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General Relativity | Lecture 7

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get back this would still netted cause what your Member plumage American before my time by those and those that do really well for a few equations that I have no permanently fixed in my head self let's write down and I want a simplified before you might just be scaling the coordinates but Friday and furs and I'm going to write it down with me radius variable and the time variable little over them the primed the purpose of the prime is just so after we scalar variables I don't have to put Prime I want that I want to minimize the number a little symbols I write an insult to begin with I'm going to write the electorate in coordinates fire prime prime no saying
things last time it's a just a slight change of station will write detail squares property price is equal to 0 1 minus to what end and underwrite are defined here everybody see the prime deep-sea prime squared minus R climb square divided by this say they want to bring this to of our are time and then plus our squared times the cover unit to spheres to to see just means of two-dimensional surface of a sphere but was it was detained squared plus a cosine squared defy squared that 1st of all prefers chamber of occasional make Bozo just remember it that is Schwartz radius the radius of the horizon the place where this coefficient function here becomes euro is and our prime equals 2 and so to is the short film radius it is the radius of these horizon of the black hole so let's show dividend main tool energy just Arthur Schwartz shield 2 Ng is just watch and now I simply want to do some very simple I'm going to read the final Reese move defined honor to be our trying divided by arch watch old no normally borrowers have units of late when I divide 2 0 lengths idea something which is dimensionless so now my Our which is not the the same on that I use less I'm busy from West but now my arm is dimensionless and can rewrite all 1 other sorry will go and do the same thing would take that but what was 1 of the units of tea but the speed of light is equal to 1 which were always assuming sequels 1 then the units of time in units of Of the same meters away OK I also want to in in the something dimension with some of the same thing prime divided by the same march toward show us what field is both a distance and at times if you let alone what sensitive time it's just the kind that it takes a light rain could go the distance of one's what you radio OK so now let's rewrite this method Kia 1st of all 1 Our of Sildomar prime just 1 minus 1 are French erred Our swat field of our prime is just 1 of are now team times Square D keep trying squared looked so what's about equation from minute that did squared colleges are structural squared 1st next Jeremy my that figure the data was exactly the same thing as the numerator over here so to get 1 minus 1 of our and now we have the aren't Times Square are crime is times are far too so this is just DR squared times again are structure squared what about the last years a Western plaster should be prime this should be off prime squared off time beginners are brush chilled services again are squared the Omega square columns again are structural squared the talks still radius of a square of it just
factors out from everything but put on the outside for a big bucket around that is because squared all you have to remember is this simplified version of course was what is what appears in a bracket that is metric that we would have written down if 2 GE was equal to 1 of 2 1 g happened Tobin equaled a warm we would have written them 1st place while we find out from this exercise is that basically the the metric Of all the short field black holes is except for overall Bush a factor which is proportional to a square of the radius of the black hole another words everything in here is dimensionless carries no dimensions no needed a anything else all of the inventions are in here are short square and for the most for most purposes studying the geometry of the of the black hole really Lee set our swatch equal the 100 yards like like studying the metric of sphere of different sizes the Metrodome as no years is the same X said the biggest fears small spheres we simply have to multiplied by the square of the radius of the beer both men also spheres of the same well only short too black holes of the same except for those overall factor here so Malama going ways from most part tonight will be just writing this thing memorably Nixon's Liberty easier should last probably yes that letter address a question that we haven't before it died as they did last visited a a a a white cell what about the curvature you they all curvature tensor curvature tensor is the thing which tells us that is based on his not flat I told you that it's analogous to worm 5 forces tendency Ferrer geometry to distort objects and indeed it is fact fact rival forces are curvature how big it is other components of the curvature tensor let's say near the horizon the horizon correspondence stock are equals 1 well the 1st question is whether the unit of curvature 3rd just go back on his command I realize you probably don't remember me of formula for the curvature of just very briefly remind you it looks like the most main thing is to read off from it to you all we want the Met that cancer gene you know that doesn't carry a new dimensions normally it's the DAX Mueller DX know these things have length units of lying and forth this is supposed to be the square from Lenka the square of this as a junior who is a thing doesn't carry any units of all expenditure now but go back to remember the Kra staff symbols I'm not going to write Christopher Christoval symbols of any details I'm just going to remind you that they contain derivatives of G with respect to acts they also university but Zia's dimensionless Jean vs. dimensionless saw what are the units of across powerful symbol there 1 Overlake G is dimensionless X Kelly's units of and so ludicrous powerful symbol has U.S. units In the 1st like that if you remember if you don't I'll tell you the curvature tensor is contained killed kinds of terms which contains derivatives of the offal symbol end it contains Chris powerful symbol square don't try to remember the formula that some portent missed Sharon will have units of wobbling Square if I differentiator Crist awful symbol which already has units or than this units of square but some of this the Gestapo symbol has units of Warner Lankans Cuevas Chrysophyllum symbol of want so you'll curvature the units of the curvature tensor usually right at the square brackets stands for units all of the units of the the curvature tensor on 1 over square 1 over meters quartered amount comeback what s and given this concern given all are in particular where where Mr. will where's the horizon units the Horizon these units is are equals 1 that says What is the curvature are equals 1 that goes 1 only a quantity in the metric which has units of length is what should be all possibilities the only possibility Is that the curvature is going to be proportional to the inverse Publish What you radius square only possible formula that it could possibly get by doing derivatives and everything else since us what shows no within with unit here so I'll be curvature at the horizon curvature at the horizon of called horizon is the couple number and France it is not import curvature at the horizon is approximately equal 1 over the Schwartz chilled radius square as you move around in the black hole geometry Lee curvature at the horizon will always be 1 of which what you squared something which may depend on where you are for example equals instead of Oracle's 1 Goto articles ahead of you but the articles hand the curvature will still be proportional to the same hand but in fact it will be bigger numerically bigger maybe twice as big or something like that and easy to read off and as you go the articles 0 that's where the singularity the curvature becomes infinite but at the horizon at Oracle's 1 curvature 1 of our squared the meaning of that is the curvature for the bigger the black hole more massive than the number are was proportional command of a black hole the bigger the man the bigger the Achille radius and the smaller the curvature of the horizon the tidal forces associated with a big black hole at the horizon at sunrise tidal forces associated with a large black hole are less severe the tidal forces and a small black hole a small black hole as a much worse Merci Maine New Horizons very big black or find that 1 observation that to reel off directly from this that the bigger the black hole more massive the last curve theaters and particular but now let's concentrate on but this geometry but throwaway Our squared here it's there but the let's concentrate on a on the geometry of a unit black holes are black coal for which is which you have set to Cooper 1 let's concentrate on that study
at bay OK so let's not go back to something we talked about last time robots we talked about last time I only and outlined for you but would do it again and let's ask for I want no coordinator no I apologize for the fact that we keep changing coordinates but that's that's what road reviews all about changing coordinates OK Logan intruders I no coordinate to replace all are Auriesville be replaced it's draw they are axis and someone else is Oracle's 1 hears equals 0 and far away as far away from the black all articles 1 is the horizon in these court right news it is right so right was the horizon and I want to invent a new coordinate which is just distance proper distance from the horizon distance measured from the horizon by Holly calculate the distance from the horizon but go back to the formula up here remember that deep house Square is proper time D best school where it is proper distance and these are just equal come mightiness each other b Tower squared and squared minus each other proper distance is the same formula with plus signs here and now let's move a little bit away from the horizon radially outward and a half ago little differential distance away from the horizon here will not all of the Texan differential distance over here and how and what the proper distance between these 2 points the proper distance between those 2 points urgent B.S now I'm moving radially outward black hole and moving from he radially outward and I'm asking how much distance that corresponds but 1st of all and moving not literally molding and justice placing the point without changing the kind 1st of all is comparing 2 points at 2 different distances from the black hole at the same time so deeply from year to year 0 and further that changing angle just moving radially so the Omega is equal to 0 no change in angle then means that DES squared is just able to DAR squared all over 1 of minus 1 laws war b s the distance between these 2 points is just Dr the movie are divided by the square root of the square root could also write this is our way of life and the right back here write multiply and divide by little are this becomes little are minus warned warmer this 1 0 are armored 1 over on this also Lou armor this 1 or more are are upstairs courtesy what I did I just wrote that 1 minus 1 of what are is equipped are minus 1 or Over here or here I just turned the Ibarra OK so now here's my formula a career as rival the differently square root all are crimes are minus 1 let's call the distance the proper distance from the horizon let's give it a name let's call it wrote the proper distance from the horizon will call world that's not unlike this war our minus 1 is this a certain kind of distant but it's not the proper distance from the horizon had only find it well this is the differential displacement in distant and moving the radial distance br the actual distance from the horizon is integral square root of all are our minus 1 Dr from where to where it is the this is definitely the world they wish he closed prior since just an integration variable number saying crimes before this is called what limits of integration that I want to calculate the distance from the horizon for some particular point at 2 at quarter net all are played the great From the horizon which is 1 2 point are it's better call was primed for alone abuse mathematical terminology integral over DR Ms. fears some functions of art we can we actually cow were property know you could do the integral part of the road a doable and to grow by but tell you some things From what but role is some functional art scored swirl of art now what nor are what the met Brecht over here is also just a function of wrote we know the realized a relationship between Rowena the relationship between role in horror an hour favored functional Our minus 1 over all our that's also a function of wrote Klitschko obscure birth Stoops truck from Roland pitcher reduced let's call it f of Rome f now emerge a little but that's not quite what I'm also going to change the definition of tea but only in a very very simple way extremely simple worried I'm going to define this definition Omega Omega is just equal to 2 year figures divided by P divided by no big deal just for convenience I'm going to use on variable which is the kind of I'd like to say that this is a sort of artifact of a particular original definition of time it might nicer if I would've used to to review steel but it the stock were all made it is equal to field and let's write the metric in the uniform and World plans rose squared no and wrote times rose winners just some function the fact that I wrote several times squared no content and that it's just some function of Rose squared times deep-sea T-square but he's glad I'm willing to replaced by deal maker square PT squared is just for all deal maker squares so I can absorb that extra factor of Tourette and he I have really done anything I just use the fact about far Enron are related
if I know are them and role with final road all of them so I can rewrite this thing over here as a function of rock and here I've done it somewhat arbitrarily dividing it into a row square the overall freezes become clear OK next what about this piece over here this piece over here looks like it's gonna be a mess but it's not a message very simple it is just that to warrant just minus D Rose squared why has it might be rose squared the reason is because rolled was defined the be proper distance and guest is defined To be proper distance winners we did it the whole point of changing variable to Rome areas but stake his definition of role here was 0 0 by Dr Dee role by is just square root of oral law on minus 1 if you differentiate an integral you just get me into grand now square that's squared they are square as Justice thing here and now multiply by Dr square Bureau square is precisely the thing which appears here that was the whole point of changing variables Rome so that this term he would just be might be it is measures distance along the radial direction property young bad road Roselia now it is time that he's a yes vote yes so far yes yes there's no can't think never gonna be any content in the division but will find it convenient a right that's true but I was actually writing Tao scored yes you're right but depending on whether we write cow Square D square What about the last turn molesters is are squared my guests are square but honor is also some functions of raw level worked out what is exactly but is a function of world arms maker square erred it is possible to work out what are as a function of Raul are and what role a functional on his but we don't need to be a form of this method justice just exactly as it is here that's interesting parts a little bit of knowledge about what of role and honor role are like we actually worked out last time when you near the horizon when you Rose world where is horizon this form here in terms of rock was a horizon Michael Iraq because what is wrong with the distance from the horizon equals 0 is the horizon and if you want this In the growth it's not hard also if it's a hard in the world but if you work it out in the vicinity of the horizon of the woods near articles warned you could do Vatican approximating you find serviceable formula you find a number of things 1st of all you find that the limits as role goes to infinity large or far away overall several square no 1 is that what is that basically just the coefficient punctured here far away what is the coefficient functions far away its are minus 1 Orange which is 1 far away not quite 1 because make square is not quite he squared it differs by a factor of what you find is that this is equal to the number 4 so far away half of wrote terms rose Square is just equal before Baptist reflects the Met far from the black hole far from the black or several rows and rows squares just before 1 thing next limits as row goes to 0 logos euro is getting right up against the horizon they were really interested and find which corner arrives and several is equal to 1 so cluster memorizing after rose just equaled the 1 thing you just have rose squared the Omega square you may find that familiar roh squared the Omega square we've seen that before so close memorize an effort rose just equaled the 1 we just get rose squared steelmakers Crivello denied that was the reason incidentally but dividing it up in this wary because closer so the horizon several times rose squared is equal to destroy a square or as just a 1 now ordering or everything else yup we know that the limits off an aura of role equals 0 that we couldn't work out what is forum row right now roh equals 0 Roybal 0 means at the horizon articles 1 1 right disabled 1 we kindle a little bit better if we could do a little bit better it's plus rose squared Over for please don't really all we need now on Monday another throws Gredel for a moment might use of but this Im basically the properties a reading here now the most important thing is to look at this when role was small small watches row squared Iomega squared minus Bureau square that's exactly flat space and Iranians in hyperbolic pole a quarter of a review 4 minutes hyperbolic Republic would we know I don't but but that's light Call of flat space and bomb rose squared the Omega square not Afro will later back but but those are the hyperbolic Court hyperbolic public coordinates where rolled is constant along hyperboloid this could be row equals 1 as commuter artists other words or along the trajectory year this point unions we want unit of rolled away from the year arrives horizons right the center wrote who 0 misses road was too oracle's 3 or 4 and so forth and that's just looking at this electric year-end noticing that apart from the of Rome near the horizon it just has the form of flat based in polar coordinates had about for work were Omega the lines of constant Omega they were like this make equals 0 bomb-making equals 1 for example or maybe Coast tool on May giggles 3 wears omega equals infinity make equals infinity is right along like all here on Day
equals in city that will make comedy is just TO overtook solely discover a some coordinates in which time was like a hyperbolic angle and time Infinity is just for white shown here OK served 1 fact what here In here and 6 we talk about in here is a place that's that's what are his less than 1 hour is less than 1 in here as less than 1 we are as less than 1 sign of these interchanges the storm becomes negative this 1 becomes positive rose squared is actually negative forces 0 in Euro squares is less than 0 you graduates to see that yah yah when Rose where it is positive Boras greater than 1 that means outside the horizon when Rose who it is negative to get behind the horizon you want are could be less than 1 and natural square article be behind the horizon is in here escorted negative and the lungs of constant role in here book autograph warns of Co the lines of constant homemaker still angles look like there's nothing crazy is going on here space is not become prime time is not become space we just introduced coordinates which have a funny property that when you go from here here on what was space like out he becomes prime like in here complete coordinate 3rd part of fact and yet the something interesting about the real far away if we go far away we have to remember that f Rose not just 1 Favre was not just 1 so that means the metric all this Omega direction here that's what this From wrote ponderous square it does differ from flat they does differ from space but the differences show up in the way of Rome varies as you move away from the black hole and the way this thing varies the cause of black hole of this with the geometry looks like now this was role equals 0 my although few quarters what happens as we move up hear what happens is removal printing here is that are decreases rose quit negative in that region so as we move into this region are gets smaller and smaller and smaller we get 1 plus a negative quantity that negative quantities increases are gets smaller and smaller eventually we hit the point man St . where Oryza quota 0 Madison St . that was not just a court would need articles Iran is yes it's a hyperboloid in these quarter and if you would if we to calculate the curvature Freewood to calculate the curvature it's not too hot believe on what's happening here as those 0 this is getting smaller and smaller and smaller curvature is getting larger and larger as we the point of the curvature becomes bigger and bigger and practical infinite on the Green Line this is a true singularity it's a place of infinite tidal forces enter don't basically a place you would want a the problem is you can avoid it it's not really a place discussed last time but really a place that really in a sense prime so many a self imagine finding yourself over here somehow let's not go there yet he is a black hole he is an orbiting space station far away where is that that's very far away attire it's out here said Oracle's a 100 thousand 400 a million that's way way out here and don't do do anything stupid namely jump off the allow yourself who to get sucked in by gravity you're safe enough our here nothing bad will happen to you but if you foolish enough to to go and say I want to explore what India Long in the past jump off the space station authority in here you might find itself and that you find yourself in the car there's no way out you can't get out you've passed a point where in order to get out you have to exceed the speed of light the speed of light corresponds 45 degrees lines in this diagram all icons like Collins of 45 deg columns so the best you could do it in trying to escape it is dotted line he the matter what you do and so inevitably you to hit a singular once you behind there is no charts on the matter how hard you try to accelerate from here I'm less it could exceed the speed of light which I assume you cannot you're going to wander pitting singularity so you're stuck where is the point of no return the point of no return is right over here that this that that like called What is that like all well Pete toward the hyperboloid is a different value of Rome heroical for over 3 road goes who wrote was 1 what is what was the point of view Michael 0 right but role equal to 0 all along just imagine a hyperboloid vary very close to year essentially all along this line here in this line roh is equal to 0 no equals the less equals X was a trainee kind number Robert was number is hyperboloid very very close to here this something very different about this kind of geometry than ordinary geometry of by orderly geometry and I said Where is the point are equal 0 what we know where that is a threat every year where is the point Oracle some kind number what's not point it's a tiny little circles but that circle gets closer and closer to the center is I make the distance smaller and smaller quite different than here where it is rolled equals something kind his whole hyperboloid here as role gets smaller and smaller it tends icon here not not just to to this point we've meaning of that was really cool 0 rose 0 0 was From horizons so the horizon is not just his point the sense of the horizon all along this line anybody who find themselves beyond this point is inside a black hole let there are less than 1 inside the interior of the horizon and as Yukos say just looking at the diagram they don't they can't get out of its singularity so the point of no return Is it be row equals 0 but not disappointed says hold on this whole thing is the horizon of the black hole I want to pass it your test a point of no return when you look at baby questions
mail that at 1st From probable summer work for the state of art at the last week to use those 4 volume related but bowls at but others still cast vote yes they are right but that places where rows with negative pay their imaginary values rose to light the place is what Aura's Leiston warned role in honor are functionally connected you know are you know wrote in a row you know on OK so are also lines of Princeton but there are real perfectly good are but lesson 1 and the red lines all lines of cars treat her look like it but they are look at these coordinates call admits that I've been constructed over here they the called Chris Gulf coordinates job history left maybe should spend 5 minutes from you the history of a black hole physics when Swatzell up down the shore chilled that he didn't know that we horizon was a horizon as far as he could tell starring people who studied these things for a while these Oracle's 1 was a nasty singular place they look at this functions and they dear is is divergent that Oracle's want something bad happening there and so the conclusion was we horizon was very bad waste armed if took the axis of what's frivolous said is that viruses it is a kind of variable but it's not the same kind that occurs the kind that up there a war in which involves along here address referred to that it is reference frame of a in falling observer as opposed to somebody who stays outside the somebody stays outside the black hole lives on 1 of these hyperboloid some money flowing in for example might find it convenient to use a rectangular court here to be questioned by so while others said to be armed the history was the 1st people do not realize that horizon was a smooth nonsingular ordinary price it was sometime in 1950 further Mark Davis Finkel Starring road what essentially coordinates of the height not exactly these coordinates recorded of these types and realized that the horizon was a point of no return and not at significantly David dangerous place the core admits you he invented himself but they would have been previously invented by Eddington by Arthur Eddington an enormous pickles Friday Eddington Finkelstein coordinates the not exactly be used somewhat later few years later gentleman by the name of course call who was not a relativity expert at all it was a plasma physicist very good equations a very good at changing quarter this was the thing you love to change coordinates the somebody tell the showed him the thing he changed the coordinates and Cheney coordinates and eventually found these coordinates step of that I've indicated with Demetra Kansas nice form where of picture looks exactly like it is a cold Cresco corn Chris golf armed they turned off of its was I think I know what it was called just a massive star a collapsed star people was called collapsed star John Wheeler who called 1st a black hole John Wheeler was very nice this week was a good friend of mine told me had a kind of side to him which I didn't approve of having a year of politically very conservative side his political content conservative this incident we did not extend the social issues as political conservatives this had to do with 1 thing in only 1 thing on he was afraid of the Soviet Union having nuclear weapons so was very and by Soviet Union and especially by Soviet expansionism and so we have a very strong politically conservative side to him and will argue about it but not so much need the summer friends and but there he was a very thoughtful and gentle and nice person did not have a young idea a socially conservative side and the effect FX remember once we were sitting in a cafe in Valparaiso Chile and John Williams about 85 years old my wife and sitting around and you but look very agitated mistletoe John you feel well want to get fickle Walker with new look at you only what not mom delivered what myself from saluted the giant capped bikini so he was not a social conservative and factor in fact when he wrote a paper his 1st paper on what attracts he coined the term black hole the result of a sincere big trouble with physical Review with a physical Review did not want the publisher the reason I knew this today was a problem as was before Mike prime actually but nevertheless and quite but there and new there'd been a problem I know it was it was largest a physical Review being it considers self but not below hyper conservative but the term black coal was up scenes and survey refused the publisher 1st fought before before them eventually won but they just to get back with his next papal was called black holes have no here and there I was ready to wake incidentally what what it means for black clothes not having him play with me if you take a among rotating black hole gravity is so strong There will always pull it together so strong that eventually after a very small amount of time the horizon even it starts out would be a spherically very asymmetric could be tool rocks coming together theory unsymmetric might start out looking like that in a very small amount of turning it will pull itself interlaced and there become indistinguishable from a fear John call that property the property of not having any on any visible structure any visible defects on equal that not having stating the Black Course have no Harris just they the gravity so strong that call any black hole the horizon of the perfect sphere there is rotating then the CEO can get performed biking get into an oblique furore but have these nature of the 0 blades right depends only on the angular momentum Michael schedule here OK but I got some bad roads where can I can write but Nordic Savo square negative the row was but at the not because normally didn't intend to use Our Nora's Rosiclare bar minus 1 square disarm OK now let's talk about the things fully into a black hole the the thing falling Blackall incidentally you might ask about the other half of his diagrams that he would like to see that the other half of the here has no real physical with only to come for the moment
mainly interested in the upper right here in part where is theory of the black hole the exterior of the black holes out here the interior of the horizon here OK so let's redraw picture and let's talk about somebody falling in volume as Alice Our fault and what we draw some lines on here falls in north lines of constant part but I'm more interested right around lines of constant time so his lawyers a constant the this following year is sequels affinity among awards the horizon people's infinity very strange but this place that you've thought was was a place the horizon really also has the character of being a time that's time equals infinity all make equals infinity strictly speaking it's or maybe because of from AT&T closely connected the seller to look is you concede that Alice Falls in she doesn't pass the horizon until equals infinity that's stint in which in falling object never passes the horizon it does test arrives here as you and sailing right through encountered nothing crazy but it's just a strange set of coordinates the visit assist equal 0 . equals infinity sup might say it doesn't really mean anything to anybody the fact that I was never gets in but really into barbarity and barbers watching Bob was watching allies but put a little closer Bobby Moore's Anaheim Bob is not going to go into the black hole that is not to stay out of fixed distance away news watch ally and let's see what it sees about our simply lost like can topple poor areas or yes was promote yet it is Bob really looks Back was look back he looks in the past he can't see Alice at instant that looks at currently locks our past he sees light coming from ours so this is our sovereignty but member like traveling on 45 degree angles later on earlier he looks back again and Alice over here he looks back again as Is he stays outside the black Colo a new looks back what he was is Alice getting closer and closer to the horizon but never passes so in fact it's not just a coordinator artifact that Alice doesn't go through the horizon until 2 equals infinity it's actually what Bob Estes as far as Bob is concerned about the past the horizon if Alice in own frame of reference passes the horizon she can't send signal the Bob to say that I pass the horizon sort of everything that Bob measures everything that Bob cares about everything that he conceived His entire physical love our observations all involved hours outside her eyes she gets closer and closer to you might say that Our skits closer and closer as she gets closer and closer all of the parts get closer and closer and the sheet but when a man as far she's concerned nothing very special happens she just go through the horizon next question 1 Alice's watching bomb protest it's it's quite apparent from this diagram the when Alan Bob watches Our seasonal slowdown Alice simply takes an infinite amount of time to get to the horizon all of her Fox as far as he can tell all of our clocks just slow down slow down slow down go so seems that asymptotically slow down and approach the horizon people generally tend to think that that must mean the palaces watching mom she sees Bob speed up but that's not the case she does not see Bob speed up here's Alice over here and she looks back at she doesn't see anything very special or let's put in the let's put the singularity singularities here that means Alice is terminated Alice is terminated at that point no more hours beyond this point probably before she gets that point terminated but above mathematically terminal so she looks back let's say from this vantage point over here and she sees Bob I Bob all boxes followed now will later she looked but issues Bobby near the fact she passes the horizon over here Bond can no longer see her Bob could never see her beyond this point there's no point on Bob trajectory here Bob looks back to conceal paralysis absolutely no problem seeing Bob policies Bob similar over does nothing that would make it think that she sees Bob speed up and she doesn't want does happen it is our last . com Bob's trajectory that Alaskan embassy polish Ken Nazi Bob Costas point why not because she's dead that's what so the complete asymmetry biopsies Alice down sees a flat now by a pancake things infinite time for get anywhere Alice looks back and is nothing special about that genetically there's a special point between where they look like where she all his way in the past right now Bob it's her going to arrive at everybody else a rare but needed juncture where majority such that special point where nothing but so this baggage is never going to see bobcat PC her goal profit right right so you will never see Bob at the and what he sees I called arise got partly because he never seasonal to the horizon but right more America became members she could never see them after that point throw but again not because he's hiding orbit because it accelerating just because you sees to 3rd and if they see the dollar normal there's no sense his most centers in which she has fallen it looks have not only the says in his fish does year you go back but if a L evolving go back to back not 1 passenger was because of what we get that is what happened this summer following us talk let God let us in the following hours somebody following that person following hours recently and John Alice's if she was moving not ours is server he asserted there's following our right sees nothing special sees Alice over here sees us over she's out of here she's out of here nothing particular likewise Alice watching Was this Charlie that's Charlie Alice watching Charlie sees nothing special because both of them eventually get terminated but Charlie is following Alice in nothing special at this point there's only bomb and this is a consequence of the fact that he's a very has accelerate the state's stay outside or if Al falling in free fall and
Charlie is following her just behind her medicine free fall they talk to each other they are taken hold music and they can have a conversation Conversation can go back and forth Alysa something Bob Bob something back Dallas our system the Bob and of course of course our stock that's Bob and salary and then Charlie Allison Charlie have nice conversation now Alaskan look back and here Bob's half of the conversation but maybe not here but please see Bob said of the conversation but once our skipped past the point mark cannot who cannot receive anything from out completely asymmetric but said the best for quite a walk for what in this is that awarded because I'm getting to work on a radio I'm not truly a basketball hoop on was Al shot of right at border else are who we about Charlie Bob who's doing work at all Jack Straw follows John Paul Boles Rideout right what it appears all here's 1 state past the horizon mother doesn't see them all together with the ball will always the direct Bob Livesey ICI right yet so it looks like bomb heroes it is at last year's Charlie and they both moving car the horizon 1st so she gets contracted according to him and 1st Charlie is still mulling easily will be contracted and there will be later hours is getting squashed flat maybe a lot later gives me a lot later is getting squashed Charlie is right behind her and we just see the fall in with Charlie closer and closer and coarser right behind her never catching watching get flat 5 that after the guessed that last year the bigger bar or thinking about right reserve important reserve an important point here where we're talking about all of this in all of this stuff happening you the horizon we're assuming the black hole is big enough that all or Allison Charlie small enough that tidal forces there is a prime force mentioning a very very big Michael incidentally that black or would have to be very big if you took us solar mass black called which has a fresh or chilled radius of about a kilometer has incredibly fears tidal forces of the horizon that's not very big memory you would get squeezed to like toothpaste arrives but if you want to take a black hole as big as the 1 that December galaxy the tidal forces of the horizon would be pretty weak theater corn which is as big as any better known anywhere in the universe and the memory of solar masses and when ordinary person would be small enough that they would not experienced serious troubled with for battle with thinking about right differing rewrite the tidal forces would have been you on giving effect pledged that shot what are you doing that is a real effect the forest Allison Charlie pile forces is not perceived about feel this effect we don't feel their meterstick get squashed their clocks get slowed down including their internal clocks biology those exactly what's necessary for them and not exponents anything at the yacht Charles that too all of a car bomb under the weight of where will Charlie us both cost her job you left outside a broker look at not not not reps it is you capital told drop picture every question like this drop picture is our going so use Charlie Charlie is of U.S. but Charlie looking elsewhere he looks here Caesar here but they hear the season he looks here Cesar here where does he see hope pass the the horizon at exactly the same time that he passes the horizon sees sees her past the horizon just as he passes the right you expect French after he is passed horizon he looks back and she's it she's inside if a little while ago a saying but people thought % here the same and that goes for the same time but of course you learned the Charlie sees Allison the packed in his past so Charlie Joey visually sees past horizon is not not signposts but it happened all by calculation and by idea of being a physicist but whatever it is really happens to Norway exactly when she's pesto horizon he will see higher at that point at the same at the time and at the point when he passes so there is no the people separated by light falling left for the all that was just a little thing now Charlie is over here he's is watching perilous but really decides at the very last minute no I don't wanna follow our I'm going to accelerate my way out of his black hole and accelerates stay out of a black hole this he looks back again he never gets to see Alice Krauss arises from this point he sees a getting close horizon graffiti changes his mind at this point and accelerates himself out of a black hole then you will not see youse gross verizon only if he quoted a price self-pity gets the our script ours How but they also grew from the such guess what the what she thought it seemed it is 1st there will be 8 else that they resolved there this rise they say well Alice doesn't see Bob go through the horizon at the same time she sees Bob will fill the Verizon over when she's over here must if there is a greater both U. an Eds the Bagram says it all the diagrams are proof of death if it Thursday looked at it and see can the rest of the cast every day and all nite sold off like that of was a lot of Georgia a Ferrari's looked almost like a stable line pretty much approximately not exactly but approximately 3 lifted same like the remit Democrat that look what can I
say that unify our or in like on mechanics general candidate we used to examine that are less than a cup of hot trips fervor that members of vary very car question fortunes being the baby like are met at the present time the term Burgos where you Mary supple would very hard but it unless your this is a list of rivalrous clockers think think think think the fit they they lost it was the loss in her own reference frames are velocities can't picture he pretty much the wrong reference frame with constant velocity and if you measure off proper time it'll just look like that she practically that this is practically flat space-time at least near the horizon you don't get too near the singularity what goes on here is pretty much closed the flat based turned so it keeper away from singularity who were too much for back but as if also here with pretty much free fall her Clark a motorist fix are kind of those death toll for the 1st year if is not a bad shot across the bow of how Europe salary but not that tried by man involved the for you might wind up for over rapid fall still late enough also vary very late but Posey falls where he then he looks back when he sees it is analysts just AFDC passes the horizon here doesn't have much he is terminated pretty quickly up here so he doesn't get much of Alice after suppresses the horizon do not he's watching her nothing strange about her but he gets terminated very quickly so from our from Alice in watching Bob will just barely see across the horizon not before get killed but before he gets killed the if you're so you get of what get sooner rather than later so now you have to you have to remember that a distinct kinds on the diagram it has to be look cell back we don't not what a because of feeder and so well that there will be no credible evidence that 1 got I don't feel like is nothing thought of but but me but in the immediate area the plan is to enable Yemen have admitted that that the bus happened the parade here we we've been through this we spend time discussing Kirby Canyon laws of physics and the force in southern Italy the northern Italy but the sun a Class a French may opt fervor nor new whom on what we are both of the Mont Blanc tunnel anymore but there is a lot of state How that Dallas makes a sudden change motion is not even clear what it means to say that Ekwok synchronize courtesy the analysis on Hurricane Ike you could think of the zillion questions but draw diagrams draw diagrams and answer the quest you got the tools now into the question yourself Our assured you had a answer a few questions and at hand or all you really need OK if there are no other urgent questions appeared roared across the board he has got some that I love my life is sorry if he does or doesn't bricks or if it does not be those extort we both across remember us and she'd are not yet is not so much she said that it is not for abuse From his perspective he sees suffered a certain amount of time but he sees a slowdown he sees a slogan he will see of buying number of heart beats out then well I don't know what the reality is a loaded word it's a loaded word and weakened come back to question of what is real and what is not real but the figure for example tells us there is a sense in which Bob Estes Alice died think about is watching her and he sees a finite a number of heart beat it takes it In the amount of time and over that is the amount of time he sees a finite number of heart beat that means in here watching her most of of the time they are quite doesn't B so on some still since he might conclude that our horror amassing a the dorms all I know she's crossing of rising but just in terms of watching her she slows down and our heart stops not quite but the slows down almost a so you might say Bob cities Our died at the horizon but of course that's that some extent just an artifact the fact that he's accelerating away from eyes see as perfectly happy want us to get near the singularity OK I got out of the house she sold falls below that it was all just well you're rescue were seized get up exactly the same as Charlie following Alice Charlie following hours killed Charlie here has arrived and he sees hours she was just that's right so she falls through but still has anything anything very unusual Charlie doesn't say anything and usually about a about hours it is was going about their business is at point where he crosses the horizon nothing special happens recourse Verizon not signed he just goes on as merry way she goes on our merry way and he sees exactly what it will that there was no real black hole where it is true that he does not see her across the horizon to appoint recourse arrives in game of special happens he added that we didn't hear about us His bond house looks back and sees Bob Estes Bob's is Bob nothing special on Tuesday it's a singularity mention anything 5
when I was just outside all all in it falls on the rise and also gets out of jail time didn't see any big students in things been exactly the same as Charlie sees nothing happened hours when he crossed the horizon G and special there is that she was seen as flung he sees this fall when Estes falling in across the horizon With with she sees it you look back in time she said she says he knows that things don't fast speed of light so she looks back she watches the stuff only on she says you know I know that stuff fell in a long time ago because it's been traveling with the speed of light is not perceived it that what you might be deceived she knows anything about lights she will know the stuff fell in before she did OK we happy How do you have the tools the answer any questions about your soul who sees what drug warnings and the German and if I were a movies it sure that you need to know movies of somebody falling into a black there are and I don't are you make try and Joe Hamilton Hamilton I think made some simulations of war looks like performed for Butler looks and feels like fallen told black hole of course say nothing special happens but of course that's right but no light that's coming at you is coming at you in a very peculiar and funny ways I don't blame me if you watch when these simulations and get very noisy I'm and here I have seen some of them I can't remember where they have illuminating you won't get much out of but found the watch 1 but they really are disorienting effusive if you're big room with a forum for review richer they Coke OK let's move on what should be be well within the what will want or we can ask more questions but the something repetitive about the questions that normality was Bob without biopsy results what's what break loose from that offered here is a not if nothing March because that a real black coal it's formed for example by stellar collapse of something like that this is even on the diagram only a portion of the Bagram appears so that is where I would go right now if rural forward or next week Tom this is a sort of idealized black holes but a real black coal which forms part collapsed only a portion of this diagram means everything particular apart apart can young Best of both the last fall for the better the bike albeit yes yes yes yes yes that From world worst that right is getting a lot of these was there was a lack of but the broader none of that it is matter where just figure out where your on the picture that I wrote here were appropriate for over Florida Alice and Bob store much lighter than the black hole and don't have that much react River black Blackall react very strongly and if you look at tank for example another heavy mansion drop billions and really follow it what you would see it is at 1st the horizon would bulge out some be attracted by the other objects and that they would merge but form 1 horizon may end because black holes now nor hair it would pull itself together into a sphere of the would be slightly larger can calculate how much larger if we know how much less was a member the radius toe injury sorry to 0 . so if we had some energy in the form of mayors than the radius responds Amir appropriate ways which were to reduce responsive appropriate were reason man nature of a leaders of the UAL deal solely on the food they eat led the like that you see from a black hole is not coming from the horizons it's coming from all sorts of stuff prompt stuff the circulating around the horizon and that's very heated up by EU member number for example that a certain distance around you can have photons of light just moving around circles all sorts of very complicated the collisions take place we're not in a real black all this stuff out there it's not just a black hole in empty space it would pretty much talking about a black hole in the bulk the stuff out there is an accretion disk all sorts of things falling in and what you say it is not really coming from precisely the the horizon coming from outside the horizon as for the it was such a of essentially they vote showed that most of the prewar Europe Mary Arafat the kind that would take if you were to make tools solar mass black collide that they formed their your serve dumbbell at 1st like that at the time that it would take for it to equilibrate is not much longer than a light transit time across this is a terrorist small fraction of a 2nd bomb a baby a hundred times of like transit time with select transit time across a 1 kilometer black hole well like travels at 3 times can be meters serve 3 times and 5 kilometers per 2nd the sick and the 5 kilometers per 2nd of 10 to the minus 5 seconds for a light to travel across a storm rest black hole maybe Cantor minus 3 2nd millisecond or something like that for those great big massive object to equilibrate to the point where it's the shiny is most perfect sphere the 2 command it is very cold starting work while now buyers got a the euro as a come together the event terrorized will look like that and then they required rat Ralph Lauren peanut shaped almost
radical but not quite the nonspherical all within about and the minus 3 seconds here's what life d stay time OK if you make the black hole big enough you could make a transit time across it anything you like but there are the biggest black hole I think it's something like about 10 of the 9th solar masses remember that would take about 20 minutes light across so 20 minutes for to equilibrate video Blair 1st buff heads of each now that but that's the what is Her eyes it ahead of us all summer said that what will happen if Bob Carey row hours carries a heavy dare was Alice carries a heavy object but the cost of the horizon over here it was not some Ellis also wanted for lot work gloves but where about last year in some quarters as I think they managed he only can do what they had know we always have that much have a bit of flexibility drawing pictures so let's decide the rural hours before Martha hours of the 4 3 is Alice right OK now we hadn't got to the point where we can talk about what happened black hole something falls in but I will tell you no you know I want and what I want and what this is keep the question in mind for next week next week with talk exactly about how black holes are formed how their horizons form and if you throwing more material however horizon respondents and we will we will get there we'll get there and I'll show you surprising different than you expect but firm all very logical let's look that's combined not far at their like that what is more here OIC of well known all we did was we scale out with Gail about for the diamond doesn't change but the meaning of distances on the diagramed changes the diagrams stays exactly the same but the proper distance of say between this point and this point is proportional to the Swat radius of diagram exactly the same it's exactly it's just like having this year supposing you have sphere to spheres of the moon and here and here but Geronimo both exactly the same way they look like spheres does it take to change our goal from a from more aware that there is no if I have scaled the radius of the earth and or more of a problem with this show about is the same diagram that this on the moon and might b a 100 miles on the other both the same angular distance would be 102 is 4 times 4 times bigger so America would be 400 miles same thing is true here if we had a black hole ciller buckle 1 kilometer radius black hole then this distance might Media say half a kilometer on the other hand we withdraw exactly the same picture Ferretti Hulu can 9th solar a monster Blackall and instead of being . 5 kilometers might be . 5 times in the 9th kilometers but the diagram would be exactly the same just the meaning of distances the actual number of leaders the Member meters connecting those 2 points would depend on which short you just exactly the same as if we them Earth do exactly the same picture of fears of fear is but they remember the distances between points was scale as the radius of the year if it is save or not much known it it becomes a much more complicated problem to to be quantitative about exactly what he sees but not all of the nothing really changes no have work for the that people it is do that said is that the room room still occur asking not the distribution of merits it is not on the horizon who young that so the city will be that would go as you said that lives of insider horizon no not the kind of gravity goes 0 inside an object is to you have a shell of when you're inside the shell now there's no shell the horizon it Bush is that each of the children they were all there is to know about is lessons that all singularity all agree on what if I can having at Siena of having a point that there was no inside and outside that Varese support failed Varese that else but that's in there gets he arm of God if you had an 8 . 2 minutes of the origins you don't get that till you get that throughout that doesn't mean that the program all the available but there might be other mass the vast majority of war good good good we'll talk about it with whimper but exactly that next time keep these questions the next time you would you are asking about the question questions all How ahead of the evolution of the black always of form will get there will make a very simple lover construction of a black hole for name and be able see everything happened Gold no the in the of where do they go but I showed that whereby is a lot of things but we gave a good idea because all of them are black hole rotating because deformed now and you could even see it from the outside of black hole the he is a good guess that reaching new gravity changes because of the rotating and the overhaul geometry changes but soldier retreat changes that we have time they rotating black hole is much more complicated than that the dregs based around with it maybe up top will discuss a little bit but the arms she material with angular momentum coming together and form in a black hole it doesn't lose its angular momentum and the angular momentum is permanently their effect of
the angular momentum performed the black hole exactly like exactly like the angular momentum of the earth the formed the surface of the earth makes through an old Blake all Blake's right yup you say it back the Journal on Tuesday that sales already near a some these contract will be just state the bigger begun a smaller role down those figures a small part of its Edwards function that's a thought murderer but say that the presence of a big source of energy complex the face of a fair that real that if you like no 1 knows where they got from what was 102 set is as I stopped to with right as is 1 of the other side of was but it was where based on what it's dead 8 . normal yet yes yes yes yes but would still go to order from as far away as what is this wonder what is it is based on UN No no not withdrawn again withdraw again here's going through Bob out here there is upside from our point of view sees Bob she sees Bob nothing special bond is to switch it gets stay here to see Bobby after that but because anything at the ball because something happened that's it drug diagram trouble icons she sees him over here I am seasonally Zamora here if she never gets the CMO over here again not because anything happened to him but because she's not all left don't perform with they a like from all history like all from Bob about alike but from what we got about having code of the areas that had not music or not cool In a court but returned to that yes but a contract to build and that the other side of the court given a geometry given a solution bearing signs equations kind of money that analyze it think what those people see worried for the other side of the calling this saying if you had some material solve Einstein's equations mass collapsing solve Einstein's equations to find out what the geometry is that's usually COM West Pleasant exercise now acceptance on very special case it will build a special case but while state arrives for about 1 of the protest leaders well I happened to be near the singularity and Verizon emerge Everything is shrinks shrink French francs and shrink stamp a point where the horizon just From from factor in your area a solid well that's it that's a good question on very very unlikely are only black holes are there well as 10 to the 22 stars altogether enter the 11 galaxies I will not and I actually don't normally black holes Aron from somewhere between the 2 of them and the and the 15 Our of course no no I object is gonna have strictly 0 angular momentum when you say 0 angular momentum you mean within some tolerant it's pretty unlikely that a black hole would have such a small angular momentum that he would not reason as its 1st All-Star over almost anything will have some renewable and when it collapses you know you've seen a number of years skaters polling arms in speed up same thing happens when a black hole collapse even a relatively small amount of rotational motion will get escalated into a very rapid rotation so it it's very unlikely that a black hole out there With such small angular momentum that would be undetectable singular set it singularity of rotating black coal it is different than we drawn here the missing you have a rotating black holes at different things
solo I would say no that's not rotating velocity and turned it into a renewable Michael is they are not that holds all of the economy is slowing down into existence by collapsing ordinary material is existing that is rapidly Greg when Astana runs out of fuel the thing which is supporting and keeping gravity from contracting it is heat falling out through it heat flowing out through it is created in the interior of the star by burning them by burning going hydrogen helium whatever it when the star eventually runs out of fuel before the radiation pressured keeping it from collapsing disappears contracts not depending on how heavy it is my contractile white dwarf but femoralis Everything pretty is pretty bad but it still made up of nuclei Adams and so forth it yet had feared might collapse and delayed neutron star that's a very very compact object but it still supported free the material was still strong enough to support itself against gravity has yet heavier then they can't support itself against gravity interschool and false Pass almost what shows radius wants wanted False Pass the college watch over a and then know that is don't want any part of it torrent want a part of it gets within the Schweiger radius then adjust collapses of planters it exists is like a great if they with that because the reason I will be great if they were their depending on the the time scale for the evaporation of a black hole is very very long extremely long it's fast of a small black coals and slower for big black holes if you would like to see a black coal evaporated and be small and store black holes were formed in particular a black hole of of about the man of Mount Everest or something like that would have a life time before it completely evaporates which would be about the age of the universe so if there was a mechanism perform such black hole we might be seeing them evaporate as very hard to imagine given they may ask the size of Mount Everest was simply not enough have slipped at the course the contracting collapse the whole earth will not form black hole years does not heavy enough to to do very much of anything when it cools the son will not form a black hole it's not thought that will form a black called but there will form a white dwarf 0 are or what would be the mechanism for creating these things you go imagine very violent things happening in the early universe which would slash stuff together so wouldn't be the limit to things that you could think of that could help create a black hole what is making things so massive a contractual Raton wait the collapse of Chong wake the other is just a motion velocity velocity commit place gravity and slam stuff together and her wasn't of violent were sick contracting stuff it could form a black hole even know was massive enough to do it on its own let us if there is a long time if a ship is she was the smallest black holes small work or mix for called plant has struck 10 to the minus 5 grams to create particles colliding with that much energy would with fewer to pick slackened just link you have to make about as big as a galaxy of something like that it but yes in principle in principle the answer is yes in principle by slamming particles together hard enough you could make a black or if gave us a had a danger is that if there was a 1st down it is a kind of a undergo would get things Zurich Durand 3rd quarter bosie we go ago last closed but that was not how Shea I can't tell you who were go from week to week give the former took about quarter of that my that I and and it's good it is good finally got that is problems as long as everybody is learning something OK look for more please visit
us at stanford . EDU
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Metadaten

Formale Metadaten

Titel General Relativity | Lecture 7
Serientitel Lecture Collection | General Relativity
Teil 7
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/15037
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (November 5, 2012) Leonard Susskind continues the discussion of black holes in depth using coordinate transformations and diagrams to develop an intuitive understanding of black hole physics. This series is the fourth installment of a six-quarter series that explore the foundations of modern physics. In this quarter Susskind focuses on Einstein's General Theory of Relativity.

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