects a race of the black vote we don't need I don't want there could this you see
something a little bit strange the Mr. Lim OMG if will hours a large this is and what's in here not only is close to 1 but even more report positive what's he is also positive so we haven't done anything really weird like changing signs of coefficients and so forth by the Norris something unusual does happen at the point where it's small enough that this gets bigger than 1 with doesn't happen that happens in from here on get rid of this see squares are we get confused me forget about what 1 where there is a cross over With this gets bigger than that and that is what art is equal to 2 0 and G something happens when or or when to 20 geological toward the same thing when becomes more foreigners Europe inside the black coal your inside wristwatch radius and or equals OMG something really weird happens the coefficient of deep square becomes 0 sounds like no time no time when the coefficient 0 a proper time from 1 value of time to a neighboring value of time is 0 and even waters the coefficient square gets Senate that sounds like something rather dramatic happens at the horizon at the Schwartz you'll radius of the black hole that any other come back that were not going to analyze that tonight tonight area will lower just takers net and something interesting is happening Oracle's to MG but as another place was something interesting is happening however winter resting her firm's weird and that's a articles an auto equals 0 0 not only is his negative but also likewise this thing has some mascot behavioral so are equals 0 some kind of singular behavior happens and Oracle's 2 some kind of other kind of singular behavior hat quite extremely different believe they have very little relationship to each our I can tell you right now what the difference if you are falling into a black hole where when any of falling into a gravitational object you experience tidal forces everybody know what tidal forces are not what forces are experience tidal forces after the big enough these gradient of the gravitational field polls on you differently in different parts of your body your feet Pearl Harbor their heads stretched by area also squeezed inward to some extent political tidal forces you could feel very tough glass you're feelin jumping off a diving board because too small by comparison would be scaled over which the gravitational field of changes but it is mental block called to field a title for the kind for at the horizon is finite and the Michael Hall get bigger and bigger and bigger the tidal force at the horizon gets less and less and less so nothing dramatic what happened you wouldn't get squeeze sufficiently big black hole you don't get squish that the horizon of black hole and went back out with machine that you don't get squeeze like toothpaste into the into a black perfectly denying placed falling as we will see even the mathematics books like something very bad happens on the other hand at articles that's where you would really feel pain that's where the tidal forces become infinite that's where are all hell breaks loose and very nasty place articles you're always call the singularity equals to induce call the horizon let's talk about light race now let's talk about light gray moving radially outward at summit angle on the sphere longer baby along the north pole of the sphere of the East pole of the year the south pole of the year some poll be of some of their fear the writer is moving radially outward that means along the right light Ray there is no change of the angles the angle to right out the angle is constant that's moving in and out of along a light ray deal makers square 0 DTE is not 0 0 long-life ready as you move from 1 . ameliorative next time changes a little bit not proper time but the ordinary time and also the radial distance changes so you are watching a lightweight your clock is a little bit deep-sea ability to DR otherwise what equation what they would rule for a break through an awful lot about light gray zone like Ray the proper time 0 so it's work that out 1 nite is Georgie or a T square must equal arms settings us we don't eat away about I'm sending the rest of it equal to 0 in on doing that by setting a deeply squared equal to the DR squared equals 1 divided by 1 minus 2 ng or are they are squared that's emotional like Ray we can multiply both sides of the equation by the thing that the nominating here just squares this and removed this now I can take the square root of I write it just as the DDR distance along the radial direction is equal to 1 minus 2 M G are 50 hour farther away from the black hole this is close to 1 and just as the equal to deep feelings of classified but the speed of light Back in that would be a speed of life here and I would say that the article the C D 2 but as just statement that the library moves with the speed of light and that the whole state is really the statement of appointment the speed of light but you could see that as you move in closer to the black hole the mathematical description of the motion of light ray gets modified a little bit and contact looks the vibe IDT west of Ivory Tree also incidentally I did take a square root nite so that means there are took possible solution 1 corresponds to an outgoing library and corresponds to it in falling like those of the 2 branches of the solution but let's take you the city of Gori ready and if if
you want being going on Regis changes are OK this of course is these kind rate of change of the radial coordinate we could call at the velocity we could call the varsity in an hour Cohen the particular Cole admits that were using the IDT rate of change of with time that's eagled the 1 to endure has so far away nothing unusual but what happens when you get a very very near beyond a somewhat of a singularity tonight but only worry about the horizon at and 1 minus 2 and you are ghost 0 are becomes Eagle to 2 Angie at the horizon and 1 Lens 1 0 so very close to the horizon right brains slow down pace slow down the closer you get to the horizon so long as they move outward and right at the horizon like reduced don't move or they just sit there static and they don't girls so it could draw a picture of a less time his time but strong arm this way as some value of autumn namely articles to energy but that he and that's the horizon of the black hole that's analyzer by and now what happened Lewis state outgoing libraries will find a job IDT is close to 1 that is the light rail as diagram woes with 45 degree axis but what happens as we get closer to a black or Missouri closer to the black all pro-life arrays more much more slowly in a given amount of time they move us smaller distance being applied here means moving slowly and as you get very close to the horizon is hardly move at all and what I find the horizon the just stand still other life is trying to get out just stand still think so get out thing inside the horizon we haven't analyzes tell me tell you what happens inside horizon inside the horizon or like rays when you think that directed would not falling into the black hole you try this and the light break out for you try this and the like In exactly who are not exactly the same thing but in either case polite arranges for inward write articles 0 Raleigh equal 0 is where the singularity like is the originate inside the black hole simply can't escape light that originate class the horizon pick very very very long time to escape life is right on the horizon just sit there and libraries far away do what libraries do you know if it was no black hole OK so that's basic framework of the torture of black coal that's really all there is to it there's a course a lot more toward there were going to do a lot more about what they like to pay for it a belief that might raise move the speed of light is depends on it was still light 3 times and it's something but I change from 7 meters 2 meters of Musa centimeters because retires at the end of while amused comet 3 times and the bad what's going on here as effectively as you move into the black hole is cut from moving change of Unix on is being measured in different units at different places but the but the real point is the real point is busy universal statement about the speed of light and that alone light ready the proper time is 0 that is what replaces the statement that the speed of light is 3 times tend to be used for 2nd Frost speeds of our all light rays move the same way they move out what we call on another trajectory of normal trajectory means that the proper time along the trajectory is exactly 0 that's the universal statement for or libraries which is In special relativity becomes a statement bail moves to share a lot a that is that the a chaotic 1st Araujo about the reward going like rages so I started very close to the horizon here it is Le Gougne Hayward them eventually goes off the end the incoming race does just exactly the opposite starts the following and then asymptotically gets closer and closer to the horizon never quite getting so what 1 lesson is that even a light shining tore the horizon takes in In the amount of this kind of time here the quote called quote time it takes it into the amount of coordinate time to get the horizon because of the slowdown effect OK so they have now the basic set up the middle jump from the geometry of a black coal which will come back to its very interesting in the many many things to explore about it in particular the most interesting of 1 of which is to explore the nature of the horizon what kind of thing is the horizon what kind of funny kind of thing is going on the horizon is anything funny going on the horizon we're going to explore to explore question how you make a black hole but who will come back but lift but I wasn't really jump into for tonight and maybe for next time is a quantum mechanics a black holes black holes are objects now the question is how are they objects like any other kind of object or Ave something new and special which violate the laws of physics from remind you what Professor Hawking said about black holes you may decline and claim was very well reasoned claimed extremely well recent course he said that anything that falls onto the surface of a black hole he said anything that falls into a black hole we will come to the question of whether things door don't fall into black holes looks from this point of view that Dixon and the amount of time a black hole This is a contentious question that exist contentious subtle question they will come back too but for the moment let's just think about things which fall
on the horizon black or persons who may be closer and closer and closer to the horizon end this August stuck on the horizon from the point of view of outside they don't get back out there get back and that means that all the information they brought into the black hole is lost to the outside observer in the form of either depending on how will discuss a greater depending on whether you envision was falling into the black hole adjust progressively closer and closer to the horizon of black hole it becomes unavailable to the outside but that's not such a bad thing you just say well OK but the information of what is simply getting closer and closer to the horizon is no big deal is not that hasn't disappeared of hasn't gone anywhere just squeezed tighter and tighter on the horizon the problem is going to seek a might broad tonight and poverty the next time there's reason for their memories extremely complete consensus about it black holes in time evaporate black holes in time evaporate then the question is what happened all of you itsy-bitsy bits of information or whatever it is that fell on the horizon was just disappear Adams world or is it somehow or stored inside a little kind he rented the black hole it refuses to disappear on the circuit radiated out would be evaporation products exactly what happens this August was a big puzzle in physics but before we can address the puzzle and it was string theory in part which largely resolve this problem before we do that we have to understand why people thought but black or white people think that holes evaporate now a black hole it is the most objects in the universe if you create a black-owned lever for a while it settles settles down toward perfectly spherical coupling on interesting object nothing comes out of it it's totally black probably black in sense that a mole light comes out of it and it's very uninteresting it messages Eveready females but turns out not to to be quite true black holes I'm not as the political and they did in the center that they are incidentally colder instantly yarns which we say really black but has 2 meanings incident with black neither refer to admit in the snow like all all our can refer to the idea of the blackbody radiation blackbody radiation thermal radiation heat radiation so we just black holes are black in the sense that they emit he radiation there was lot earlier that black holes were black in the sense that they needed no radiation so we need to come understanding of what it was made talking back in Stein of the people think that black holes has some he could not completely dead they have some heat because he have some heat the glad because they grow they give off light because they give off light or radiation they give energy that means kind their energy must decrease well energy and ask the same thing equals and squared and so time the man's should decrease as it gives office radiation the this decreases this watch over 2 MG decreases wants expect time if black holes really do have active up he and that eventually they will simply disappeared because of a radiator Waelder energy that happens as I said the the puzzle of what happened everything a fell basically OK To understand and we're not going to do any fancy mathematics of studying quantum field theory in the background of a black hole don't really understand the concept you don't have to there I don't feel up to teaching and advanced course in quantum field theory in curved space tonight with more than 1 nite so you do something with little food 2nd signs arguments about why black holes have chirping now 1st before we talk about why black also should be let's talk about feet in general what is entropy and her have you find we did the failure with precision or we can define a qualitatively preferred defined qualitatively understood and took years you need to understand what information that may be vary very soon but simplistic about what information information you could be information about a system the information about a system must not less about what is art How do you quantify How much information Is there in a system or better yet how many how much information seperate 1 system from another let's what's the way that you X was explained to somebody the difference between systems while you ask a bunch of questions about the system and try their best to quite well Liao Hui we could do we could we could use it as a system that's a big box the boxes broken up into little boxes of boxes are about as big as a Adam and you could know everything about what's in this box of I'd tell you yes know whether Adam and I would have to tell you what kind of Adam would simplify our work to be just 1 kind of out of the world advances and theorem boxes full and I got through the boxes and tell you whether this enemy or not I have told you just about everything about that is an avenue avenues here the gravity of the around here here here here mother Adams well there is a block a lump of stuff in the middle maybe that was missing a lump of stuff well the funny shape they've got the listen with a bunch of yes no questions is there is Adams this box that box so for another describing information who's to say if I want to describe the system what I need to do is give a card food to describe it and Carlos might simply be a series of zeros and ones will you go through this thing 0 0 0 1 0 1 1 0 another word it could be a long binary digit how long now depends on how many bits of information you try describe
a yes-no questions it takes to describe a system completely number of no questions or the number of binary digits that you would have their prescribed to describe a system to distinguish it from other systems is called the information from our information serious use it all the time and say they characterize it in terms of it it's a bit is a yes no question and no the quantity of information is characterized by the number of bits in this system here and the number of bids would just be the number of boxes number boxes will you might or might not have very are as we clearly I mean we could got everything in this room by AER by breaking it up and a sufficiently small boxes if we didn't want to deal with the fact that there are different kinds of elements we could break it up and even small boxes simply escrow whether there where I suppose we would have to distinguish electrons from quirks that we can break it down into tour yes-no questions is Watkins boxes and a report by a lecture on the box box and eventually we would learn everything there is to know about this room by asking a series of questions the minimum number that would take her described the Roman to distinguish it from every other room would be the quantity of information that they it could be it could be the instantaneous a description yacht young yard at a moment of silence a moment of time right another factor information never disappears the basic idea physics probably more basic and anything else in the meaning of that is the forum that if you start with 2 different configuration through different configurations but say here's 1 here's another 1 a fairly similar was a matter of fact they just have a couple of items out of place Olaf but in different did the state the distinguishable distinct and a new experiment principal of difference between and then if you let them run run means let them evolve with time they will stay different of course that will change relative to to what they started with this 1 might move to hear this from might move to himself 4 of you start with 2 distinctly different configurations they a different whichever other for example you never have a situation where as you run these things they evolve the same configuration at some very very bid that such a deep rule of physics that people forget the stated when this stating what the laws of physics are that the amount of information that it takes to describe the system is constant and and that we distinctions between sisters never disappear orchid now sometimes it looks like information disappears you drop a series or what molecules into your bathtub and my dripped mammalian and some very very specific way the drops of water in with Morse code messages on them to different 2 different bathtubs for warrant and 1 the messages the some great literary masterpiece on some of the great literary masterpiece finger coded in the dripping of before and better still bad tabs for water started about differently they sure look the same after a few minutes after a few minutes to let that water settled down and they should look the same but has only because you don't look carefully enough you could look carefully enough that every single molecule than those bathtubs you would find the configurations of them stay the same way I say that effectively information doesn't disappear but it gets it gets hidden costs after just because From gets hidden by the fact that there are just so many degrees of freedom and this all small but you can see them that in effect information which is there is effectively lost just because stored in degrees of freedom which are too small and too numerous to keep track of soil in Prague this ever since information those disappeared but not in principle there's not enough hidden information information that inaccessible to you for 1 reason or another now Of course and the notion of inaccessible could depend just how good your your ability to study and manipulate the system might depend on how would you microscopic Mike that depend on quick are in your ability to measure molecules so full of but subject to ever limitations are there's a notion of inaccessible or hidden information was entropy the simplest statement of entropy is that it is the number of hidden bits of information the number of things which in principle are there to distinguish things but which had just unavailable to you because they're stored in things which is too small and too numerous to keep track of or so in the case of is that the full was so the only thing I really care about is how how public only duty and I wanna make sure there's enough water the bed of that I could make it nice that I want to lure warehouse on should water not hydrochloric acid on a did others in he for the things that I want to know about that water and furthermore the handful of things like can know about the water the amount of energy that's in it that former its thermal energy fact water and a few other things and that's a pretty much complete description of water where might have rebels on and I don't care much about the rebels and Vicki nevertheless I could see the ripples and I could distinguish them factor that was filled up 5 minutes ago in half an hour ago but few way out while those ripples go away and not even any ripples on the surface tell you where any detailed information it's all a hidden In the microstructure for molecules so all that information is hidden information was cold the entropy rats entropy the number I'm bit of information richer mother a more technical definitions that for the moment I think that the better it is good enough for us incidentally arm entropy and energy of the basis for thermal Memorex normally when you learned thermodynamics of statistical mechanics pick of thermodynamics you sell the idea temperature temperatures a highly derived concept while you amounted to measure with thermometer very simple
Gallagher measure with thermometer it was very very highly evolved thing it's easy to measure but it is very hard to define the definition of temperature was a definition of temperature anybody know the definition of temperature now of course you don't because the definition of temperatures are highly contrived not consulates contrived it's was sorry the derived concept derived from the ideas of entropy temperature OK Telenet right rib energy I assume we all know it energy use and we have entropy Melanie given example Anteby might the amount of information and the atmosphere I'll tell you right now where temperatures Veteran equation 1st myself just so I get right OK temperature the change in entropy of a system if you and 1 bit of information if you chain and that he is here's an example argue an example of where this comes up you take you computer your computer's full bits of information that's a started individually in the form of a form of what their dates yeah whenever there is some little thing inside computer is either switched on or switched off and that's yes or no question value computers for a a lot of information that you put in their supposing I want to get rid of a bit of information I want to eliminate some are not latest circuit I just want to send back to its original love of starting point that we take that computer and rebooted sister I get back to its original starting configurations in that case I will lose a lot of information all the information that's in the computer what cycle that far just you release 1 bit of information if you re race 1 bit of information I just told you it's impossible to EU race bits of information that's that's basic law of physics what really happens is that bit of information whatever it is is displaced from inside the computer start out into the atmosphere the distinction between 1 configuration of a computer another configuration which differ by 1 will buy 1 that distinction gets taken from the computer into the atmosphere so that they atmosphere becomes a little bit different by 1 bit of information rather than it was before you we raced debate that process of transferring a bit From the computer into the atmosphere he'd be atmosphere it adds a little bit of energy to the atmosphere and inevitable heeding that happens because you we raise bits from your computer what is energy transferred to be to the atmosphere when you transfer 1 bit the answer is the temperature of the definition of temperature the change in the energy of a system when you get 1 bit of information to it is called the temperature you go write another way you could write a change in the energy when you change the number of bits of our by some number is the temperature transit chain or correct such slots on the Slovak notation entropy we need a notation entropy and I wanna say entropy every kind underwater right the word entropy on the blackboard equations we need a symbol for entropy standard symbol for entropy is X as there's entropy OK now do is that the a change in the energy what's court felt that he when I 1 bit that means when Delta S. Warne Delta as is equal to 1 of the best is 1 unit and the amount of energy that's could in the system is called the temperature what happens if you a to bits the atmosphere the energy of the atmosphere changes by twice the temperature what happens if you live rebates 3 times of temperature what happens if you intend s bits then the change in energy his tee times the change in entropy change in and then energy is proportional the number of bits that you transfer to the atmosphere the you transfer to anything of that thoughtful of hot water another Woods its change in the energy when your heart made a certain number of bits of information to remove them from someplace and hide them in euro in your whatever it is hiding your heat bath basically visited basic formula thermodynamics extremely confusing formula if you start With temperature as the basic concept but very simple concept if you start with entropy is the basic concept and temperature is said energy that you Edward USA 1 bit and this is usually written differentially DES thus ID equals PBX Sephora fundamental laws of thermodynamics change in energy temperature change in entropy are it that's the permissive entropy the new information but the can easily see I did a calculation of character changed it to be booked for private you is a gigabytes of how much that picture would be for a glass of water that the 15 degrees loose fact that that's that's vault change Chavira cannot about 100 gigabytes now that whole here a bit a bit of information Out of the ordinary room temperature at room temperature very well for 1 thing yeah room temperature rise so room temperature in any kind of reasonable units is very small and a bit Is there a typically have a very small amount of energy for the year rose almost the wall computers that's a close land hours the land around Bowers rule about you are a little amazingly it means were raised you haven't erased the object which held the bay but let's say that you do you remember what ever the configuration of the computers you limit turned that bit top off so that so you will have forgotten what was there because you turn or what and that since you've erased the knowledge that that but that will circuit ahead now and eventually could turn off you're a turned off by Brooks by rebuilding the computer by studying the computer over and bringing it back
to the configuration of when you bought it there's nothing left in that computer remembers what you did to it so you erased numerous and there are a number of its security awards the number that you brought back neutral is a change in the Inter b of the computer and that or a few like the number of bits that had to be taken from the computer put into the atmosphere and that's what would you calculated renew calculated but it he is a yacht now rest you do a lot of stuff besides erased a bit normally there you you rub your finger on keyboard will kinds of whether they are higher the hot gas Europe that will heat will heat persist over will only increase the energy of the heat backed by more the higher the temperature more energy you will be putting when you hide that big now but that's exactly what's but but in end that's the definition of the temperature so it's a question of course before car before about it that is what this that yes uh chit prices decreases the amount of information and a computer but that information can not disappear out of this world so you on that putting it into the atmosphere but Tinonee the atmosphere OK that's the basic idea venture while my teacher didn't teach me that will thermodynamics in 1959 and are now but I around asking my colleagues when you teach statistical mechanics he said and basically what I've found Barrett is none my colleagues it but occurred but now the information the computers I entered the computer scientists or right yes yes yes but these ideas about these ideas go back a his ideas are older than computer science actually quite not mine In statistical mechanics and physics they go back aboard the 19th century in on computer science they go back the 14 a Claude Shannon was the 1 who formulated India's uptake enough history OK so now let's let's talk about information that hidden in a black hole I said you could I don't think of it as hidden in the black hole although you could think about it as he did on the surface of a black hole you keep flowing south into the black hole and it accumulates closer and closer to the horizon in some sense and becomes unavailable 12 from a stop on the horizon calm takes light forever and ever and ever to get out of the horizon so it's effectively lost Why did back in stunning rioted beckon starring around think that black holes have entered well I'm gonna tell you on the way I think about it but I was also that stand for about it but I tell you why photos entropy by computing the entropy our by computing amount of information that you could store and a black hole when going to those imagined building up the black hole by flowing bits of information into it and I want to know after differ only in a certain number of bits of information how big is the black hole a logic is very very similar they're asking How will the better a B FOIA filet by dropping in Adam by Adam and tell you after Adams how big food better be fully figure it out you say that has a certain volume every time you drop anatomy and periods of the volume of water and if you each each differential drop changes the volume by a little bit and raises the level of the water by a certain amount of wonder how high the water will be for a given number drops you don't you just count how much will be raised by dropping in 1 drop but but Of course you have to be that you could be smarter but that would be 1 way to do it just drop drop by drop and discover each drop that you put in there's a certain volume therefore rate the over a certain amount of FBI in drops the water has a certain height of the same thing by dropping in of information the former black hole for use by the law I have a book lot shares they throw it may OK so we get that but basically we going to something in which were pretty sure has no other information in it under then the question of whether it's In the end there or not another words Phyllis black hole up by a series of processes for which each 1 of them has no other information stored in other then that a photon went into the black hole you can't simply asked the defunct or didn't apart from the rim yes and no question and all that but I think long trip to get squashed on the horizon becomes available yup but young but serve our example or polarization of the photon polarization of the photon is related to the upward down along the x-axis what's 1 bit of information but you have to be careful on his return should actually reduce let's take a black hole as a black hole that has made us it has a radius all of our Verizon has radius Our which is twice Ng already made it we've got now want the a throw-in 1 more object and that should be the simplest possible kind of object for which we can simply ask Is there hours of not there 1 bit of information but sick of a photon already careful differ throw photon from he opposes black or a kilometer radius away from any file time along a beam of laser light where the photon has a wavelength witches the answer microscopic Waverly that is a lot more information it has and the arrival of that full-time just whether garden
my kids ample did coming from he and coming from he added that come in from he and mosey angle on the horizon where it arrived from so clearly going to a few of the maybe a lot of bits of information in describing the angle of the various amounts of precision so throwing in a photon is not by any means necessarily the same as throwing in 1 bit of information is a lot more information or you could ask you could begin Iast of question his followers is it in upper hemisphere the king yes OK is it in the upper half right but not to the left and then imagine dividing up to be a horizon of the black hole and asking a series of yes no questions on till you localize without a vote he needed for the maximum degree that it be localized a kid in a localized license wavelength is fuzzy outwards wavelength but it's clear that is a lot more yes-no questions you could ask about that time communion and photon is not 1 bit of information it's a lot of information on the grass you do something that from time you deal with what time which is some more uncertainty about where it is because photons are always answers brain but about unrest in principle there isn't uncertainty uncertainty in the sense of Heisenberg unless there is a maximum amount of uncertainty about where that photon entered the black hole now what does that mean in practice practice that reusable wavelength Of the photons the way like the photon is a measure of how deal localized the position of the photon theaters in quantum mechanics with talking about mechanics on the wavelength of the photon is as big as the whole black hold them at photon arrives to right away where are there's complete uncertainty about where it ended so the 1st thing you do is you want to work with photons which have the longest possible wavelength but can still get into the black hole is a restriction on not being able to enter the black called yes Eds a fact that can known basically from classical physics that if you shined a light I'm a black hole of Waverly longer than the size of the black hole just reflects off likely only wavelengths which our no longer than the radius of the black hole horizon to have any chance of penetrating through to eyes so what's the best you can do you could work with photons whose Waverly is comparable to the radius of the black hole there are uncertainty in a location that you can't tell where they into the black hole on the other hand sure enough wavelength that they can get in a black holes to working just on the margin on the edge with photon or go into the black hole but where would provide no further information other than that the garden or didn't get it OK so lots it didn't a full-time black all year that's more clout and it is really isn't at the wavelength is very long because they go the way of so long that the electric field is very very slowly across a black hole and it doesn't matter which comes from but your there's a question but I think the answer is no better way with long enough the present was providing information OK but wasn't a photon of wavelength lambda equal to 2 and Judy now 1st question was energy of such a full time limit it would be a getting some imaging now it is as business that that whenever you were hired piece of information you'll wind-up by adding a little bit of energy to the system that hiding information and so how much energy Have you are hidden inside a black hole how much energy to enter the black all the energy of the Photon Sources right down a to about energies of Tom right the former you energy of a photon or any light is equal the speed of light turns its momentum P is the momentum of a photon know what's the connection between momentum and wavelength they bear number but h bottled water Bora talks In this constant right this is B class would dealing with full because in front of mechanics information is quantized the photons are this is a real quantum problem Seaway wines up will be interesting to see where each by warns of for a while so what's the wavelength of the wave among however momentum is each board over land Land the being the wavelength shorter the labeling the higher the momentum so that I can say that energy of the photon we through and his speed of life times plants can't stand the thought that for over a quarter of the box which Obama cannot tracked her city when there's a yardstick of and that's our the energy is a total that I can it is see Times p Good the that's the important market auto parties with such forget to prize port magnitude energy is C H Orlando and thus ERP P is a teacher at age 4 times the wave number and tow by you right Mike OK but this is a total amount fought a on the use used a storm now lender we're going to be choosing equal to the card to be the size of the black called Malika right now that these energy that we put in applied from factors of 2 0 and uninterested in fact to here we kinda miscalculations Lockerbie that precise with with that of factors to price of energy the way a heading for the black all is a speed of light arms plants constant divided by G and that adds 1 a bit of information the black or Michael stop right here I have not told you how much information it cost energy cause behind 1 bit but I tell you that temperature is the temperature of the black hole C H of Ng that's a Hawking temperature another some factors a foreign pies and things there were not getting it right but little argument but let's let's let's go the further misses 1 bit of information that we put in let's see how much bigger it makes the black hole facilities damaged be replaced a black hole
in makes a black hole bigger it figures for you and energy you and mass equals M C squared right so easy Eagles MC Square so this could also be written as the change in the May what corporate change in the mayor's can't see square energy that I edited goes and the increase in the last and the changes the masses by this formula yes but cleanup up the formula will better survived by C squared and that warns of putting the city of the nominated here that's a change in the merits of the minimum change and the rest were new 1 bit of information is each bar and OK let's go further let's calculate the change in the radius of the black hole calculate the change in the radius of the black hole had I do that I do it because I know the radius is proportional to end times G. so I just multiply had multiplied both sides of this equation a doing yeah heard you aware that we did all of this was a guy was a wavelength downstairs so we're multiply by GE and this is a change in the radius of a black or with where do a rut the wavelength on over radius age see always watch radius the change In energy as Delta E equals M C squared Stauffer want change unless I think I wanted provided by C squared change in Maris physical C a noisy Quebec H Of all are C agreed computer thoroughly against that that's a change in mass enough multiplied by by G. it gives me change a radius this is the change in a radius of a black hole OK so every time I only in 1 bit of information I change the radius by an amount which is in most of the radius itself With the G H overseas I really think I've lost some seized Russia where well maybe not where you although you write you right here right what good good that this watch over radius is not ingenious Ng overseas square at that's where we're Ng overseas squared so I think it should be but now seek cube yet has cities people 1 part of the thing appeal the price gap and its Yadav's G H lawyer C-Cube that's the change in radius now multiplied by the radius terms a change in the radius what's odd trends a change in radius R D R as dealers acquired the change in our apart from a factor of 2 to are br Elementary Calculus so all times a small change in art is proportional the change in the square of the radius for us is whether radius the square of the radius of the horizon is the area of the horizon so what this formula says it's is every time you drop a bit of information in you change the area on the horizon by a universal on which doesn't depend on the merits of the black hole it doesn't depend on the radius of the black hole at the instant did you did it you will always wind-up changing area by the same amount it's kind of like saying that every time drop a drop into a bath tub assuming that have had straight walls that you always raise the level of the better by the same and it doesn't matter how good is you always read the volume by the same amount in this case you would increase area of the black hole by a community where you it means the area GE h CQ OK this isn't a small area every snarling area is the square of some every area square of some weight of course so this is a square of some lady is the square of the square root this quantity has units of lane let's it's it's Parkway score the popcorn and very small to begin estimated OK GATT In ordinary units units means 5 kilograms meters 2nd G is about 7 . 7 times tend to amend the 11th plants constant but I think that's H bathing was anchoritic 600 and another 7 OK 7 times 10 to the minus 34 this and CQ CQ at square with take a square in minute boss a square of the speed of light tube of speed of light kilos this bill 27 times can the 24th right yell 27 times can 24 OK we are a small numbers upstairs ally big numbers downstairs this is a very small number of our ideas about 10 to the minus 68 69 or something like that and the minors 69 square meters or something like that I don't remember exactly IT Avenue memberships 10 to the minus 66 66 some was about that square send native the plunk late itself square this is about Pinto my 35 those sorts smaller than anything to container access in any way an experimental physics says his every time you drop a bit on the bits In the think about bits so impenetrable and they expect further horizon not find that amount of time we have to get past the horizon a stick of allies and they have shut down the boson they simply because they are the ones on the way each 1 occupying about him but it goes right now right well pastures pushes the
other ones and the way makes the area bigger In the Delaware Bay Area biggest video free prior that None None are saying they never get at DVD she added that the life cycle of the metal in the yard we will will will go through a bit that we have to define what the horizon the careful where rift the the precise count but the horizon which we haven't done yet we normally do this the problem is that it is but 1 of the areas that were saying Fernandez where's my period 1 that services but promised to share clear of all or player black waiting world what is it all all radio missile radio the color but that's what there was rat all the at OK but that that was the magic go that's our Jacob back in starring argued that in this way argued that we information but you can hide in a black hole is proportional to its area and plunk as it was finally eventually derived where I was sequels 1 they'll try to seize and with CIA I hesitate sees in somewhere going somewhere we or the hidden information and a black hole is equal to the area on the horizon figure about building up these are the black Colbert dropping in this debate each time you 1 unit of area provided by GE In the vibe H-bomb bartenders a C-Cube upstairs he Fiat C-Cube upstairs this means the entropy of a black holes and modest by comparison with the kinds of entry fees ordinary systems of the same size have C-Cube that's a huge number you have G and H ball which I pray numbers in the nominator the entropy of a black hole is bigger than the entropy anything else of comparable size by a lot of mean anything else ordinarily would think about so huge numbers of bits gets stored on the surface of the black hole proportional to the area not the volume of information you could keep track of black hole that needs to be kept track of it is less than you might expect it my follows the volume of the black or but still huge this is probably 1 of the most important principles of physics that is for sure I think it's time had this formula with an unknown coefficient he tried to estimate the coefficient getting all sorts of arguments but about with peculiar numbers like lager rhythms of To win of the things which are now is just for it was Hawking who figure out before but in Burma good but rare whatsoever picture while there are some interesting as 1 very interesting thing about this for and that's that each bar is in the denominator what's the meaning of that will you might have thought Red and could be the existence of entropy for a black hole is a quantum mechanical effect has something to do with quantum mechanics and she might think that the b is proportional Planck's constant because Fox constant which is a measure of the quantum this system but it goes in the denominator if talks constant woody get smaller and smaller as it dozen classical physics and classical physics Planck's constant 0 it says peas it as just another way of saying that in classical physics Ucon Hinds since information is not quantized it doesn't come and photons you put as much information in tow as small and energy is you light arbitrarily small amount of energy you could put and much information on that always even electromagnetic waves you like but quantum mechanics information comes in discrete quanta on information entropy of the black hole is finite because of quantum mechanics it would be infinite in classical physics you could simply throw arbitrary amounts of information onto that black hole without increasing its mast by almost anything so this is an interesting fact that although all of this is quantum effects its quantum mechanics which prevents the entropy from being too big not the other way makes a fine From what else can we say about this well we could say a lot more about it we will getting a little tired there is a real full 8 bar Is To constant which has certain units units angular movement through it comes from our point of view of affray Otto a private convert their angular momentum and moment and differ by a power of a distance angular momentum as momentum terms distance momentum is H Over a Alam Bell land a distance Pete how they use slammed momentum kind distances units of H and P slammed also has units of angular momentum or cross distance tournament OK for
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Titel Topics in String Theory | Lecture 2
Serientitel Lecture Collection | Topics in String Theory
Teil 2
Anzahl der Teile 9
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/15115
Herausgeber Stanford University
Erscheinungsjahr 2011
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (January 17, 2011) Leonard Susskind describes the special theory of relativity and focuses on showing how it connects to string theory. He considers concepts such as space-time and some of Einstein's original concepts. 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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