Lecture 2 | Topics in String Theory

Stanford University we'll move on and talk about cosmology and what string theory has to offer our cosmology I suppose I could spend two minutes just telling you what I think the accomplishments and the failures of string theory have been I think we'll do that another time it's getting late and we want to get on with some some substance all right let's go back to the special theory of relativity for a moment and just remember what it's all about it's about thinking of space and time as a single entity that we call space-time the space-time is geometry the geometry is described by a metric make you think I better leave it on it's described by a metric metric tensor special relativity and special relativity the metric is especially simple and the metric is just if we take two points in space and time two neighboring points in space and time can you see them no it doesn't matter let's take two points in space and time any two points in space in time they're separated let's call this the x axis let's call this the T axis and there's two more axes coming out of blackboard are these are two arbitrary but neighboring points and they're separated by some DX DT dy DZ separation here and we define a new concept the new concept is the relativistic distance between them the proper time and the proper time is defined to be the T squared minus one over the speed of light squared the x squared plus dy squared plus DZ squared different than you might have written down if this were ordinary Euclidean four-dimensional space if we were Euclidean four-dimensional space you probably would have put a plus sign here you certainly would have put a plus sign there and you might have left out the 1 over c-squared although the one over C squared is just a rather arbitrary factor you could always get rid of it by rescaling the space dimensions or the time dimensions appropriately in other words by choosing units for space and time in which the velocity of light is 1 you can always do that let's leave it for a moment this is the proper time between two points along the trajectory it's a nice concept as long as DT is bigger than the x squared plus dy squared plus DZ squared you the proper time the square of the proper time in particular ought to be a positive thing but notice that if the x squared plus dy squared plus DZ squared is bigger than DT squared something bad happens to this expression namely the tau squared is negative but never mind nevertheless this is the definition of proper time between any two points the more the most important thing for us tonight is how light rays move how two light rays move let's suppose a light ray passes through those two points then the rule the rule for a light wave let's forget the X the Y squared and DZ squared for a moment let's just take the x squared here so the light ray is moving along the x axis little infinitesimal separations along the light ray have only a DX and a DT the rule about light rays is that along the trajectory of the light ray the proper time is zero somebody traveling with a light ray if you could what you can't would discover that the clock stands still which is another way of saying that proper time along the trajectory of a light wave zero so if you want to know what kind of trajectory a light ray consists of you simply set the left or the right hand side of this equal to zero so this is equal to zero for a for the trajectory of a light ray and another way to say the same thing is that the t squared well C squared is T squared is equal to the x squared I just multiplied by C squared and that transposed one side to the right or taking the square root and sends that C DT is equal to DX or what's the other possibility or C DT is minus DX this is just another way of saying that a light wave moves with the speed of light that the X by DT that's the velocity of the light ray is equal to C or DX by DT is minus C one of the light rays with the plus sign moves to the right the other light ray with the minus sign here moves to the left alright the main point here is just that the way you diagnose a geometry geometry like this to find out how light rays move is you solve the equation that says that the proper time along the light trajectory is zero all right now we're going to come to the more complicated geometries of general relativity general relativity the metric is more complicated it varies from place to place the geometry varies from place to place and it's described by a more complicated object the metric tensor which itself can vary from place to place and we've done these things before so I assume you know a little bit about it we have a metric tensor GI or let's call it G mu nu just remember our notation mu and nu are variables which run over the four dimensions of space-time usually we take them to go from zero to three zero being time and X 1 X 2 X 3 being the ordinary components of space the S squared is Jean you know the times the X mu the X mu X mu and X nu are the cordon the four coordinates of space-time and as always in an expression like this it is automatic that we sum over the repeated index here there are two repeated indices so this stands for things like G naught naught D T squared X naught is T plus G naught 1 DT DX blah blah blah blah blah all possible combinations with the metric coefficients again if we want to know how the light ray moves we simply in other words if we want to find the little infinitesimal gaps that correspond to a light ray moving from one to the other we set this equal to zero that's the rule we set that equal to zero and that gives us constraints on on the allowable trajectories of light rays that's the content incidentally of saying that light rays move with the speed of light whether the speed of light is 1 or C or if you use some other units something else D that doesn't matter the invariant statement is that light rays move on trajectories of zero this should be D tau square excuse me trajectories which correspond to zero papa all right now what we're going to do is we're going to examine a particular metric it's a metric of a black hole we're not going to solve Einstein's equations we're just going to write down the answer and we're going to inspect it for a while and learn about it tonight I'm going to only spend a few minutes with a metric showing you basically one property about the way light rays move in that metric but we're going to get right to the point tonight about the quantum mechanics of black holes the quantum mechanics of black holes is very strange in some ways it's very strange and in some ways it's extremely ordinary in fact it's very strange how ordinary it is ah given how bizarre black holes are as classical none quantum mechanical entities okay so let's write down let's just explicitly write down the metric first let's write down the metric of flat space I'm going to set from here on in I'm going to set the speed of light equal to one in that case what I would write down for the metric of flat space would be DT squared minus DX squared plus dy square plus minus the X square minus the Y squared minus DZ squared right let's write it minus the Y squared minus DZ squared or this way now this here is nothing but the ordinary metric of ordinary flat Euclidean space here and we can write in another way we can write it in polar coordinates let's think about polar coordinates for a moment polar coordinates are the coordinates that I would use if I'm standing in the centre which I am I'm always at the center I'm very self-centered and I'm always in the center and instead of using coordinates X Y & Z I'm going to use coordinates which are the distance from me our let's call it our the distance from me and two angles to represent the direction that I might be examining so our and two angles in fact the two angles are Auden are often summarized by the symbol capital Omega that symbol actually stands for a unit sphere a unit sphere why a unit sphere because I can think of myself surrounded by a unit sphere and then a point on the unit sphere determines the direction the symbol Omega often stands for a unit sphere and the symbol the Omega squared stands for the metric on the unit sphere maybe we should write down the metric on a unit sphere to just be really calm our concrete to represent the unit sphere you need two angles or they call them as amutha land pull polar angles or longitude and latitude okay two angles theta and five and the metric of the unit sphere is the length element on the unit sphere is just D theta square theta is the polar angle plus sine squared theta D Phi squared that's the metric on the unit sphere Phi Phi is the angle around the pole and theta is the distance along an angular distance from let's say from the North Pole or maybe it says no I think this is actually our distance from the equator and this is the thing that's usually called just just on abbreviated it's just an abbreviation that's called the Omega squared and it means the metric of a two dimensional ordinary two-dimensional sphere but of unit length if the sphere had some other unit the radius if the sphere had some other radius then the metric the metric of it would simply have the square of the radius multiplying the whole thing but we're going to be drawing a unit sphere yeah okay let's let's put in what happens if the radius of the sphere is not one if the radius of the sphere is not one then we would write R squared where R is the radius of that sphere times D Omega T theta squared plus sine squared theta D Phi squared or just R squared D Omega squared okay so R squared the Omega squared is the metric of a sphere of radius R now I stand at the center and I look around me and I see two points they're separated you could think of them as separated by an ad the X in V Y and a Z but now I'm going to think of them as separated by a dr a D theta and a D Phi he are different this different distance to me different angle and so forth what's the distance between those two neighboring points the distance between those Navy neighboring points in ordinary flat conventional space in other words the thing which really replaces this thing this piece of it in polar coordinates it is the R squared plus that's the difference in radial distance along the along the radial distance and then a contribution coming from the distance R in angular space and that's just this plus R squared the Omega squared shruts shield s.h.i.e.l.d mushy oh no t d oh okay okay okay oh no I don't think so it's our shield as far as show yeah there isn't okay um how does this get modified in the presence of a black hole here's a black hole some coordinates and we're going to describe the metric again in polar coordinates it's a nice round object nice spherically symmetric object polar coordinates of a convenient coordinates to write it in terms of in fact that's generally true when you have something which has nice vertical symmetry then polar coordinates are usually the best coordinates to describe it in okay so how do we do what's first of all if we go very far from the black hole the influence of the black hole should be it should go should the fade should fade as you move away that's the first thing that means that far from the black hole this should be pretty good I'll write down the answer and then we'll just check that the answer is one - now black hole has a mass let me just write down over here what variables are interesting is the mass of the black hole M there is something that will call this either the Schwarzschild radius or the radius of the horizon in core R sub s that's for SWAT shield the Swart shield and that happens to be related to m is twice mg and just once I will put the speeds of light in after that will set the speed of light equal to one see I'm a I'm a I'm a here and there decide to put the C's back in just for illustrative purposes ah but if I write that our Schwarz shield is 2mg you know that I'm setting the speed of light equal to 1 all right so it's ah 2 mg of a C squared little R little R is the radial coordinate little R is the radial coordinate we can in brackets we can also write this as 1 minus R Schwarzchild over R the ratio of the distance to the point where there are little R here is the point of interest our short Shield is the radius of the horizon we're interested in points far from the black hole for a moment or at least out beyond the horizon of the black hole so little R is bigger than big R here all right that gets multiplied by DT squared replacing this now this looks good when little R gets big and we go far from the black hole this becomes this here becomes small and we just get back R 1 DT squared and there's - the same thing except in the denominator one minus two mg over C squared R times the R squared again when little R gets large the denominator becomes one and this just becomes the R squared and then the last term here that's easy just plus R squared the Omega squared this is the metric of a Schwarzschild black hole memorize it it's a great thing for for cocktail parties and yeah thank you good good for you all right and two mg over C squared is just our Schwarz shows oh did my clone okay okay all right so let's erase off the blackboard what we don't need now when you look at this you see something a little bit strange one minus two mg if little R is large this is small and what's in here not only is close to one but even more important that's positive what's here is also positive so we haven't done anything really weird like changing signs of coefficients and so forth but notice that something unusual does happen at the point where all it gets small enough that this gets bigger than one where does that happen that happens you know what from here on I'm going to get rid of this is C squares they're only going to confuse me I'm going to forget about them just let's get rid of them set C equal to one where is the crossover where this gets bigger than this and that happens when R is equal to two mg something happens when R or when two mg over R is equal to 1 the same thing when all becomes smaller than this you're inside the black hole you're inside the Schwarzschild radius and R equals to 2 mg something really weird happened the coefficient of DT squared becomes zero sounds like no time no time when the coefficient here is zero a proper time from one value of time to a neighboring value of time this is zero all right and even worse the coefficient of D R squared gets infinite it sounds like something rather dramatic happens at the horizon or at the Schwarzschild radius of the black hole and we're going to come back to that we're not going to analyze that tonight tonight or will we'll just take this metric and say all right something interesting is happening at R equals to mg but there's another place where something interesting is happening I don't know interesting weird and that's R equals 0 and R equals 0 not only is this negative but it's also infinite likewise this thing has some nasty behavior in it so R equals 0 some kind of singular behavior happens and it R equals to mg some kind of other kind of singular behavior happens they're quite different extremely different they have a very little relationship to each other I can tell you right now what the difference is if you are falling into a black hole well whenever you're falling into a gravitational object you experience tidal forces everybody know what tidal forces are anybody not know what tidal forces are ok you experienced tidal forces and if you're big enough the gradient of the gravitational field pulls on you differently in different parts of your body your feet are pulled harder than your head so you get stretched you're also squeezed inward to some extent those are called tidal forces and you can feel them of course you wouldn't feel them jumping off a diving board because you small by comparison with the the scale over which the gravitational field of the earth changes but if you fell into a black hole you'll feel a tidal force the tidal force at the horizon is finite and as the black hole gets 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 squiffy there's a sufficiently big black hole you don't get squished at the horizon of a black hole and we're going to work that out we're going to see that you don't get squeezed like toothpaste into the into the black hole perfectly benign place to fall into as we will see even though the mathematics looks like something fairly bad happens on the other hand at R equals zero that's where you would really feel pain that's where the tidal force has become infinite that's where all hell breaks loose and the nasty place R equals zero is called the singularity R equals two mg is called the horizon okay let's talk about light rays now let's talk about a light ray moving radially outward at some angle on the sphere in other words the longer may be along the North Pole of the sphere or East Pole of the sphere or the South Pole of the sphere or some Pole of the sphere some direction on the sphere the light ray is moving radially outward that means along the right light ray there is no change of the angles the angle as I shoot the light ray out the angle is constant that it's moving in and so along the light ray D Omega squared is zero D T is not zero along the light ray as you move from one point on the light rate to the next time changes a little bit not proper time but ordinary time and also the radial distance changes so you're watching the light ray you have your clock you see a little bit of a little bit of ddr how does the light rain move what's the equation what's the what's the rule for the right rate the rule for the right light ray is that a long light ray the proper time is zero so let's work that out that's one minus two mg DT squared must equal I'm setting this we don't need to worry about I'm setting the rest of it equal to zero and I'm doing that by setting the DT squared equal to the D R squared term equals one divided by one minus two mg of R the R squared that's the motion of a light ray we can multiply both sides of the equation by the thing in the denominator here and that just squares this and removes this and now I can take the square root if I like it just says that D are the distance along the radial direction that it moves is equal to one minus two mg over R DT if you fall if you fall away from the black hole this is close to one and it just says dr is equal to DT of course if I put the speed of light back in there would be a speed of light here and it would say that the R is equal to C DT but that's just a statement that the light ray moves with the speed of light in fact the whole statement is really the statement that the light ray moves of the speed of light but you can see that as you move in closer to the black hole the mathematical description of the motion of the light ray gets modified a little bit and in fact let's divide by DT let's divide by DT Oh incidentally I did take a square root didn't I yeah so that means there are two possible solutions one corresponds to an outgoing light ray and other corresponds to an in falling light ray those are the two branches of the solution but let's take the let's take the outgoing light ray and if you want the in going light ray you just change the sign okay this of course is the time rate of change of the radial coordinate we could call it the velocity we could call it the velocity in our and the particular coordinates that we're using the R DT that's the rate of change of the radial coordinate with time and that's equal to one minus two mg over R as I said far away nothing unusual but what happens when you get in very very near the let's not worry about the singularity tonight let's only worry about the horizon at the horizon one minus two mg over R goes to zero R becomes equal to 2 mg at the horizon and one minus one is zero so very close to the horizon the light rays slow down they slow down the closer you get to the horizon the slower they move outward and right at the horizon the light rays don't move at all they just sit there static and they don't move at all so we could draw a picture of this let's draw time his time let's draw our this way and that some value of R namely R equals to mg let's put that in over here that's the horizon of the black hole that's the horizon of the black hole and now what happens - let's take outgoing light rays when we're far away dr by dt is close to 1 that means the light ray on this diagram moves with close to the 45 degree axis here but what happens is we get closer to the black hole as we get closer to the black hole the light rays move much more slowly in a given amount they move a smaller distance being upright here means moving slowly and as you get very close to the horizon the light rays hardly move at all and right on the horizon the light rays just stand still or the light rays that are trying to get out just stand still okay so they'll get out anything that's inside the horizon we haven't analyzed it in detail but let me tell you what happens inside the horizon inside the horizon all light rays whether you think they're directed outward or not you're falling into the black hole you try to send the light ray out or you try to send the light ray in exactly we're not exactly the same thing but in either case the light rays fall inward right to our equals zero where R equals zero is where the singularity is light rays that originate inside the black hole simply cannot escape light rays that originate close to the horizon take a very very very long time to escape light rays right on the horizon just sit there and light rays far away do what light rays do in if there was no black hole there okay so that's the basic framework of the short show black hole that's really all there is to it this would cost a lot more to it and we're going to do a lot more but good luck could you clarify the statement light slows down what exactly yeah mr. Ward I'm just that state levels various for this team yeah we rule that light rays move of the speed of light is depends on what is the speed of light three times ten to the eighth something but if I change from centimeters to meters or meters to centimeters becomes three times ten to the tenth or or meters to kilometers becomes 3 times 10 to the fifth what's going on here is effectively as you move in toward the black hole this kind of a moving change of units are is being measured in different units at different places but but the real point is the real point is there's a universal statement about the speed of light and it's that a long a light ray the proper time is 0 that is what replaces the statement that the speed of light is 3 times 10 to the 8th meters per second for all speeds of light all light rays move the same way they move on what we call an old trajectory an ultra jek turi means that the proper time along the trajectory is exactly zero that's the universal statement for all light rays which is in special relativity becomes a statement that they all move with speed of light exchange true for regard no no that's it yeah okay well first of all I do an outgoing light ready and an outgoing light they just sort of started really close to the horizon here at sticks it doesn't want to go anywheres and then eventually it goes off the end of the incoming light ray does just exactly the opposite it starts to fall in and then asymptotically gets closer and closer to the horizon never quite getting there so one lesson is that even a light ray that's shined in toward the horizon takes an infinite amount of this kind of time here what's called coordinate time it takes an infinite amount of coordinate time to get to the horizon because of the slowdown effect okay so we have now the basic setup and I actually want to jump from the geometry of a black hole which we're going to come back to it's very interesting and many many things to explore about it in particular the most interesting of Y 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 at the horizon is anything funny going on at the horizon we are going to explore that we're going to explore the question how do you make a black hole but we'll come back to this later well it allows us to really jump into for tonight and maybe for next time is the quantum mechanics of black holes black holes are objects now the question is are they objects like any other kind of object or are they something new and special which violates the laws of physics I'll remind you what professor Hawking said about black holes he made a claim and a claim was a very well reasoned claim an extremely well reasoned claim he said that anything that falls onto surface of a black hole he said anything that falls into a black hole we will come to the question whether things do or don't fall into black holes looks from this point of view that it takes an infinite amount of time to fall into the black hole this is a contentious question that the it's not contentious it's a subtle question that we'll come back to but for the moment let's just think about things which fall onto the horizon of a black hole or a sense here they they get closer and closer and closer to the horizon and they sort of get stuck on the horizon from the point of view of outside they don't get back out if they don't get back out that means that all the information that they brought into the black hole is lost to the outside observer in the form of either depending on it we will discuss it later but depending on whether you envision it is falling into the black hole or just progressively closer and closer to the horizon of the black hole it becomes unavailable to the outside all right that's not such a bad thing you just say well okay but the information of what fell in is simply getting closer and closer to the horizon no big deal it's not not it hasn't disappeared it hasn't gone anywhere is it's just gotten squeezed tighter and tighter onto the horizon the problem is is we're going to see tonight or tonight and partly the next time this reason that I have not a reason it's it's extremely complete consensus about it black holes in time evaporate if black holes in time evaporate then the question is what happened to all the little itsy-bitsy bits of information or whatever it is that fell onto the horizon this is just disappear out of this world or is it somehow stored inside a little tiny remnant of the black hole it refuses to disappear or does it get radiated out with the evaporation products exactly what happens this this was a big puzzle in physics but before we can address the puzzle and it was string theory in part which largely resolved this before we do that we have to understand why people thought that black hole why people think that black holes evaporate now a black hole is the most dull object in the universe if you create the black hole and leave it for a while it settle settles down to a perfectly spherical totally uninteresting object nothing comes out of it it's totally black totally black in the sense that the the no light comes out of it and it's very uninteresting I can't get messages out of it or anything else but it turns out not to be quite true black holes are not as some dead let's call them dead dead in the sense that that they are infinitely cold infinitely what should we say infinitely black black has two meanings incidentally black can either refer to them it's emits no light at all or it can refer to the idea that it emits blackbody radiation blackbody radiation is thermal radiation heat radiation so let me just black holes are black and the sense that they emit heat radiation it was thought earlier that black holes were black in the sense that they emitted no radiation so we need to come to an understanding of what it was it made the Hawking bekenstein other people think that black holes have some heat in them they're not completely dead they have some heat and because they're here they have some heat they glow because they glow they give off light because they give off light or radiation they give off energy that means with time their energy decrease well energy and mass are the same thing equals mc-squared and so with time the mass should decrease as it gives off this radiation and if the mass decreases then this what chilled radius 2 mg decreases and one should expect in time if black holes really do have some active 3rd active heat in them that eventually they will simply disappear because they'll radiate away all their energy if that happens as I said there's the puzzle of what happened - everything is fell in 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 to really understand the concepts you don't have to and I don't feel up to teaching an advanced course in quantum field theory and curved space tonight it would take more than one night so we're going to do something simpler we're going to go through beckon Stein's arguments about why black holes have to have entropy now first before we talk about why black holes have entropy let's talk about entropy in general what is entropy and have you define it we can define it either with precision or we can define it qualitatively I prefer to define it qualitatively to understand what entropy is you need to understand what information is and let me be very very simplest simplistic about what information is information you can the information about a system the information about a system let's not ask about what it is but how do you quantify it how much information is there in a system or better yet how many how much information separates one system from another what's the what's the way that you X what's the way you explain to somebody the difference between systems well you ask a bunch of questions about the system I'm not even going to try to ask the question well yeah we could we could we could here's a system here's a system it's a big box the box is broken up into little boxes those little boxes are about as big as an atom and you can know everything about what's in this box if I tell you yes or no whether there's an atom I would have to tell you what kind of atom but let's simplify it there ought to be just one kind of atom in the world and either a box is empty or a box is full if I go through the boxes and tell you whether there's an atom there or not I have told you just about everything about that boxes and atom here there's an atom here there's an atom here there's an atom here there's an atom here here here here no other atoms well then there's a block or a lump of stuff in the middle okay maybe that one's missing now it's a lump of stuff for with a funny shape so you can go through this thing with a bunch of yes/no questions is there or isn't there an atom in this box that box and so forth another way of describing information is to say if I want to describe the system what I need to do is give a code to describe it and the code might simply be a series of zeros and ones where you go through this thing 0 0 0 1 0 1 1 0 in other words it could be a long binary digit how long well it depends on how many bits of information you're trying to describe how many yes/no questions it takes to describe a system completely the number of yes/no questions or the number of binary digits that you would have to prescribe to describe a system or to distinguish it from other systems is called the information in the system information theorists use it all the time and as I say they characterize it in terms of bits a bit is a yes/no question and okay so the quantity of information is characterized by the number bits in this system here the number of bits would just be the number of boxes the number of boxes where you might or might not have a an atom that's pretty clear I mean we could describe everything in this room by by breaking it up into 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 into even smaller boxes and simply ask whether they're well I suppose we would have to distinguish electrons from quarks but we can always break it down into and two yes/no questions is there a quark in this boxes into a quark in this box is there an electron in this box isn't there an electron in this box and eventually we would learn everything there is to know about this room by asking that series of questions the minimum number that it would take to describe the room and the distinguish it from every other room would be the quantity of information that describes that room no it could be it could be an instantaneous description yeah yeah yeah at a moment of time at a moment of time right right another fact information never disappears this is the basic idea in physics it's probably more basic than anything else in the meaning of that is the following that if you start with two different configurations two different configurations let's say here's one here's another one fairly similar as a matter of fact they just have a couple of atoms out of place are really different they're distinct the distinguishable distinct and then you can do an experiment in principle to tell the difference between them then if you let them run run means let them evolve with time they will stay different of course they will change relative to what they started with this one might move to here this one might move to here and so forth but if you start with two distinctly different configurations they stay different they don't run into each other for example you never have a situation where as you run these things they will evolve to the same configuration that's a very very bit that's such a deep rule of physics that people forget to state it when they're stating what the what the laws of physics are that the amount of information that it takes to describe a system is constant in time and that the distinctions between systems never disappear okay now sometimes it looks like information disappears you drop a series of water molecules into your bathtub and you might drip them in in some very very specific way you might drip the drops of water in with a Morse code messages on them to different to different bathtubs full of water and one of the messages I know some great literary masterpiece and the other one some other great literary masterpiece being coated in the dripping of a faucet and those two bathtubs full of water started out differently they sure look the same after a few minutes after a few minutes you let that water settle down and they surely look the same but that's only because you don't look carefully enough if you could look carefully enough at every single molecule in those bathtubs you will find that the configurations of them stay the same we might say that effectively information doesn't disappear but it gets hidden it gets hidden because after just because it gets hidden by the fact that there are just so many degrees of freedom and they're so small that you can't see them that in effect information which is there is effectively lost just because it's stored in degrees of freedom which are too small and too numerous to keep track of so in practice and the sense information does disappear but not in principle there's a notion of hidden information information that's inaccessible to you for one reason or another now of course the notion of inaccessible can depend just on how good your your ability to study and manipulate the system is it might depend on how good your microscope it is it might depend on how quick you are and your ability to measure molecules and so forth but subject to whatever limitations there are there's a notion of inaccessible or hidden information what is entropy the simplest statement of entropy is that it is the number of hidden bits of information a number of things which in principle are there to distinguish things but which are just unavailable to you because they're stored in things which is too small and new too numerous to keep track of so in the case of this bathtub full of water so the only thing I really care about is how hot the bathtub is I don't want to burn myself when I get in I want to make sure there's enough water in the bathtub that I can take a nice bath I want to what else I want to make sure its water not the hydrochloric acid but you know there's a handful of things that I want to know about that water and furthermore is a handful of things that I can know about the water the amount of energy that's in it that in the form of its thermal energy the fact that its water and and a few other things and that's a pretty much complete description of the water well what I might have ripples on it I don't care much about the ripples that I'm thinking about but nevertheless I could see the ripples and I could distinguish the fact that the bathtub was filled up five minutes ago than half an hour ago but if you wait a while even those ripples go away and there's not even any ripples on the surface to tell you any detailed information it's all hidden in the microstructure of the molecules so all that information is hidden information and it's called the entropy that's what entropy is the number of bits of information which are unavailable there are more technical definitions but for the moment I think that that is good enough for us incidentally entropy and energy are the basis for thermodynamics normally when you learn thermodynamics so statistical mechanics particular mo dynamics you start with the idea of temperature temperature is a highly derived concept well you know it's what you measure with a thermometer very simple yeah would you measure where the thermometer is a very very highly evolved the thing it's easy to measure but it's very hard to define the definition of temperature what is the definition of temperature anybody know the definition of temperature no of course you don't because the definition of temperature is a highly contrived thing not not so much contrived it's a it's a a what's the right way a derived concept derived from the ideas of entropy and temperature okay so I'm going to tell you now our we have energy I assume we all know what energy is and we have entropy now let me give an example entropy might be the amount of hidden information in the atmosphere okay I'll tell you right now what temperature is I have to write an equation first for myself just so I get it right okay temperature is the change in entropy of a system if you add one bit of information to it if you chant now here's here's an example I'll give you an example of where this comes up you take your computer your computer is full of bits of information the bits are stored individually in the form of in the form of what them bits yeah I mean whatever some little thing inside the computer is either switched on or switched off and that's your yes or no question now your computer is full of information that you put in there supposing I want to get rid of a bit of information in other words I want to eliminate some I'm not going to eliminate the circuit I just want to set it back to its original starting point I want to take the computer and reboot it just to get it back to its original starting configuration in that case I will lose a lot of information all the information that's in the computer well let's not go that far let's just erase one bit of information if you erase one bit of information I just told you it's impossible to erase bits of information that's that's the basic law of physics what really happens is that bit of information whatever it is is displaced from inside the computer to out into the atmosphere the distinction between one configuration of the computer and another configuration which differ one or by one bit that distinction gets taken from the computer into the atmosphere so that the atmosphere becomes a little bit different by one bit of information are than it was before you erased the bit that process of transferring a bit from the computer into the atmosphere heats the atmosphere adds a little bit of energy to the atmosphere it's an inevitable heating that happens because you erase bits from your computer what is the energy that's transferred to the to the atmosphere when you transfer one bit the answer is the temperature that's the definition of temperature the change in the energy of a system when you add one bit of information to it is called the temperature you could write this another way you could write the change in the energy when you change the number of bits but by some number is the temperature times the chain well okay let's let's let's um let's slow down notation for entropy we need a notation for entropy I don't want to say entropy every time and I don't want to write the word entropy on the blackboard in equations we need a symbol for entropy the standard symbol for entropy is s s is entropy okay now what I told you is that the change in the energy let's call it Delta e when I add one bit that means when Delta s when Delta s is equal to 1 if Delta s is one unit then the amount of energy that's added to the system is called the temperature what happens if you add two bits for the at to the atmosphere then the energy of the atmosphere changes by twice the temperature what happens if you add three bits three times a temperature what happens if you add s bits then the change in energy is T times the change in entropy the change in energy is proportional to the number of bits that you transfer to the atmosphere that you transfer to anything a bathtub full of hot water in other words it's the change in the energy when you hide a certain number of bits of information when you remove them from someplace and hide them in your in your whatever it is you're hiding them in in your heat bath basically this is a basic formula of thermodynamics extremely confusing formula if you start with temperature as the basic concept but it's a very simple concept if you start with entropy is the basic concept then temperature is just said the energy that you add when you add one bit and this is usually written differentially D s sorry de equals TDS this is one of the fundamental laws of thermodynamics the change in energy is the temperature times the change in entropy that's the definition of entropy hidden information stuff that you can't easily see can i I did a calculation comparing a change in entropy of a hard drive if you erase it 100 gigabytes and how much that trip cure would be for a glass of water it's about 10 to minus 15 degrees who is it back that's that small change of temperature is how much about 100 gigabytes of information for hard drive yeah oh yeah a bit of information at an ordinary room temperature that's in room temperature is very low for one thing yeah every room temperature right so room temperature in any kind of reasonable other units is very small and yeah a bit is a very typically a very small amount of energy at ordinary temperatures that's a real fundamental law of computers that's some or who's our land ours Lendl and ours of rule about the ratio of information erasing it means what eraser you haven't erased the object which held a bit all right but let's say you see here's what you're going to do you're going to whatever the configuration of the computer is you're going to turn that bit to off so that so you will have forgotten what was there because you turned it off in that sense you've erased the knowledge that that that that little circuit had you've and eventually you could turn off your turn off by books by rebuilding to the computer or by starting the computer over and bringing it back to the configuration as when you bought it there's nothing left in that that remembers what you did to it so you've erased it you've erased it and the number of bits that you've erased in other words the number that you brought back to neutral is the change in the entropy of the computer and that or if you like that's the number of bits that have to be taken from the computer put into the atmosphere and that's what what you calculated when you calculated there yeah yeah yeah yeah yeah yeah that's right that's right you do a lot of other stuff besides erase the bit normally like you rub your finger on the keyboard and all kinds of other I hide in one bit of information horse origin castle yes well you're right will heat will heat the system well we'll increase the energy of the heat bath by more the higher the temperature the more energy you will be putting in when you hide that bit yeah that's that's exactly what it says but in the sense that's the definition of the temperature so it's a question of horse before the cart or cart before the horse yeah and yes yes changes the amount of information decreases the amount of information in the in the computer but that information cannot disappear out of this world so you end up putting it into the atmosphere but it's hidden in the atmosphere and so it's entropy okay that's the basic idea of entropy why my teacher didn't teach me that when I studied thermodynamics in 1959 I don't know I went around asking my colleagues when you teach a statistical mechanics that you teach that and basically what I found out is none of my colleagues knew it not good hmm hidden information record the computer science the computer scientists all know it right yes yes yes but these ideas that's right but these ideas go back to Boltzmann these ideas are older than computer science actually there was quite uh no all right in statistical mechanics and in physics they go back to Boltzmann in the 19th century in computer science they go back to the 40s Shannon Claude Shannon was the one who formulated these ideas okay enough history okay so now let's let's talk about the information that's hidden in a black hole as I said you can either think of it as hidden in the black hole or you can think about it as hidden on the surface of the black hole you keep throwing stuff into the black hole and it accumulates closer and closer to the horizon in some sense and becomes unavailable it's all smooshed up onto the The Horizon it takes light forever and ever and ever to get out of the horizon so it's effectively lost why did bekenstein why did beckon Stein think that black holes had entropy well I'm going to tell you the way I think about it but it was also the way back in Stein thought about it but I'm going to tell you why he thought that was entropy by computing the entropy by computing the amount of information that you can store in a black hole what I'm going to do is imagine building up the black hole by flowing bits of information into it and I want to know after I have thrown in a certain number of bits of information how big is the black hole the logic is very very similar to asking how full will a bathtub be if I fill it by dropping in atom by atom and I tell you after in atoms how big alpha will the bathtub be all the way you figure it out as you say an atom has a certain volume every time you drop an atom in it adds to the volume of the water and if you Inc and each each differential drop changes the volume by a little bit and raises the level of the water by a certain amount so if you want to know how high the water will be for a given number of drops what do you do you just count how much will the water be raised by dropping in one drop of course you don't have to be that dumb you could be smarter than that but that would be one way to do it just drop by drop by drop and discover each drop that you put in has a certain volume therefore it will raise the water a certain amount after n drops the water has a certain height going to do the same thing by dropping in bits of information to form a black hole we can start with a small how does throwing in photons electrons chairs would lead to throwing in information okay so are we going to we're going to come to that but basically we're going to we're going to throw something in which we're pretty sure has no other information in it other than the question of whether it's there in the in there or not in other words we're going to fill this black hole up by a series of processes for which each one of them has no other information stored in it other than that a photon went into the black hole you can simply ask did the photon go in or didn't the photon go in that's a yes or no question and but it doesn't take very long for it to get squashed onto the horizon where it becomes an unavailable yeah right boo-yeah that sort of other example or the polarization of the photon the polarization of the photon is it's either upward down along the x axis so it's one bit of information but you have to be careful ah and here's what you have to do a lot so we're going to start let's take a black hole there's a black hole it has mass in it has radius R or the horizon has radius R which is twice mg we've already made it we've got it there now we want to throw in one more object and that should be the simplest possible kind of object for which we can simply ask is it there or is it not there one bit of information let's think of as a photon boy Anthony careful if we throw in a photon from here let's suppose this black hole is a kilometer in radius and we throw in a photon along a beam of laser light where the photon has a wavelength which is you know some microscopic wavelength then there's a lot more information in that photon in the arrival of that photon than just whether it got in or not for example did it come in from here that it come in from here that I come in from here there's the angle on the horizon where it arrived from that's clearly going to there may be a lot of bits of information in describing the angles to various amounts of precision so throwing in a photon is not by any means necessarily the same as throwing in one bit of information there's a lot more information you know you could ask you could begin a yes/no question as follows is it in the upper hemisphere that it came in yes okay is it in the upper half of hand and the sphere to the right or no okay it's to the left and now imagine dividing up the horizon of the black hole and asking a series of yes/no questions until you've localized where that photon came in to the maximum degree that it can be localized it can't be more localized in its wavelength it's fuzzy out to its wavelength but it's clear that there's a lot more yes/no questions you can ask about that photon coming in and so that photon is not one bit of information it's a lot of information unless you do something to that photon or you deal with a photon which is so uncertain about where it is because photons are always uncertain they don't have any brain but unless in principle there is an 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 in practice that means that the wavelength of the photon the wavelength of the photon is a measure of how delocalized the position of the photon is in quantum mechanics we're talking about quantum mechanics now if the wavelength of the photon is as big as the whole black hole then when that photon arrives it arrives in a way where there's complete uncertainty about where it entered from so the first thing you want to do is you want to work with photons which have the longest possible wavelength but can still get into the black hole is there a restriction our not being able to get into the black hole yes it's a fact that can that's been known basically from classical physics that if you shine light on a black hole of wavelength longer than the size of the black hole it just reflects off it will not go in only wavelengths which are no longer than the radius of the black hole horizon have any chance of penetrating through to the horizon so what's the best you can do you can work with photons whose wavelength is comparable to the radius of the black hole they are so uncertain in their location that you can't tell where they enter the black hole on the other hand they're short enough wavelength that they can get into the black hole so you're working just on the margin on the edge where the photon will go into the black hole but where it will provide no other information other than it either got in or it didn't get in okay so let's add a photon to this black hole I mean it seems like that's more information well and it it really isn't that the wavelength is very long because the wavelength is so long that the electric field is varying very slowly across the black hole and it doesn't matter which direction it comes in from but but you'll like to ask the question but I think the answer is no that if the wavelength is long enough that it doesn't it doesn't provide any information okay good so we want to send in a photon of wavelength lambda equal to 2m g now first question what is the energy of such a photon we're going to add we're going to be adding some energy now is this business that that whenever you hide a piece of information you'll wind up adding a little bit of energy to the system that you're hiding the information in so how much energy have you hidden inside the black hole or how much energy have you added in the black hole the energy of the photon right so we'll just write down a formula or two about energies of photon all right then just to remind you the energy of a photon or any light wave is equal to the speed of light times its momentum P is the momentum of the photon now what's the connection between momentum and wavelength anybody remember h-bar over lambda H bar plunks famous constant all right this is because we're dealing with photons because in quantum mechanics information is quantized into photons this is a real quantum problem we're going to see where h-bar winds up it will be interesting to see where h-bar winds up in this formula all right so what's the wavelength the wavelength or the mum of the momentum is h-bar over lambda lambda being the wavelength the shorter the wavelength the higher the momentum so now we can say that the energy of the photon that we threw in is the speed of light times Planck's constant divided by lambda is it is it h bar over lambda or H bar over lambda bar which is same as H over lambda I'm not I'm not keeping track of two PI's teach PI over lambda isn't it yeah H bar Valen H over lambda that's are the energy is H over Lin I can it is C times peak that's not important I don't care about two PI's let's let's forget two PI's not important order magnitude energy is CH over lambda and ya P is P is H h-bar times the wave number and there's a 2pi you write this is 2 pi okay but there's this little H over them non non point H over lambda the energy is H over lambda now lambda we're going to be choosing equal to the cop to the to the size of the black hole so now we can write down that the energy that we've put in apart from factors of 2 I'm not interested in factors of 2 here we can't that this calculation is not going to be that precise with within a factor of 2 all right so the energy that we're adding to the black hole is the speed of light times Planck's constant divided by M over G and that adds one bit of information to the black hole now I could stop right here I have now told you how much information it cost how much energy it cost to hide one bit what do I told you the temperature here is the temperature of the black hole see H over mg that's the Hawking temperature now there are some factors of 4 and PI's and things that we're not going to get right by this little argument but let's let's let's go a little bit further this is one bit of information that we put in let's see how much bigger it makes the black hole let's see if we can see how much bigger it makes a black hole it makes the black hole bigger because when you add energy you add mass equals MC squared all right so e equals MC squared so this can also be written as the change in the mass let's call it the change in the mass times C squared the energy that I added goes into increasing the mass and it changes the mass by this formula here let's clean up the formula a little bit let's divide by C squared and that lines up putting the C in the denominator here that's the change in the mass that's the minimum change in the mass when you add one bit of information is H bar over mg ok let's go a little further let's calculate the change in the radius of the black hole let's calculate the change in the radius of the black hole how do I do that I do it because I know that the radius is proportional to M times G so I just multiply I multiply both sides of this equation let's see am i doing am i umm yeah RG we do this was a yeah this was the wavelength downstairs so we multiplied by G and this is the change in the radius of the black hole let's see where did I do that right the wavelength over the Schwarzschild radius HC over the Schwarzschild radius is the change in energy that's Delta e e equals MC squared so if I want the change in mass I think I want to divide it by C squared the change in mass is equal to C H now is equal to H over R C agreed have I confused you thoroughly okay good that's the change in mass and now if I multiply it by G it gives me the change in radius this is the change in radius of the black hole okay so every time I throw in one bit of information I change the radius by an amount which is inverse to the radius itself with the gh over C I really think I've lost some C's but I'm not sure where well maybe not hmm where oh oh oh you're right you're right you're right the Schwartz good good good the Schwarzschild radius is not mg it's mg over C squared that's where it were an mg over C squared so I think this should be by now C cubed if I'm not mistaken yeah that's it thanks Eddie see who wanted part of the thing not the other part yeah I think it's the I think it's G H over RC cubed that's the change in radius now multiplied by the radius R times the change in the radius what's R times the change in radius RDR that's d R squared alright the change in R squared apart from a factor of 2 is 2r they are just elementary calculus so R times a small change in R is proportional to the change in the square of the radius but what's the square of the radius the square of the radius of the horizon is the area of the horizon so look what this formula says it says every time you drop a bit of information in you change the area of the horizon by a universal amount which doesn't depend on the mass of the black hole it doesn't depend on the radius of the black hole at the instant that you did it you always wind up changing the area by the same amount it's kind of like saying that every time you drop a drop into a bathtub assuming the bathtub have nice straight walls that you always raise the level of the bathtub by the same amount and it doesn't matter how full it is you always raise the volume by the same amount in this case you increase the area of the black hole by one unit where unit means the area G H over C cubed okay this is a very small area every small area is the square of some of every area is the square of some length of course so this is the square of some length it's the square of the square root this quantity has units of length let's say it's the plunk length it's called the clock length and it's very small let's see if we can estimate it okay G in ordinary units units means kilograms meters seconds G is about seven about seven times 10 to the minus 11th how about Planck's constant I think well that's h-bar I think is that H or H bar six Adam remember another seven that okay seven times 10 to the minus 34 that's this and C cubed a let's EQ left square we're going to take the square root in a minute but what's the square of the speed of light the cube of the speed of light Cuba the speed of the speed of 27 times 10 to the 24th all right so we got all 27 times 10 whether I say 24th okay we got a lot of small numbers upstairs and a lot of big numbers downstairs this is a very small number I think it's about 10 to the minus 60 eighth or 69th or something like that 10 to the minus 69th square meters or something like that I don't remember exactly okay I happen to remember it's 10 to the minus 6666 someone's about that square centimeters okay the Planck length itself the square root of this is about 10 to the minus 35 centimeters so it's smaller than anything that you can access in any way in experimental physics and what it says is every time you drop a bit on the bits I mean the way to think about it is the bits are impenetrable and they stick to the horizon in no finite amount of time do they ever get past the horizon they stick the horizon they have sharp elbows and they simply push the other ones out of the way each one occupying about a plank area the first one was sent in its approaching horizon now the next one comes into the prize and moves out well it pushes it pushes the other ones out of the way and it makes the area bigger and the only way to make the area bigger is to make it bigger it doesn't just be situation out the prior one is inside not in science no no no they never get past the horizon even if you added a huge mass - - like - colliding black holes that's when they come here yeah that's or we will will will go through a bit of that we have to define what the horizon is in a careful way we have to define the precise concept of a horizon which we haven't done yet we've only yeah this has been uh theistic probably using significant but on a the area that we're saving plank that Square is a blank area on a flat surface but Thomas was here a circle so for your corner line black hole you waive ones do you want this photon is eight o'clock right long radio nice long radio wave if you took a basketball around the spinning round all and you do the same thing or seem to see white the same distance regularly okay but that that was the magic of death jacob bekenstein argued and in this way argued that the information that you can hide in a black hole is proportional to its area and plunk units the formula as it was finally eventually derived and i'm going to write it with C equals one maybe I'll try to put the C's in wait let's see I I know I hesitate to try to put the season there's something they go in somewhere else but though the entropy or the hidden information in the black hole is equal to the area of the horizon we think about building up the the black hole by dropping in bit after bit each time you add one unit of area divided by G and divided by H bar I think there's a C cubed upstairs yeah C cubed upstairs okay this means the entropy of a black hole is enormous by comparison with the kinds of entropy x' and ordinary systems of the same size you have C cube that's a huge number you have G and H bar which are tiny numbers in the denominator the entropy of a black hole is bigger than the entropy of anything else of comparable size by a lot I mean anything else that you ordinarily would think about so huge numbers of bits get stored on the surface of the black hole proportional to the area not the volume so the amount of information you can keep track of in a black hole that needs to be kept track of is less than you might have expected you might have thought it was the volume of the black hole but it's still huge all right this is probably one of the most important principles of physics sorry I miss something for ya bekenstein had this formula with an unknown coefficient he tried to estimate the coefficient they need all sorts of arguments but the wound upper with peculiar numbers like logarithms of 2 and other things which were are not there it's just four it was talking who figured out the four and good all right well let's or whatever there are some interesting there's one very interesting thing about this formula and that's that the H bar is in the denominator what's the meaning of that well you might have thought that entropy the existence of entropy for a black hole is a quantum mechanical effect has something to do with quantum mechanics and so you might think that the entropy is proportional to Planck's constant because it's Planck's constant which is the measure of the quantumness of a system but it goes in the denominator if Planck's constant would they get smaller and smaller as it does in classical physics in classical physics Planck's constant is zero it says the entropy is infinite that's just another way of saying that in classical physics you can hide since information is not quantized it doesn't come in photons you can put as much information into as small and energy as you like take an arbitrarily small amount of energy you can put as much information on to that or wave an electromagnetic wave as you like but in quantum mechanics information comes in discrete quanta the information or the 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 mass by almost anything so this is an interesting fact that although all of this is quantum effect it's quantum mechanics which prevents the entropy from being too big not the other way makes it finite um what else can we say about this well we could say a lot more about it we will I think I'm getting a little tired so we'll well h-bar is simply a constant which has certain units as units of angular momentum it's true but it comes up in other quantities you have used action no it's not H over 2 pi the convert Center no no no angular momentum and momentum differ by a power of a distance angular momentum is momentum times distance okay momentum is H over lambda lambda is a distance P times lambda momentum times distances units of H bar and P times lambda also has units of angular momentum R cross P distance times momentum okay see you next week for more please visit us at stanford.edu

Channel: Stanford

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Keywords: physics, science, math, thought, theory, string theory, relativity, special, general, space-time, space, time, quantum mechanics, atoms, universe, particle physics, theoretical, philosophy, dark matter, gravity, cosmic

Id: dmS5J6xM5s0

Length: 94min 37sec (5677 seconds)

Published: Thu Jun 02 2011