Einstein's General Theory of Relativity | Lecture 12

this program is brought to you by stanford university please visit us at stanford.edu okay so we were about to embark on a distant journey into a journey into a large black hole before we do let me remind you of one or two facts i just hope we have enough oh i actually brought some pens anyway let me just go back for a moment the accelerated reference frame and just remind you of what the various facts about it are excuse me the accelerated reference frame was a frame in which every observer is moving on a hyperbolic trajectory as you get closer and closer to the light cone here the trajectory has an increased acceleration and you can see that from the fact that it has to turn the corner in a short in a smaller amount of uh space so it's not like an accelerated coordinate system in newtonian mechanics where everybody part takes of the same exact acceleration and all move off simultaneously but rather the acceleration gets larger and larger as you move forward is this an example of why special relativity does not apply to accelerated normally this is an accelerated reference plan yes but that's also true and that's also true in the vicinity of a black hole that this this point or this whole line in fact it replaces the black hole horizon and if you're standing if you're supported 10 meters 100 meters away from the black hole horizon you have one acceleration well let's first take the earth if you lower down to the surface of the earth well from a large distance and not lower down but you're supported of course your acceleration is larger the closer you are to the center of the earth so it's not that surprising uh in that sense the accelerated reference frame is not a very precise version of a gravitational field because gravitational field varies here in special relativity we already have this property that the closest thing to a uniformly accelerated reference frame already has this fact that the gravitational strength or the acceleration that you feel varies from distance from this point over here okay we also discussed um these are the different observers they could be characterized by a coordinate r which at least along here just represents how far they are from the origin of coordinates over here a characterized band are and they're also characterized by a hyperbolic angle the hyperbolic angle did i call it omega last time what did i call it omega such that if you want the precise relationship with the original coordinates on the blackboard to call them x and t then to be to make the relationship precise x is equal to r times the hyperbolic cosine cosine of the hyperbolic angle omega and t is equal to r times the hyperbolic sine of omega very very much like polar coordinates in which x and y completely different uh system now but just to just by analogy i don't use x and t just x and y we would write that a point is given by an r and not an omega but a theta i use omega for hyperbolic angle but it's a kind of angle it doesn't go from zero to two pi sorry from yeah it doesn't go from zero to two pi goes from minus infinity to infinity and it's a time-lag variable it's obviously moving more in a time direction than it is in a space direction and it's some sort of time along these trajectories but the analog here would just be x equals r cosine theta ordinary cosine not hyperbolic and y equals r sine of theta okay that's just to keep in mind the analogy and if we work out well first let's write the um the metric in polar coordinates the ordinary metric the x squared plus the y squared just becomes the r squared plus r squared d theta squared there might be other directions sticking out of the blackboard we would add them in i'll do that over here likewise the metric over here or similar and similar spirit the metric over here just becomes d r squared well let's be careful signs signs are important the r squared with a minus sign this is proper time the tau squared and we have r squared d omega squared which is the analog of the r squared the theta squared term here omega is like time so it comes in with a plus sign here r is like space so it comes in with a minus sign this is equal to d t squared minus the x squared now if there happened to be and of course there are additional coordinates the additional coordinates can be thought of as coming out of the blackboard what would those additional coordinates be let's call them as y and z then we would also write minus d y squared minus dz squared and we do the same thing here so it's mainly r and omega which are the new things and they're kind of hyperbolic polar coordinates lines of constant omega which mean lines of constant time in the accelerated coordinate system look like this this is omega equals zero for example we also can have negative omega omega equals one omega equals two omega equals three omega equals four dot dot dot dot and omega equals infinity the end of time from the point of view of the accelerated reference frame is just this light cone here which is x equals t x equals t of course means x equals c t if i put back the speeds of light and so this is just a motion of a light ray okay that was the that was the hyperbolic polar coordinates we also pointed out that anybody who finds himself back here cannot send a message out to to the observers who are out here of course an observer out here can jump in you can just let go of whatever the accelerating mechanism is let go and fall into here in which case he can make contact but if he stays outside which means if he stays on these hyperbolic trajectories but in particular stays does not cross this line then uh let's give them names this is alice over here alice who falls through the horizon and bob well bob over here who continues to accelerate after a point bob cannot see alice anymore no that's not quite true bob looks back let's ask what bob sees what does c mean c means to look with uh with light bob can see everything backward along his own trajectory can see everything backward along light like directions and it takes an infinite amount of bob's time omega goes to infinity before he's in a position to actually see alice right at that light cone there so from bob's perspective he sees alice asymptotically approaching this line we might as well call it the horizon it would be improper to call it a black hole horizon there's no black hole here sometimes it's called an acceleration horizon or a rindler horizon so bob sees alice asymptotically approach the middle horizon relative to bob alice is moving faster and faster particularly as bob it's bob who's doing the moving from our point of view but bob is moving outward with an enormous velocity that means that he sees alice moving with a large velocity what happens when you have somebody with a large velocity they get lawrence contracted so bob will see alice getting thinner and thinner in the vertical direction not in the horizontal direction and as she gets sort of squeezed onto the horizon but alice doesn't see anything out of the ordinary she just goes right through all right so that was a setup which was intended yeah so will bob see alice's time seem to slow down well well bob see al yes yes bob we'll see alice's time slow down what will alice see here she just looks back and she sees somebody scooting along an accelerated trajectory that okay keep in mind there's some asymmetry here when you talk about what bob sees and what alice sees let's draw the diagram again there's some asymmetry the question is where the asymmetry comes from but let's let's see what the asymmetry is okay so here here's bob here's alice and for bob to see alice he has to look backward along light-like directions like that he looks backward we all look backward i mean when backward in time uh we can never see anything at the same moment that we're looking we see things in the past he looks back and he sees alice at that point a little bit later he sees alice at that point a little bit later he sees alice at that point and so forth alice looks at bob she doesn't look in this direction she looks back this way and she never sees bob disappear in fact she just sees bob accelerating away from her this means she does see bob's motions slow down because bob is moving away from her but something moves away from you they seem to slow down but alice in this picture never sees bob disappear or anything else she just keeps watching him as he recedes further and further into the distance along the accelerated trajectory i have a little confusion i i see that the regular space is a there's a transform of minkowski yeah it's just coordinated it is a coordinate transfer but now we're imagining i don't see the fact i don't see the uniform acceleration no we're imagining now a collection of observers this is not just a coordinate transformation it is a coordinate transformation but now we're imagining observers who are moving at a fixed r so imagining a collection of observers each one of which is at a fixed r that means they're accelerating off and we're asking what they see it is a coordinate transformation but at the same time we're imagining that there are people located at fixed values of the spatial coordinates r means a uniform arrangement the omega squared uh it's not quite proper times no it's not proper time sorry squared over rod d omega squared is not let's say r equals fixed r fixed r right thick star ds uh what are they right here d tau along one of these trajectories along one of these trajectories the r is zero the x and the y is zero so along one of the trajectories d tau is just equal to r d omega and r is fixed but the rate of proper time relative to the rate of omega differs from r to r from one r to another okay so if you had exactly the same kind of clock over here as over here as over heroes over here namely clocks which tick off proper time they would not be ticking off omega if you want clocks which tick off omega then those clocks have to have uh different rates uh of reading off proper time okay but that's that's just a question of coordinates uh you can imagine clocks which are built to tick off omega you'd have to have them built differently at each point here but so let it be for example the proper time between here and here is longer than the proper time between here and here okay so if we wanted clocks which built or which read off omega they would have to run at a different rate each one would run at a different rate suppose you had an observer um who was uh going along one of those paths uh one of those trajectories only their actually their their real x component r i'm just trying to put him in a different point in space with the same trajectory in other words you want to put another trajectory for space yes but which happens at the same shape yes same acceleration so you have okay good yes i have heard someone argue that they have different gravitational potential because of the fact that one has the displeased remember the gravitational principle is not an invariant thing you can add a constant to the gravitational potential it doesn't change anything you could ask whether they feel different force on the bottoms of their feet they don't they feel exactly the same force on the bottoms of their feet if they're being supported by so they don't see each other moving one direction or another no they don't stay the same distance apart from each other in their own frames of reference since they're since they're parallel trajectories let's draw them they correspond to shifting the origin by drawing the same curve okay now in the original coordinates this distance is equal to this distance is equal to this distance is equal to this distance just by we've always done is translated the hyperbola but in in their own reckoning they don't measure distance along here well at this point he does measure but here if all of these observers are maintaining constant relative distance in their own frame of reference sees the one ahead of him always being the same distance away natural here because of lorentz contraction right so you could call this an accelerated coordinate system in which you just displaced each one by the same uh by the same amount but it would have the property that an observer moving along here would see the guy in front of him receding away and not staying the same distance well you got to figure it out i got to figure it out i'm not sure what each one sees but um but whatever they see the distance that this person over here sees to him c is perhaps the wrong word the distance that would be measured by a measuring rod along here would not be the same as the distance along a measuring rod along here where this line here was chosen to be the space-like surface for a moving observer over here would not be the same so i believe that i believe he would see him recede but i it's something we got to figure out and uh right uh the properties of what each one sees would depend on time in particular so it wouldn't be a static description from the point of view of the accelerated reference frame okay so that's uh good now let me write down i think i did write down the short shield black hole metric last time did i not the main thing i want to do today is to show you the connection between this and the schwarzschild metric once you know that connection between that picture and the schwarzschild then there are many many questions that you can answer simply by thinking of the corresponding question here okay so let's uh let's write down the schwarzschild metric which looks entirely different here it is the tau squared equals 1 minus 2 m over r i'm setting the speed of light as usual equal to 1. g thank you dt squared minus 1 divided by 1 minus 2 mg over r the r squared plus r squared not plus minus minus r squared times the metric of a two-dimensional unit sphere y this is what we're using basically is um spherical polar coordinates ordinary spherical polar coordinates just wouldn't have these mg's here would be dt squared minus the r squared minus r squared times i think i called it last time the omega squared and the omega squared is um d theta squared or is it the phi squared the phi squared i guess plus either cosine or sine depending on where you measure the angle from sine phi squared d theta squared where theta and phi just looking around you and there's a polar angle and a and an azimuthal angle that is the metric of the schwarzschild black hole that's all there is to it and analyzing this taking apart and understanding it uh is will teach you about as much about black holes as well it'll teach you a lot about black holes okay incidentally just to remind you this is also the metric of the earth's gravitational field but only out beyond the surface of the earth in other words this is the solution the general solution of the problem of a spherically symmetric distribution of mass outside of the place where that mass is in other words for r larger than the boundaries of the of the chunk of material as we take as we take the object smaller and smaller and smaller eventually it tends to be this everywheres but of course there are some problems in this metric some singularities and there are two kinds of singularities in this metric one of them is a really nasty singularity where if you encountered it you would not be very comfortable the other is a coordinate artifact in other words a funniness of coordinates okay the horizon is the place where one minus two mg over r becomes equal to zero two mg equals r that's the horizon and why do i say something singular is happening well first of all there's something happening to the dt squared notice that if you're sitting right at r equals 2 mg then the proper time and the ordinary schwarzschild time this is called short shield time have an infinite ratio of the rate at which they run any amount of ordinary time corresponds to zero proper time so that's a little bit funny but it really we've already seen it we see it right here the tau squared at r equals 0 which is right at this point over here this is a different r incidentally i think we are going to run into trouble if i continue to call this r over here the same r is over here but uh no i'm going to call it row row i'm used to calling it row so i'm going to call it roll d row squared of course it's it's bound to happen that we'll get into these kind of conflicts of notation i called it r originally because i wanted to compare it with ordinary polar coordinates well we always use r on the other hand um that r we will see is not quite the same as this r it's related to it but it's a little different and this thing has been called r also from time immemorial so we have to change notation we have to give way to some other notation either on that side or that side and i prefer to do it over here okay now remember though that r and rho are related to each other they may not they may or may not be exactly the same thing we'll find out but notice over here here we have this omega which is the time from the point of view of these accelerated observers notice that at rho equals zero that any amount of d omega corresponds to no amount of d tau at rho equals zero something like that is going on over here at 2 mg equals r arbitrarily large amounts of dt correspond to no proper time at all okay so these are funny points now there's nothing physically important going on here it's just that we've chosen coordinates which have a particular center over here such that i mean it's exactly if we went to polar coordinates ordinary polar coordinates it would just be the statement that at the center any amount of angle at all right at the center corresponds to no circumference okay as you move out a little bit of angle corresponds to a larger and larger piece of um of a perimeter of distance along the perimeter so this is kind of stagnation point in the coordinates over here but it's not anything physical going on uh so you see there are situations where the coordinates can do something funny where there really is nothing going on or at least nothing that the observer at that point would notice okay we're going to try to find out is this now that's first of all one thing happens at the 2 mg equals r the coefficient of dt squared goes to 0. that's got to do with the fact that clocks slow down as you get closer and closer to the horizon we'll come back to it i think but uh but and that wouldn't be so bad but this also does something bad at r equals 2 mg at r equals 2 mg this goes to 1 1 minus 1 goes to 0 and even worse than going to zero the coefficient of dr squared becomes infinite question all right so it looks like there's something nasty going on at r equals 2 mg well if not from this term over here which is doing something fairly similar to what's going on over here the the r squared term is blowing up in our face so from one point of view that sounds like there's something nasty there we'll see that there isn't next at r equals zero there's also something funny happening at r equals zero the coefficient of dt squared blows up but even worse it blows up with a negative sign when r is equal to zero this becomes big 1 minus something big is negative something big so the coefficient of dt squared blows up with a negative sign and the coefficient of dr squared goes to zero why no is that right let's see yes i think so at r equals zero what's in here becomes infinite and therefore the coefficient of the r squared gets smaller and smaller but also with the wrong sign so something is happening at r equals 2 mg and in fact it's interchanging the signs of dt squared and dr squared in other words let's uh let's imagine that r is less than 2 mg then this is negative and this one is positive this term has negative this one has been interchanged of what was space-like with what was time-like now does that mean that space has become time and time has become space no it just means that the coordinates are doing something funny and we'll come back to what it is all right let me change let me take apart this metric a little more instead of working with row i want to invent another variable i don't know what to call it i've run out of radial variables i hate to introduce another coordinate okay i'm going to do it i have to i have no choice i'm going to take rho squared and give it a new name okay let's see yeah give me a name for rho squared capital r okay rho squared equals capital r then we have of course that this is equal to r d omega squared that goes to zero when r goes to zero that's not surprising so did rho squared d omega squared but how about this d rho squared what is that let's calculate that let's calculate the relationship between d rho and d r what is d r d capital r d capital r is equal to twice rho d rho right okay let's divide that and notice that d rho is the r over twice rho now let's put in d rho squared the rho squared is minus the r squared divided by in this case it happens if there's a 4 there the 4 is not the interesting thing we could get rid of the 4 by a trick but uh with rho squared row squared in the denominator right everybody see what i did i just wrote d rho is equal to d r divided by 2 rho and i stuck it into there what is rho squared rho squared is r forgetting about the four the four the four we could actually soak up into a into a uh into a change of units of r i simply actually chose the wrong uh the wrong definition i should have put a one half in here or something would have gotten rid of this four but there it is let's look at this metric this now has the omega squared term goes to zero just like this one does at r equals zero at r equals zero it has the same property as this namely something goes to zero on the other hand look what's happening over here also at r equals 0 the coefficient of the dr squared term is getting infinite what was in the numerator here appears in the denominator here what's in the numerator here appears in the denominator over here there's nothing pathological in the space time over here i've just used some pathological coordinates which have the property that the metric has in terms of the time component it has something which goes to zero at r equals zero and in terms of the space component that has something which goes to infinity there's something shall i slow down here okay everybody with me all right so we see in itself there's nothing pathological in the space just because the coefficient here goes to zero and a coefficient here goes to infinity in the same way as i said the the relative four which appears here is not is not an important issue now there's something else if we looked at this metric here and we thought of r as our coordinate notice what happens when r goes from positive to negative when r goes from positive to negative this term becomes negative and this term becomes positive whatever r less than zero means if it means anything at all it has this interchange of space and time just as this does just as this does so the question is can we understand what r less than zero r is rho squared how can rho squared be less than zero okay the point is rho squared less than zero is up in here let's see if we can understand why that is each one of those hyperboloids or hyperbolas corresponds to let's see what it is it's x squared minus t squared equals rho squared each one of those hyperbolas corresponds to a different value of rho and precisely the same way as for a circle we would have x squared plus y squared equals rho squared here we have x squared minus t squared equal to rho squared which happens to be what i called r also known as rho squared what happened when r got negative look at this what happens when r gets negative let's suppose r gets to b goes from plus one to minus one so x squared minus t squared equals one that's a hyperbola like this what happens when it gets to be minus one then it becomes x squared minus t squared equals minus one or t squared minus x squared equals one that's another hyperbola where is that hyperbola 90 degrees yeah that's a hyperbola up here let's look at it right at this point over here at that point over here t is equal to one x is equal to zero and t squared minus x squared is equal to one so it's not that anything crazy has happened over here that space has become time and time has become space or anything like that it's just that the coordinate that we're using has a funny property let's think about what let's think about what r equals zero let's see r equals zero means on this side over not r equals zero sorry um not r equals zero uh yeah omega equals zero omega equals zero omega equals 0 is this line over here okay that's what it is when r is bigger than when r is positive when r is negative it just becomes this line here so it's crazy coordinates in which what we're calling time makes a right angle turn over here all right here's our here's omega equals one omega equals one looks like that and then when it gets to this point it makes a turn and it goes over here so the main point here is that coordinates if they're chosen in some funny way can make it look like something very dramatic is happening at a point when in fact there's nothing dramatic happening over there it's just that we've chosen coordinates that do uh funny things okay how do we actually go from yeah how do we actually go from this metric is there some approximation in which it is just this metric over here can we see some approximation in which these two things are the same and if we can we will have learned something important about the black hole we would have found an approximation certain situations where the black hole is really approximately flat space time yes that's right right okay so let's take this metric and fiddle with it a little bit we're going to be interested in it very near the horizon we're going to try to analyze it very near the horizon near r equals 2 mg let's rewrite this this is r minus 2 mg over r times dt squared minus the r squared divided by or multiplied by r divided by r minus 2 mg i've done nothing to get the r squared now we'll do something with the r squared the omega squared in a minute all i've done was write 1 minus 2 mg over r is r minus 2 mg over r that's nothing at all and now i'm interested in the vicinity of the horizon r very close to 2 mg well i can't set r equals to 2mg over here that's too extreme i do want to keep track of the fact that we're not right at the horizon but on the other hand the r over here there i can set it equal to 2mg near the horizon this doesn't as i move near the horizon a little closer a little further r doesn't change very much let's suppose r is a kilometer and i move in and out capital r r schwarzschild supposing the schwarzschild radius 2mg is a kilometer and i move in and out by a millimeter what happens to r well it changes from a kilometer to a kilometer plus a millimeter a negligible change in r all right so i might as well over here just write that r is equal to 2mg and that's a good approximation i don't want to do it in the numerator because although r doesn't change very much percentage-wise r minus 2 mg does change a lot when i go from r from uh near the horizon i can actually change sign so i don't want to muddle with this but in the denominator not much happens when i change r a little bit near the horizon likewise over here 2mg that's a good approximation to the metric very near the horizon notice it still has the property that it goes to zero r goes to 2 mg and this one goes to 0 in the denominator when r goes to 2 mg and i've made a small approximation near the horizon now let's see what's going on over here minus r squared again what we have is yeah we move in a little bit closer or a little bit further from the horizon r squared doesn't change very much it changes from a kilometer squared to a kilometer plus a millimeter squared it doesn't change much and so very very close to the horizon we might as well call this also 2mg squared the omega squared okay we'll come back to this over here this is what we're doing let me tell you what we're doing here's the here's the horizon of the black hole r equals 2mg we're interested in the neighborhood of a point near the horizon i'm interested in what somebody sees who moves in and out well if they move in they don't move out again but if they're moving a little bit near the vicinity of his horizon and they simply want to blow this up they want to blow it up look at it under a microscope and see what the horizon looks like okay so apart from the pieces here where we have to keep the difference between r and 2 mj we've simply set r equal to 2 mg now what's going on over here what is the omega squared if i look only in the immediate vicinity of this point if i look only in the immediate vicinity of this point it's like looking at the immediate vicinity of a point on a sphere a point on a sphere the immediate vicinity of a point on a big sphere just looks like a flat plane right right this whatever it is it's a sphere of radius 2mg that's what it is it's a sphere of radius 2mg as i move around on here but a sphere of radius 2mg it doesn't matter how big the sphere is as long as it's big enough when i move around on it all i see is a flat plane so i can actually replace this by a flat plane it's just a tangent plane to the sphere just like we replaced the tangent plane to the earth by flat space the flat space approximation it's not good for the entire horizon but it's good for the neighborhood of the horizon so actually we can replace this by d y squared plus dz squared the two coordinates this coordinate here we call whatever r the other two coordinates i'm calling y and z and this can just be called d y squared plus dz squared why because in the immediate vicinity of a point it can be approximated by a plane if we approximate it by a plane it's just the y squared plus dz squared so now we're getting there we have the metric right at that point in the immediate vicinity of that point here it is r minus 2 mg over 2mg the r squared 2 mg over r minus 2 mg and the other pieces of it very simple almost trivial just the x squared plus the y squared the x squared minus theta sorry dz squared plus the y squared now i just want to emphasize this is not the exact metric of the black hole it is the metric of a piece of space very close to a particular point what we're trying to find out is what is actually going on in a small neighborhood of that horizon is there something nasty and vicious because r is going to 2mg or is it more like this over here where there was in fact nothing nasty or vicious going on it was just the flat region of flat space flat space time okay so this has vague similarities whereas with this something in the numerator the same thing in the denominator a factor of four which i wasn't very careful to i could have defined as i said i could have defined things in a way to get rid of it i didn't so there it is okay let's change coordinates again now this whole game is a game of changing coordinates until you find coordinates where you recognize what's going on where you recognize with some simplicity what's going on i don't want to change to this capital r coordinate this capital r coordinate was just a contrivance to show you that there are situations where a numerator of one thing can become the will be the denominator of the other when this goes to zero this one goes to infinity that was just a contrivance to make it look as much as possible like this but i'm really more interested in the original polar coordinates here the original polar coordinates and my claim is that by a smart change of variables i can make this metric over here these two pieces of it look exactly the same as this close to the horizon all right how can we do that well the trick is to realize what row is rho is the proper distance from this point over here spatial distance it's the proper distance how do i know it's the proper distance d tau for move out along here the omega is equal to zero if i move horizontally and d d tau is just equal to minus the tau squared is minus d rho squared that's definition of proper distance along here okay um let's see if we can find the proper distance from the horizon to an arbitrary point r is an arbitrary point r and then use that to replace the coordinate r the proper distance from the r from the horizon to that point i want to use that as a new variable i'll call it row again how do we find the proper distance from this point over here which is r equals 2 mg to an arbitrary point r well anybody have a suggestion yeah we'll plug in zero for the other coordinates but then we have to do a little bit of work yeah we're going to move out straight out without changing time without changing x or out changing z and and y and see how much proper distance there is between here and here the proper distance is determined by this term in the metric if we move out without changing time and without changing dz and dy then the proper distance that we move out let's call it d s the s squared is the same as d t tau squared except for the minus sign the proper distance the s squared is just equal to d r squared times 2 mg over r minus 2 mg okay we take the square root of that now how do we find the distance from this point to that point we have to do an integral the distance is the integral of this okay this is the differential distance when you change r by a little bit if we actually want the distance from r equals 2 mg to an arbitrary point we have to integrate so let's call that integral rho it's the proper distance to an arbitrary point and it's the integral d rho root of 2 mg that's a constant that comes out divided by r minus 2 mg square root there we are this is the proper distance where do we integrate from we integrate sorry this is uh dr the r we integrate from r yeah from r equals 2mg to r i'm going to stop for questions at this point because this is quite critical to understanding what's going on but uh what's that that's just the point r okay let's call this r prime the integration variable right right and we integrate from r equals 2mg to the moving point or to the arbitrary point r right or outside the horizon to be specific okay can anybody do well let's say first of all just take the 2mg outside the integral that's easy that part is easy yes there's a 2mg of course so we take out the 2mg in the numerator here and we have an integral to do it's a square root integral it's a simple square root integral one over square root of something and what's the integral of one over square root uh yeah who can do one over square root uh x to the minus one half integrated what is that it's x to the three halves uh sorry x to the plus one half and then uh a two uh yeah two right the integral of x to the minus one half is twice x to the one half how do i prove that i differentiate this i get x to the minus one half and one half times two is one good okay so here we are rho the distance between the horizon and a point r is given by uh did i lose i think i did you're absolutely right right we have to take the square root of this whole thing good thank you there's a square root of 2 mg that was in the numerator and then the square root of r prime minus 2 mg in the denominator the square root came from taking the square root of the s squared the r square root square root okay so what do i have we have then that row is equal to twice that's the 2 from here then there's the square root of 2 m g that's from here and then the square root in the numerator square root of r minus 2 m g where i've plugged in the end point into the integral that's rho again what is it it's the proper distance from the horizon to an arbitrary point at distance r it's also a change of variables it's also a change of variables where we've where i have now imagine a change of variables which i substitute for the variable r the schwarzschild coordinate r i substitute the coordinate row notice there's a square root let's see did we have uh if you remember r was equal to rho squared uh again there was a variable over here which was related to a variable over here by a square root rho is equal to the square root of r here we have rho related to the square root of r minus 2 mg so they do look similar okay let's let's rewrite the metric let's rewrite the metric it's a bit of a nuisance it's not very hard i'm taking you through all the steps i could just write down the answer but i want to take you through the steps there we are no coordinate coordinate transformation the proper distance which is just equal to square root of 8 mg i got the 8 by bringing the 2 inside times square root of r minus 2 mg that's the proper distance from the horizon and all we have to do now to rewrite the metric is to well we have to figure out what it is first of all rho over 8 m g is r minus 2 mg is square root of r minus 2 mg so let's square it sorry square root right that's rho divided by square root of 8 mg is equal to the square root of what's on this side let's square it rho squared divided by 8 mg is equal to r minus 2 mg well that's convenient because we have r minus 2 mg all over the place in this metric here so it's convenient that we know what r minus 2 mg is let's now rewrite the metric first of all is this term i realize that this is a tedious uh thing to sit through you don't need to sit through it you can do it yourself but let's do it together all right here we are r minus 2 mg over 2 mg well here it is r minus 2 mg is rho squared over 8 mg then i believe there's another 2 2mg down squares that makes that 16 m squared g squared right times what times dt squared right now what about this monstrosity over here anybody there's a quick answer to this a really quick answer to it what it is in terms of rho anybody yeah but then we'd have to also convert d rho d r to d rho right what was the definition of rho no no the definition was the proper distance from the horizon if rho is the proper distance from the horizon then this is just d rho squared it is the proper distance from the horizon the way we got rho was basically writing that d rho squared is d r squared over 2mg is just equal to this thing here so we don't have any work to do to figure out the row part of the metric it is just the rho squared it is the proper distance minus minus minus minus good all the x the z squared and the y squared which go along for the ride we're almost there what are we comparing this with let's go back where is my rindler metric my polar coordinate metric my polar coordinate metric was rho squared the omega squared minus d rho squared and if there were other coordinates coming out of the blackboard if there were other coordinates coming out of the blackboard then we would put them in by hand minus dz squared minus dy squared is this the same as this not quite what's the difference 16 mg squared okay but now there's a simple thing we can do we can just make a change another change of variables namely let the omega equal or just let omega equal t over 4 mg omega squared is well let's just do d omega d omega is equal to one over four mg dt and guess what the omega squared is just dt squared over 16 m squared g squared so i just look at this and i say oh my goodness this is nothing but d t over 4 mg squared so all i have to do is change variables from t to omega and this becomes rho squared d omega squared blah blah blah the same thing as this here okay so let's just go through what i did again without actually doing it i changed well the first thing i did was i approximated near the horizon it's called the near horizon approximation and it's not just the near horizon it's near a specific point on the horizon we substituted for 1 minus 2 mg over r r minus 2 mg over r and then said r is approximately equal to 2mg so that just gave us a simple expression over here same thing upside down over here and then this metric on the sphere was approximated by the tangent space which just means it's the same operation as when i do on the surface of the earth i use x and y for a small patch on a farmer's field he doesn't need to keep track of the curvature of the earth he just needs x and y or north and east or whatever and uses the flat space metric okay that was the first step simplify this way next step was to compute the proper distance that corresponds in other words it really simply stated it was just setting the r squared over 2 mg times i'm not just equal to d rho squared and figure out what rho is let's write it that way let's just think of it that way d r square root of 2 mg over r minus 2 mg equals d rho definition of d rho i took the square root of this square root of this square root of this and then i integrated to find out that rho was nothing but where is it here it is that gave me a new coordinate row which simplified this term the main point was to simplify this term why did it simplify it well d r times this piece of stuff over here is just equal to d rho so this just becomes d rho squared this on the other hand was also not too complicated it just turned out to be rho squared divided by 16 m squared g squared why what who said d rho equals the r yeah i didn't say that the rho squared is equal to d r squared times 2 energy over r minus 1 over the other side here so again second term right top of your right hand squared right there where you the universe yeah yeah but you didn't replace the r squared with that no no i didn't i replaced the r squared divided by this okay look here we are let's let's go back one more step do it once more here we have this the r squared over 1 minus 2 mg well 2 mg over r minus 2 mg okay whoever asked the question are you happy with that i didn't do anything i just rewrote it i'm setting that equal to d rho squared now you can't ask me why afterwards y d rho squared is equal to this it just is okay definition of d rho squared i take the square root of both sides and i integrate to find out that rho is equal to this now there's no question that this thing over here is equal to just 0 squared it's the way it was defined it was just a trick to redefine a coordinate to simplify this thing over here that's all always do it you can always make a change of variables to change uh from the r to d row where by definition the r times the square root here is equal to 0. okay so by the time we're finished we get back to here well that's pretty interesting tells us what does it tell us it tells us that the geometry the near horizon geometry very near this point is really no different than the geometry of flat space time in polar coordinates you can't tell the difference very very close to the horizon of a black hole you can't tell the difference between flat space and the black hole number one number two this row variable which was proper distance from the horizon and which enters into the metric in this form over here is obviously close to the horizon the same variable as the accelerated coordinate system uses to distinguish one observer from the other just the ray or i don't know what to call it the variable from one coordinate to another so the horizon of the black hole the horizon of the black hole and different observers stationed at different distances are just like the observers stationed at different distances in the accelerated reference frame why accelerated because gravity and acceleration are the same thing if i station somebody over here near the horizon of the black hole but stationery i mean hold them there hold them there by a rope around their neck they're gonna be if they're gonna feel like they're being pulled they're gonna feel like they're being accelerated in that direction the further out we are the less the acceleration the less stretching of the neck same thing here to keep somebody on one of these accelerated coordinate systems you might imagine pulling them along by a rope around their neck the closer they are to the horizon here the harder you have to pull them so it's exactly the same geometry the near horizon geometry is the accelerated coordinate system geometry because all it is is good old flat space time but the coordinates in this region here are not ten after nine no i keep making it give me a mistake right okay it's time dilation near a black hole or something like that it's going too fast okay so that uh that what does it tell us it tells us that as i said repeat it over and over again the horizon of the black hole is not a singular place now in particular again the fact that the relative sign change between here and here that makes it look like space becomes time and time becomes space there's nothing really happening what it corresponds to to see what it corresponds to we have to go back to that capital r coordinate and i'll just remind you the capital r coordinate which was r equals rho squared is that what it was yeah the metric in terms of that was r dt squared minus and there was a silly one over four that doesn't mean anything uh r downstairs the uh the r square the r squared so that's very much like what's going on here okay if people on this side of the blackboard the uh the accelerated object gets closer and closer to the verizon would never get different infiniti it's not a defining amount of time same on this side of the so how does that neighborhood do a black hole all i can do is point you to this diagram from somebody looking from outside this line here is omega equals infinity right omega z 1 2 3 4 omega equals infinity out there from the reckoning of the accelerated coordinate system first of all nothing passes the horizon until omega equals infinity and furthermore somebody watching they can wait forever and ever they go way out here that's a long time in the future according to this fellow here he looks back and he still doesn't see anything having crossed the horizon here it is here's the horizon it looks way back at even after a very long time he may be close to that horizon it looks he has not crossed it he looks back and he sees poor alice having not yet crossed the horizon now um exactly the same thing happens here exactly the same thing and this is the picture to come to when you're uh when you're i know there is a difference there is a difference the difference is far from the horizon far from the horizon the metrics are quite different close to the horizon it looks like good old flat space but flat space in these strange accelerated coordinates and strange i mean they have a singularity over here time is this hyperbolic angle but far away when r is very large this is small 2 mg over r is negligible 2 mg over r is negligible and it just looks again like flat space but flat space in which time is time and space are in which with ordinary coordinates r and t so there's a transition from very close to the horizon where it looks like flat space and polar coordinates they're very far away where it just looks like flat space in ordinary coordinates far away you would not be tempted to call t a hyperbolic angle but close in to the horizon t or at least t over 4 mg where is it t over 4 mg omega was t over 4 mg yes t over 4 mg close in becomes a hyperbolic angle so there's a characteristi a characteristic difference in these coordinates close to the horizon and far from the horizon i can give you a well i won't give you another analogy for it it'll just get any observer are as something in falls towards the event horizon doesn't matter how far where close it is because it's continuous um no one will ever see it actually get be mentalized nobody inside the event horizon no one will ever see in the end of the black hole how can the black hole ever make larger because nothing ever goes into it it gets larger by depositing its energy very very close to the horizon and causing the horizon to grow a little bit it's it's questionable whether you should distinguish the stuff which is very close to the horizon from the horizon itself the horizon grows not because anything goes into the black hole it just merges with the horizon what is that distance can it be particular yeah it's the conquest does that relate to this or doesn't it no no no at some point as you watch imagine imagine what happens as you watch something fall through the horizon that something is sending out signals right sending out radio signals or sending out no let's say it's sending out light let's see if we can see what happens what's observed from the outside we can do it by drawings or we can do it by equations i prefer to do it like drawings oh because radio signals i wanted to start with something shorter wavelength and then get the longer and longer wavelength radio there's nothing longer than i don't know any name for a thing longer than a radio signal so if i want to start with something short wavelength i'll start with what i say optical light optical light what comes after optical light infrared after infrared is microwaves after microwaves and so forth i wanted to have a range that i could talk about okay so let's uh first yeah let's um so here's alice alice is going in and she's sending out in her own reference frame she's got a little oscillating dipole which is sending out light a little atomic dipole which is oscillating and here are the oscillations one oscillation per unit of proper time in some units which is appropriate to the emission of light so a wave crest from here a trough from here a crest from here a trough from here and so forth and that's alice here's bob and alice is sending out i wish i had another color i don't have another let me see if i do have another color no i'm not going to leave these psychologists to my pins okay so let's see what bob sees bob sees a wave the first wave hit him over here the second wave hits him over here the third wave hits him over here the 25th wave hits him over here how much proper time is there between the waves as they hit bob that gets drawn out longer and longer one way to see it is just to say that bob is receding and therefore a doppler shift from this point of view there's no real difference between doppler shift and gravitational redshift as alice gets closer and closer to the horizon here she sends out waves and the last wave that she sends out over here gets to bob at a very very late stretched out time if the waves actually have finite frequency then there's going to be a basically a last wave that she sends this means that bob sees alice more and more redshifted she's sending out light signals but bob sees after a short amount of time or a certain amount of time he's receding away and he sees infrared then he sees microwave then he sees radio waves of a meter and then he sees radio waves of a kilometer and then he sees radio waves of a million kilometers and radio waves with a million kilometers don't have very much energy they're sending out you know one photon or some number of photons per unit time alice is sending out bob is receiving those photons slower and slower and they're longer and longer wavelengths so the energy that he's receiving is getting lower and lower what he sees is alice's dipole slowing down in fact everything about alice because he sees it through light he sees slowing down and you don't need black holes to see it you just see it from this picture here that that bob sees alice slowing down and slowing down and slowing down and finally she sends out her last wave okay why is it the last way well if she had higher frequencies she could fit some more in there and yet higher frequencies she could fit even more in there but with any finite frequency oscillator to send out the rate of the radiation there's going to be a last wave that she sends out and then bob doesn't see her anymore she's gone as far as bob is concerned what does happen is the remnant of alice is is merged with the horizon and in fact makes the black hole a little bit bigger now to see that we would have to solve einstein's equations with in-falling material and so forth that's a little bit hard uh and but it's yeah black holders that's about ground that's right the longest language you can put out is about the side of the black hole does alex see bob slowing down okay so there's alice see bob what alice sees is bob receding away and moving closer and closer to the speed of light now when something goes past you with the speed of light yes it is slowed down it's slowed down by lorentz contraction so yes alice watching bob well yes okay alice watching bob sees bob recede away and therefore yes slow down there's a problem for alice though this is the near horizon geometry and we haven't talked about what happens at r equals zero something really bad happens at r equals zero that is not a coordinate singularity if we were to calculate the curvature in particular let's say the curvature scalar well no not the curvature scalar some some measure of the curvature of this geometry we would find there's nothing radical happening at the horizon but at r equals zero remember r equals zero was the place where let's go back to newton what happens at r equals zero gravity becomes infinite um tidal forces become infinite right if we calculate the tidal forces associated with this thing here we will find they become infinite at r equals zero okay so r equals 0 is a real genuine singularity a place where curvature becomes import infinite it's a place where tidal forces become infinite it is not a place which is anything like flat space but the question is where is it on this diagram let's let's redraw the diagram and try to figure out where did you say that this is not the case when the mass but now we've had a complete gravitational collapse and please collapse to a point collapse to a point this mathematics doesn't know about anything don't go out to alice but alice isn't here right right right right behind me alice has some friends that are having a conversation doesn't matter which direction they are relative to each other they just have a conversation with well bob bob uh they're both following freely alice and her friends are all falling freely through there yeah they just talk to each other just as if nothing was happening until they get near the singularity well if one friend is ahead of the other then they can't talk then we just want to get better minded horizon quicker than the other what's that again if one is a little ahead of the other then they can't communicate when one gets behind the horizon you have to still be fine here they are they're both falling okay they're communicating back and forth freely nothing funny happening now it is true that once alice is alice and here's shirley alice is ahead of shirley she's fallen in first here she is just as she gets near the horizon she sends a signal out to shirley shirley gets the signal once she passes the horizon she cannot get a signal to the shirley who is before the horizon there's no way she can get a signal from here to here but she can get a signal from here to here even if there was no black hole she couldn't have gotten the signal from here to here that would have exceeded the speed of light so alice sends shirley signals shirley sends alice signals nothing funny about them now if shirley would have decided not to go into the black hole at the last minute here she is and then she decides not to go into the black hole not to cross the horizon but to get onto one of these accelerated trajectories maybe she was connected to the space station by a cable and the cable only let her go down so far and then got pulled taut she's supported away from the horizon then alice can see shirley but shirley cannot see alice behind the horizon there's no way she can see behind the horizon so if shirley freely falls her relationship to alice is not unusual back and forth they can communicate if shirley at the last minute decides not to fall through the horizon then indeed alice falls out of her experience yeah so uh um doesn't occur at the horizon what kind of thing uh atoms on ionization horizon uh well now you talk now we're getting into quantum mechanics which we're not going to do this quarter but uh do i do atoms ionize nearly right that depends on who's watching it if an object is moving along yeah the problem is that these trajectories that move along here are going to be space-like and why is that supposing supposing we have a trajectory very close to the horizon okay that's moving with a dx or sorry a d y or a d z okay so let's see so that means d y or dz is not equal to zero right but we're extremely close to the horizon let's say practically at the horizon then this term is zero and this term is zero oh sorry well this term is zero because dr is equal to zero we're not moving in and out we're only moving along the horizon right okay so this one's not there because dr is equal to zero along that motion and this one's zero because the coefficient of dt is equal to zero second one zero over zero we're slightly away from the horizon slightly away from the horizon just outside okay but the main thing is dr is exactly equal to zero was skirting the horizon at a fixed r that was the question if you skirt the horizon at a fixed r that means this is exactly zero all right and r minus 2 mg is just a small number on the other hand this is not zero but it's very very small in particular the coefficient here is so small that if d if we if we get close enough if we get close enough this will eventually get smaller than dz so if we make a little excursion along dz here by a given distance no matter how long we take to do it if we're close enough to the horizon this term is zero that means the trajectory here is space-like and it means it's exceeding the speed of light so anything that tries to skirt along the horizon is exceeding the speed of light and that's because time has slowed down so much or the coefficient of the dt square to slow down so much that the trajectory becomes space-like you understand the difference between understand what a space like trajectory is right so skirting along the horizon is forbidden motion it's motion faster than the speed of light uh supposing all right let's talk about a thing well i think that'll that'll do it right so you can't skirt along the horizon like that ah let's get to the singularity now the singularity is at r equals zero here now let's remember the connection r equals zero or r equals small let's take small values of r for small values of r that's where the interchange between positive and negative has taken place okay for small values of r we're in here smaller values of r don't correspond fixed values of r don't correspond to time-like trajectories trajectories of observers but they correspond to space-like surfaces like this just to see what's going on let's go back to this metric over here supposing r if r is positive then we're on the outside if r is negative we're on the inside all right so here's the inside here's the outside in this case r being negative that's different than little r being negative here it's like r minus 2 mg being negative big r is like r minus 2 mg all right when big r changes sign the hyperboloids x square minus t squared equals r become hyperboloids like this so fixed r outside corresponds to the motion of an accelerated observer inside it corresponds to a kind of space like surface another way to say it is fixed r inside the black hole is some time not some place it's a time and not a place fixed r okay now what about r equals zero well r equals zero is sort of like capital r equals negative as i said r minus 2 mg is sort of like capital r what does r equal something negative correspond to it corresponds to a surface like this so little r equals zero is a surface like this this is little r equals zero like that that's the singularity and it's not a place so much as it is a time once you fall through here notice once you fall through here you have no way of avoiding this singularity the speed of light is 45 degree angles you cannot move faster than the speed of light when you're out here and you send a light ray in the outward direction it might escape if you send it in the inward direction it might hit the singularity will hit the singularity if you're inside it doesn't matter which direction you send the light ring in you're going to have to say that yeah if you're free falling there's no reason for you to use these funny coordinates well free flowing just means you're moving straight lines you don't have to use any coordinates here's the geometry the whole idea that the x and t coordinates have switched because of these funny questions the coordinates are funny but the geometry this is the geometry this this picture doesn't require let's take the r 0 away that's it that's the geometry that's the geometry that's what it looks like i don't need to put any coordinates on it that's what it looks like and once you're in here you can't escape falling into the singularity here well i keep telling you time and space don't interchange it's just you've drawn coordinate axes which make a right angle right so basically everything is other than except for title effects the free-falling observer doesn't mean so at a certain proper time at a certain proper time you've crossed the horizon but crossing the horizon is not in itself a dangerous event you're doomed but you're not there you have a particular expense by proper time and then you will reach the price right you might not be going with that at 45 less if you just fall in and you pass the horizon you have some particular amount of time left over yeah yeah yeah that's right the maximum amount of time that you have is basically approximately equal to the time that it takes to cross the horizon of a black hole it's just a numerical fact if you measure distances and time in the same units c equals one then the schwarzschild radius becomes a time schwarzschild radius of a star would be about a meter that's a kilometer excuse me light takes seconds to cross a kilometer one one over three uh one over three times ten to the fifth order of magnitude outer magnitude order of magnitude it takes about one transit time across the black hole everything becomes infinitely dense infinitely distorted infinitely quantum mechanics becomes important everything is torn apart into its constituents and worse and uh we're into uh real terror incognita but really you're never going to get far enough to experience uh that singularity and so yeah does the seniority actually i mean does the infinity actually exist in real life or is it is there something right before you get there that sort of uh mitigates that well nobody really knows nobody really knows my own feeling is no it's it's an ultimate end of things but but let's put it this way let's just say nobody really knows but we know how to follow it for example into a region where the curvature might be the radius of curvature might be no bigger than the radius of a proton easy to follow it there nothing we know very very well what physics is like down to those distances so you would not like to be squeezed into a radius equal to the size of a proton actually yeah tidal forces would be of that magnitude that they would uh so does it seem like no sense of impact really live with a large mass with the horizon there's no sense of impact with the horizon there is something dreadful that happens at the singularity but the singularity as i said is not so much a place a singularity if the singularity was a place a spot over here and you're falling in while you can go around it if the singularity is the end of time as i've drawn here there's no way to avoid it so there is something i should tell you about real about genuine black holes you would not survive the transit across the horizon of a stellar mass black hole the tidal forces are just too large if you were to estimate then this is a newtonian calculation if you were to take a stellar mass and compress it down to the size of a um of a uh compress it to the size of a black hole which would be about a kilometer well a kilometer doesn't seem very small it seems like you could pass through it safely but the tidal forces just the gradient of the gravitational field would uh you know squeeze you out like toothpaste through uh so that would be it would be very dramatic going through the horizon of a on the other hand if you were as small as a paramecium then the gravitational gradients across your body would be negligible and you would fall right through the horizon without without any trouble um similarly if we took a black hole which was i don't know a billion solar masses a billion solar masses the horizon is so big and so flat that the tidal forces are not very strong on your body at a billion solar masses it's a little bit um how shall i say a little bit counterintuitive you might think the bigger the mass of the black hole the worse it is to go through the horizon of the black hole but that's not true the point is that the bigger the mass of the black hole the bigger the horizon and the less curved and the less tidal forces the less you would be squeezed going through the horizon so if a black hole were a billion solar masses you would go through the horizon pretty easily i think i once estimated i can't remember whether let's see a billion solar masses so uh it's a billion uh billion kilometer how long how long does it take to go a billion kilometers about an hour right is that right a billion billion kilometers is like the radius of the solar system yeah there's about a radius of solar system which is about a light hour so right so with the biggest black holes that are found at the centers of the biggest galaxies you could probably last for half an hour or something like that after falling through the horizon not longer than that uh that's how much time it would take light to go across it take a black hole of a trillion solar masses there are no such things probably there are no such things you would last another thousand times longer so going through the horizon of a black hole for a stellar mass black hole is not as innocent as i've made it sound but the principle is there a sufficiently small living creature would survive going through the horizon even of a solar mass block it wouldn't last very long hit the singularity in a fraction of a second but still it would be at the singularity not at the horizon okay um i don't know are there any any things left to uh yeah um locations outside accelerating coordinates is a series of rods and clocks which are undergoing acceleration relative to one relative to one life okay accelerated relative to what yeah relative to alice okay now if alice falls through the horizon of a black hole believe me bob is accelerated relative the understanding of the coordinates over uh near the black hole we're assuming that the we have a bunch of points scattered around in a small cube and they're clocked there and they're running now when you get inside the black of course you can't do that actually on the surface of the black hole itself you can't stand there yeah but but you could extend that coordinate system across the boundary of a black hole and you still get a series of points and and clocks and the clocks are still measuring time aren't they not very very slow the timer out there slows down there's nothing to prevent you in this picture over here from using a set of coordinates and related to real clocks and rods which look like that all right this is um alice's freely falling reference frame okay it's just a reference frame which freely falls through the horizon of the black hole okay and the one we're talking about with the t's and inside the black hole if we just extend the coil mix they look like this okay right so we can we can imagine a series of rods and frames which actually extends across you can imagine but they'd have to be falling you cannot imagine anything standing still behind the horizon why not see what the variables are producing anything that's standing still that means that a fixed r fixed linear arms you can't move along one of those coordinates the way you were able to move it because if you tried to make something in a fixed r here it would be exceeding the speed of light something that a fixed r behind the horizon is exceeding the speed of light but isn't that what they want with right behind the horizon it's exceeding this big yeah so no there can't be rods which are go through here with these rods just standing still statically there they're going so it's basically a notional coordinate system that you can't imagine actually constructing well no well you can imagine constructing it but in here the things that would measure r would be clocks and the things which would measure let's see this is this was time the time coordinate becomes a coordinate like that so behind the horizon these but they're only coordinates keep in mind that they're rather arbitrary we've just drawn some coordinates behind the horizon if you wanted to measure r you would use clocks and if you wanted to measure t you'd use meter sticks all right t flows along here the same coordinate same mathematical coordinate flows along here you measure things in here with meter sticks well keep in mind this is not a problem of general relativity this is a problem of special relativity this is just ordinary flat space-time no gravitating objects except as seen by an observer who is being pulled along at a constant acceleration what the the one thing which is not there in that case would be that singularity the singularity is really the gravitating object right that uh that's the real signal that there's title forces for example can we go back to the way back to where the metric comes from is any any as long as it's enclosed within here sort of right all right so we start with einstein's equations let's make some assumptions i'll tell you what assumptions we first started landslides equations now we're studying a region outside of a place where there is any energy distribution and that means that we're studying the solutions to this equation knowing full well that inside here there may be some material but outside here sorry do you need an r equals zero those are einstein's equations in the absence of any matter it's exactly the same as newton newton you would write del squared of the potential equals the energy distribution the matter distribution mass distribution from outside the mass distribution you would write zero you know that inside something's happening but outside there's nothing happening and so you ask what are the general solutions of this equation under certain conditions all right so i'll tell you what the conditions are what are the general solutions of this equation which are radially symmetric in other words which are only a function of r let's let's start with this one what is the general solution of the equation del squared y equals zero where phi is only a function of r and not a function of theta phi and time for example well there's a general um one of the restrictions five goes to zero far away okay five goes to zero far away you just told me in fact what the general solution is ax squared is by well quadratic form uh no no no no linear form quadratic form would be constant on the right hand side but assuming also that it goes to zero far away well the answer is there are none there are no solutions of this equation where phi goes to zero far away but now supposing you say what are the solutions to this equation outside a certain sphere not caring whether you solve the equations on the inside not worrying about whether you solve the equations on inside but only pairing about whether you start with these solvent equations on the outside then there are solutions in general only one form of solution the one form of the solution is that phi is a constant over r only mathematical solution now what determines this constant in the real world the constant is the mass contained within that sphere so it's mg minus sign the physics uh tells you that it's a minus sign and the mg is minus mq that's the most general solution of the equation of l squared by zero if you don't care if the equation is solved on the enzyme now supposing we take this sphere i sort of understood this answer but i was gonna then ask was what happened to the mass inside well let's continue to follow okay look we can think we can ask what happens if we think a very small skier well the answer is exactly the same except the mass is inside that small sphere some place what happens if you go to the limit of a point particle you go to a point particle and the equation is satisfied everywhere except at the point where the probability of the point particles when a point particle is there's a singularity that's where r is equal to zero okay so what if you like you can think of the mass that's falling in and all accumulating at a point a point of space where is the material that falls into a black hole what's on here it falls in here ultimately arrives on singularity the singularity is also r equals zero so all the material fill in is located in some sense at r equals zero but r equals zero is now spaceline in that timeline so where is the material it's here is required to exist that being r equals zero everywhere is outside of here which means everywhere is out of here einstein's equations are solved with zero right hand side the only place where there's a source but that's the end of time anyway that's just a singularity for newton it's also a singularity but with a very different character uh does that answer the question um yeah but a bunch of mass points in newtonian physics will not collapse to a point will not collapse to a singularity or let's put it this way it's a real set of measures zero initial conditions which allows it to collapse to a singularity and the reason is that the tiniest bit of angular momentum a one particular relative to the other will have to miss each other just a little bit of angular momentum will provide enough centrifugal force if they will uh so unless they are really very very symmetrically distributed with no angular momentum at all they will just go past each other and come and uh miss each other and not form a singularity this was known this was a theorem about classical mechanics because of it einstein assumed that a black hole could never form he assumed that somehow the same thing would happen and it would be impossible is that the same thing that makes globular clusters basically stable as a whole i mean they live for a long time it's different here i'll come here like a single point particle supposedly was supposed it was for a moment a singularity newtonian physics a point particle and then you throw another one in unless you happen as a pointless a point partner here unless you're happy to aim it with infinite precision it will miss this point and just go back out what happens here what happens here once a particle passes the horizon there's no way that it can escape that's because the singularity is a different character it's the end of time rather than being a pointless face so einstein was wrong his logic was wrong you have a picture that the singularity is a point and that anything that tries to add some mass to it will miss it and go back out whereas singularity is a real track he didn't understand the idea of a horizon and the horizon was really a point of no return that once you passed it i mean it's not good enough to have a collapsing star providing a lot of pressure to because otherwise the whole argument gave that otherwise electrons or singularities i mean it's not clear that there exists a singularity you know the same thing leave you by after that you're saying even if there were singularity you couldn't handle it and nobody expected it even if it was so even if you could get one star in the tag one you couldn't add to it if you can't and for one having one started obviously you can't ever make one yeah but let's say in terms of face space and picture change from a classical picture later on the mechanical picture the incompressibility uncertainty relation prevent any further compression so to speak in the face space if you are looking at the black holes so the problem is that in order to have a resistance against compression you have to have a reasonable density of matter uh and a reasonable amount of kinetic energy and so forth and then you can't squeeze it because you can't squeeze it in face case but if you look at how thin something has to be in order for it to form a horizon here the answer is if it's heavy enough the density of it can be negligible um what kind of forces you're talking about which prevent collapse it does have to do with the uh with the incompressibility of base space but just think of the incompressibility of phase space of providing spatial force uh which keeps things from collapsing and you get into trouble typically in compressing when pressure gets too big when density gets too large how large you have to make the density you can make a black hole it's fantastic you have a given amount of matter row mass is equal to 2 mg over r mass um okay let's divide this by r cubed okay why because i'm interested in m over r cubed which is density the bigger the black hole the more massive the black hole the smaller the density of a collection of material would have to be before it forms a black hole why take enough mass and you put it in the volume the density only has to be on order does this mean and ours of water is equal to 2mg 2mg squared so if you have a given amount of mass in order to make it form a black hole the density only has to be 1 over 2mg squared this is why people are very very certain of black holes form they were very certain before before there were astronomical observations you don't have to make it you have enough mass arbitrarily low density arbitrarily low density let's say it another way let's take an arbitrarily low density row grow as small as you like particles every 60 meters how much mass let's take let's take a radius r that's the radius r how much mass is there well rho times r cubed some four pies and so forth i don't know four this is order of magnitude the amount of mass is there right so r even for this very low density r the radius of the ball of mass is m over row and row of space one is just this very low density to the one third power no matter how small row is if you make m big enough this will be less than twice mg why because this increases linearly with m and this only increases as n to the one third power so for a fixed but tiny row you can eventually make m or the rope to the one-third smaller than the short shield radius once it's smaller than the schwarzschild radius it's a black hole so no matter how rarefied the material is if you create enough people if you take the vacuum energy has a certain energy density what would be the associated um for shaw radius to create a black hole with that density would be uh so i think we can be pretty sure that in our universe it cannot be a black hole of size bigger than handling my ears so can we can we consider ourselves inside a black hole no not inside a black hole but inside an inside out black hole it's an inside out black hole in that they're surrounded by a horizon but the analog of the inside of the horizon is the outside of the uh of the region we on the inside of this sphere are as if we were on the outside of the horizon and outside our eyes things fall forward things flow away from us they don't fall toward us on a big enough scale because of the dark energy things fall away from us they don't fall toward us that's the that's the accelerated expansion of the universe things fall away from us and because they fall away from us eventually if you go far enough out you will find it moving faster than the speed of light relative to you and that becomes exterior horizon but it's not like being on the inside of a black hole being on the inside of a black hole to crash into a singularity basically being on the inside of a cosmic you just stay there forever if singularity is the end of time how does it ever happen that you do reach it oh at the end of time is probably wrong it's the end of time in the sense that that was the end of time just proper time to change that was your proper time with the singularities so when we have a fixed r at that point and and that's after singularity even though r is zero you would still have a fixed r which would put you traveling in theory faster than c to play at that singularity straight again when well as i said i mean the physics of what happens when something gets the singularity is um obscure there's no sense in which it stops it just i don't know what to say about it physics doesn't know how to get there density has become so large um depression has become so large that no matter how much physics you know it will always go beyond that no physics the temperature will become infinite pressures will become infinite densities will become infinite so if you don't think you understand physics some desired density that means that you can understand what happens give a singularity to some shell but it'll always get worse wait long enough not very long it'll be worse and you'll always be driven beyond what we know about uh matter so is it the problem that we don't know enough to figure out what happened to the singularity well i don't think so i don't think that's the point i think nothing survives to find out it seems to me as long as you stay outside it's also irrelevant but that's true because nothing like nothing goes on on the other side that's right that's right but you could jump in to find out you jump into the black hole to try to find out what happens in the singularity you're let's take two atoms falling toward the singularity here's the singularity now atoms can send back and forth messages messages can simply be the photons that make up a coulomb field but uh the fact that this this atom is influenced by that atom well we know atoms influence each other and so forth but when they get too close to the singularity this atom over here cannot send the signal to the other one why there's not enough time to send the signal they fall out of causal contact with each other they can no longer influence each other and so as you get closer and closer to the singularity the atoms of your body fall out of contact with each other and you just disintegrate into a collection of uncoupled atoms that have no connection between them but it's worse than that the electrons in the atom fall out of contact out of communication with the nucleus and protons and neutrons inside nucleus fall out of contact with each other so everything just disintegrates into completely uncoupled some things bits which don't talk to each other is that really the approximation that you made on the measure to make a correspondence to the regular geometry only work where r is near the workshop rates yeah so by going near the singularity you're going far from the shoreship radius at some level right so i've drawn the picture of what that what that really does look like within the horizon also and that's what we see okay so there's some other approximations that that make it right on the uh yeah yeah so that's more like a diagram than every word i'd say it's just not defined on the other side of it or if it is defined nobody knows well you simply go through uh physics which is totally unknown but i mean the coordinate is exists right there's a key that's that's larger than that well how do you how do you all right so if you're mathematician you could ask can you analytically continue the metric past that point uh and then the other question is so what happens if you try to continue this to r negative uh does this become positive this becomes negative it is the bottom right and is it obvious that that i mean there's no mass action there's no right there's no matter here look you could draw like trajectory but we can't find out even if we're willing to jump into the black hole we can't find out you mentioned one time if i really heard the last touch plunger kind of close the argument let's let's not get into that now that's okay when alice goes through the horizon and she's carrying a meter stick does the readership suddenly become a pop no alice here's shirley sterling is at one end of the music and alice is at the nothing happens to it it just goes straight through there's nothing fancy happening at the horizon no the stick does not become a clock moreover alex's clock called something else when it gets behind the horizon just because you call it r doesn't mean there is no difference how do we measure time if you think about how we measure time you can use meter sticks to help you measure your time is a meter stick nothing else there's nothing else that is non-mechanical or physical that defines something nothing dramatic happens absolutely now we do have to keep track of the fact that bob is accelerating away from uh but that's okay so already alice is here right sorry alice can't keep falling in she's still outside yeah she's outside that time she sends a live theme here she is all right crashed into the singularity over here alice sends a message and the message comes back yep alice sends a message from over here and the message comes back behind the right but from here alice can no longer send a message but she wouldn't know that while she's passing the horizon that's right that's right so so what she sees is when she sends a message the messages take longer and longer to get back she sends a message from here it just takes a very very let's forget the singularity if she sends a message from here it takes a very very long time for it to get back and what's more it's back very very rich so as she gets closer and closer to the horizon and sends out her just usual radio waves those radio waves have to get the ball which takes a very long time and they have to come back which takes the same time and they have the moving mirror has a redshifted devops so the photons get very very redshifted and so what alice would see would be a very very red-shifted version of what she's saying but and but she cannot see any image from here so why is the maximum wavelength the size of the black hole from that analysis you're taking too literally the idea that gravitational force comes from uh that's uh that or if you like you can think of the gravitons as being emitted from near the surface of the black hole you can it's perfectly okay to imagine that all the material of the black hole accumulates since it never gets to the horizon it must accumulate at the horizon it must keep sinking in closer and closer asymptotically getting closer and closer but all of the mass is concentrated in a little little thin shell from the point of view of somebody outside is concentrated in a very thin shell in your pricing uh if you like to think of gravitons being jumping back and forth okay do it but they're jumping back and forth from the surface just above the horizon so outside i just think of the material is collecting kind of cinnamon three layers which are forever falling closer and closer this is a possible statement which are forever falling closer and closer and thinner and thinner sedimentary layers never quite getting there but their mass is there their mass is there you can tell the mess is there because you put a surface around there and this gravitational field and you can always think of it as a mass being localized just in a very very thin layer black holes which are not spherically symmetric that might for example might be rotating or something would be flattened out or something that's extremely flattened out there's a limit on how fast it can rotate and how asymmetrical they decided that parts rotating around each other for example a binary star that's collapsed a binary uh bible yeah some kind of a binary object that was that fit inside a black hole the the pieces would collapse and there was singularity and it would stop that yeah yeah we have a couple of binary things going around each other inside black hole okay that's really not very different than an atom with the electron volume and so so so there's no need for example for gravity to escape the black hole and tell us what's going on inside we don't need it it's not possible the preceding program is copyrighted by stanford university please visit us at stanford.edu

Channel: Stanford

Views: 136,897

Rating: 4.8471336 out of 5

Keywords: Science, physics, cosmology, black, holes, Einstein, Newton, accelerated, reference, frame, horizon, coordinates, Rindler, Lorentz, contraction, acceleration, Schwarzchild, metric, geometry, polar, angles, Azimuthal, gravitational, fields, flat, sp

Id: fVqYlSNqSQk

Length: 140min 45sec (8445 seconds)

Published: Fri May 08 2009